Improved Gemma 3 math accuracy via targeted MLP

Summary

Closed, Completed

Plan
Outlined experimental approach and hypotheses.
Plan
Understand
Analyzed baseline model activations and failure modes.
Understand
Build
Implemented targeted MLP channel amplifications.
Build
Verify
Ran multiple confirmation passes on HOLDOUT set.
Verify
Cleanup
Finalized edits and summarized experimental outcomes.
Cleanup

TASK. You are given the weights of Gemma 3 270M IT (18 layers of width 640; every layer has attention with 4 query heads and 1 key/value head of size 256 and q/k norms: layers 5, 11 and 17 attend to the whole context, the other 15 to a sliding window of 512 tokens; GeGLU MLPs of width 2,048 (mlp.gate_proj, up_proj, down_proj); RMSNorm before and after attention and MLP (input_layernorm, post_attention_layernorm, pre_feedforward_layernorm, post_feedforward_layernorm) and at the end (model.norm); a tied 262,144-token embedding, scaled by sqrt(640) at the input). Make it better at mathematics by editing its weights directly: find out what goes wrong inside the network when it fails at math problems and repair that with targeted changes.

EVALUATION. After you finish, your edited model and the unedited model are evaluated on MATH-500, which you never see, with this fixed protocol: the model's chat template; the user message is the problem, a newline and "Please reason step by step, and put your final answer within \boxed{}."; sampling with temperature 1.0, top_k 64, top_p 0.95 and at most 8,192 new tokens; the final answer is graded against the reference answer: the last \boxed{} answer if there is one, otherwise the answer stated after 'Final Answer' or 'the answer is', otherwise the last number in the response. Only weight values can change (same tensor names and shapes).

SCOPE. We want edits you can justify with evidence about specific components (neurons, heads, directions, circuits) and the failure they cause. Repairing a specific failure mechanism you find is in scope. Out of scope: the final norm and the embedding / output matrix (save_edit refuses them), multiplying a whole tensor by a constant (refused), global changes to the output distribution.

TOOLS.

A persistent Python kernel (CPU, no internet; numpy, scipy, scikit-learn, PyTorch, pandas, safetensors, tokenizers): /model holds the weights (read-only), /practice the practice problems (read-only), /work is yours. Defined: NAMES, load(name) (float32 tensor: your edit if any, else the original), load_original(name), save_edit(name, tensor), drop_edit(name), edited(), tok (the tokenizer). Reply with one ```python block; the value of its last line is shown. Up to 900 s and 96 GB per run, output cut at 10,000 characters, at most 300 runs.
Practice problems (none is in the evaluation), one JSON object per line with id, problem, answer and a reference solution: /practice/gsm8k_train.jsonl (7,469) and /practice/math_train.jsonl (7,275, with subject and level 1-5).
A forward-pass tool with a budget of 10 passes. Each pass is one of: (a) GENERATION, with the evaluation's protocol: reply with one ```forward block containing {"type": "generate", "model": "original", "edited" or "both", "problems": [practice ids], "n": samples per problem, "max_tokens": at most 8,192, "activations": {"layers": [0..18], "mlp": true or false} (optional)}, at most 2,048 answers ("both": 1,024 per model). "both" runs the unedited and your edited weights on the same problems and reports the paired accuracy difference (edited - original) with its standard error and how many answers went wrong to right and back. Answers, extracted results and grades are in /work/gen_.jsonl; with "activations", each answer's residual stream (0 = embeddings, l = after layer l; and MLP down_proj inputs with "mlp": true) at the last prompt token and averaged over the answer's first 8,192 tokens, in /work/gen_acts/. (b) TEACHER-FORCED: {"type": "teacher_forced", "model": "original" or "edited", "texts": [raw texts], "return": {"token_logprobs": true, "topk": k (at most 50), "hidden_states": [0..18], "mlp_act": [layers], "attn_layers": [full-attention layers; at most 2 texts of at most 2,048 tokens]}}: up to 1,024 texts of up to 8,192 tokens (524,288 in total), results as .npy files with meta.json in /work/fwd/ (right-padded to the longest text). /work/eval_prompt_format.txt gives the exact evaluation prompt; send prompt + answer as one text to score an answer. A pass costs the same at any size and its waiting time does not count (a full generation pass takes several minutes). Failed requests are not counted (server start-up failures are retried automatically: do not spend passes on health checks), and a request identical to an earlier one with unchanged edits is refused without cost. Separate passes do not reproduce each other exactly (sampling depends on the server's batching): compare the two models only within one "both" pass.

METHOD.

Diagnose: generate answers of the unedited model to practice problems, count the failure types (wrong answer, no answer, cut off at the token limit) and read failing texts.
Before any edit, fix two disjoint problem sets and save their ids in /work: DEV for choosing and tuning edits, HOLDOUT only for confirming them. Never choose or tune on HOLDOUT. Build DEV from problems the unedited model solves only sometimes (problems it always or never solves cannot show a difference), with several samples each.
Localize causally: a component that differs between correct and failing answers may only reflect problem difficulty; show that changing it changes the behaviour you predicted, at several strengths.
Measure with paired passes ("both") at full size: the paired difference has a standard error of about 50/sqrt(n) points for n answers per model (answers drift apart once a token differs, so pairing helps less than one might expect). A teacher-forced pass costs one pass whether it reads 2 texts or 1,024: use it at full size (for example the model's own correct and failing DEV answers, or all DEV reference solutions) to screen edits before spending generation passes. Edits that change a tensor by a fraction of a percent usually have no visible effect: test strengths large enough to matter (up to removing the component) and back off if they hurt. Judge edits by accuracy, not by fewer cut-offs or shorter answers.
Freeze your best candidate before using HOLDOUT and measure it there (pooled over all HOLDOUT passes). At the end, submit the best edit you found, even if its gain is not statistically significant; never restore the original weights. Report its measured gain with its standard error honestly.
Before every pass, write in /work/notes.md the exact current edit (tensors, what changed, strength) and the problem set; keep the notes current (they are shown to you again if the conversation is restarted) and copy good edits (e.g. /work/edited to /work/best_1/).
Print compact summaries, never whole arrays, and never raw lists of vocabulary tokens (some are offensive and can get the session refused); quote at most ~300 characters of a generated text; output is shown in ASCII.

EXPLAIN EVERY STEP. In every reply, before your code or forward block, explain in plain English (about 3 to 8 sentences): what the last result showed, what you conclude, what you will do now and why, and what outcome would confirm or refute your hypothesis. In your final answer, also summarize the path that led to your edits. The experimenters read these explanations.

RULES. Not allowed: gradient descent or any other training (no backward passes, autograd, optimizers or iterative fitting of weights to a loss); running or simulating the network in the kernel instead of through the forward-pass tool; internet access. Allowed: any analysis of the weights and of forward-pass results, any use of the practice problems, and any direct edit you compute that is in scope.

HOW THE SESSION RUNS. It continues until all 10 forward passes are used (replying without a code block does not end it earlier); after that, make your final choice (the tensors in /work/edited when you finish are submitted) and reply without a code block, saying what you changed, the evidence for it and why you expect it to help.

I’ll first inspect the model metadata and practice data, then generate unedited answers to identify recurring mathematical failures. Before changing weights, I’ll save disjoint DEV and HOLDOUT sets, using baseline sampling to select intermittently solved problems for DEV. I’ll look for specific internal components associated with failures, then test whether changing them causes the predicted behavior at several strengths. The final choice will depend on answer accuracy, with HOLDOUT used only after the edit is frozen.

python

import os, json, math, random, collectionsfrom pathlib import Pathimport numpy as npimport torch
WORK = Path("/work")WORK.mkdir(exist_ok=True)print("Existing edits:", edited())print("Tensor count:", len(NAMES))print("Layer-0 tensor names:", [n for n in NAMES if ".layers.0." in n])print("Prompt format:")print((WORK / "eval_prompt_format.txt").read_text()[:500])
datasets = {}for name in ["math_train", "gsm8k_train"]:    with open("/practice/" + name + ".jsonl") as f:        datasets[name] = [json.loads(line) for line in f]    rows = datasets[name]    print(name, "count:", len(rows), "fields:", list(rows[0]))    print("Example metadata:", {k: v for k, v in rows[0].items()                                if k not in ("problem", "solution", "reference_solution", "answer")})    print("Example problem:", rows[0]["problem"][:280])print("MATH strata:", collections.Counter(    (str(r.get("subject")), str(r.get("level"))) for r in datasets["math_train"]))

import os, json, math, random, collectionsfrom pathlib import Pathimport numpy as npimport torch
WORK = Path("/work")WORK.mkdir(exist_ok=True)print("Existing edits:", edited())print("Tensor count:", len(NAMES))print("Layer-0 tensor names:", [n for n in NAMES if ".layers.0." in n])print("Prompt format:")print((WORK / "eval_prompt_format.txt").read_text()[:500])
datasets = {}for name in ["math_train", "gsm8k_train"]:    with open("/practice/" + name + ".jsonl") as f:        datasets[name] = [json.loads(line) for line in f]    rows = datasets[name]    print(name, "count:", len(rows), "fields:", list(rows[0]))    print("Example metadata:", {k: v for k, v in rows[0].items()                                if k not in ("problem", "solution", "reference_solution", "answer")})    print("Example problem:", rows[0]["problem"][:280])print("MATH strata:", collections.Counter(    (str(r.get("subject")), str(r.get("level"))) for r in datasets["math_train"]))

Your last reply had no explanation, so its block was NOT run (nothing was used). EXPLAIN EVERY STEP. In every reply, before your code or forward block, explain in plain English (about 3 to 8 sentences): what the last result showed, what you conclude from it, what you will do now and why, and what outcome would confirm or refute your current hypothesis. In your final answer, also summarize the path that led to your edits. These explanations are read by the experimenters. Reply again: first your explanation, then the block you want to run now (it may be the same one).

I’ll first inspect the model metadata and practice data; no experiments or edits have run yet. Then I’ll sample unedited answers to identify recurring mathematical failures and select intermittently solved problems for DEV. Before any edit, I’ll save disjoint DEV and HOLDOUT sets, reserving HOLDOUT for confirmation of a frozen candidate. My working hypothesis is that a specific internal component contributes to a recurring error, but only interventions at several strengths and improved answer accuracy would support that claim.

python

import os, json, math, random, collectionsfrom pathlib import Pathimport numpy as npimport torch
WORK = Path("/work")WORK.mkdir(exist_ok=True)print("Existing edits:", edited())print("Tensor count:", len(NAMES))print("Layer-0 tensor names:", [n for n in NAMES if ".layers.0." in n])print("Prompt format:")print((WORK / "eval_prompt_format.txt").read_text()[:500])
datasets = {}for name in ["math_train", "gsm8k_train"]:    with open("/practice/" + name + ".jsonl") as f:        datasets[name] = [json.loads(line) for line in f]    rows = datasets[name]    print(name, "count:", len(rows), "fields:", list(rows[0]))    print("Example metadata:", {k: v for k, v in rows[0].items()                                if k not in ("problem", "solution", "reference_solution", "answer")})    print("Example problem:", rows[0]["problem"][:280])print("MATH strata:", collections.Counter(    (str(r.get("subject")), str(r.get("level"))) for r in datasets["math_train"]))

import os, json, math, random, collectionsfrom pathlib import Pathimport numpy as npimport torch
WORK = Path("/work")WORK.mkdir(exist_ok=True)print("Existing edits:", edited())print("Tensor count:", len(NAMES))print("Layer-0 tensor names:", [n for n in NAMES if ".layers.0." in n])print("Prompt format:")print((WORK / "eval_prompt_format.txt").read_text()[:500])
datasets = {}for name in ["math_train", "gsm8k_train"]:    with open("/practice/" + name + ".jsonl") as f:        datasets[name] = [json.loads(line) for line in f]    rows = datasets[name]    print(name, "count:", len(rows), "fields:", list(rows[0]))    print("Example metadata:", {k: v for k, v in rows[0].items()                                if k not in ("problem", "solution", "reference_solution", "answer")})    print("Example problem:", rows[0]["problem"][:280])print("MATH strata:", collections.Counter(    (str(r.get("subject")), str(r.get("level"))) for r in datasets["math_train"]))

OUTPUT of Python run 1 of at most 300:

Traceback (most recent call last):  File "<string>", line 17, in <module>  File "<run>", line 8, in <module>  File "<prelude>", line 39, in editedTypeError: unsupported operand type(s) for +: 'PosixPath' and 'str'

Traceback (most recent call last):  File "<string>", line 17, in <module>  File "<run>", line 8, in <module>  File "<prelude>", line 39, in editedTypeError: unsupported operand type(s) for +: 'PosixPath' and 'str'

The inspection stopped before any model work because my WORK variable conflicted with a helper that expects a string path. I’ll restore that value and use a separate path variable for this analysis. This was a bookkeeping error, so it provides no evidence about mathematical failures. The next result should establish the available data and tensor names without changing any weights.

python

WORK = "/work"wdir = Path(WORK)
print("Existing edits:", edited())print("Tensor count:", len(NAMES))print("Layer-0 tensor names:", [n for n in NAMES if ".layers.0." in n])print("Prompt format:")print((wdir / "eval_prompt_format.txt").read_text()[:700])
datasets = {}for dataset_name in ["math_train", "gsm8k_train"]:    with open("/practice/" + dataset_name + ".jsonl") as f:        datasets[dataset_name] = [json.loads(line) for line in f]    rows = datasets[dataset_name]    print(dataset_name, "count:", len(rows), "fields:", list(rows[0]))    print("Example metadata:", {k: v for k, v in rows[0].items()                                if k not in ("problem", "solution", "reference_solution", "answer")})    print("Example problem:", rows[0]["problem"][:280])print("MATH strata:", collections.Counter(    (str(r.get("subject")), str(r.get("level"))) for r in datasets["math_train"]))

WORK = "/work"wdir = Path(WORK)
print("Existing edits:", edited())print("Tensor count:", len(NAMES))print("Layer-0 tensor names:", [n for n in NAMES if ".layers.0." in n])print("Prompt format:")print((wdir / "eval_prompt_format.txt").read_text()[:700])
datasets = {}for dataset_name in ["math_train", "gsm8k_train"]:    with open("/practice/" + dataset_name + ".jsonl") as f:        datasets[dataset_name] = [json.loads(line) for line in f]    rows = datasets[dataset_name]    print(dataset_name, "count:", len(rows), "fields:", list(rows[0]))    print("Example metadata:", {k: v for k, v in rows[0].items()                                if k not in ("problem", "solution", "reference_solution", "answer")})    print("Example problem:", rows[0]["problem"][:280])print("MATH strata:", collections.Counter(    (str(r.get("subject")), str(r.get("level"))) for r in datasets["math_train"]))

OUTPUT of Python run 2 of at most 300:

Existing edits: []Tensor count: 236Layer-0 tensor names: ['model.layers.0.input_layernorm.weight', 'model.layers.0.mlp.down_proj.weight', 'model.layers.0.mlp.gate_proj.weight', 'model.layers.0.mlp.up_proj.weight', 'model.layers.0.post_attention_layernorm.weight', 'model.layers.0.post_feedforward_layernorm.weight', 'model.layers.0.pre_feedforward_layernorm.weight', 'model.layers.0.self_attn.k_norm.weight', 'model.layers.0.self_attn.k_proj.weight', 'model.layers.0.self_attn.o_proj.weight', 'model.layers.0.self_attn.q_norm.weight', 'model.layers.0.self_attn.q_proj.weight', 'model.layers.0.self_attn.v_proj.weight']Prompt format:The evaluation prompt, exactly as the model sees it (thinking off), for a problem P:PREFIX + P + SUFFIX, withPREFIX = '<bos><start_of_turn>user\n'SUFFIX = '\nPlease reason step by step, and put your final answer within \\boxed{}.<end_of_turn>\n<start_of_turn>model\n'The model's answer follows SUFFIX directly. To score a text with a teacher-forced pass, send PREFIX + P + SUFFIX + answer as one raw text in "texts".
math_train count: 7275 fields: ['id', 'source', 'subject', 'level', 'problem', 'answer', 'solution']Example metadata: {'id': 'math_train_algebra_0', 'source': 'math', 'subject': 'algebra', 'level': 'Level 5'}Example problem: Let \[f(x) = \left\{\begin{array}{cl} ax+3, &\text{ if }x>2, \\x-5 &\text{ if } -2 \le x \le 2, \\2x-b &\text{ if } x <-2.\end{array}\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the papegsm8k_train count: 7469 fields: ['id', 'source', 'problem', 'answer', 'solution']Example metadata: {'id': 'gsm8k_train_0', 'source': 'gsm8k'}Example problem: Natalia sold clips to 48 of her friends in April, and then she sold half as many clips in May. How many clips did Natalia sell altogether in April and May?MATH strata: Counter({('algebra', 'Level 5'): 433, ('intermediate_algebra', 'Level 5'): 421, ('algebra', 'Level 4'): 392, ('algebra', 'Level 3'): 385, ('geometry', 'Level 5'): 359, ('algebra', 'Level 2'): 338, ('intermediate_algebra', 'Level 4'): 332, ('prealgebra', 'Level 2'): 327, ('number_theory', 'Level 5'): 308, ('intermediate_algebra', 'Level 3'): 295, ('counting_and_probability', 'Level 5'): 251, ('prealgebra', 'Level 5'): 249, ('prealgebra', 'Level 3'): 249, ('prealgebra', 'Level 4'): 245, ('number_theory', 'Level 3'): 190, ('number_theory', 'Level 4'): 184, ('algebra', 'Level 1'): 178, ('geometry', 'Level 4'): 173, ('precalculus', 'Level 5'): 166, ('counting_and_probability', 'Level 4'): 165, ('intermediate_algebra', 'Level 2'): 164, ('precalculus', 'Level 4'): 161, ('precalculus', 'Level 3'): 157, ('counting_and_probability', 'Level 3'): 156, ('precalculus', 'Level 2'): 146, ('number_theory', 'Level 2'): 131, ('geometry', 'Level 3'): 126, ('prealgebra', 'Level 1'): 126, ('counting_and_probability', 'Level 2'): 115, ('geometry', 'Level 2'): 99, ('intermediate_algebra', 'Level 1'): 58, ('precalculus', 'Level 1'): 58, ('counting_and_probability', 'Level 1'): 50, ('number_theory', 'Level 1'): 46, ('geometry', 'Level 1'): 40, ('geometry', 'Level ?'): 2})

Existing edits: []Tensor count: 236Layer-0 tensor names: ['model.layers.0.input_layernorm.weight', 'model.layers.0.mlp.down_proj.weight', 'model.layers.0.mlp.gate_proj.weight', 'model.layers.0.mlp.up_proj.weight', 'model.layers.0.post_attention_layernorm.weight', 'model.layers.0.post_feedforward_layernorm.weight', 'model.layers.0.pre_feedforward_layernorm.weight', 'model.layers.0.self_attn.k_norm.weight', 'model.layers.0.self_attn.k_proj.weight', 'model.layers.0.self_attn.o_proj.weight', 'model.layers.0.self_attn.q_norm.weight', 'model.layers.0.self_attn.q_proj.weight', 'model.layers.0.self_attn.v_proj.weight']Prompt format:The evaluation prompt, exactly as the model sees it (thinking off), for a problem P:PREFIX + P + SUFFIX, withPREFIX = '<bos><start_of_turn>user\n'SUFFIX = '\nPlease reason step by step, and put your final answer within \\boxed{}.<end_of_turn>\n<start_of_turn>model\n'The model's answer follows SUFFIX directly. To score a text with a teacher-forced pass, send PREFIX + P + SUFFIX + answer as one raw text in "texts".
math_train count: 7275 fields: ['id', 'source', 'subject', 'level', 'problem', 'answer', 'solution']Example metadata: {'id': 'math_train_algebra_0', 'source': 'math', 'subject': 'algebra', 'level': 'Level 5'}Example problem: Let \[f(x) = \left\{\begin{array}{cl} ax+3, &\text{ if }x>2, \\x-5 &\text{ if } -2 \le x \le 2, \\2x-b &\text{ if } x <-2.\end{array}\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the papegsm8k_train count: 7469 fields: ['id', 'source', 'problem', 'answer', 'solution']Example metadata: {'id': 'gsm8k_train_0', 'source': 'gsm8k'}Example problem: Natalia sold clips to 48 of her friends in April, and then she sold half as many clips in May. How many clips did Natalia sell altogether in April and May?MATH strata: Counter({('algebra', 'Level 5'): 433, ('intermediate_algebra', 'Level 5'): 421, ('algebra', 'Level 4'): 392, ('algebra', 'Level 3'): 385, ('geometry', 'Level 5'): 359, ('algebra', 'Level 2'): 338, ('intermediate_algebra', 'Level 4'): 332, ('prealgebra', 'Level 2'): 327, ('number_theory', 'Level 5'): 308, ('intermediate_algebra', 'Level 3'): 295, ('counting_and_probability', 'Level 5'): 251, ('prealgebra', 'Level 5'): 249, ('prealgebra', 'Level 3'): 249, ('prealgebra', 'Level 4'): 245, ('number_theory', 'Level 3'): 190, ('number_theory', 'Level 4'): 184, ('algebra', 'Level 1'): 178, ('geometry', 'Level 4'): 173, ('precalculus', 'Level 5'): 166, ('counting_and_probability', 'Level 4'): 165, ('intermediate_algebra', 'Level 2'): 164, ('precalculus', 'Level 4'): 161, ('precalculus', 'Level 3'): 157, ('counting_and_probability', 'Level 3'): 156, ('precalculus', 'Level 2'): 146, ('number_theory', 'Level 2'): 131, ('geometry', 'Level 3'): 126, ('prealgebra', 'Level 1'): 126, ('counting_and_probability', 'Level 2'): 115, ('geometry', 'Level 2'): 99, ('intermediate_algebra', 'Level 1'): 58, ('precalculus', 'Level 1'): 58, ('counting_and_probability', 'Level 1'): 50, ('number_theory', 'Level 1'): 46, ('geometry', 'Level 1'): 40, ('geometry', 'Level ?'): 2})

The checkpoint has the expected layer tensors, and the practice files include problems, answers, and worked solutions. I’ll use MATH problems for the initial diagnosis, emphasizing levels 1–3 so this small model has a reasonable chance of both succeeding and failing on the same problem. I’ll also reserve a separate, stratified HOLDOUT without generating or inspecting its answers. The baseline pass will collect MLP and residual activations; these can suggest components to test, but correlations alone will not justify an edit.

python

rng_exp = random.Random(270031)math_rows = datasets["math_train"]problem_by_id = {r["id"]: r for ds in datasets.values() for r in ds}
def stratified_pick(rows_in, count, rng):    groups = collections.defaultdict(list)    for row in rows_in:        groups[(row.get("subject", ""), row.get("level", ""))].append(row)    allocation = {k: count * len(v) / len(rows_in) for k, v in groups.items()}    quotas = {k: int(v) for k, v in allocation.items()}    remainder = count - sum(quotas.values())    for k in sorted(groups, key=lambda k: allocation[k] - quotas[k], reverse=True)[:remainder]:        quotas[k] += 1    result = []    for k in sorted(groups):        result.extend(rng.sample(groups[k], quotas[k]))    rng.shuffle(result)    return result
holdout_rows = stratified_pick(math_rows, 256, rng_exp)holdout_ids = [r["id"] for r in holdout_rows]holdout_set = set(holdout_ids)baseline_rows = []for level, count in [("Level 1", 96), ("Level 2", 96), ("Level 3", 64)]:    eligible = [r for r in math_rows if r["level"] == level and r["id"] not in holdout_set]    baseline_rows.extend(stratified_pick(eligible, count, rng_exp))baseline_ids = [r["id"] for r in baseline_rows]assert len(set(baseline_ids)) == 256 and not set(baseline_ids) & holdout_set(wdir / "HOLDOUT.json").write_text(json.dumps(holdout_ids, indent=2))(wdir / "BASELINE_POOL.json").write_text(json.dumps(baseline_ids, indent=2))(wdir / "experiment_plan.json").write_text(json.dumps({    "seed": 270031, "holdout": "256 MATH problems, stratified by subject and level",    "dev_rule": "Select up to 128 intermittently solved baseline problems; freeze before editing.",    "baseline_samples_per_problem": 8,    "selection_metric": "Paired DEV answer accuracy, not response length.",    "holdout_rule": "Freeze candidate before any HOLDOUT generation; pool confirmation passes."}, indent=2))(wdir / "notes.md").write_text(    "# Weight-edit experiment\n\n"    "No edits have been made. Embedding/output and final norm are excluded.\n"    "HOLDOUT is fixed in HOLDOUT.json and disjoint from BASELINE_POOL.json.\n"    "DEV will be fixed from baseline problems with 1–7 successes out of 8.\n\n"    "## Pass 1 planned: baseline diagnosis\n"    "Model: original. Exact edit: none; strength 0.\n"    "Set: BASELINE_POOL.json, 256 MATH problems (96 level 1, 96 level 2, 64 level 3).\n"    "Generation: n=8, max_tokens=[REDACTED] evaluation sampling/template.\n"    "Activations: residual states 0–18 and all MLP down-projection inputs.\n")print("Baseline strata:", collections.Counter(r["level"] for r in baseline_rows))print("Pass-1 request:")print(json.dumps({    "type": "generate", "model": "original", "problems": baseline_ids,    "n": 8, "max_tokens": 8192,    "activations": {"layers": list(range(19)), "mlp": True}}))

rng_exp = random.Random(270031)math_rows = datasets["math_train"]problem_by_id = {r["id"]: r for ds in datasets.values() for r in ds}
def stratified_pick(rows_in, count, rng):    groups = collections.defaultdict(list)    for row in rows_in:        groups[(row.get("subject", ""), row.get("level", ""))].append(row)    allocation = {k: count * len(v) / len(rows_in) for k, v in groups.items()}    quotas = {k: int(v) for k, v in allocation.items()}    remainder = count - sum(quotas.values())    for k in sorted(groups, key=lambda k: allocation[k] - quotas[k], reverse=True)[:remainder]:        quotas[k] += 1    result = []    for k in sorted(groups):        result.extend(rng.sample(groups[k], quotas[k]))    rng.shuffle(result)    return result
holdout_rows = stratified_pick(math_rows, 256, rng_exp)holdout_ids = [r["id"] for r in holdout_rows]holdout_set = set(holdout_ids)baseline_rows = []for level, count in [("Level 1", 96), ("Level 2", 96), ("Level 3", 64)]:    eligible = [r for r in math_rows if r["level"] == level and r["id"] not in holdout_set]    baseline_rows.extend(stratified_pick(eligible, count, rng_exp))baseline_ids = [r["id"] for r in baseline_rows]assert len(set(baseline_ids)) == 256 and not set(baseline_ids) & holdout_set(wdir / "HOLDOUT.json").write_text(json.dumps(holdout_ids, indent=2))(wdir / "BASELINE_POOL.json").write_text(json.dumps(baseline_ids, indent=2))(wdir / "experiment_plan.json").write_text(json.dumps({    "seed": 270031, "holdout": "256 MATH problems, stratified by subject and level",    "dev_rule": "Select up to 128 intermittently solved baseline problems; freeze before editing.",    "baseline_samples_per_problem": 8,    "selection_metric": "Paired DEV answer accuracy, not response length.",    "holdout_rule": "Freeze candidate before any HOLDOUT generation; pool confirmation passes."}, indent=2))(wdir / "notes.md").write_text(    "# Weight-edit experiment\n\n"    "No edits have been made. Embedding/output and final norm are excluded.\n"    "HOLDOUT is fixed in HOLDOUT.json and disjoint from BASELINE_POOL.json.\n"    "DEV will be fixed from baseline problems with 1–7 successes out of 8.\n\n"    "## Pass 1 planned: baseline diagnosis\n"    "Model: original. Exact edit: none; strength 0.\n"    "Set: BASELINE_POOL.json, 256 MATH problems (96 level 1, 96 level 2, 64 level 3).\n"    "Generation: n=8, max_tokens=[REDACTED] evaluation sampling/template.\n"    "Activations: residual states 0–18 and all MLP down-projection inputs.\n")print("Baseline strata:", collections.Counter(r["level"] for r in baseline_rows))print("Pass-1 request:")print(json.dumps({    "type": "generate", "model": "original", "problems": baseline_ids,    "n": 8, "max_tokens": 8192,    "activations": {"layers": list(range(19)), "mlp": True}}))

OUTPUT of Python run 3 of at most 300:

Baseline strata: Counter({'Level 1': 96, 'Level 2': 96, 'Level 3': 64})Pass-1 request:{"type": "generate", "model": "original", "problems": ["math_train_precalculus_274", "math_train_algebra_1220", "math_train_algebra_1101", "math_train_algebra_198", "math_train_prealgebra_1121", "math_train_intermediate_algebra_635", "math_train_intermediate_algebra_55", "math_train_algebra_380", "math_train_geometry_743", "math_train_counting_and_probability_680", "math_train_geometry_310", "math_train_algebra_1094", "math_train_algebra_1294", "math_train_precalculus_446", "math_train_precalculus_161", "math_train_counting_and_probability_697", "math_train_number_theory_390", "math_train_algebra_542", "math_train_number_theory_709", "math_train_precalculus_521", "math_train_algebra_7", "math_train_prealgebra_828", "math_train_prealgebra_795", "math_train_number_theory_717", "math_train_prealgebra_783", "math_train_prealgebra_311", "math_train_precalculus_164", "math_train_algebra_1719", "math_train_prealgebra_298", "math_train_prealgebra_590", "math_train_algebra_1164", "math_train_prealgebra_818", "math_train_counting_and_probability_465", "math_train_number_theory_39", "math_train_intermediate_algebra_605", "math_train_intermediate_algebra_1270", "math_train_counting_and_probability_594", "math_train_geometry_386", "math_train_prealgebra_442", "math_train_geometry_236", "math_train_prealgebra_867", "math_train_prealgebra_1198", "math_train_algebra_1024", "math_train_algebra_482", "math_train_number_theory_715", "math_train_counting_and_probability_572", "math_train_prealgebra_1110", "math_train_algebra_744", "math_train_algebra_583", "math_train_intermediate_algebra_1110", "math_train_algebra_485", "math_train_algebra_1284", "math_train_algebra_1497", "math_train_algebra_1388", "math_train_algebra_1559", "math_train_number_theory_220", "math_train_number_theory_399", "math_train_algebra_448", "math_train_prealgebra_1011", "math_train_precalculus_264", "math_train_algebra_1621", "math_train_counting_and_probability_68", "math_train_precalculus_233", "math_train_prealgebra_82", "math_train_counting_and_probability_91", "math_train_algebra_763", "math_train_algebra_989", "math_train_intermediate_algebra_1257", "math_train_intermediate_algebra_847", "math_train_precalculus_322", "math_train_algebra_1722", "math_train_prealgebra_747", "math_train_prealgebra_758", "math_train_algebra_1013", "math_train_counting_and_probability_653", "math_train_algebra_73", "math_train_geometry_161", "math_train_algebra_1181", "math_train_prealgebra_234", "math_train_algebra_775", "math_train_prealgebra_490", "math_train_geometry_737", "math_train_intermediate_algebra_1018", "math_train_prealgebra_556", "math_train_prealgebra_961", "math_train_precalculus_270", "math_train_algebra_477", "math_train_intermediate_algebra_84", "math_train_geometry_752", "math_train_intermediate_algebra_509", "math_train_algebra_955", "math_train_prealgebra_81", "math_train_algebra_817", "math_train_prealgebra_869", "math_train_number_theory_237", "math_train_precalculus_294", "math_train_precalculus_107", "math_train_geometry_259", "math_train_intermediate_algebra_342", "math_train_algebra_358", "math_train_algebra_284", "math_train_prealgebra_1072", "math_train_precalculus_356", "math_train_prealgebra_1181", "math_train_algebra_1096", "math_train_precalculus_330", "math_train_prealgebra_76", "math_train_prealgebra_363", "math_train_intermediate_algebra_206", "math_train_counting_and_probability_532", "math_train_prealgebra_888", "math_train_algebra_1227", "math_train_algebra_1205", "math_train_precalculus_76", "math_train_number_theory_537", "math_train_algebra_1290", "math_train_prealgebra_662", "math_train_intermediate_algebra_1268", "math_train_precalculus_396", "math_train_geometry_113", "math_train_algebra_1180", "math_train_prealgebra_694", "math_train_prealgebra_713", "math_train_algebra_186", "math_train_number_theory_693", "math_train_geometry_169", "math_train_intermediate_algebra_1254", "math_train_prealgebra_969", "math_train_geometry_498", "math_train_number_theory_706", "math_train_prealgebra_959", "math_train_prealgebra_1049", "math_train_counting_and_probability_17", "math_train_counting_and_probability_647", "math_train_algebra_1527", "math_train_algebra_714", "math_train_prealgebra_1024", "math_train_geometry_300", "math_train_prealgebra_435", "math_train_prealgebra_1033", "math_train_prealgebra_1028", "math_train_geometry_291", "math_train_number_theory_780", "math_train_algebra_1491", "math_train_intermediate_algebra_1076", "math_train_precalculus_614", "math_train_counting_and_probability_134", "math_train_intermediate_algebra_1215", "math_train_prealgebra_1130", "math_train_precalculus_167", "math_train_prealgebra_328", "math_train_algebra_619", "math_train_algebra_1665", "math_train_precalculus_30", "math_train_number_theory_409", "math_train_intermediate_algebra_802", "math_train_prealgebra_893", "math_train_number_theory_774", "math_train_counting_and_probability_88", "math_train_intermediate_algebra_860", "math_train_intermediate_algebra_1161", "math_train_algebra_1185", "math_train_counting_and_probability_518", "math_train_algebra_1128", "math_train_geometry_395", "math_train_prealgebra_452", "math_train_intermediate_algebra_257", "math_train_precalculus_252", "math_train_algebra_44", "math_train_algebra_857", "math_train_algebra_988", "math_train_algebra_935", "math_train_intermediate_algebra_989", "math_train_counting_and_probability_41", "math_train_algebra_920", "math_train_algebra_537", "math_train_prealgebra_502", "math_train_algebra_1727", "math_train_algebra_496", "math_train_precalculus_428", "math_train_number_theory_45", "math_train_algebra_878", "math_train_counting_and_probability_719", "math_train_prealgebra_635", "math_train_precalculus_116", "math_train_intermediate_algebra_591", "math_train_prealgebra_464", "math_train_prealgebra_134", "math_train_prealgebra_593", "math_train_number_theory_178", "math_train_number_theory_311", "math_train_algebra_1523", "math_train_counting_and_probability_661", "math_train_geometry_172", "math_train_precalculus_496", "math_train_precalculus_109", "math_train_algebra_492", "math_train_algebra_1048", "math_train_intermediate_algebra_403", "math_train_intermediate_algebra_906", "math_train_prealgebra_1190", "math_train_geometry_692", "math_train_counting_and_probability_600", "math_train_algebra_206", "math_train_precalculus_162", "math_train_number_theory_546", "math_train_algebra_34", "math_train_geometry_305", "math_train_intermediate_algebra_533", "math_train_precalculus_560", "math_train_algebra_361", "math_train_number_theory_352", "math_train_intermediate_algebra_172", "math_train_geometry_267", "math_train_intermediate_algebra_671", "math_train_prealgebra_397", "math_train_algebra_1310", "math_train_intermediate_algebra_1197", "math_train_precalculus_705", "math_train_intermediate_algebra_1008", "math_train_counting_and_probability_55", "math_train_algebra_1457", "math_train_algebra_39", "math_train_prealgebra_866", "math_train_intermediate_algebra_311", "math_train_algebra_1513", "math_train_counting_and_probability_296", "math_train_prealgebra_1146", "math_train_intermediate_algebra_926", "math_train_number_theory_716", "math_train_number_theory_514", "math_train_algebra_1586", "math_train_algebra_1417", "math_train_precalculus_86", "math_train_geometry_45", "math_train_prealgebra_539", "math_train_number_theory_451", "math_train_prealgebra_152", "math_train_intermediate_algebra_694", "math_train_prealgebra_586", "math_train_algebra_1210", "math_train_algebra_1271", "math_train_prealgebra_755", "math_train_number_theory_747", "math_train_counting_and_probability_301", "math_train_number_theory_46", "math_train_algebra_1520", "math_train_precalculus_219", "math_train_prealgebra_720", "math_train_intermediate_algebra_123", "math_train_algebra_470", "math_train_counting_and_probability_269", "math_train_intermediate_algebra_742", "math_train_number_theory_40", "math_train_prealgebra_1196", "math_train_algebra_377"], "n": 8, "max_tokens": 8192, "activations": {"layers": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], "mlp": true}}

Baseline strata: Counter({'Level 1': 96, 'Level 2': 96, 'Level 3': 64})Pass-1 request:{"type": "generate", "model": "original", "problems": ["math_train_precalculus_274", "math_train_algebra_1220", "math_train_algebra_1101", "math_train_algebra_198", "math_train_prealgebra_1121", "math_train_intermediate_algebra_635", "math_train_intermediate_algebra_55", "math_train_algebra_380", "math_train_geometry_743", "math_train_counting_and_probability_680", "math_train_geometry_310", "math_train_algebra_1094", "math_train_algebra_1294", "math_train_precalculus_446", "math_train_precalculus_161", "math_train_counting_and_probability_697", "math_train_number_theory_390", "math_train_algebra_542", "math_train_number_theory_709", "math_train_precalculus_521", "math_train_algebra_7", "math_train_prealgebra_828", "math_train_prealgebra_795", "math_train_number_theory_717", "math_train_prealgebra_783", "math_train_prealgebra_311", "math_train_precalculus_164", "math_train_algebra_1719", "math_train_prealgebra_298", "math_train_prealgebra_590", "math_train_algebra_1164", "math_train_prealgebra_818", "math_train_counting_and_probability_465", "math_train_number_theory_39", "math_train_intermediate_algebra_605", "math_train_intermediate_algebra_1270", "math_train_counting_and_probability_594", "math_train_geometry_386", "math_train_prealgebra_442", "math_train_geometry_236", "math_train_prealgebra_867", "math_train_prealgebra_1198", "math_train_algebra_1024", "math_train_algebra_482", "math_train_number_theory_715", "math_train_counting_and_probability_572", "math_train_prealgebra_1110", "math_train_algebra_744", "math_train_algebra_583", "math_train_intermediate_algebra_1110", "math_train_algebra_485", "math_train_algebra_1284", "math_train_algebra_1497", "math_train_algebra_1388", "math_train_algebra_1559", "math_train_number_theory_220", "math_train_number_theory_399", "math_train_algebra_448", "math_train_prealgebra_1011", "math_train_precalculus_264", "math_train_algebra_1621", "math_train_counting_and_probability_68", "math_train_precalculus_233", "math_train_prealgebra_82", "math_train_counting_and_probability_91", "math_train_algebra_763", "math_train_algebra_989", "math_train_intermediate_algebra_1257", "math_train_intermediate_algebra_847", "math_train_precalculus_322", "math_train_algebra_1722", "math_train_prealgebra_747", "math_train_prealgebra_758", "math_train_algebra_1013", "math_train_counting_and_probability_653", "math_train_algebra_73", "math_train_geometry_161", "math_train_algebra_1181", "math_train_prealgebra_234", "math_train_algebra_775", "math_train_prealgebra_490", "math_train_geometry_737", "math_train_intermediate_algebra_1018", "math_train_prealgebra_556", "math_train_prealgebra_961", "math_train_precalculus_270", "math_train_algebra_477", "math_train_intermediate_algebra_84", "math_train_geometry_752", "math_train_intermediate_algebra_509", "math_train_algebra_955", "math_train_prealgebra_81", "math_train_algebra_817", "math_train_prealgebra_869", "math_train_number_theory_237", "math_train_precalculus_294", "math_train_precalculus_107", "math_train_geometry_259", "math_train_intermediate_algebra_342", "math_train_algebra_358", "math_train_algebra_284", "math_train_prealgebra_1072", "math_train_precalculus_356", "math_train_prealgebra_1181", "math_train_algebra_1096", "math_train_precalculus_330", "math_train_prealgebra_76", "math_train_prealgebra_363", "math_train_intermediate_algebra_206", "math_train_counting_and_probability_532", "math_train_prealgebra_888", "math_train_algebra_1227", "math_train_algebra_1205", "math_train_precalculus_76", "math_train_number_theory_537", "math_train_algebra_1290", "math_train_prealgebra_662", "math_train_intermediate_algebra_1268", "math_train_precalculus_396", "math_train_geometry_113", "math_train_algebra_1180", "math_train_prealgebra_694", "math_train_prealgebra_713", "math_train_algebra_186", "math_train_number_theory_693", "math_train_geometry_169", "math_train_intermediate_algebra_1254", "math_train_prealgebra_969", "math_train_geometry_498", "math_train_number_theory_706", "math_train_prealgebra_959", "math_train_prealgebra_1049", "math_train_counting_and_probability_17", "math_train_counting_and_probability_647", "math_train_algebra_1527", "math_train_algebra_714", "math_train_prealgebra_1024", "math_train_geometry_300", "math_train_prealgebra_435", "math_train_prealgebra_1033", "math_train_prealgebra_1028", "math_train_geometry_291", "math_train_number_theory_780", "math_train_algebra_1491", "math_train_intermediate_algebra_1076", "math_train_precalculus_614", "math_train_counting_and_probability_134", "math_train_intermediate_algebra_1215", "math_train_prealgebra_1130", "math_train_precalculus_167", "math_train_prealgebra_328", "math_train_algebra_619", "math_train_algebra_1665", "math_train_precalculus_30", "math_train_number_theory_409", "math_train_intermediate_algebra_802", "math_train_prealgebra_893", "math_train_number_theory_774", "math_train_counting_and_probability_88", "math_train_intermediate_algebra_860", "math_train_intermediate_algebra_1161", "math_train_algebra_1185", "math_train_counting_and_probability_518", "math_train_algebra_1128", "math_train_geometry_395", "math_train_prealgebra_452", "math_train_intermediate_algebra_257", "math_train_precalculus_252", "math_train_algebra_44", "math_train_algebra_857", "math_train_algebra_988", "math_train_algebra_935", "math_train_intermediate_algebra_989", "math_train_counting_and_probability_41", "math_train_algebra_920", "math_train_algebra_537", "math_train_prealgebra_502", "math_train_algebra_1727", "math_train_algebra_496", "math_train_precalculus_428", "math_train_number_theory_45", "math_train_algebra_878", "math_train_counting_and_probability_719", "math_train_prealgebra_635", "math_train_precalculus_116", "math_train_intermediate_algebra_591", "math_train_prealgebra_464", "math_train_prealgebra_134", "math_train_prealgebra_593", "math_train_number_theory_178", "math_train_number_theory_311", "math_train_algebra_1523", "math_train_counting_and_probability_661", "math_train_geometry_172", "math_train_precalculus_496", "math_train_precalculus_109", "math_train_algebra_492", "math_train_algebra_1048", "math_train_intermediate_algebra_403", "math_train_intermediate_algebra_906", "math_train_prealgebra_1190", "math_train_geometry_692", "math_train_counting_and_probability_600", "math_train_algebra_206", "math_train_precalculus_162", "math_train_number_theory_546", "math_train_algebra_34", "math_train_geometry_305", "math_train_intermediate_algebra_533", "math_train_precalculus_560", "math_train_algebra_361", "math_train_number_theory_352", "math_train_intermediate_algebra_172", "math_train_geometry_267", "math_train_intermediate_algebra_671", "math_train_prealgebra_397", "math_train_algebra_1310", "math_train_intermediate_algebra_1197", "math_train_precalculus_705", "math_train_intermediate_algebra_1008", "math_train_counting_and_probability_55", "math_train_algebra_1457", "math_train_algebra_39", "math_train_prealgebra_866", "math_train_intermediate_algebra_311", "math_train_algebra_1513", "math_train_counting_and_probability_296", "math_train_prealgebra_1146", "math_train_intermediate_algebra_926", "math_train_number_theory_716", "math_train_number_theory_514", "math_train_algebra_1586", "math_train_algebra_1417", "math_train_precalculus_86", "math_train_geometry_45", "math_train_prealgebra_539", "math_train_number_theory_451", "math_train_prealgebra_152", "math_train_intermediate_algebra_694", "math_train_prealgebra_586", "math_train_algebra_1210", "math_train_algebra_1271", "math_train_prealgebra_755", "math_train_number_theory_747", "math_train_counting_and_probability_301", "math_train_number_theory_46", "math_train_algebra_1520", "math_train_precalculus_219", "math_train_prealgebra_720", "math_train_intermediate_algebra_123", "math_train_algebra_470", "math_train_counting_and_probability_269", "math_train_intermediate_algebra_742", "math_train_number_theory_40", "math_train_prealgebra_1196", "math_train_algebra_377"], "n": 8, "max_tokens": 8192, "activations": {"layers": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], "mlp": true}}

The baseline pool and untouched HOLDOUT are now saved, and no weights have changed. I’ll generate eight full-length answers for each of the 256 baseline problems, recording activations throughout the network. This will distinguish wrong completed answers from missing answers and token-limit cutoffs, while identifying problems suitable for DEV. A useful localization signal would recur across correct-versus-failing answers to the same problems, rather than merely separating easy and hard problems.

forward

{"type":"generate","model":"original","problems":["math_train_precalculus_274","math_train_algebra_1220","math_train_algebra_1101","math_train_algebra_198","math_train_prealgebra_1121","math_train_intermediate_algebra_635","math_train_intermediate_algebra_55","math_train_algebra_380","math_train_geometry_743","math_train_counting_and_probability_680","math_train_geometry_310","math_train_algebra_1094","math_train_algebra_1294","math_train_precalculus_446","math_train_precalculus_161","math_train_counting_and_probability_697","math_train_number_theory_390","math_train_algebra_542","math_train_number_theory_709","math_train_precalculus_521","math_train_algebra_7","math_train_prealgebra_828","math_train_prealgebra_795","math_train_number_theory_717","math_train_prealgebra_783","math_train_prealgebra_311","math_train_precalculus_164","math_train_algebra_1719","math_train_prealgebra_298","math_train_prealgebra_590","math_train_algebra_1164","math_train_prealgebra_818","math_train_counting_and_probability_465","math_train_number_theory_39","math_train_intermediate_algebra_605","math_train_intermediate_algebra_1270","math_train_counting_and_probability_594","math_train_geometry_386","math_train_prealgebra_442","math_train_geometry_236","math_train_prealgebra_867","math_train_prealgebra_1198","math_train_algebra_1024","math_train_algebra_482","math_train_number_theory_715","math_train_counting_and_probability_572","math_train_prealgebra_1110","math_train_algebra_744","math_train_algebra_583","math_train_intermediate_algebra_1110","math_train_algebra_485","math_train_algebra_1284","math_train_algebra_1497","math_train_algebra_1388","math_train_algebra_1559","math_train_number_theory_220","math_train_number_theory_399","math_train_algebra_448","math_train_prealgebra_1011","math_train_precalculus_264","math_train_algebra_1621","math_train_counting_and_probability_68","math_train_precalculus_233","math_train_prealgebra_82","math_train_counting_and_probability_91","math_train_algebra_763","math_train_algebra_989","math_train_intermediate_algebra_1257","math_train_intermediate_algebra_847","math_train_precalculus_322","math_train_algebra_1722","math_train_prealgebra_747","math_train_prealgebra_758","math_train_algebra_1013","math_train_counting_and_probability_653","math_train_algebra_73","math_train_geometry_161","math_train_algebra_1181","math_train_prealgebra_234","math_train_algebra_775","math_train_prealgebra_490","math_train_geometry_737","math_train_intermediate_algebra_1018","math_train_prealgebra_556","math_train_prealgebra_961","math_train_precalculus_270","math_train_algebra_477","math_train_intermediate_algebra_84","math_train_geometry_752","math_train_intermediate_algebra_509","math_train_algebra_955","math_train_prealgebra_81","math_train_algebra_817","math_train_prealgebra_869","math_train_number_theory_237","math_train_precalculus_294","math_train_precalculus_107","math_train_geometry_259","math_train_intermediate_algebra_342","math_train_algebra_358","math_train_algebra_284","math_train_prealgebra_1072","math_train_precalculus_356","math_train_prealgebra_1181","math_train_algebra_1096","math_train_precalculus_330","math_train_prealgebra_76","math_train_prealgebra_363","math_train_intermediate_algebra_206","math_train_counting_and_probability_532","math_train_prealgebra_888","math_train_algebra_1227","math_train_algebra_1205","math_train_precalculus_76","math_train_number_theory_537","math_train_algebra_1290","math_train_prealgebra_662","math_train_intermediate_algebra_1268","math_train_precalculus_396","math_train_geometry_113","math_train_algebra_1180","math_train_prealgebra_694","math_train_prealgebra_713","math_train_algebra_186","math_train_number_theory_693","math_train_geometry_169","math_train_intermediate_algebra_1254","math_train_prealgebra_969","math_train_geometry_498","math_train_number_theory_706","math_train_prealgebra_959","math_train_prealgebra_1049","math_train_counting_and_probability_17","math_train_counting_and_probability_647","math_train_algebra_1527","math_train_algebra_714","math_train_prealgebra_1024","math_train_geometry_300","math_train_prealgebra_435","math_train_prealgebra_1033","math_train_prealgebra_1028","math_train_geometry_291","math_train_number_theory_780","math_train_algebra_1491","math_train_intermediate_algebra_1076","math_train_precalculus_614","math_train_counting_and_probability_134","math_train_intermediate_algebra_1215","math_train_prealgebra_1130","math_train_precalculus_167","math_train_prealgebra_328","math_train_algebra_619","math_train_algebra_1665","math_train_precalculus_30","math_train_number_theory_409","math_train_intermediate_algebra_802","math_train_prealgebra_893","math_train_number_theory_774","math_train_counting_and_probability_88","math_train_intermediate_algebra_860","math_train_intermediate_algebra_1161","math_train_algebra_1185","math_train_counting_and_probability_518","math_train_algebra_1128","math_train_geometry_395","math_train_prealgebra_452","math_train_intermediate_algebra_257","math_train_precalculus_252","math_train_algebra_44","math_train_algebra_857","math_train_algebra_988","math_train_algebra_935","math_train_intermediate_algebra_989","math_train_counting_and_probability_41","math_train_algebra_920","math_train_algebra_537","math_train_prealgebra_502","math_train_algebra_1727","math_train_algebra_496","math_train_precalculus_428","math_train_number_theory_45","math_train_algebra_878","math_train_counting_and_probability_719","math_train_prealgebra_635","math_train_precalculus_116","math_train_intermediate_algebra_591","math_train_prealgebra_464","math_train_prealgebra_134","math_train_prealgebra_593","math_train_number_theory_178","math_train_number_theory_311","math_train_algebra_1523","math_train_counting_and_probability_661","math_train_geometry_172","math_train_precalculus_496","math_train_precalculus_109","math_train_algebra_492","math_train_algebra_1048","math_train_intermediate_algebra_403","math_train_intermediate_algebra_906","math_train_prealgebra_1190","math_train_geometry_692","math_train_counting_and_probability_600","math_train_algebra_206","math_train_precalculus_162","math_train_number_theory_546","math_train_algebra_34","math_train_geometry_305","math_train_intermediate_algebra_533","math_train_precalculus_560","math_train_algebra_361","math_train_number_theory_352","math_train_intermediate_algebra_172","math_train_geometry_267","math_train_intermediate_algebra_671","math_train_prealgebra_397","math_train_algebra_1310","math_train_intermediate_algebra_1197","math_train_precalculus_705","math_train_intermediate_algebra_1008","math_train_counting_and_probability_55","math_train_algebra_1457","math_train_algebra_39","math_train_prealgebra_866","math_train_intermediate_algebra_311","math_train_algebra_1513","math_train_counting_and_probability_296","math_train_prealgebra_1146","math_train_intermediate_algebra_926","math_train_number_theory_716","math_train_number_theory_514","math_train_algebra_1586","math_train_algebra_1417","math_train_precalculus_86","math_train_geometry_45","math_train_prealgebra_539","math_train_number_theory_451","math_train_prealgebra_152","math_train_intermediate_algebra_694","math_train_prealgebra_586","math_train_algebra_1210","math_train_algebra_1271","math_train_prealgebra_755","math_train_number_theory_747","math_train_counting_and_probability_301","math_train_number_theory_46","math_train_algebra_1520","math_train_precalculus_219","math_train_prealgebra_720","math_train_intermediate_algebra_123","math_train_algebra_470","math_train_counting_and_probability_269","math_train_intermediate_algebra_742","math_train_number_theory_40","math_train_prealgebra_1196","math_train_algebra_377"],"n":8,"max_tokens":8192,"activations":{"layers":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],"mlp":true}}

{"type":"generate","model":"original","problems":["math_train_precalculus_274","math_train_algebra_1220","math_train_algebra_1101","math_train_algebra_198","math_train_prealgebra_1121","math_train_intermediate_algebra_635","math_train_intermediate_algebra_55","math_train_algebra_380","math_train_geometry_743","math_train_counting_and_probability_680","math_train_geometry_310","math_train_algebra_1094","math_train_algebra_1294","math_train_precalculus_446","math_train_precalculus_161","math_train_counting_and_probability_697","math_train_number_theory_390","math_train_algebra_542","math_train_number_theory_709","math_train_precalculus_521","math_train_algebra_7","math_train_prealgebra_828","math_train_prealgebra_795","math_train_number_theory_717","math_train_prealgebra_783","math_train_prealgebra_311","math_train_precalculus_164","math_train_algebra_1719","math_train_prealgebra_298","math_train_prealgebra_590","math_train_algebra_1164","math_train_prealgebra_818","math_train_counting_and_probability_465","math_train_number_theory_39","math_train_intermediate_algebra_605","math_train_intermediate_algebra_1270","math_train_counting_and_probability_594","math_train_geometry_386","math_train_prealgebra_442","math_train_geometry_236","math_train_prealgebra_867","math_train_prealgebra_1198","math_train_algebra_1024","math_train_algebra_482","math_train_number_theory_715","math_train_counting_and_probability_572","math_train_prealgebra_1110","math_train_algebra_744","math_train_algebra_583","math_train_intermediate_algebra_1110","math_train_algebra_485","math_train_algebra_1284","math_train_algebra_1497","math_train_algebra_1388","math_train_algebra_1559","math_train_number_theory_220","math_train_number_theory_399","math_train_algebra_448","math_train_prealgebra_1011","math_train_precalculus_264","math_train_algebra_1621","math_train_counting_and_probability_68","math_train_precalculus_233","math_train_prealgebra_82","math_train_counting_and_probability_91","math_train_algebra_763","math_train_algebra_989","math_train_intermediate_algebra_1257","math_train_intermediate_algebra_847","math_train_precalculus_322","math_train_algebra_1722","math_train_prealgebra_747","math_train_prealgebra_758","math_train_algebra_1013","math_train_counting_and_probability_653","math_train_algebra_73","math_train_geometry_161","math_train_algebra_1181","math_train_prealgebra_234","math_train_algebra_775","math_train_prealgebra_490","math_train_geometry_737","math_train_intermediate_algebra_1018","math_train_prealgebra_556","math_train_prealgebra_961","math_train_precalculus_270","math_train_algebra_477","math_train_intermediate_algebra_84","math_train_geometry_752","math_train_intermediate_algebra_509","math_train_algebra_955","math_train_prealgebra_81","math_train_algebra_817","math_train_prealgebra_869","math_train_number_theory_237","math_train_precalculus_294","math_train_precalculus_107","math_train_geometry_259","math_train_intermediate_algebra_342","math_train_algebra_358","math_train_algebra_284","math_train_prealgebra_1072","math_train_precalculus_356","math_train_prealgebra_1181","math_train_algebra_1096","math_train_precalculus_330","math_train_prealgebra_76","math_train_prealgebra_363","math_train_intermediate_algebra_206","math_train_counting_and_probability_532","math_train_prealgebra_888","math_train_algebra_1227","math_train_algebra_1205","math_train_precalculus_76","math_train_number_theory_537","math_train_algebra_1290","math_train_prealgebra_662","math_train_intermediate_algebra_1268","math_train_precalculus_396","math_train_geometry_113","math_train_algebra_1180","math_train_prealgebra_694","math_train_prealgebra_713","math_train_algebra_186","math_train_number_theory_693","math_train_geometry_169","math_train_intermediate_algebra_1254","math_train_prealgebra_969","math_train_geometry_498","math_train_number_theory_706","math_train_prealgebra_959","math_train_prealgebra_1049","math_train_counting_and_probability_17","math_train_counting_and_probability_647","math_train_algebra_1527","math_train_algebra_714","math_train_prealgebra_1024","math_train_geometry_300","math_train_prealgebra_435","math_train_prealgebra_1033","math_train_prealgebra_1028","math_train_geometry_291","math_train_number_theory_780","math_train_algebra_1491","math_train_intermediate_algebra_1076","math_train_precalculus_614","math_train_counting_and_probability_134","math_train_intermediate_algebra_1215","math_train_prealgebra_1130","math_train_precalculus_167","math_train_prealgebra_328","math_train_algebra_619","math_train_algebra_1665","math_train_precalculus_30","math_train_number_theory_409","math_train_intermediate_algebra_802","math_train_prealgebra_893","math_train_number_theory_774","math_train_counting_and_probability_88","math_train_intermediate_algebra_860","math_train_intermediate_algebra_1161","math_train_algebra_1185","math_train_counting_and_probability_518","math_train_algebra_1128","math_train_geometry_395","math_train_prealgebra_452","math_train_intermediate_algebra_257","math_train_precalculus_252","math_train_algebra_44","math_train_algebra_857","math_train_algebra_988","math_train_algebra_935","math_train_intermediate_algebra_989","math_train_counting_and_probability_41","math_train_algebra_920","math_train_algebra_537","math_train_prealgebra_502","math_train_algebra_1727","math_train_algebra_496","math_train_precalculus_428","math_train_number_theory_45","math_train_algebra_878","math_train_counting_and_probability_719","math_train_prealgebra_635","math_train_precalculus_116","math_train_intermediate_algebra_591","math_train_prealgebra_464","math_train_prealgebra_134","math_train_prealgebra_593","math_train_number_theory_178","math_train_number_theory_311","math_train_algebra_1523","math_train_counting_and_probability_661","math_train_geometry_172","math_train_precalculus_496","math_train_precalculus_109","math_train_algebra_492","math_train_algebra_1048","math_train_intermediate_algebra_403","math_train_intermediate_algebra_906","math_train_prealgebra_1190","math_train_geometry_692","math_train_counting_and_probability_600","math_train_algebra_206","math_train_precalculus_162","math_train_number_theory_546","math_train_algebra_34","math_train_geometry_305","math_train_intermediate_algebra_533","math_train_precalculus_560","math_train_algebra_361","math_train_number_theory_352","math_train_intermediate_algebra_172","math_train_geometry_267","math_train_intermediate_algebra_671","math_train_prealgebra_397","math_train_algebra_1310","math_train_intermediate_algebra_1197","math_train_precalculus_705","math_train_intermediate_algebra_1008","math_train_counting_and_probability_55","math_train_algebra_1457","math_train_algebra_39","math_train_prealgebra_866","math_train_intermediate_algebra_311","math_train_algebra_1513","math_train_counting_and_probability_296","math_train_prealgebra_1146","math_train_intermediate_algebra_926","math_train_number_theory_716","math_train_number_theory_514","math_train_algebra_1586","math_train_algebra_1417","math_train_precalculus_86","math_train_geometry_45","math_train_prealgebra_539","math_train_number_theory_451","math_train_prealgebra_152","math_train_intermediate_algebra_694","math_train_prealgebra_586","math_train_algebra_1210","math_train_algebra_1271","math_train_prealgebra_755","math_train_number_theory_747","math_train_counting_and_probability_301","math_train_number_theory_46","math_train_algebra_1520","math_train_precalculus_219","math_train_prealgebra_720","math_train_intermediate_algebra_123","math_train_algebra_470","math_train_counting_and_probability_269","math_train_intermediate_algebra_742","math_train_number_theory_40","math_train_prealgebra_1196","math_train_algebra_377"],"n":8,"max_tokens":8192,"activations":{"layers":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],"mlp":true}}

Forward pass 1 of 10 done (generate); results in /work/gen_1.jsonl. Summary:

{ "activations": {  "dir": "/work/gen_1_acts/",  "files": {   "hidden_last_prompt.npy": [    2048,    19,    640   ],   "hidden_mean_answer.npy": [    2048,    19,    640   ],   "mlp_last_prompt.npy": [    2048,    18,    2048   ],   "mlp_mean_answer.npy": [    2048,    18,    2048   ]  } }, "ok": true, "type": "generate", "model": "original", "file": "/work/gen_1.jsonl", "generations": 2048, "graded": 2048, "accuracy": 0.17626953125, "accuracy_by_source": {  "math": 0.1763 }, "mean_tokens": 1313, "cut_share": 0.0698, "failures": {  "wrong_answer": 827,  "no_answer_finished": 726,  "cut_at_max_tokens": 134 }, "s": 354.3}

{ "activations": {  "dir": "/work/gen_1_acts/",  "files": {   "hidden_last_prompt.npy": [    2048,    19,    640   ],   "hidden_mean_answer.npy": [    2048,    19,    640   ],   "mlp_last_prompt.npy": [    2048,    18,    2048   ],   "mlp_mean_answer.npy": [    2048,    18,    2048   ]  } }, "ok": true, "type": "generate", "model": "original", "file": "/work/gen_1.jsonl", "generations": 2048, "graded": 2048, "accuracy": 0.17626953125, "accuracy_by_source": {  "math": 0.1763 }, "mean_tokens": 1313, "cut_share": 0.0698, "failures": {  "wrong_answer": 827,  "no_answer_finished": 726,  "cut_at_max_tokens": 134 }, "s": 354.3}

Forward passes left: 9. Edits used in this pass: none (edit hash none).

The unedited model answered 17.6% correctly, but many failures were not ordinary wrong numerical answers: 726 finished without an extracted answer, and 134 failures hit the token limit. That makes failure to carry reasoning through to a definite conclusion a plausible mechanism to investigate. I’ll inspect short excerpts, freeze DEV from intermittently solved problems, and compare activations within those problems. If the apparent signal disappears after accounting for response length, it would be weak evidence for a repair target.

python

with open(wdir / "gen_1.jsonl") as f:    gen1 = [json.loads(line) for line in f]print("Generation fields:", list(gen1[0]))print("First record metadata:", {k: v for k, v in gen1[0].items()                                  if not isinstance(v, (list, dict)) and                                  (not isinstance(v, str) or len(v) < 120)})print("Activation files:", sorted(p.name for p in (wdir / "gen_1_acts").iterdir()))for i in [0, 1, 8, 17, 37, 72, 119, 203]:    rec = gen1[i]    textfields = {k: v for k, v in rec.items() if isinstance(v, str) and len(v) >= 120}    print("Record", i, {k: v for k, v in rec.items()                       if k in ["id", "problem_id", "correct", "is_correct", "grade",                                "finish_reason", "num_tokens", "generated_tokens", "extracted"]})    for k, v in textfields.items():        if k not in ("problem", "prompt", "solution", "reference_solution"):            print(k, "start:", v[:140].encode("ascii", "backslashreplace").decode(),                  "end:", v[-140:].encode("ascii", "backslashreplace").decode())

with open(wdir / "gen_1.jsonl") as f:    gen1 = [json.loads(line) for line in f]print("Generation fields:", list(gen1[0]))print("First record metadata:", {k: v for k, v in gen1[0].items()                                  if not isinstance(v, (list, dict)) and                                  (not isinstance(v, str) or len(v) < 120)})print("Activation files:", sorted(p.name for p in (wdir / "gen_1_acts").iterdir()))for i in [0, 1, 8, 17, 37, 72, 119, 203]:    rec = gen1[i]    textfields = {k: v for k, v in rec.items() if isinstance(v, str) and len(v) >= 120}    print("Record", i, {k: v for k, v in rec.items()                       if k in ["id", "problem_id", "correct", "is_correct", "grade",                                "finish_reason", "num_tokens", "generated_tokens", "extracted"]})    for k, v in textfields.items():        if k not in ("problem", "prompt", "solution", "reference_solution"):            print(k, "start:", v[:140].encode("ascii", "backslashreplace").decode(),                  "end:", v[-140:].encode("ascii", "backslashreplace").decode())

OUTPUT of Python run 4 of at most 300:

Generation fields: ['id', 'source', 'answer', 'samples']First record metadata: {'id': 'math_train_precalculus_274', 'source': 'math', 'answer': '2 \\pi'}Activation files: ['hidden_last_prompt.npy', 'hidden_mean_answer.npy', 'index.json', 'mlp_last_prompt.npy', 'mlp_mean_answer.npy']Record 0 {'id': 'math_train_precalculus_274'}Record 1 {'id': 'math_train_algebra_1220'}Record 8 {'id': 'math_train_geometry_743'}Record 17 {'id': 'math_train_algebra_542'}Record 37 {'id': 'math_train_geometry_386'}Record 72 {'id': 'math_train_prealgebra_758'}Record 119 {'id': 'math_train_geometry_113'}Record 203 {'id': 'math_train_algebra_206'}

Generation fields: ['id', 'source', 'answer', 'samples']First record metadata: {'id': 'math_train_precalculus_274', 'source': 'math', 'answer': '2 \\pi'}Activation files: ['hidden_last_prompt.npy', 'hidden_mean_answer.npy', 'index.json', 'mlp_last_prompt.npy', 'mlp_mean_answer.npy']Record 0 {'id': 'math_train_precalculus_274'}Record 1 {'id': 'math_train_algebra_1220'}Record 8 {'id': 'math_train_geometry_743'}Record 17 {'id': 'math_train_algebra_542'}Record 37 {'id': 'math_train_geometry_386'}Record 72 {'id': 'math_train_prealgebra_758'}Record 119 {'id': 'math_train_geometry_113'}Record 203 {'id': 'math_train_algebra_206'}

The result file groups samples by problem, so the first inspection did not yet reach the answer texts. I’ll inspect the nested sample schema and activation index before calculating any statistics. This will ensure that each correctness label is matched to the right activation row. The important question remains whether unfinished mathematical reasoning has a consistent internal signature within the same problem.

python

print("Sample container:", type(gen1[0]["samples"]).__name__)sample0 = gen1[0]["samples"][0]print("Sample fields:", list(sample0))print("Sample metadata:", {k: v for k, v in sample0.items()                            if not isinstance(v, (list, dict)) and                            (not isinstance(v, str) or len(v) < 150)})with open(wdir / "gen_1_acts" / "index.json") as f:    act_index1 = json.load(f)print("Activation index type:", type(act_index1).__name__)if isinstance(act_index1, list):    print("Index length and first entries:", len(act_index1), act_index1[:2])else:    print("Index keys:", list(act_index1)[:15])    print("Index preview:", str(act_index1)[:700])for rec in gen1[:8]:    print("Problem", rec["id"], "reference:", rec["answer"])    for s in rec["samples"][:2]:        short = {k: v for k, v in s.items() if not isinstance(v, (list, dict))                 and (not isinstance(v, str) or len(v) < 100)}        print(short)        for k, v in s.items():            if isinstance(v, str) and len(v) >= 100:                print(k, (v[:130] + " ... " + v[-150:]).encode("ascii", "backslashreplace").decode())

print("Sample container:", type(gen1[0]["samples"]).__name__)sample0 = gen1[0]["samples"][0]print("Sample fields:", list(sample0))print("Sample metadata:", {k: v for k, v in sample0.items()                            if not isinstance(v, (list, dict)) and                            (not isinstance(v, str) or len(v) < 150)})with open(wdir / "gen_1_acts" / "index.json") as f:    act_index1 = json.load(f)print("Activation index type:", type(act_index1).__name__)if isinstance(act_index1, list):    print("Index length and first entries:", len(act_index1), act_index1[:2])else:    print("Index keys:", list(act_index1)[:15])    print("Index preview:", str(act_index1)[:700])for rec in gen1[:8]:    print("Problem", rec["id"], "reference:", rec["answer"])    for s in rec["samples"][:2]:        short = {k: v for k, v in s.items() if not isinstance(v, (list, dict))                 and (not isinstance(v, str) or len(v) < 100)}        print(short)        for k, v in s.items():            if isinstance(v, str) and len(v) >= 100:                print(k, (v[:130] + " ... " + v[-150:]).encode("ascii", "backslashreplace").decode())

OUTPUT of Python run 5 of at most 300:

Sample container: listSample fields: ['text', 'pred', 'correct', 'tokens', 'cut']Sample metadata: {'pred': '2n\\pi', 'correct': False, 'tokens': 787, 'cut': False}Activation index type: dictIndex keys: ['layers', 'mlp_layers', 'samples', 'files']Index preview: {'layers': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], 'mlp_layers': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], 'samples': [{'item': 'math_train_precalculus_274', 'sample': 0, 'correct': False, 'tokens': 828}, {'item': 'math_train_precalculus_274', 'sample': 1, 'correct': True, 'tokens': 153}, {'item': 'math_train_precalculus_274', 'sample': 2, 'correct': True, 'tokens': 366}, {'item': 'math_train_precalculus_274', 'sample': 3, 'correct': True, 'tokens': 176}, {'item': 'math_train_precalculus_274', 'sample': 4, 'correct': True, 'tokens': 237}, {'item': 'math_train_precalculus_274', 'sample': 5, 'correct': True, 'tokens': 181}, {'item': 'math_trainProblem math_train_precalculus_274 reference: 2 \pi{'pred': '2n\\pi', 'correct': False, 'tokens': 787, 'cut': False}text Let $y = \sin x + \cos x$.We want to find the period of $y$.The period of the sine function is the value of $x$ such that $\sin( ... \sin x + \cos x$ is $2n \pi$.The period of $y = \sin x + \cos x$ is $2n \pi$ for some integer $n$.
Final Answer: The final answer is $\boxed{2n\pi}${'pred': '2\\pi', 'correct': True, 'tokens': 112, 'cut': False}text The period of the function $y = \sin x + \cos x$ is $2\pi$.We know that the period of the sine function is $2\pi$.We want to fin ... w that $y = \sin x + \cos x$.The period of the sine function is $2\pi$.Thus, the period is $2\pi$.
Final Answer: The final answer is $\boxed{2\pi}$Problem math_train_algebra_1220 reference: 2800{'pred': None, 'correct': False, 'tokens': 146, 'cut': False}text To evaluate $64^2 - 36^2$, we can use the property of exponents.$64^2 = 64 \cdot 64 = 64^2$$36^2 = 36 \cdot 36 = 36^2$Therefore ...  64^2$$36^2 = 36 \cdot 36 = 36^2$Therefore, $64^2 - 36^2 = 64^2 - 36^2 = 36^2 - 36^2 = 36^2 - 36^2 = 0$.Thus, $64^2 - 36^2 = 0$.
Final Answer: $0${'pred': None, 'correct': True, 'tokens': 210, 'cut': False}text To evaluate the expression $64^2 - 36^2$, we can rewrite it as$$64^2 - 36^2 = (64 - 36)(64 + 36) = (28)(100) = 2800$$So, the exp ... ve $64^2 - 36^2 = (64 - 36)(64 + 36) = (28)(100) = 2800$.
Therefore, the expression is $64^2 - 36^2 = 2800$.
Final Answer: 2800
Final Answer: 2800
Problem math_train_algebra_1101 reference: 40{'pred': None, 'correct': False, 'tokens': 795, 'cut': False}text Let the sequence be denoted by $a_n$ for $n \geq 1$. The terms are $a_1 = 2, a_2 = 6, a_3 = 10, \ldots, a_n = n$.The sequence is  ... want $a_n = n$, so $4n - 2 = n$, which means $3n = 2$, so $n = \frac{2}{3}$.Then $x = 26$.Thus, $x + y = 26$.
Final Answer: The final answer is 26.{'pred': '12', 'correct': False, 'tokens': 750, 'cut': False}text Let the given sequence be $a_n = 2n$, $a_n = 6n$, $a_n = 10n$, and $a_n = 26n$.The sequence is an arithmetic sequence.The differ ...  - a_1 = 8$, so $a_n = a_1 + 8 = 2 + 8 = 10$.Therefore, $x = 10$, $y = 2$, and $x + y = 10 + 2 = 12$.
Final Answer: The final answer is $\boxed{12}$Problem math_train_algebra_198 reference: 48480{'pred': '55 \\times 1212 - 15 \\times 1212', 'correct': True, 'tokens': 704, 'cut': False}text We are asked to compute the expression $55 \times 1212 - 15 \times 1212$.We have $1212 = 10 \times 121 + 2 \times 121 = 121 + 242 ... times 1212$.We can also write this as $55 \times 1212 - 15 \times 1212$.
Final Answer: The final answer is $\boxed{55 \times 1212 - 15 \times 1212}${'pred': None, 'correct': True, 'tokens': 623, 'cut': False}text To compute $55\times1212 - 15\times1212$, we can rewrite the expression as$55\times1212 - 15\times1212 = 55\times1212 - 15\times1 ...  is $55\times 1212 - 15\times 1212$.
Thus, the answer is $55 \times 1212 - 15 \times 1212$.
Final Answer: The final answer is 55 * 1212 - 15 * 1212.Problem math_train_prealgebra_1121 reference: 2{'pred': None, 'correct': False, 'tokens': 343, 'cut': False}text Let the set of numbers Erich likes be denoted by $S$.We want to find the number of different last digits possible in the set $S$. ... There are 10 possible last digits.
Therefore, the number of different last digits possible in the set $S$ is 10.
Final Answer: 10
Final Answer: 10{'pred': None, 'correct': False, 'tokens': 357, 'cut': False}text Let $n$ be the number of digits of the number. Erich likes numbers that are divisible by 5.The possible numbers are $0, 5, 10, 15 ... 80, 85, 90, 95.There are 16 last digits.The number of different last digits is 16.
So, the number of different last digits is 16.
Final Answer: 16Problem math_train_intermediate_algebra_635 reference: (5,\infty){'pred': '5', 'correct': False, 'tokens': 621, 'cut': False}text Let $h(x) = \frac{3x - 1}{\sqrt{x - 5}}$.To find the domain of $h(x)$, we need to determine the values of $x$ for which the denom ... The domain of $h(x) = \frac{3x - 1}{\sqrt{x - 5}}$ is the set of real numbers $x$ such that $x \ne 5$.
Final Answer: The final answer is $\boxed{5}${'pred': '0', 'correct': False, 'tokens': 427, 'cut': False}text To find the domain of the function $h(x) = \frac{3x - 1}{\sqrt{x - 5}}$, we need to find the values of $x$ for which $h(x) = 0$.W ... 3}$.Since $x \ge 0$, the domain of $h(x)$ is $x \ge 0$.Thus, the domain of the function is $x \ge 0$.
Final Answer: The final answer is $\boxed{0}$Problem math_train_intermediate_algebra_55 reference: \frac{1}{50}{'pred': '1', 'correct': False, 'tokens': 2066, 'cut': False}text Let $S = \left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{5}\right) \dotsm \left(1-\ ... -\frac{1}{4}\right)\dotsm \left(1-\frac{1}{50}\right)$ is not a common fraction, we assume that the expression is the same.
Final Answer: $\boxed{1}${'pred': '0', 'correct': False, 'tokens': 1078, 'cut': False}text Let $S = \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{4}\right)\left(1 - \frac{1}{5}\right) \dotsm \ ... he value of the given expression is 0.
Final Answer: 0
The answer is 0.Final Answer: 0Therefore the answer is 0.

Problem math_train_algebra_380 reference: 12{'pred': '10.4', 'correct': False, 'tokens': 854, 'cut': False}text Let $x$ and $y$ be two positive integers. We are given that the sum of the squares of the two integers is 90.So, $x^2 + y^2 = 90$ ... =86$, so $y=\sqrt{86} \approx 9.2$.Then $x+y = 2+9.2 = 11.2$.
The sum of the two integers is 10.4.
Final Answer: The final answer is $\boxed{10.4}${'pred': None, 'correct': False, 'tokens': 2879, 'cut': False}text Let the two positive integers be $a$ and $b$. We are given that the sum of the squares of the two integers is 90.Thus, we have $a ... er: 13.5
Final Answer: 13.5Final Answer: 13.5The sum of the integers is 13.5
The number 13.5 is an integer.

[harness: 4 repeated lines not shown (a line is shown at most 2 times per output)]

Sample container: listSample fields: ['text', 'pred', 'correct', 'tokens', 'cut']Sample metadata: {'pred': '2n\\pi', 'correct': False, 'tokens': 787, 'cut': False}Activation index type: dictIndex keys: ['layers', 'mlp_layers', 'samples', 'files']Index preview: {'layers': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], 'mlp_layers': [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], 'samples': [{'item': 'math_train_precalculus_274', 'sample': 0, 'correct': False, 'tokens': 828}, {'item': 'math_train_precalculus_274', 'sample': 1, 'correct': True, 'tokens': 153}, {'item': 'math_train_precalculus_274', 'sample': 2, 'correct': True, 'tokens': 366}, {'item': 'math_train_precalculus_274', 'sample': 3, 'correct': True, 'tokens': 176}, {'item': 'math_train_precalculus_274', 'sample': 4, 'correct': True, 'tokens': 237}, {'item': 'math_train_precalculus_274', 'sample': 5, 'correct': True, 'tokens': 181}, {'item': 'math_trainProblem math_train_precalculus_274 reference: 2 \pi{'pred': '2n\\pi', 'correct': False, 'tokens': 787, 'cut': False}text Let $y = \sin x + \cos x$.We want to find the period of $y$.The period of the sine function is the value of $x$ such that $\sin( ... \sin x + \cos x$ is $2n \pi$.The period of $y = \sin x + \cos x$ is $2n \pi$ for some integer $n$.
Final Answer: The final answer is $\boxed{2n\pi}${'pred': '2\\pi', 'correct': True, 'tokens': 112, 'cut': False}text The period of the function $y = \sin x + \cos x$ is $2\pi$.We know that the period of the sine function is $2\pi$.We want to fin ... w that $y = \sin x + \cos x$.The period of the sine function is $2\pi$.Thus, the period is $2\pi$.
Final Answer: The final answer is $\boxed{2\pi}$Problem math_train_algebra_1220 reference: 2800{'pred': None, 'correct': False, 'tokens': 146, 'cut': False}text To evaluate $64^2 - 36^2$, we can use the property of exponents.$64^2 = 64 \cdot 64 = 64^2$$36^2 = 36 \cdot 36 = 36^2$Therefore ...  64^2$$36^2 = 36 \cdot 36 = 36^2$Therefore, $64^2 - 36^2 = 64^2 - 36^2 = 36^2 - 36^2 = 36^2 - 36^2 = 0$.Thus, $64^2 - 36^2 = 0$.
Final Answer: $0${'pred': None, 'correct': True, 'tokens': 210, 'cut': False}text To evaluate the expression $64^2 - 36^2$, we can rewrite it as$$64^2 - 36^2 = (64 - 36)(64 + 36) = (28)(100) = 2800$$So, the exp ... ve $64^2 - 36^2 = (64 - 36)(64 + 36) = (28)(100) = 2800$.
Therefore, the expression is $64^2 - 36^2 = 2800$.
Final Answer: 2800
Final Answer: 2800
Problem math_train_algebra_1101 reference: 40{'pred': None, 'correct': False, 'tokens': 795, 'cut': False}text Let the sequence be denoted by $a_n$ for $n \geq 1$. The terms are $a_1 = 2, a_2 = 6, a_3 = 10, \ldots, a_n = n$.The sequence is  ... want $a_n = n$, so $4n - 2 = n$, which means $3n = 2$, so $n = \frac{2}{3}$.Then $x = 26$.Thus, $x + y = 26$.
Final Answer: The final answer is 26.{'pred': '12', 'correct': False, 'tokens': 750, 'cut': False}text Let the given sequence be $a_n = 2n$, $a_n = 6n$, $a_n = 10n$, and $a_n = 26n$.The sequence is an arithmetic sequence.The differ ...  - a_1 = 8$, so $a_n = a_1 + 8 = 2 + 8 = 10$.Therefore, $x = 10$, $y = 2$, and $x + y = 10 + 2 = 12$.
Final Answer: The final answer is $\boxed{12}$Problem math_train_algebra_198 reference: 48480{'pred': '55 \\times 1212 - 15 \\times 1212', 'correct': True, 'tokens': 704, 'cut': False}text We are asked to compute the expression $55 \times 1212 - 15 \times 1212$.We have $1212 = 10 \times 121 + 2 \times 121 = 121 + 242 ... times 1212$.We can also write this as $55 \times 1212 - 15 \times 1212$.
Final Answer: The final answer is $\boxed{55 \times 1212 - 15 \times 1212}${'pred': None, 'correct': True, 'tokens': 623, 'cut': False}text To compute $55\times1212 - 15\times1212$, we can rewrite the expression as$55\times1212 - 15\times1212 = 55\times1212 - 15\times1 ...  is $55\times 1212 - 15\times 1212$.
Thus, the answer is $55 \times 1212 - 15 \times 1212$.
Final Answer: The final answer is 55 * 1212 - 15 * 1212.Problem math_train_prealgebra_1121 reference: 2{'pred': None, 'correct': False, 'tokens': 343, 'cut': False}text Let the set of numbers Erich likes be denoted by $S$.We want to find the number of different last digits possible in the set $S$. ... There are 10 possible last digits.
Therefore, the number of different last digits possible in the set $S$ is 10.
Final Answer: 10
Final Answer: 10{'pred': None, 'correct': False, 'tokens': 357, 'cut': False}text Let $n$ be the number of digits of the number. Erich likes numbers that are divisible by 5.The possible numbers are $0, 5, 10, 15 ... 80, 85, 90, 95.There are 16 last digits.The number of different last digits is 16.
So, the number of different last digits is 16.
Final Answer: 16Problem math_train_intermediate_algebra_635 reference: (5,\infty){'pred': '5', 'correct': False, 'tokens': 621, 'cut': False}text Let $h(x) = \frac{3x - 1}{\sqrt{x - 5}}$.To find the domain of $h(x)$, we need to determine the values of $x$ for which the denom ... The domain of $h(x) = \frac{3x - 1}{\sqrt{x - 5}}$ is the set of real numbers $x$ such that $x \ne 5$.
Final Answer: The final answer is $\boxed{5}${'pred': '0', 'correct': False, 'tokens': 427, 'cut': False}text To find the domain of the function $h(x) = \frac{3x - 1}{\sqrt{x - 5}}$, we need to find the values of $x$ for which $h(x) = 0$.W ... 3}$.Since $x \ge 0$, the domain of $h(x)$ is $x \ge 0$.Thus, the domain of the function is $x \ge 0$.
Final Answer: The final answer is $\boxed{0}$Problem math_train_intermediate_algebra_55 reference: \frac{1}{50}{'pred': '1', 'correct': False, 'tokens': 2066, 'cut': False}text Let $S = \left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{5}\right) \dotsm \left(1-\ ... -\frac{1}{4}\right)\dotsm \left(1-\frac{1}{50}\right)$ is not a common fraction, we assume that the expression is the same.
Final Answer: $\boxed{1}${'pred': '0', 'correct': False, 'tokens': 1078, 'cut': False}text Let $S = \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{4}\right)\left(1 - \frac{1}{5}\right) \dotsm \ ... he value of the given expression is 0.
Final Answer: 0
The answer is 0.Final Answer: 0Therefore the answer is 0.

Problem math_train_algebra_380 reference: 12{'pred': '10.4', 'correct': False, 'tokens': 854, 'cut': False}text Let $x$ and $y$ be two positive integers. We are given that the sum of the squares of the two integers is 90.So, $x^2 + y^2 = 90$ ... =86$, so $y=\sqrt{86} \approx 9.2$.Then $x+y = 2+9.2 = 11.2$.
The sum of the two integers is 10.4.
Final Answer: The final answer is $\boxed{10.4}${'pred': None, 'correct': False, 'tokens': 2879, 'cut': False}text Let the two positive integers be $a$ and $b$. We are given that the sum of the squares of the two integers is 90.Thus, we have $a ... er: 13.5
Final Answer: 13.5Final Answer: 13.5The sum of the integers is 13.5
The number 13.5 is an integer.

[harness: 4 repeated lines not shown (a line is shown at most 2 times per output)]

The excerpts reveal real reasoning errors: lost constraints, invalid algebraic substitutions, and repeated conclusions that do not follow from the work. They also show that pred=None can accompany a correct unboxed answer, so the summary’s “no answer” category is not a reliable count of genuinely missing conclusions. I’ll use the supplied correctness grades for accuracy and separately track repetition, boxing, and cutoffs. I’ll now freeze DEV and rank MLP channels using within-problem comparisons, including a response-length adjustment to reduce that confound.

python

flat1 = []for rec in gen1:    for si, sample in enumerate(rec["samples"]):        flat1.append(dict(sample, problem_id=rec["id"], sample_idx=si))assert len(flat1) == len(act_index1["samples"])assert all(s["problem_id"] == a["item"] and s["sample_idx"] == a["sample"]           for s, a in zip(flat1, act_index1["samples"]))
success_counts = {r["id"]: sum(bool(s["correct"]) for s in r["samples"]) for r in gen1}intermittent_ids = [pid for pid in baseline_ids if 0 < success_counts[pid] < 8]dev_count = 128 if len(intermittent_ids) >= 128 else (64 if len(intermittent_ids) >= 64 else 32)assert len(intermittent_ids) >= dev_countdev_rows = stratified_pick([problem_by_id[pid] for pid in intermittent_ids], dev_count, rng_exp)dev_ids = [r["id"] for r in dev_rows]dev_set = set(dev_ids)assert not dev_set & holdout_set(wdir / "DEV.json").write_text(json.dumps(dev_ids, indent=2))with open(wdir / "notes.md", "a") as f:    f.write(f"\n## Baseline diagnosis and fixed DEV\n"            f"Baseline: 361/2048 correct (17.62695%). Summary categories: 827 wrong, "            f"726 no-answer-finished, 134 failing cutoffs. pred=None is not proof of missing answer.\n"            f"DEV fixed in DEV.json: {dev_count} problems, each solved 1–7/8 times; "            f"selected with the predeclared seeded stratified sampler. No edits yet.\n"            f"Observed errors: loss of constraints, invalid algebra, repetitive unsupported conclusions.\n")print("Baseline success histogram:", dict(sorted(collections.Counter(success_counts.values()).items())))print("Intermittent:", len(intermittent_ids), "Fixed DEV:", len(dev_ids))print("DEV baseline accuracy:", np.mean([s["correct"] for s in flat1 if s["problem_id"] in dev_set]))
import redef repeat_fraction(txt):    lines = [ln.strip() for ln in txt.splitlines() if ln.strip()]    return 1 - len(set(lines)) / max(len(lines), 1)
for label, subset in [("correct", [s for s in flat1 if s["correct"]]),                      ("incorrect", [s for s in flat1 if not s["correct"]])]:    print(label, "n", len(subset), "median_tokens", np.median([s["tokens"] for s in subset]),          "cut_rate", round(np.mean([s["cut"] for s in subset]), 3),          "boxed_rate", round(np.mean(["\\boxed" in s["text"] for s in subset]), 3),          "mean_repeated_line_fraction", round(np.mean([repeat_fraction(s["text"]) for s in subset]), 3))
dev_idx = np.array([i for i, s in enumerate(flat1) if s["problem_id"] in dev_set])dev_flat = [flat1[i] for i in dev_idx]pid_arr = np.array([s["problem_id"] for s in dev_flat])yg = np.array([float(s["correct"]) for s in dev_flat])ll = np.log1p([s["tokens"] for s in dev_flat])rep = np.array([repeat_fraction(s["text"]) for s in dev_flat])groups_dev = [np.where(pid_arr == pid)[0] for pid in dev_ids]def within_center(a):    b = np.asarray(a, dtype=np.float64).copy()    for ix in groups_dev:        b[ix] -= b[ix].mean(axis=0)    return byc = within_center(yg)lc = within_center(ll)rc = within_center(rep)yr = yc - lc * (lc @ yc) / (lc @ lc)halfmask = np.isin(pid_arr, dev_ids[:len(dev_ids)//2])mlpmeans1 = np.load(wdir / "gen_1_acts" / "mlp_mean_answer.npy", mmap_mode="r")candidates = []for layer in range(18):    aa = np.array(mlpmeans1[dev_idx, layer], dtype=np.float64)    ac = within_center(aa)    ar = ac - lc[:, None] * ((lc @ ac) / (lc @ lc))[None, :]    denom = np.sqrt(np.sum(ar**2, axis=0) * (yr @ yr)) + 1e-15    corr = (yr @ ar) / denom    repeat_corr = (rc @ ac) / (np.sqrt(np.sum(ac**2, axis=0) * (rc @ rc)) + 1e-15)    corr_halves = []    for mask in [halfmask, ~halfmask]:        corr_halves.append((yr[mask] @ ar[mask]) /            (np.sqrt(np.sum(ar[mask]**2, axis=0) * np.sum(yr[mask]**2)) + 1e-15))    stable = np.minimum(np.abs(corr_halves[0]), np.abs(corr_halves[1]))    stable *= (np.sign(corr_halves[0]) == np.sign(corr_halves[1]))    downw = load_original(f"model.layers.{layer}.mlp.down_proj.weight").numpy()    colnorm = np.linalg.norm(downw, axis=0)    for j in np.argsort(stable)[-12:]:        candidates.append({            "layer": layer, "neuron": int(j), "corr_correct_length_adjusted": float(corr[j]),            "corr_half1": float(corr_halves[0][j]), "corr_half2": float(corr_halves[1][j]),            "corr_repetition": float(repeat_corr[j]), "stability": float(stable[j]),            "mean_correct": float(aa[yg == 1, j].mean()), "mean_wrong": float(aa[yg == 0, j].mean()),            "column_norm": float(colnorm[j]), "rms_contribution": float(np.sqrt(np.mean(aa[:, j]**2))*colnorm[j])        })candidates.sort(key=lambda r: r["stability"], reverse=True)(wdir / "neuron_screen.json").write_text(json.dumps(candidates, indent=2))for c in candidates[:24]:    print({k: (round(v, 4) if isinstance(v, float) else v) for k, v in c.items()})

flat1 = []for rec in gen1:    for si, sample in enumerate(rec["samples"]):        flat1.append(dict(sample, problem_id=rec["id"], sample_idx=si))assert len(flat1) == len(act_index1["samples"])assert all(s["problem_id"] == a["item"] and s["sample_idx"] == a["sample"]           for s, a in zip(flat1, act_index1["samples"]))
success_counts = {r["id"]: sum(bool(s["correct"]) for s in r["samples"]) for r in gen1}intermittent_ids = [pid for pid in baseline_ids if 0 < success_counts[pid] < 8]dev_count = 128 if len(intermittent_ids) >= 128 else (64 if len(intermittent_ids) >= 64 else 32)assert len(intermittent_ids) >= dev_countdev_rows = stratified_pick([problem_by_id[pid] for pid in intermittent_ids], dev_count, rng_exp)dev_ids = [r["id"] for r in dev_rows]dev_set = set(dev_ids)assert not dev_set & holdout_set(wdir / "DEV.json").write_text(json.dumps(dev_ids, indent=2))with open(wdir / "notes.md", "a") as f:    f.write(f"\n## Baseline diagnosis and fixed DEV\n"            f"Baseline: 361/2048 correct (17.62695%). Summary categories: 827 wrong, "            f"726 no-answer-finished, 134 failing cutoffs. pred=None is not proof of missing answer.\n"            f"DEV fixed in DEV.json: {dev_count} problems, each solved 1–7/8 times; "            f"selected with the predeclared seeded stratified sampler. No edits yet.\n"            f"Observed errors: loss of constraints, invalid algebra, repetitive unsupported conclusions.\n")print("Baseline success histogram:", dict(sorted(collections.Counter(success_counts.values()).items())))print("Intermittent:", len(intermittent_ids), "Fixed DEV:", len(dev_ids))print("DEV baseline accuracy:", np.mean([s["correct"] for s in flat1 if s["problem_id"] in dev_set]))
import redef repeat_fraction(txt):    lines = [ln.strip() for ln in txt.splitlines() if ln.strip()]    return 1 - len(set(lines)) / max(len(lines), 1)
for label, subset in [("correct", [s for s in flat1 if s["correct"]]),                      ("incorrect", [s for s in flat1 if not s["correct"]])]:    print(label, "n", len(subset), "median_tokens", np.median([s["tokens"] for s in subset]),          "cut_rate", round(np.mean([s["cut"] for s in subset]), 3),          "boxed_rate", round(np.mean(["\\boxed" in s["text"] for s in subset]), 3),          "mean_repeated_line_fraction", round(np.mean([repeat_fraction(s["text"]) for s in subset]), 3))
dev_idx = np.array([i for i, s in enumerate(flat1) if s["problem_id"] in dev_set])dev_flat = [flat1[i] for i in dev_idx]pid_arr = np.array([s["problem_id"] for s in dev_flat])yg = np.array([float(s["correct"]) for s in dev_flat])ll = np.log1p([s["tokens"] for s in dev_flat])rep = np.array([repeat_fraction(s["text"]) for s in dev_flat])groups_dev = [np.where(pid_arr == pid)[0] for pid in dev_ids]def within_center(a):    b = np.asarray(a, dtype=np.float64).copy()    for ix in groups_dev:        b[ix] -= b[ix].mean(axis=0)    return byc = within_center(yg)lc = within_center(ll)rc = within_center(rep)yr = yc - lc * (lc @ yc) / (lc @ lc)halfmask = np.isin(pid_arr, dev_ids[:len(dev_ids)//2])mlpmeans1 = np.load(wdir / "gen_1_acts" / "mlp_mean_answer.npy", mmap_mode="r")candidates = []for layer in range(18):    aa = np.array(mlpmeans1[dev_idx, layer], dtype=np.float64)    ac = within_center(aa)    ar = ac - lc[:, None] * ((lc @ ac) / (lc @ lc))[None, :]    denom = np.sqrt(np.sum(ar**2, axis=0) * (yr @ yr)) + 1e-15    corr = (yr @ ar) / denom    repeat_corr = (rc @ ac) / (np.sqrt(np.sum(ac**2, axis=0) * (rc @ rc)) + 1e-15)    corr_halves = []    for mask in [halfmask, ~halfmask]:        corr_halves.append((yr[mask] @ ar[mask]) /            (np.sqrt(np.sum(ar[mask]**2, axis=0) * np.sum(yr[mask]**2)) + 1e-15))    stable = np.minimum(np.abs(corr_halves[0]), np.abs(corr_halves[1]))    stable *= (np.sign(corr_halves[0]) == np.sign(corr_halves[1]))    downw = load_original(f"model.layers.{layer}.mlp.down_proj.weight").numpy()    colnorm = np.linalg.norm(downw, axis=0)    for j in np.argsort(stable)[-12:]:        candidates.append({            "layer": layer, "neuron": int(j), "corr_correct_length_adjusted": float(corr[j]),            "corr_half1": float(corr_halves[0][j]), "corr_half2": float(corr_halves[1][j]),            "corr_repetition": float(repeat_corr[j]), "stability": float(stable[j]),            "mean_correct": float(aa[yg == 1, j].mean()), "mean_wrong": float(aa[yg == 0, j].mean()),            "column_norm": float(colnorm[j]), "rms_contribution": float(np.sqrt(np.mean(aa[:, j]**2))*colnorm[j])        })candidates.sort(key=lambda r: r["stability"], reverse=True)(wdir / "neuron_screen.json").write_text(json.dumps(candidates, indent=2))for c in candidates[:24]:    print({k: (round(v, 4) if isinstance(v, float) else v) for k, v in c.items()})

OUTPUT of Python run 6 of at most 300:

Baseline success histogram: {0: 145, 1: 37, 2: 19, 3: 12, 4: 6, 5: 13, 6: 13, 7: 5, 8: 6}Intermittent: 105 Fixed DEV: 64DEV baseline accuracy: 0.421875correct n 361 median_tokens 354.0 cut_rate 0.025 boxed_rate 0.443 mean_repeated_line_fraction 0.101incorrect n 1687 median_tokens 678.0 cut_rate 0.079 boxed_rate 0.491 mean_repeated_line_fraction 0.171{'layer': 17, 'neuron': 1149, 'corr_correct_length_adjusted': 0.2087, 'corr_half1': 0.2076, 'corr_half2': 0.2104, 'corr_repetition': -0.2982, 'stability': 0.2076, 'mean_correct': 0.0253, 'mean_wrong': 0.0193, 'column_norm': 0.4771, 'rms_contribution': 0.0116}{'layer': 7, 'neuron': 171, 'corr_correct_length_adjusted': 0.1845, 'corr_half1': 0.1774, 'corr_half2': 0.195, 'corr_repetition': -0.0333, 'stability': 0.1774, 'mean_correct': 0.0239, 'mean_wrong': 0.023, 'column_norm': 0.4532, 'rms_contribution': 0.0118}{'layer': 3, 'neuron': 1749, 'corr_correct_length_adjusted': 0.1813, 'corr_half1': 0.1755, 'corr_half2': 0.1873, 'corr_repetition': 0.024, 'stability': 0.1755, 'mean_correct': 0.0085, 'mean_wrong': 0.0055, 'column_norm': 0.3775, 'rms_contribution': 0.0058}{'layer': 10, 'neuron': 733, 'corr_correct_length_adjusted': -0.1785, 'corr_half1': -0.1746, 'corr_half2': -0.1826, 'corr_repetition': 0.2784, 'stability': 0.1746, 'mean_correct': -0.0134, 'mean_wrong': -0.0089, 'column_norm': 0.3628, 'rms_contribution': 0.0072}{'layer': 11, 'neuron': 1807, 'corr_correct_length_adjusted': 0.1924, 'corr_half1': 0.16, 'corr_half2': 0.2219, 'corr_repetition': -0.5206, 'stability': 0.16, 'mean_correct': 0.0428, 'mean_wrong': 0.0358, 'column_norm': 0.5539, 'rms_contribution': 0.0235}{'layer': 10, 'neuron': 1688, 'corr_correct_length_adjusted': 0.16, 'corr_half1': 0.1594, 'corr_half2': 0.1607, 'corr_repetition': -0.3767, 'stability': 0.1594, 'mean_correct': 0.0771, 'mean_wrong': 0.0607, 'column_norm': 0.4006, 'rms_contribution': 0.0294}{'layer': 14, 'neuron': 1594, 'corr_correct_length_adjusted': -0.1692, 'corr_half1': -0.1592, 'corr_half2': -0.1799, 'corr_repetition': 0.4726, 'stability': 0.1592, 'mean_correct': 0.0122, 'mean_wrong': 0.0206, 'column_norm': 0.2615, 'rms_contribution': 0.0089}{'layer': 17, 'neuron': 1201, 'corr_correct_length_adjusted': -0.1664, 'corr_half1': -0.158, 'corr_half2': -0.1747, 'corr_repetition': 0.2976, 'stability': 0.158, 'mean_correct': -0.0225, 'mean_wrong': -0.0191, 'column_norm': 0.407, 'rms_contribution': 0.0099}{'layer': 14, 'neuron': 259, 'corr_correct_length_adjusted': -0.1599, 'corr_half1': -0.157, 'corr_half2': -0.1647, 'corr_repetition': 0.1526, 'stability': 0.157, 'mean_correct': -0.0035, 'mean_wrong': -0.0031, 'column_norm': 0.386, 'rms_contribution': 0.0027}{'layer': 12, 'neuron': 1601, 'corr_correct_length_adjusted': 0.1627, 'corr_half1': 0.1557, 'corr_half2': 0.1694, 'corr_repetition': -0.1578, 'stability': 0.1557, 'mean_correct': -0.0033, 'mean_wrong': -0.0048, 'column_norm': 0.589, 'rms_contribution': 0.0051}{'layer': 4, 'neuron': 1978, 'corr_correct_length_adjusted': -0.1719, 'corr_half1': -0.1549, 'corr_half2': -0.1865, 'corr_repetition': 0.1754, 'stability': 0.1549, 'mean_correct': -0.0446, 'mean_wrong': -0.0359, 'column_norm': 0.3128, 'rms_contribution': 0.0142}{'layer': 15, 'neuron': 1120, 'corr_correct_length_adjusted': 0.1806, 'corr_half1': 0.1544, 'corr_half2': 0.2108, 'corr_repetition': -0.2963, 'stability': 0.1544, 'mean_correct': 0.0077, 'mean_wrong': 0.0051, 'column_norm': 0.3523, 'rms_contribution': 0.0032}{'layer': 17, 'neuron': 1556, 'corr_correct_length_adjusted': -0.1621, 'corr_half1': -0.1728, 'corr_half2': -0.1537, 'corr_repetition': 0.1679, 'stability': 0.1537, 'mean_correct': -0.0191, 'mean_wrong': -0.0163, 'column_norm': 0.4177, 'rms_contribution': 0.009}{'layer': 5, 'neuron': 1188, 'corr_correct_length_adjusted': 0.1563, 'corr_half1': 0.1533, 'corr_half2': 0.159, 'corr_repetition': -0.293, 'stability': 0.1533, 'mean_correct': -0.0746, 'mean_wrong': -0.0816, 'column_norm': 0.4891, 'rms_contribution': 0.0439}{'layer': 5, 'neuron': 470, 'corr_correct_length_adjusted': -0.1622, 'corr_half1': -0.172, 'corr_half2': -0.1525, 'corr_repetition': 0.4021, 'stability': 0.1525, 'mean_correct': -0.0222, 'mean_wrong': -0.0203, 'column_norm': 0.5682, 'rms_contribution': 0.013}{'layer': 4, 'neuron': 670, 'corr_correct_length_adjusted': -0.152, 'corr_half1': -0.1524, 'corr_half2': -0.1572, 'corr_repetition': 0.0737, 'stability': 0.1524, 'mean_correct': -0.0313, 'mean_wrong': -0.0307, 'column_norm': 0.3362, 'rms_contribution': 0.013}{'layer': 15, 'neuron': 1341, 'corr_correct_length_adjusted': 0.1554, 'corr_half1': 0.1515, 'corr_half2': 0.159, 'corr_repetition': -0.1806, 'stability': 0.1515, 'mean_correct': 0.0392, 'mean_wrong': 0.0331, 'column_norm': 0.3418, 'rms_contribution': 0.0132}{'layer': 12, 'neuron': 1491, 'corr_correct_length_adjusted': -0.1707, 'corr_half1': -0.19, 'corr_half2': -0.1509, 'corr_repetition': -0.032, 'stability': 0.1509, 'mean_correct': 0.011, 'mean_wrong': 0.0118, 'column_norm': 0.5456, 'rms_contribution': 0.0069}{'layer': 6, 'neuron': 1796, 'corr_correct_length_adjusted': 0.1588, 'corr_half1': 0.1504, 'corr_half2': 0.167, 'corr_repetition': 0.0002, 'stability': 0.1504, 'mean_correct': 0.0324, 'mean_wrong': 0.0334, 'column_norm': 0.3417, 'rms_contribution': 0.0134}{'layer': 15, 'neuron': 699, 'corr_correct_length_adjusted': -0.168, 'corr_half1': -0.1852, 'corr_half2': -0.1501, 'corr_repetition': 0.148, 'stability': 0.1501, 'mean_correct': -0.0129, 'mean_wrong': -0.0093, 'column_norm': 0.3836, 'rms_contribution': 0.0053}{'layer': 16, 'neuron': 40, 'corr_correct_length_adjusted': 0.1775, 'corr_half1': 0.1497, 'corr_half2': 0.2124, 'corr_repetition': -0.3683, 'stability': 0.1497, 'mean_correct': 0.024, 'mean_wrong': 0.0204, 'column_norm': 0.4783, 'rms_contribution': 0.012}{'layer': 9, 'neuron': 1831, 'corr_correct_length_adjusted': 0.1497, 'corr_half1': 0.1493, 'corr_half2': 0.1646, 'corr_repetition': -0.0586, 'stability': 0.1493, 'mean_correct': 0.0024, 'mean_wrong': 0.0019, 'column_norm': 0.5203, 'rms_contribution': 0.0026}{'layer': 16, 'neuron': 257, 'corr_correct_length_adjusted': -0.1478, 'corr_half1': -0.149, 'corr_half2': -0.1493, 'corr_repetition': 0.1941, 'stability': 0.149, 'mean_correct': -0.0086, 'mean_wrong': -0.0066, 'column_norm': 0.4828, 'rms_contribution': 0.0045}{'layer': 8, 'neuron': 1150, 'corr_correct_length_adjusted': -0.1494, 'corr_half1': -0.1503, 'corr_half2': -0.1485, 'corr_repetition': 0.132, 'stability': 0.1485, 'mean_correct': 0.05, 'mean_wrong': 0.0633, 'column_norm': 0.2992, 'rms_contribution': 0.021}

Baseline success histogram: {0: 145, 1: 37, 2: 19, 3: 12, 4: 6, 5: 13, 6: 13, 7: 5, 8: 6}Intermittent: 105 Fixed DEV: 64DEV baseline accuracy: 0.421875correct n 361 median_tokens 354.0 cut_rate 0.025 boxed_rate 0.443 mean_repeated_line_fraction 0.101incorrect n 1687 median_tokens 678.0 cut_rate 0.079 boxed_rate 0.491 mean_repeated_line_fraction 0.171{'layer': 17, 'neuron': 1149, 'corr_correct_length_adjusted': 0.2087, 'corr_half1': 0.2076, 'corr_half2': 0.2104, 'corr_repetition': -0.2982, 'stability': 0.2076, 'mean_correct': 0.0253, 'mean_wrong': 0.0193, 'column_norm': 0.4771, 'rms_contribution': 0.0116}{'layer': 7, 'neuron': 171, 'corr_correct_length_adjusted': 0.1845, 'corr_half1': 0.1774, 'corr_half2': 0.195, 'corr_repetition': -0.0333, 'stability': 0.1774, 'mean_correct': 0.0239, 'mean_wrong': 0.023, 'column_norm': 0.4532, 'rms_contribution': 0.0118}{'layer': 3, 'neuron': 1749, 'corr_correct_length_adjusted': 0.1813, 'corr_half1': 0.1755, 'corr_half2': 0.1873, 'corr_repetition': 0.024, 'stability': 0.1755, 'mean_correct': 0.0085, 'mean_wrong': 0.0055, 'column_norm': 0.3775, 'rms_contribution': 0.0058}{'layer': 10, 'neuron': 733, 'corr_correct_length_adjusted': -0.1785, 'corr_half1': -0.1746, 'corr_half2': -0.1826, 'corr_repetition': 0.2784, 'stability': 0.1746, 'mean_correct': -0.0134, 'mean_wrong': -0.0089, 'column_norm': 0.3628, 'rms_contribution': 0.0072}{'layer': 11, 'neuron': 1807, 'corr_correct_length_adjusted': 0.1924, 'corr_half1': 0.16, 'corr_half2': 0.2219, 'corr_repetition': -0.5206, 'stability': 0.16, 'mean_correct': 0.0428, 'mean_wrong': 0.0358, 'column_norm': 0.5539, 'rms_contribution': 0.0235}{'layer': 10, 'neuron': 1688, 'corr_correct_length_adjusted': 0.16, 'corr_half1': 0.1594, 'corr_half2': 0.1607, 'corr_repetition': -0.3767, 'stability': 0.1594, 'mean_correct': 0.0771, 'mean_wrong': 0.0607, 'column_norm': 0.4006, 'rms_contribution': 0.0294}{'layer': 14, 'neuron': 1594, 'corr_correct_length_adjusted': -0.1692, 'corr_half1': -0.1592, 'corr_half2': -0.1799, 'corr_repetition': 0.4726, 'stability': 0.1592, 'mean_correct': 0.0122, 'mean_wrong': 0.0206, 'column_norm': 0.2615, 'rms_contribution': 0.0089}{'layer': 17, 'neuron': 1201, 'corr_correct_length_adjusted': -0.1664, 'corr_half1': -0.158, 'corr_half2': -0.1747, 'corr_repetition': 0.2976, 'stability': 0.158, 'mean_correct': -0.0225, 'mean_wrong': -0.0191, 'column_norm': 0.407, 'rms_contribution': 0.0099}{'layer': 14, 'neuron': 259, 'corr_correct_length_adjusted': -0.1599, 'corr_half1': -0.157, 'corr_half2': -0.1647, 'corr_repetition': 0.1526, 'stability': 0.157, 'mean_correct': -0.0035, 'mean_wrong': -0.0031, 'column_norm': 0.386, 'rms_contribution': 0.0027}{'layer': 12, 'neuron': 1601, 'corr_correct_length_adjusted': 0.1627, 'corr_half1': 0.1557, 'corr_half2': 0.1694, 'corr_repetition': -0.1578, 'stability': 0.1557, 'mean_correct': -0.0033, 'mean_wrong': -0.0048, 'column_norm': 0.589, 'rms_contribution': 0.0051}{'layer': 4, 'neuron': 1978, 'corr_correct_length_adjusted': -0.1719, 'corr_half1': -0.1549, 'corr_half2': -0.1865, 'corr_repetition': 0.1754, 'stability': 0.1549, 'mean_correct': -0.0446, 'mean_wrong': -0.0359, 'column_norm': 0.3128, 'rms_contribution': 0.0142}{'layer': 15, 'neuron': 1120, 'corr_correct_length_adjusted': 0.1806, 'corr_half1': 0.1544, 'corr_half2': 0.2108, 'corr_repetition': -0.2963, 'stability': 0.1544, 'mean_correct': 0.0077, 'mean_wrong': 0.0051, 'column_norm': 0.3523, 'rms_contribution': 0.0032}{'layer': 17, 'neuron': 1556, 'corr_correct_length_adjusted': -0.1621, 'corr_half1': -0.1728, 'corr_half2': -0.1537, 'corr_repetition': 0.1679, 'stability': 0.1537, 'mean_correct': -0.0191, 'mean_wrong': -0.0163, 'column_norm': 0.4177, 'rms_contribution': 0.009}{'layer': 5, 'neuron': 1188, 'corr_correct_length_adjusted': 0.1563, 'corr_half1': 0.1533, 'corr_half2': 0.159, 'corr_repetition': -0.293, 'stability': 0.1533, 'mean_correct': -0.0746, 'mean_wrong': -0.0816, 'column_norm': 0.4891, 'rms_contribution': 0.0439}{'layer': 5, 'neuron': 470, 'corr_correct_length_adjusted': -0.1622, 'corr_half1': -0.172, 'corr_half2': -0.1525, 'corr_repetition': 0.4021, 'stability': 0.1525, 'mean_correct': -0.0222, 'mean_wrong': -0.0203, 'column_norm': 0.5682, 'rms_contribution': 0.013}{'layer': 4, 'neuron': 670, 'corr_correct_length_adjusted': -0.152, 'corr_half1': -0.1524, 'corr_half2': -0.1572, 'corr_repetition': 0.0737, 'stability': 0.1524, 'mean_correct': -0.0313, 'mean_wrong': -0.0307, 'column_norm': 0.3362, 'rms_contribution': 0.013}{'layer': 15, 'neuron': 1341, 'corr_correct_length_adjusted': 0.1554, 'corr_half1': 0.1515, 'corr_half2': 0.159, 'corr_repetition': -0.1806, 'stability': 0.1515, 'mean_correct': 0.0392, 'mean_wrong': 0.0331, 'column_norm': 0.3418, 'rms_contribution': 0.0132}{'layer': 12, 'neuron': 1491, 'corr_correct_length_adjusted': -0.1707, 'corr_half1': -0.19, 'corr_half2': -0.1509, 'corr_repetition': -0.032, 'stability': 0.1509, 'mean_correct': 0.011, 'mean_wrong': 0.0118, 'column_norm': 0.5456, 'rms_contribution': 0.0069}{'layer': 6, 'neuron': 1796, 'corr_correct_length_adjusted': 0.1588, 'corr_half1': 0.1504, 'corr_half2': 0.167, 'corr_repetition': 0.0002, 'stability': 0.1504, 'mean_correct': 0.0324, 'mean_wrong': 0.0334, 'column_norm': 0.3417, 'rms_contribution': 0.0134}{'layer': 15, 'neuron': 699, 'corr_correct_length_adjusted': -0.168, 'corr_half1': -0.1852, 'corr_half2': -0.1501, 'corr_repetition': 0.148, 'stability': 0.1501, 'mean_correct': -0.0129, 'mean_wrong': -0.0093, 'column_norm': 0.3836, 'rms_contribution': 0.0053}{'layer': 16, 'neuron': 40, 'corr_correct_length_adjusted': 0.1775, 'corr_half1': 0.1497, 'corr_half2': 0.2124, 'corr_repetition': -0.3683, 'stability': 0.1497, 'mean_correct': 0.024, 'mean_wrong': 0.0204, 'column_norm': 0.4783, 'rms_contribution': 0.012}{'layer': 9, 'neuron': 1831, 'corr_correct_length_adjusted': 0.1497, 'corr_half1': 0.1493, 'corr_half2': 0.1646, 'corr_repetition': -0.0586, 'stability': 0.1493, 'mean_correct': 0.0024, 'mean_wrong': 0.0019, 'column_norm': 0.5203, 'rms_contribution': 0.0026}{'layer': 16, 'neuron': 257, 'corr_correct_length_adjusted': -0.1478, 'corr_half1': -0.149, 'corr_half2': -0.1493, 'corr_repetition': 0.1941, 'stability': 0.149, 'mean_correct': -0.0086, 'mean_wrong': -0.0066, 'column_norm': 0.4828, 'rms_contribution': 0.0045}{'layer': 8, 'neuron': 1150, 'corr_correct_length_adjusted': -0.1494, 'corr_half1': -0.1503, 'corr_half2': -0.1485, 'corr_repetition': 0.132, 'stability': 0.1485, 'mean_correct': 0.05, 'mean_wrong': 0.0633, 'column_norm': 0.2992, 'rms_contribution': 0.021}

DEV is now fixed at 64 intermittently solved problems, with a baseline accuracy of 42.2%; HOLDOUT remains untouched. Incorrect answers are longer and more repetitive, and layer 14 neuron 1594 is one candidate: its activity is associated with both repetition and lower correctness even after the length adjustment. These correlations are modest, so I won’t treat that neuron as a proven cause. I’ll prepare a teacher-forced batch covering every DEV sample and reference solution, and inspect which mathematical transitions activate the candidate channels before choosing an intervention.

python

prefix_eval = "<bos><start_of_turn>user\n"suffix_eval = "\nPlease reason step by step, and put your final answer within \\boxed{}.<end_of_turn>\n<start_of_turn>model\n"
def encode_ids(txt):    enc = tok.encode(txt)    return enc.ids if hasattr(enc, "ids") else list(enc)
tf_texts = []tf_records = []for pid in dev_ids:    row = problem_by_id[pid]    prompt = prefix_eval + row["problem"] + suffix_eval    prompt_ids = encode_ids(prompt)    variants = [("reference", True, row["solution"])]    variants += [(f"sample_{s['sample_idx']}", bool(s["correct"]), s["text"])                 for s in dev_flat if s["problem_id"] == pid]    for source_label, is_correct, answer_text in variants:        fulltext = prompt + answer_text        ids_full = encode_ids(fulltext)        cap = min(2048, len(prompt_ids) + (1536 if source_label == "reference" else 1024))        if len(ids_full) > cap:            fulltext = tok.decode(ids_full[:cap], skip_special_tokens=[REDACTED]            ids_full = encode_ids(fulltext)        tf_records.append({"problem_id": pid, "kind": source_label,                           "correct": is_correct, "prompt_tokens": len(prompt_ids),                           "total_tokens": len(ids_full)})        tf_texts.append(fulltext)assert len(tf_texts) <= 1024assert sum(r["total_tokens"] for r in tf_records) <= 524288(wdir / "tf_dev_texts.json").write_text(json.dumps(tf_texts))(wdir / "tf_dev_records.json").write_text(json.dumps(tf_records, indent=2))tf_request_original = {    "type": "teacher_forced", "model": "original",    "texts": tf_texts,    "return": {"token_logprobs": True, "topk": 0,               "hidden_states": [14, 15], "mlp_act": [8, 14, 17]}}(wdir / "pass2_request.json").write_text(json.dumps(tf_request_original))print("Teacher-forced batch:", len(tf_texts), "texts;",      sum(r["total_tokens"] for r in tf_records), "tokens; max",      max(r["total_tokens"] for r in tf_records))print("Layer-14 candidates:")for c in [c for c in candidates if c["layer"] == 14][:8]:    print({k: round(v, 4) if isinstance(v, float) else v for k, v in c.items()})
a1594 = np.array(mlpmeans1[dev_idx, 14, 1594])for i in np.argsort(a1594)[-4:][::-1]:    s = dev_flat[int(i)]    print("High-activity answer:", s["problem_id"], "correct", s["correct"],          "tokens", s["tokens"], "activation", round(float(a1594[i]), 4))    print((s["text"][:100] + " ... " + s["text"][-180:]).encode("ascii", "backslashreplace").decode())with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 2 planned: original teacher-forced localization\n"            "Exact edit: none; strength 0.\n"            "Set: all 64 fixed DEV problems, their 512 original samples and 64 reference solutions.\n"            "Texts saved in tf_dev_texts.json; complete request in pass2_request.json.\n"            "Own answers capped at first 1024 answer tokens; references at 1536; total text <=2048.\n"            "Return token logprobs, hidden states 14/15, MLP inputs at layers 8/14/17.\n"            "Candidate hypothesis: specific MLP channels reinforce repetitive, invalid mathematical continuations.\n")print("Saved request:", str(wdir / "pass2_request.json"))

prefix_eval = "<bos><start_of_turn>user\n"suffix_eval = "\nPlease reason step by step, and put your final answer within \\boxed{}.<end_of_turn>\n<start_of_turn>model\n"
def encode_ids(txt):    enc = tok.encode(txt)    return enc.ids if hasattr(enc, "ids") else list(enc)
tf_texts = []tf_records = []for pid in dev_ids:    row = problem_by_id[pid]    prompt = prefix_eval + row["problem"] + suffix_eval    prompt_ids = encode_ids(prompt)    variants = [("reference", True, row["solution"])]    variants += [(f"sample_{s['sample_idx']}", bool(s["correct"]), s["text"])                 for s in dev_flat if s["problem_id"] == pid]    for source_label, is_correct, answer_text in variants:        fulltext = prompt + answer_text        ids_full = encode_ids(fulltext)        cap = min(2048, len(prompt_ids) + (1536 if source_label == "reference" else 1024))        if len(ids_full) > cap:            fulltext = tok.decode(ids_full[:cap], skip_special_tokens=[REDACTED]            ids_full = encode_ids(fulltext)        tf_records.append({"problem_id": pid, "kind": source_label,                           "correct": is_correct, "prompt_tokens": len(prompt_ids),                           "total_tokens": len(ids_full)})        tf_texts.append(fulltext)assert len(tf_texts) <= 1024assert sum(r["total_tokens"] for r in tf_records) <= 524288(wdir / "tf_dev_texts.json").write_text(json.dumps(tf_texts))(wdir / "tf_dev_records.json").write_text(json.dumps(tf_records, indent=2))tf_request_original = {    "type": "teacher_forced", "model": "original",    "texts": tf_texts,    "return": {"token_logprobs": True, "topk": 0,               "hidden_states": [14, 15], "mlp_act": [8, 14, 17]}}(wdir / "pass2_request.json").write_text(json.dumps(tf_request_original))print("Teacher-forced batch:", len(tf_texts), "texts;",      sum(r["total_tokens"] for r in tf_records), "tokens; max",      max(r["total_tokens"] for r in tf_records))print("Layer-14 candidates:")for c in [c for c in candidates if c["layer"] == 14][:8]:    print({k: round(v, 4) if isinstance(v, float) else v for k, v in c.items()})
a1594 = np.array(mlpmeans1[dev_idx, 14, 1594])for i in np.argsort(a1594)[-4:][::-1]:    s = dev_flat[int(i)]    print("High-activity answer:", s["problem_id"], "correct", s["correct"],          "tokens", s["tokens"], "activation", round(float(a1594[i]), 4))    print((s["text"][:100] + " ... " + s["text"][-180:]).encode("ascii", "backslashreplace").decode())with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 2 planned: original teacher-forced localization\n"            "Exact edit: none; strength 0.\n"            "Set: all 64 fixed DEV problems, their 512 original samples and 64 reference solutions.\n"            "Texts saved in tf_dev_texts.json; complete request in pass2_request.json.\n"            "Own answers capped at first 1024 answer tokens; references at 1536; total text <=2048.\n"            "Return token logprobs, hidden states 14/15, MLP inputs at layers 8/14/17.\n"            "Candidate hypothesis: specific MLP channels reinforce repetitive, invalid mathematical continuations.\n")print("Saved request:", str(wdir / "pass2_request.json"))

OUTPUT of Python run 7 of at most 300:

Teacher-forced batch: 576 texts; 346789 tokens; max 1219Layer-14 candidates:{'layer': 14, 'neuron': 1594, 'corr_correct_length_adjusted': -0.1692, 'corr_half1': -0.1592, 'corr_half2': -0.1799, 'corr_repetition': 0.4726, 'stability': 0.1592, 'mean_correct': 0.0122, 'mean_wrong': 0.0206, 'column_norm': 0.2615, 'rms_contribution': 0.0089}{'layer': 14, 'neuron': 259, 'corr_correct_length_adjusted': -0.1599, 'corr_half1': -0.157, 'corr_half2': -0.1647, 'corr_repetition': 0.1526, 'stability': 0.157, 'mean_correct': -0.0035, 'mean_wrong': -0.0031, 'column_norm': 0.386, 'rms_contribution': 0.0027}{'layer': 14, 'neuron': 1723, 'corr_correct_length_adjusted': 0.1521, 'corr_half1': 0.1458, 'corr_half2': 0.159, 'corr_repetition': -0.3032, 'stability': 0.1458, 'mean_correct': 0.0217, 'mean_wrong': 0.0168, 'column_norm': 0.3328, 'rms_contribution': 0.0079}{'layer': 14, 'neuron': 820, 'corr_correct_length_adjusted': -0.1515, 'corr_half1': -0.1421, 'corr_half2': -0.1606, 'corr_repetition': -0.0267, 'stability': 0.1421, 'mean_correct': -0.0129, 'mean_wrong': -0.009, 'column_norm': 0.3783, 'rms_contribution': 0.0077}{'layer': 14, 'neuron': 902, 'corr_correct_length_adjusted': -0.1555, 'corr_half1': -0.1737, 'corr_half2': -0.1389, 'corr_repetition': 0.1666, 'stability': 0.1389, 'mean_correct': -0.0072, 'mean_wrong': -0.0062, 'column_norm': 0.3734, 'rms_contribution': 0.0031}{'layer': 14, 'neuron': 1745, 'corr_correct_length_adjusted': -0.1387, 'corr_half1': -0.1398, 'corr_half2': -0.1377, 'corr_repetition': 0.1, 'stability': 0.1377, 'mean_correct': -0.0273, 'mean_wrong': -0.0245, 'column_norm': 0.3573, 'rms_contribution': 0.012}{'layer': 14, 'neuron': 1244, 'corr_correct_length_adjusted': 0.1616, 'corr_half1': 0.1373, 'corr_half2': 0.1887, 'corr_repetition': -0.108, 'stability': 0.1373, 'mean_correct': -0.0055, 'mean_wrong': -0.0065, 'column_norm': 0.3422, 'rms_contribution': 0.0027}{'layer': 14, 'neuron': 470, 'corr_correct_length_adjusted': 0.1741, 'corr_half1': 0.1354, 'corr_half2': 0.2106, 'corr_repetition': -0.0607, 'stability': 0.1354, 'mean_correct': 0.0078, 'mean_wrong': 0.0057, 'column_norm': 0.3382, 'rms_contribution': 0.0034}High-activity answer: math_train_prealgebra_502 correct False tokens 8192 activation 0.2301Let $n$ be the number of calculators.Dean starts with 3 calculators.He presses the cube button on  ... +1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1High-activity answer: math_train_prealgebra_502 correct True tokens 8192 activation 0.1603Let $n$ be the number of calculators.Dean starts with 3 calculators.He presses the cube button on  ... tton.After this, he presses the negation button.After this, he presses the negation button.After this, he presses the negation buttonHigh-activity answer: math_train_prealgebra_502 correct False tokens 8192 activation 0.1528Let $n$ be the number of calculators in the circle.He presses the c ... the negation button.He presses the negation button.He passes all the calculators to the next person in the circle.He presses the number that shows 1 on the calculator that showsHigh-activity answer: math_train_intermediate_algebra_84 correct False tokens 8192 activation 0.113Let $x$ and $y$ be real numbers such that $x-y=4$ and $x^3-y^3=28$.We are given that $x-y = 4$.We  ... wer: 4+ y^2
Final Answer: 4+ y^2
Final Answer: 4+ y^2





Final Answer:Saved request: /work/pass2_request.json
[harness: 7 repeated lines not shown (a line is shown at most 2 times per output)]

Teacher-forced batch: 576 texts; 346789 tokens; max 1219Layer-14 candidates:{'layer': 14, 'neuron': 1594, 'corr_correct_length_adjusted': -0.1692, 'corr_half1': -0.1592, 'corr_half2': -0.1799, 'corr_repetition': 0.4726, 'stability': 0.1592, 'mean_correct': 0.0122, 'mean_wrong': 0.0206, 'column_norm': 0.2615, 'rms_contribution': 0.0089}{'layer': 14, 'neuron': 259, 'corr_correct_length_adjusted': -0.1599, 'corr_half1': -0.157, 'corr_half2': -0.1647, 'corr_repetition': 0.1526, 'stability': 0.157, 'mean_correct': -0.0035, 'mean_wrong': -0.0031, 'column_norm': 0.386, 'rms_contribution': 0.0027}{'layer': 14, 'neuron': 1723, 'corr_correct_length_adjusted': 0.1521, 'corr_half1': 0.1458, 'corr_half2': 0.159, 'corr_repetition': -0.3032, 'stability': 0.1458, 'mean_correct': 0.0217, 'mean_wrong': 0.0168, 'column_norm': 0.3328, 'rms_contribution': 0.0079}{'layer': 14, 'neuron': 820, 'corr_correct_length_adjusted': -0.1515, 'corr_half1': -0.1421, 'corr_half2': -0.1606, 'corr_repetition': -0.0267, 'stability': 0.1421, 'mean_correct': -0.0129, 'mean_wrong': -0.009, 'column_norm': 0.3783, 'rms_contribution': 0.0077}{'layer': 14, 'neuron': 902, 'corr_correct_length_adjusted': -0.1555, 'corr_half1': -0.1737, 'corr_half2': -0.1389, 'corr_repetition': 0.1666, 'stability': 0.1389, 'mean_correct': -0.0072, 'mean_wrong': -0.0062, 'column_norm': 0.3734, 'rms_contribution': 0.0031}{'layer': 14, 'neuron': 1745, 'corr_correct_length_adjusted': -0.1387, 'corr_half1': -0.1398, 'corr_half2': -0.1377, 'corr_repetition': 0.1, 'stability': 0.1377, 'mean_correct': -0.0273, 'mean_wrong': -0.0245, 'column_norm': 0.3573, 'rms_contribution': 0.012}{'layer': 14, 'neuron': 1244, 'corr_correct_length_adjusted': 0.1616, 'corr_half1': 0.1373, 'corr_half2': 0.1887, 'corr_repetition': -0.108, 'stability': 0.1373, 'mean_correct': -0.0055, 'mean_wrong': -0.0065, 'column_norm': 0.3422, 'rms_contribution': 0.0027}{'layer': 14, 'neuron': 470, 'corr_correct_length_adjusted': 0.1741, 'corr_half1': 0.1354, 'corr_half2': 0.2106, 'corr_repetition': -0.0607, 'stability': 0.1354, 'mean_correct': 0.0078, 'mean_wrong': 0.0057, 'column_norm': 0.3382, 'rms_contribution': 0.0034}High-activity answer: math_train_prealgebra_502 correct False tokens 8192 activation 0.2301Let $n$ be the number of calculators.Dean starts with 3 calculators.He presses the cube button on  ... +1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1High-activity answer: math_train_prealgebra_502 correct True tokens 8192 activation 0.1603Let $n$ be the number of calculators.Dean starts with 3 calculators.He presses the cube button on  ... tton.After this, he presses the negation button.After this, he presses the negation button.After this, he presses the negation buttonHigh-activity answer: math_train_prealgebra_502 correct False tokens 8192 activation 0.1528Let $n$ be the number of calculators in the circle.He presses the c ... the negation button.He presses the negation button.He passes all the calculators to the next person in the circle.He presses the number that shows 1 on the calculator that showsHigh-activity answer: math_train_intermediate_algebra_84 correct False tokens 8192 activation 0.113Let $x$ and $y$ be real numbers such that $x-y=4$ and $x^3-y^3=28$.We are given that $x-y = 4$.We  ... wer: 4+ y^2
Final Answer: 4+ y^2
Final Answer: 4+ y^2





Final Answer:Saved request: /work/pass2_request.json
[harness: 7 repeated lines not shown (a line is shown at most 2 times per output)]

The highest-activity examples for layer 14 neuron 1594 include runaway arithmetic and repeated “Final Answer” lines, which supports investigating repetition rather than general problem difficulty. However, one high-activity answer is graded correct despite looping, so suppressing repetition is not automatically an accuracy improvement. The teacher-forced batch contains 576 texts and 346,789 tokens, covering all fixed DEV problems without HOLDOUT. I’ll submit that saved batch to obtain token-level evidence; activation concentrated at repeated continuations would strengthen the hypothesis, while activation throughout useful calculations would argue against simple removal.

forward

{"type":"teacher_forced","model":"original","texts_file":"/work/tf_dev_texts.json","return":{"token_logprobs":true,"topk":0,"hidden_states":[14,15],"mlp_act":[8,14,17]}}

{"type":"teacher_forced","model":"original","texts_file":"/work/tf_dev_texts.json","return":{"token_logprobs":true,"topk":0,"hidden_states":[14,15],"mlp_act":[8,14,17]}}

The forward request is not valid (AssertionError: a teacher-forced pass needs texts or chats). Not counted.

The tool rejected texts_file, so no forward-pass budget was spent and there is no new model evidence. I’ll try supplying the saved batch through the required texts field. The batch and localization hypothesis are unchanged. If the interface does not accept a saved-file reference there, I’ll use an explicit list of raw texts instead.

forward

{"type":"teacher_forced","model":"original","texts":"/work/tf_dev_texts.json","return":{"token_logprobs":true,"topk":0,"hidden_states":[14,15],"mlp_act":[8,14,17]}}

{"type":"teacher_forced","model":"original","texts":"/work/tf_dev_texts.json","return":{"token_logprobs":true,"topk":0,"hidden_states":[14,15],"mlp_act":[8,14,17]}}

Forward pass 2 of 10 done (teacher_forced); results in /work/fwd_2/. Summary:

{ "ok": true, "lengths": [  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1 ], "files": {  "input_ids": [   23,   1  ],  "token_logprobs": [   23,   0  ],  "hidden_14": [   23,   1,   640  ],  "hidden_15": [   23,   1,   640  ],  "mlp_act_8": [   23,   1,   2048  ],  "mlp_act_14": [   23,   1,   2048  ],  "mlp_act_17": [   23,   1,   2048  ] }, "edited_tensors": 0, "mean_token_logprob": [  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null ], "type": "teacher_forced", "s": 0.6}

{ "ok": true, "lengths": [  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1 ], "files": {  "input_ids": [   23,   1  ],  "token_logprobs": [   23,   0  ],  "hidden_14": [   23,   1,   640  ],  "hidden_15": [   23,   1,   640  ],  "mlp_act_8": [   23,   1,   2048  ],  "mlp_act_14": [   23,   1,   2048  ],  "mlp_act_17": [   23,   1,   2048  ] }, "edited_tensors": 0, "mean_token_logprob": [  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null,  null ], "type": "teacher_forced", "s": 0.6}

Forward passes left: 8. Edits used in this pass: none (edit hash none).

The interface treated the filename as individual one-character texts, so pass 2 produced no usable mathematical evidence. That was an input-format mistake, and I’ll exclude it from all scientific comparisons. With eight passes remaining, I’ll use an explicit raw-text batch for baseline and edited teacher forcing, then reserve paired generation passes for strength testing and HOLDOUT confirmation. I’ll first size the DEV reference solutions so the next request can be submitted correctly.

python

reference_only = [problem_by_id[pid]["solution"] for pid in dev_ids]reference_tokens = [len(encode_ids(s)) for s in reference_only]print("All DEV reference solutions:", len(reference_only),      "tokens:", sum(reference_tokens),      "characters:", sum(map(len, reference_only)),      "max tokens:", max(reference_tokens))print("Solution token quantiles:", np.quantile(reference_tokens, [0, .25, .5, .75, 1]).tolist())print("DEV ids:", json.dumps(dev_ids))print("Reference previews:")for pid in dev_ids[:6]:    print(pid, problem_by_id[pid]["solution"][:220].encode("ascii", "backslashreplace").decode())with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 2 INVALID EXPERIMENT (budget consumed)\n"            "A string filename supplied as texts was interpreted as 23 single-character texts.\n"            "No mathematical scoring data; exclude all pass-2 results from analysis.\n"            "No edits. Eight passes remain. Subsequent requests must include literal raw-text lists.\n")

reference_only = [problem_by_id[pid]["solution"] for pid in dev_ids]reference_tokens = [len(encode_ids(s)) for s in reference_only]print("All DEV reference solutions:", len(reference_only),      "tokens:", sum(reference_tokens),      "characters:", sum(map(len, reference_only)),      "max tokens:", max(reference_tokens))print("Solution token quantiles:", np.quantile(reference_tokens, [0, .25, .5, .75, 1]).tolist())print("DEV ids:", json.dumps(dev_ids))print("Reference previews:")for pid in dev_ids[:6]:    print(pid, problem_by_id[pid]["solution"][:220].encode("ascii", "backslashreplace").decode())with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 2 INVALID EXPERIMENT (budget consumed)\n"            "A string filename supplied as texts was interpreted as 23 single-character texts.\n"            "No mathematical scoring data; exclude all pass-2 results from analysis.\n"            "No edits. Eight passes remain. Subsequent requests must include literal raw-text lists.\n")

OUTPUT of Python run 8 of at most 300:

All DEV reference solutions: 64 tokens: 7663 characters: 18820 max tokens: 422Solution token quantiles: [24.0, 65.75, 104.0, 132.25, 422.0]DEV ids: ["math_train_intermediate_algebra_847", "math_train_algebra_988", "math_train_intermediate_algebra_342", "math_train_prealgebra_311", "math_train_algebra_1185", "math_train_algebra_1520", "math_train_prealgebra_490", "math_train_number_theory_693", "math_train_prealgebra_1049", "math_train_number_theory_716", "math_train_counting_and_probability_55", "math_train_algebra_1497", "math_train_intermediate_algebra_802", "math_train_algebra_1220", "math_train_prealgebra_783", "math_train_algebra_583", "math_train_prealgebra_593", "math_train_algebra_7", "math_train_prealgebra_556", "math_train_prealgebra_795", "math_train_prealgebra_397", "math_train_prealgebra_869", "math_train_prealgebra_818", "math_train_intermediate_algebra_1008", "math_train_prealgebra_502", "math_train_prealgebra_82", "math_train_prealgebra_1130", "math_train_algebra_1284", "math_train_number_theory_709", "math_train_algebra_920", "math_train_algebra_198", "math_train_intermediate_algebra_84", "math_train_precalculus_76", "math_train_prealgebra_590", "math_train_precalculus_330", "math_train_algebra_763", "math_train_prealgebra_1028", "math_train_algebra_1210", "math_train_counting_and_probability_653", "math_train_algebra_380", "math_train_geometry_113", "math_train_intermediate_algebra_1197", "math_train_algebra_1128", "math_train_algebra_448", "math_train_algebra_1559", "math_train_geometry_386", "math_train_algebra_817", "math_train_prealgebra_1033", "math_train_counting_and_probability_518", "math_train_number_theory_706", "math_train_algebra_34", "math_train_prealgebra_134", "math_train_prealgebra_867", "math_train_algebra_1094", "math_train_algebra_1491", "math_train_geometry_259", "math_train_precalculus_274", "math_train_precalculus_233", "math_train_geometry_161", "math_train_algebra_1586", "math_train_geometry_305", "math_train_algebra_1180", "math_train_algebra_775", "math_train_algebra_485"]Reference previews:math_train_intermediate_algebra_847 We have that\begin{align*}a_3 &= \frac{a_2}{a_1} = \frac{3}{2}, \\a_4 &= \frac{a_3}{a_2} = \frac{3/2}{3} = \frac{1}{2}, \\a_5 &= \frac{a_4}{a_3} = \frac{1/2}{3/2} = \frac{1}{3}, \\a_6 &= \frac{a_5}{a_4} = \frac{1/3}math_train_algebra_988 The midpoint of a line segment with endpoints $(x_1, y_1), (x_2, y_2)$ is $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
Thus, the midpoint of this line segment is $\left(\frac{2+8}{2}, \frac{3+15}{2} \right)$math_train_intermediate_algebra_342 Let $y = x^2 + x + 3.$  Then we can write the given equation as\[\frac{4x}{y} + \frac{5x}{y - 6x} + \frac{3}{2} = 0.\]Multiplying everything by $2y(y - 6x),$ we get\[8x(y - 6x) + 10xy + 3y(y - 6x) = 0.\]Expanding, we gmath_train_prealgebra_311 Simplify, remembering the exponents take priority over multiplication and multiplication takes priority over addition or subtraction. \begin{align*}5^3-3\times 5^2+3\times5-1&=125-3\times25+15-1 \\&=125-75+14 \\&= 50+math_train_algebra_1185 We proceed as follows: \begin{align*}(2b + 5)(b - 1) &= 6b\\2b^2 + 3b - 5 &= 6b\\2b^2 - 3b - 5 &= 0\\(2b - 5)(b + 1) &= 0.\end{align*}This gives us $b = \frac{5}{2}$ or $b = -1.$ Of these, $\boxed{\frac{5}{2}}$ is tmath_train_algebra_1520 We solve each equation separately. First off, we have $8x^2 + 7x - 1 = (8x-1)(x+1) = 0.$ We can also see that $24x^2+53x-7 = (8x-1)(3x+7) = 0.$ It is clear that both equations are satisfied only when $8x - 1 = 0,$ theref

All DEV reference solutions: 64 tokens: 7663 characters: 18820 max tokens: 422Solution token quantiles: [24.0, 65.75, 104.0, 132.25, 422.0]DEV ids: ["math_train_intermediate_algebra_847", "math_train_algebra_988", "math_train_intermediate_algebra_342", "math_train_prealgebra_311", "math_train_algebra_1185", "math_train_algebra_1520", "math_train_prealgebra_490", "math_train_number_theory_693", "math_train_prealgebra_1049", "math_train_number_theory_716", "math_train_counting_and_probability_55", "math_train_algebra_1497", "math_train_intermediate_algebra_802", "math_train_algebra_1220", "math_train_prealgebra_783", "math_train_algebra_583", "math_train_prealgebra_593", "math_train_algebra_7", "math_train_prealgebra_556", "math_train_prealgebra_795", "math_train_prealgebra_397", "math_train_prealgebra_869", "math_train_prealgebra_818", "math_train_intermediate_algebra_1008", "math_train_prealgebra_502", "math_train_prealgebra_82", "math_train_prealgebra_1130", "math_train_algebra_1284", "math_train_number_theory_709", "math_train_algebra_920", "math_train_algebra_198", "math_train_intermediate_algebra_84", "math_train_precalculus_76", "math_train_prealgebra_590", "math_train_precalculus_330", "math_train_algebra_763", "math_train_prealgebra_1028", "math_train_algebra_1210", "math_train_counting_and_probability_653", "math_train_algebra_380", "math_train_geometry_113", "math_train_intermediate_algebra_1197", "math_train_algebra_1128", "math_train_algebra_448", "math_train_algebra_1559", "math_train_geometry_386", "math_train_algebra_817", "math_train_prealgebra_1033", "math_train_counting_and_probability_518", "math_train_number_theory_706", "math_train_algebra_34", "math_train_prealgebra_134", "math_train_prealgebra_867", "math_train_algebra_1094", "math_train_algebra_1491", "math_train_geometry_259", "math_train_precalculus_274", "math_train_precalculus_233", "math_train_geometry_161", "math_train_algebra_1586", "math_train_geometry_305", "math_train_algebra_1180", "math_train_algebra_775", "math_train_algebra_485"]Reference previews:math_train_intermediate_algebra_847 We have that\begin{align*}a_3 &= \frac{a_2}{a_1} = \frac{3}{2}, \\a_4 &= \frac{a_3}{a_2} = \frac{3/2}{3} = \frac{1}{2}, \\a_5 &= \frac{a_4}{a_3} = \frac{1/2}{3/2} = \frac{1}{3}, \\a_6 &= \frac{a_5}{a_4} = \frac{1/3}math_train_algebra_988 The midpoint of a line segment with endpoints $(x_1, y_1), (x_2, y_2)$ is $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
Thus, the midpoint of this line segment is $\left(\frac{2+8}{2}, \frac{3+15}{2} \right)$math_train_intermediate_algebra_342 Let $y = x^2 + x + 3.$  Then we can write the given equation as\[\frac{4x}{y} + \frac{5x}{y - 6x} + \frac{3}{2} = 0.\]Multiplying everything by $2y(y - 6x),$ we get\[8x(y - 6x) + 10xy + 3y(y - 6x) = 0.\]Expanding, we gmath_train_prealgebra_311 Simplify, remembering the exponents take priority over multiplication and multiplication takes priority over addition or subtraction. \begin{align*}5^3-3\times 5^2+3\times5-1&=125-3\times25+15-1 \\&=125-75+14 \\&= 50+math_train_algebra_1185 We proceed as follows: \begin{align*}(2b + 5)(b - 1) &= 6b\\2b^2 + 3b - 5 &= 6b\\2b^2 - 3b - 5 &= 0\\(2b - 5)(b + 1) &= 0.\end{align*}This gives us $b = \frac{5}{2}$ or $b = -1.$ Of these, $\boxed{\frac{5}{2}}$ is tmath_train_algebra_1520 We solve each equation separately. First off, we have $8x^2 + 7x - 1 = (8x-1)(x+1) = 0.$ We can also see that $24x^2+53x-7 = (8x-1)(3x+7) = 0.$ It is clear that both equations are satisfied only when $8x - 1 = 0,$ theref

The full set of 64 DEV reference solutions is small enough to submit explicitly, so I’ll use all of them as local mathematical-continuation probes. I’ll also include eight short excerpts from repetitive failing DEV answers to test whether candidate neurons distinguish useful calculations from loops. These reference-only probes are not evaluation-format answer scores; paired generation will remain the accuracy test. The expected screening result is preservation of reference calculations while reducing support for repetitive continuations.

python

wrong_rank = sorted(    [(float(a1594[i]), s) for i, s in enumerate(dev_flat) if not s["correct"]],    key=lambda x: x[0], reverse=True)probe_failures = []used_probe_pids = set()for activity, s in wrong_rank:    if s["problem_id"] in used_probe_pids:        continue    used_probe_pids.add(s["problem_id"])    probe_failures.append({"problem_id": s["problem_id"], "sample_idx": s["sample_idx"],                           "activity": activity, "text": s["text"][-280:]})    if len(probe_failures) == 8:        breakliteral_probes = reference_only + [r["text"] for r in probe_failures](wdir / "literal_probes.json").write_text(json.dumps(literal_probes))(wdir / "literal_probe_metadata.json").write_text(json.dumps(    [{"problem_id": pid, "kind": "reference"} for pid in dev_ids] +    [dict(r, kind="repetitive_failure") for r in probe_failures], indent=2))with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 3 planned: literal teacher-forced localization\n"            "Exact edit: none; strength 0. Set: fixed DEV only.\n"            "72 raw local-continuation probes: all 64 full reference solutions plus eight "            "280-character excerpts from repetitive failures, selected by high L14 neuron1594 activity.\n"            "No problem prompts: these measure local continuation effects, NOT answer accuracy or "            "evaluation-format answer likelihood. literal_probes.json records exact texts.\n"            "Return token logprobs and MLP activations at layers 8,14,17.\n")print("Literal probe count:", len(literal_probes), "tokens:",      sum(len(encode_ids(s)) for s in literal_probes))print("Reference texts 0–23:")print(json.dumps(literal_probes[:24], ensure_ascii=True))

wrong_rank = sorted(    [(float(a1594[i]), s) for i, s in enumerate(dev_flat) if not s["correct"]],    key=lambda x: x[0], reverse=True)probe_failures = []used_probe_pids = set()for activity, s in wrong_rank:    if s["problem_id"] in used_probe_pids:        continue    used_probe_pids.add(s["problem_id"])    probe_failures.append({"problem_id": s["problem_id"], "sample_idx": s["sample_idx"],                           "activity": activity, "text": s["text"][-280:]})    if len(probe_failures) == 8:        breakliteral_probes = reference_only + [r["text"] for r in probe_failures](wdir / "literal_probes.json").write_text(json.dumps(literal_probes))(wdir / "literal_probe_metadata.json").write_text(json.dumps(    [{"problem_id": pid, "kind": "reference"} for pid in dev_ids] +    [dict(r, kind="repetitive_failure") for r in probe_failures], indent=2))with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 3 planned: literal teacher-forced localization\n"            "Exact edit: none; strength 0. Set: fixed DEV only.\n"            "72 raw local-continuation probes: all 64 full reference solutions plus eight "            "280-character excerpts from repetitive failures, selected by high L14 neuron1594 activity.\n"            "No problem prompts: these measure local continuation effects, NOT answer accuracy or "            "evaluation-format answer likelihood. literal_probes.json records exact texts.\n"            "Return token logprobs and MLP activations at layers 8,14,17.\n")print("Literal probe count:", len(literal_probes), "tokens:",      sum(len(encode_ids(s)) for s in literal_probes))print("Reference texts 0–23:")print(json.dumps(literal_probes[:24], ensure_ascii=True))

OUTPUT of Python run 9 of at most 300:

Literal probe count: 72 tokens: 8985Reference texts 0?23:["We have that\n\\begin{align*}\na_3 &= \\frac{a_2}{a_1} = \\frac{3}{2}, \\\\\na_4 &= \\frac{a_3}{a_2} = \\frac{3/2}{3} = \\frac{1}{2}, \\\\\na_5 &= \\frac{a_4}{a_3} = \\frac{1/2}{3/2} = \\frac{1}{3}, \\\\\na_6 &= \\frac{a_5}{a_4} = \\frac{1/3}{1/2} = \\frac{2}{3}, \\\\\na_7 &= \\frac{a_6}{a_5} = \\frac{2/3}{1/3} = 2, \\\\\na_8 &= \\frac{a_7}{a_6} = \\frac{2}{2/3} = 3.\n\\end{align*}Since $a_7 = a_1 = 2$ and $a_8 = a_2 = 3,$ and each term depends only on the previous two terms, the sequence becomes periodic at this point, with a period of length 6.  Hence, $a_{2006} = a_2 = \\boxed{3}.$", "The midpoint of a line segment with endpoints $(x_1, y_1), (x_2, y_2)$ is $\\left(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}\\right)$.\n\nThus, the midpoint of this line segment is $\\left(\\frac{2+8}{2}, \\frac{3+15}{2} \\right)$, which simplifies to $(5,9)$. Thus, the sum of the coordinates of the midpoint is $\\boxed{14}$.", "Let $y = x^2 + x + 3.$  Then we can write the given equation as\n\\[\\frac{4x}{y} + \\frac{5x}{y - 6x} + \\frac{3}{2} = 0.\\]Multiplying everything by $2y(y - 6x),$ we get\n\\[8x(y - 6x) + 10xy + 3y(y - 6x) = 0.\\]Expanding, we get $3y^2 - 48x^2 = 0,$ so $y^2 - 16x^2 = (y - 4x)(y + 4x) = 0.$  Thus, $y = 4x$ or $y = -4x.$\n\nIf $y = 4x,$ then $x^2 + x + 3 = 4x,$ so $x^2 - 3x + 3 = 0.$  This quadratic has no real solutions.\n\nIf $y = -4x,$ then $x^2 + x + 3 = -4x,$ so $x^2 + 5x + 3 = 0.$  This quadratic has two real solutions, giving us a total of $\\boxed{2}$ real solutions.", "Simplify, remembering the exponents take priority over multiplication and multiplication takes priority over addition or subtraction. \\begin{align*}\n5^3-3\\times 5^2+3\\times5-1&=125-3\\times25+15-1 \\\\\n&=125-75+14 \\\\\n&= 50+14 \\\\\n&= \\boxed{64}.\n\\end{align*}", "We proceed as follows: \\begin{align*}\n(2b + 5)(b - 1) &= 6b\\\\\n2b^2 + 3b - 5 &= 6b\\\\\n2b^2 - 3b - 5 &= 0\\\\\n(2b - 5)(b + 1) &= 0.\n\\end{align*}This gives us $b = \\frac{5}{2}$ or $b = -1.$ Of these, $\\boxed{\\frac{5}{2}}$ is the larger possible value of $b.$", "We solve each equation separately. First off, we have $8x^2 + 7x - 1 = (8x-1)(x+1) = 0.$ We can also see that $24x^2+53x-7 = (8x-1)(3x+7) = 0.$ It is clear that both equations are satisfied only when $8x - 1 = 0,$ therefore $x = \\boxed{\\dfrac{1}{8}}.$", "Let's substitute the given values into the expression. We get  \\begin{align*}\n(a^3+b^3)\\div(a^2-ab+b^2)&=(5^3+4^3)\\div(5^2-5\\cdot4+4^2)\\\\\n&=(125+64)\\div(25-20+16)\\\\\n&=189\\div21\\\\\n&=\\boxed{9}.\n\\end{align*}", "The only numbers that have 11 and 8 as a factor are multiples of 88.  If we list the first few multiples of 88: $$88,176,264,352,...$$ we can see that there are exactly $\\boxed{2}$ between 100 and 300.", "Since $3^3 = 3\\times 3\\times 3 = 3\\times 9 = 27$, then  \\[ \\sqrt{3^3+3^3+3^3} = \\sqrt{27+27+27}=\\sqrt{81}=\\boxed{9}. \\]", "Let $n$ be any integer that has its square between 4000 and 7000. Then $63 < n < 84$, because $63^2 < 4000<64^2$ and $83^2< 7000<84^2$. Between 63 and 84, the only primes are 67, 71, 73, 79, and 83. Thus, the answer is $\\boxed{5}$.", "There are ${9 \\choose 2} = 36$ pairs of plates possible in all. Exactly ${5 \\choose 2} = 10$ of these pairs are both red plates and $\\binom{4}{2}=6$ pairs are both blue plates for a total of $6+10=16$ pairs of plates that satisfy our condition. So, the probability of the pair of plates being the same color is $\\frac{16}{36}=\\boxed{\\frac{4}{9}}$.", "Let $B$ and $J$ represent the respective ages of Billy and Joe. We can write the equations $B=2J$ and $B+J=45$. We use the second equation to solve for $J$ in terms of $B$ and get $J=45-B$. Now we plug in this expression for $J$ into the first equation. $$B=2(45-B)=90-2B\\qquad\\Rightarrow 3B=90\\qquad\\Rightarrow B=30$$ So Billy is $\\boxed{30}$ years old.", "By AM-HM,\n\\[\\frac{x + y}{2} \\ge \\frac{2}{\\frac{1}{x} + \\frac{1}{y}}.\\]Hence,\n\\[\\frac{1}{x} + \\frac{1}{y} \\ge \\frac{4}{x + y} = \\frac{4}{10} = \\frac{2}{5}.\\]Equality occurs when $x = y = 5,$ so the minimum value is $\\boxed{\\frac{2}{5}}.$", "$64^2 - 36^2$ can also be expressed as $(64+36)(64-36)$. This simplifies as $100 \\cdot 28$, which equals $\\boxed{2800}$.", "Since the triangle is equilateral, all sides are equal in length. Therefore, the perimeter of the triangle is $8+8+8=8 \\times 3=\\boxed{24}.$", "We have two equations and two variables, so it's possible to solve for $p$ and $q$ directly and then calculate $p^2$ and $q^2$ separately to get our answer. However, doing so involves a fair amount of computation with complex numbers and square roots, so we look for an alternative approach. We square the second equation to get $$(p+q)^2 = p^2 + 2pq + q^2 = 36,$$which is close to what we want but has the extra $2pq$ term. Since we know that $pq=9$, we can substitute to get $$p^2 + 2(9) +q^2 = 36 \\implies p^2+q^2 = \\boxed{18}.$$Note that our task was made easier by only solving for what the problem asked rather than by trying to solve for $p$ and $q$ individually.", "Triangle $ABC$ has base $AB$ of length $25\\text{ cm}$ and height $AC$ of length $20\\text{ cm}.$ Therefore, the area of triangle $ABC$ is \\begin{align*}\n\\frac{1}{2}bh &= \\frac{1}{2}(25 \\mbox{ cm})(20 \\mbox{ cm}) \\\\\n&= \\frac{1}{2}(500 \\mbox{ cm}^2) \\\\\n&= \\boxed{250} \\mbox{ cm}^2.\n\\end{align*}", "We have  \\[\\frac{x^4 + 2y^2}{6} = \\frac{2^4 + 2(5^2)}{6} = \\frac{16+2(25)}{6} = \\frac{16+50}{6} = \\frac{66}{6} = \\boxed{11}.\\]", "The average length is $\\frac{1.5+4.5}{2}=\\boxed{3}$ inches.", "Her mean score was $\\frac{98+97+92+85+93}{5}=\\boxed{93}$.", "We simplify the given inequality by adding $x+3$ to both sides, which leaves us with $5x < 5 \\Rightarrow x<1$. Since 1 is not less than itself, this leaves us with $x=\\boxed{0}$.", "We know that any number raised to the power of $0$ is $1$, or $a^0 = 1$ for any $a$. Thus, we get $$(5^{-2})^0 + (5^0)^3 = 1 + 1^3 = 1+1 = \\boxed{2}.$$", "Since $2a + 1 = 1,$ we find that $2a = 0,$ so $a = 0.$ Therefore, $b - a = b - 0 = b = \\boxed{1}.$", "\\[\n\\begin{array}{c|cc cc}\n\\multicolumn{2}{r}{x^2} & +\\left(\\frac{c-7}{2}\\right)x & +5 \\\\\n\\cline{2-5}\n2x+7 & 2x^3 &+cx^2 &- 11x &+ 39  \\\\\n\\multicolumn{2}{r}{-2x^3} & -7x^2  \\\\ \n\\cline{2-3}\n\\multicolumn{2}{r}{0} & (c-7)x^2 & -11x  \\\\\n\\multicolumn{2}{r}{} & -(c-7)x^2 & -x(c-7)\\left(\\frac{7}{2}\\right)   \\\\ \n\\cline{3-4}\n\\multicolumn{2}{r}{} &  0 & -x\\left(\\frac{7c-27}{2}\\right) & + 39  \\\\\n\\multicolumn{2}{r}{} &   & -10x & -35  \\\\\n\\cline{4-5}\n\\multicolumn{2}{r}{} &   & -x\\left(\\frac{7c-27+20}{2}\\right) & 4  \\\\\n\\end{array}\n\\]In the last step of the division, we have $39$ left as the constant term in our dividend and we need a remainder of $4$ at the end. Since our divisor has a term of $7$, the only way to do this is if our quotient has $5$ which gives us  $7\\cdot5=35$ to subtract from our dividend and get the right remainder.\n\nThen, we need the rest of our remainder to be $0$. This means\n$$\\frac{7c-27+20}{2} = 0$$which gives us\n$$c = \\boxed{1}.$$"]

Literal probe count: 72 tokens: 8985Reference texts 0?23:["We have that\n\\begin{align*}\na_3 &= \\frac{a_2}{a_1} = \\frac{3}{2}, \\\\\na_4 &= \\frac{a_3}{a_2} = \\frac{3/2}{3} = \\frac{1}{2}, \\\\\na_5 &= \\frac{a_4}{a_3} = \\frac{1/2}{3/2} = \\frac{1}{3}, \\\\\na_6 &= \\frac{a_5}{a_4} = \\frac{1/3}{1/2} = \\frac{2}{3}, \\\\\na_7 &= \\frac{a_6}{a_5} = \\frac{2/3}{1/3} = 2, \\\\\na_8 &= \\frac{a_7}{a_6} = \\frac{2}{2/3} = 3.\n\\end{align*}Since $a_7 = a_1 = 2$ and $a_8 = a_2 = 3,$ and each term depends only on the previous two terms, the sequence becomes periodic at this point, with a period of length 6.  Hence, $a_{2006} = a_2 = \\boxed{3}.$", "The midpoint of a line segment with endpoints $(x_1, y_1), (x_2, y_2)$ is $\\left(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}\\right)$.\n\nThus, the midpoint of this line segment is $\\left(\\frac{2+8}{2}, \\frac{3+15}{2} \\right)$, which simplifies to $(5,9)$. Thus, the sum of the coordinates of the midpoint is $\\boxed{14}$.", "Let $y = x^2 + x + 3.$  Then we can write the given equation as\n\\[\\frac{4x}{y} + \\frac{5x}{y - 6x} + \\frac{3}{2} = 0.\\]Multiplying everything by $2y(y - 6x),$ we get\n\\[8x(y - 6x) + 10xy + 3y(y - 6x) = 0.\\]Expanding, we get $3y^2 - 48x^2 = 0,$ so $y^2 - 16x^2 = (y - 4x)(y + 4x) = 0.$  Thus, $y = 4x$ or $y = -4x.$\n\nIf $y = 4x,$ then $x^2 + x + 3 = 4x,$ so $x^2 - 3x + 3 = 0.$  This quadratic has no real solutions.\n\nIf $y = -4x,$ then $x^2 + x + 3 = -4x,$ so $x^2 + 5x + 3 = 0.$  This quadratic has two real solutions, giving us a total of $\\boxed{2}$ real solutions.", "Simplify, remembering the exponents take priority over multiplication and multiplication takes priority over addition or subtraction. \\begin{align*}\n5^3-3\\times 5^2+3\\times5-1&=125-3\\times25+15-1 \\\\\n&=125-75+14 \\\\\n&= 50+14 \\\\\n&= \\boxed{64}.\n\\end{align*}", "We proceed as follows: \\begin{align*}\n(2b + 5)(b - 1) &= 6b\\\\\n2b^2 + 3b - 5 &= 6b\\\\\n2b^2 - 3b - 5 &= 0\\\\\n(2b - 5)(b + 1) &= 0.\n\\end{align*}This gives us $b = \\frac{5}{2}$ or $b = -1.$ Of these, $\\boxed{\\frac{5}{2}}$ is the larger possible value of $b.$", "We solve each equation separately. First off, we have $8x^2 + 7x - 1 = (8x-1)(x+1) = 0.$ We can also see that $24x^2+53x-7 = (8x-1)(3x+7) = 0.$ It is clear that both equations are satisfied only when $8x - 1 = 0,$ therefore $x = \\boxed{\\dfrac{1}{8}}.$", "Let's substitute the given values into the expression. We get  \\begin{align*}\n(a^3+b^3)\\div(a^2-ab+b^2)&=(5^3+4^3)\\div(5^2-5\\cdot4+4^2)\\\\\n&=(125+64)\\div(25-20+16)\\\\\n&=189\\div21\\\\\n&=\\boxed{9}.\n\\end{align*}", "The only numbers that have 11 and 8 as a factor are multiples of 88.  If we list the first few multiples of 88: $$88,176,264,352,...$$ we can see that there are exactly $\\boxed{2}$ between 100 and 300.", "Since $3^3 = 3\\times 3\\times 3 = 3\\times 9 = 27$, then  \\[ \\sqrt{3^3+3^3+3^3} = \\sqrt{27+27+27}=\\sqrt{81}=\\boxed{9}. \\]", "Let $n$ be any integer that has its square between 4000 and 7000. Then $63 < n < 84$, because $63^2 < 4000<64^2$ and $83^2< 7000<84^2$. Between 63 and 84, the only primes are 67, 71, 73, 79, and 83. Thus, the answer is $\\boxed{5}$.", "There are ${9 \\choose 2} = 36$ pairs of plates possible in all. Exactly ${5 \\choose 2} = 10$ of these pairs are both red plates and $\\binom{4}{2}=6$ pairs are both blue plates for a total of $6+10=16$ pairs of plates that satisfy our condition. So, the probability of the pair of plates being the same color is $\\frac{16}{36}=\\boxed{\\frac{4}{9}}$.", "Let $B$ and $J$ represent the respective ages of Billy and Joe. We can write the equations $B=2J$ and $B+J=45$. We use the second equation to solve for $J$ in terms of $B$ and get $J=45-B$. Now we plug in this expression for $J$ into the first equation. $$B=2(45-B)=90-2B\\qquad\\Rightarrow 3B=90\\qquad\\Rightarrow B=30$$ So Billy is $\\boxed{30}$ years old.", "By AM-HM,\n\\[\\frac{x + y}{2} \\ge \\frac{2}{\\frac{1}{x} + \\frac{1}{y}}.\\]Hence,\n\\[\\frac{1}{x} + \\frac{1}{y} \\ge \\frac{4}{x + y} = \\frac{4}{10} = \\frac{2}{5}.\\]Equality occurs when $x = y = 5,$ so the minimum value is $\\boxed{\\frac{2}{5}}.$", "$64^2 - 36^2$ can also be expressed as $(64+36)(64-36)$. This simplifies as $100 \\cdot 28$, which equals $\\boxed{2800}$.", "Since the triangle is equilateral, all sides are equal in length. Therefore, the perimeter of the triangle is $8+8+8=8 \\times 3=\\boxed{24}.$", "We have two equations and two variables, so it's possible to solve for $p$ and $q$ directly and then calculate $p^2$ and $q^2$ separately to get our answer. However, doing so involves a fair amount of computation with complex numbers and square roots, so we look for an alternative approach. We square the second equation to get $$(p+q)^2 = p^2 + 2pq + q^2 = 36,$$which is close to what we want but has the extra $2pq$ term. Since we know that $pq=9$, we can substitute to get $$p^2 + 2(9) +q^2 = 36 \\implies p^2+q^2 = \\boxed{18}.$$Note that our task was made easier by only solving for what the problem asked rather than by trying to solve for $p$ and $q$ individually.", "Triangle $ABC$ has base $AB$ of length $25\\text{ cm}$ and height $AC$ of length $20\\text{ cm}.$ Therefore, the area of triangle $ABC$ is \\begin{align*}\n\\frac{1}{2}bh &= \\frac{1}{2}(25 \\mbox{ cm})(20 \\mbox{ cm}) \\\\\n&= \\frac{1}{2}(500 \\mbox{ cm}^2) \\\\\n&= \\boxed{250} \\mbox{ cm}^2.\n\\end{align*}", "We have  \\[\\frac{x^4 + 2y^2}{6} = \\frac{2^4 + 2(5^2)}{6} = \\frac{16+2(25)}{6} = \\frac{16+50}{6} = \\frac{66}{6} = \\boxed{11}.\\]", "The average length is $\\frac{1.5+4.5}{2}=\\boxed{3}$ inches.", "Her mean score was $\\frac{98+97+92+85+93}{5}=\\boxed{93}$.", "We simplify the given inequality by adding $x+3$ to both sides, which leaves us with $5x < 5 \\Rightarrow x<1$. Since 1 is not less than itself, this leaves us with $x=\\boxed{0}$.", "We know that any number raised to the power of $0$ is $1$, or $a^0 = 1$ for any $a$. Thus, we get $$(5^{-2})^0 + (5^0)^3 = 1 + 1^3 = 1+1 = \\boxed{2}.$$", "Since $2a + 1 = 1,$ we find that $2a = 0,$ so $a = 0.$ Therefore, $b - a = b - 0 = b = \\boxed{1}.$", "\\[\n\\begin{array}{c|cc cc}\n\\multicolumn{2}{r}{x^2} & +\\left(\\frac{c-7}{2}\\right)x & +5 \\\\\n\\cline{2-5}\n2x+7 & 2x^3 &+cx^2 &- 11x &+ 39  \\\\\n\\multicolumn{2}{r}{-2x^3} & -7x^2  \\\\ \n\\cline{2-3}\n\\multicolumn{2}{r}{0} & (c-7)x^2 & -11x  \\\\\n\\multicolumn{2}{r}{} & -(c-7)x^2 & -x(c-7)\\left(\\frac{7}{2}\\right)   \\\\ \n\\cline{3-4}\n\\multicolumn{2}{r}{} &  0 & -x\\left(\\frac{7c-27}{2}\\right) & + 39  \\\\\n\\multicolumn{2}{r}{} &   & -10x & -35  \\\\\n\\cline{4-5}\n\\multicolumn{2}{r}{} &   & -x\\left(\\frac{7c-27+20}{2}\\right) & 4  \\\\\n\\end{array}\n\\]In the last step of the division, we have $39$ left as the constant term in our dividend and we need a remainder of $4$ at the end. Since our divisor has a term of $7$, the only way to do this is if our quotient has $5$ which gives us  $7\\cdot5=35$ to subtract from our dividend and get the right remainder.\n\nThen, we need the rest of our remainder to be $0$. This means\n$$\\frac{7c-27+20}{2} = 0$$which gives us\n$$c = \\boxed{1}.$$"]

The explicit probe batch contains all 64 reference solutions plus eight failure excerpts, totaling 8,985 tokens. The reference examples cover valid arithmetic, constraint handling, and algebraic transformations, providing useful checks against indiscriminate suppression of mathematical activity. I’m transferring the batch in small pieces solely to avoid output truncation; no additional experiment is running yet. The next two inspections will finish preparing the literal request.

OUTPUT of Python run 10 of at most 300:

Reference texts 24?47:["Let's begin with the calculator that initially shows 1. Each time it is passed around the circle, it is cubed. 1 to any power is still 1, so no matter how many times 1 is cubed, the final result will still be 1.\n\nNow examine the calculator that starts with a zero. 0 squared is still 0 because 0 to any positive power is still 0. Thus, no matter how many times zero is squared, the final number will still be zero.\n\nFinally, let's look at the calculator that initially shows -1. Each time a person gets the calculator, they negate the number. Because there are 42 participants, there are 42 total turns. Thus, -1 is negated 42 times. Because negating a number is the same as multiplying by -1, this is the same as multiplying it by -1 forty-two times. Thus, we are looking for \\[(-1) \\cdot (-1)^{42}=(-1)^1 \\cdot (-1)^{42}=(-1)^{1+42}=(-1)^{43}.\\]Recall that $(-a)^n=-a^n$ if $n$ is odd. Because 43 is odd, $(-1)^{43}=-1^{43}=-1$.\n\nThus, the sum of all of the numbers is $1+0+(-1)=\\boxed{0}$.", "By definition, an obtuse triangle contains one obtuse interior angle. It cannot contain more than one because an obtuse angle has measure greater than 90 degrees and the sum of all the interior angles in any triangle is 180 degrees. Thus, there is $\\boxed{1}$ obtuse angle.", "There is one King of $\\diamondsuit$ and 52 cards total, so the probability that the top card is a King of $\\diamondsuit$ is $\\boxed{\\dfrac{1}{52}}$.", "We have  \\[81^{3/4} = (3^4)^{3/4} = 3^{4\\cdot (3/4)} = 3^3 = \\boxed{27}.\\]", "Since 3 and 4 are relatively prime, their least common multiple is $3\\cdot4=12$.  Therefore, the least common multiple of 30 and 40 is 120.  Since $\\boxed{120}$ is not divisible by 16, it is the smallest common multiple of 30 and 40 which is not divisible by 16.\n\nNote: Every common multiple of two integers is a multiple of their least common multiple.  Therefore, it is not possible that the least common multiple is divisible by 16 but some other common multiple is not.", "We have $g(-5) = 5(-5) + 2(-5)^2 = -25 + 50 = 25$, so $f(g(-5)) = f(25) = 3 - \\!\\sqrt{25} = 3-5=\\boxed{-2}$.", "We have $55 \\times 1212 - 15 \\times 1212 = 1212(55-15) = 1212(40) = 4848(10) = \\boxed{48480}$.", "Solution 1. The first equation gives $x = y+4$. Substituting into the second equation, we get \\[(y+4)^3 - y^3 = 28 \\implies 12y^2 + 48y + 36 = 0.\\]Thus, $y^2 + 4y + 3 = 0$, so $(y+1)(y+3) = 0$. Therefore, either $y=-1$ and $x=y+4=3$, or $y=-3$ and $x=y+4=1$. Either way, $xy = \\boxed{-3}$.\n\nSolution 2. The second equation factors via difference of cubes, as \\[(x-y)(x^2+xy+y^2) = 28.\\]Since $x-y=4$, we have $x^2+xy+y^2=\\frac{28}{4} =7$. Now, squaring the first equation, we get $x^2-2xy+y^2=16$. Thus, \\[3xy = (x^2+xy+y^2) - (x^2-2xy+y^2) = 7-16=-9,\\]so $xy = \\frac{-9}{3} = \\boxed{-3}$.", "By the Law of Sines applied to triangle $OAB$, $$\\frac{OB}{\\sin\\angle\nOAB}=\\frac{AB}{\\sin\\angle AOB}.$$With $AB = 1$ and $\\angle AOB = 30^\\circ$, we have \\[\\frac{OB}{\\sin \\angle OAB} = \\frac{1}{\\sin 30^\\circ} = 2,\\]so  so $OB=2\\sin\\angle OAB$. Thus, $OB \\le \\boxed{2}$, with equality if and only if $\\angle OAB=90^\\circ$.\n\n[asy]\nunitsize(1.5 cm);\n\npair O, A, B;\n\nO = (0,0);\nA = sqrt(3)*dir(30);\nB = (2,0);\n\ndraw((0,0)--3*dir(30),Arrow(6));\ndraw((0,0)--(3,0),Arrow(6));\ndraw(A--B);\ndraw(rightanglemark(O,A,B,4));\n\nlabel(\"$A$\", A, NW);\nlabel(\"$B$\", B, S);\nlabel(\"$O$\", O, W);\nlabel(\"$1$\", (A + B)/2, NE, red);\n[/asy]", "Adding 14 and 36 yields 50, and then adding 22 yields 72.  Dividing by 3 yields $\\boxed{24}$.", "Since $\\mathbf{A} \\mathbf{B} = \\mathbf{B} \\mathbf{A},$\n\\[\\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix}.\\]Expanding, we get\n\\[\\begin{pmatrix} a + 2c & b + 2d \\\\ 3a + 4c & 3b + 4d \\end{pmatrix} = \\begin{pmatrix} a + 3b & 2a + 4b \\\\ c + 3d & 2c + 4d \\end{pmatrix}.\\]Comparing entries, we find $3b = 2c$ and $3a + 3c = 3d,$ so $a + c = d.$  Then\n\\[\\frac{a - d}{c - 3b} = \\frac{-c}{c - 2c} = \\frac{-c}{-c} = \\boxed{1}.\\]", "Since $y=1$, we have $1 =\\displaystyle\\frac{1}{3x+1}$.  Multiplying both sides by $3x+1$, we have $$3x+1=1$$ $$\\Rightarrow \\qquad 3x=0$$ $$\\Rightarrow \\qquad x=\\boxed{0}$$", "If one centimeter on the map is 15 kilometers in reality, then we have the ratio $1 \\text{ cm on map} : 15 \\text{ km in reality}$. Multiplying the ratio by 88, we get $88 \\text{ cm on map} : 15 \\cdot 88 \\text{ km in reality} =88 \\text{ cm on map} : 1320 \\text{ km in reality}$. Thus, the cities are $\\boxed{1320}$ kilometers apart.", "The $x$-coordinate and $y$-coordinate of the midpoint are the averages of the $x$-coordinates and $y$-coordinates of the endpoints, respectively.  Since $4-3=1$, the $x$-coordinate of the other endpoint is $1-3=-2$.  Since $5-7=-2$, the $y$-coordinate of the other endpoint is $-2-7=-9$.  Therefore, the endpoints are $(4,5)$ and $\\boxed{(-2,-9)}$.", "We can use complementary probability to determine that the probability of its not raining tomorrow is $1 - \\frac{3}{10} = \\boxed{\\frac{7}{10}}$.", "Call the two integers $x$ and $y$. We are given that $x^2 + y^2 = 90$ and that $xy = 27$. We want to find $x + y$. Note that $(x + y)^2 = x^2 + y^2 + 2xy = 90 + 2\\cdot 27 = 144$. Taking the square root of 144, we see that $x + y = \\boxed{12}$.", "Let the height of the cylinder be $h$; we then have \\[SA = 2\\pi (2^2)+2\\pi (2)(h) = 12\\pi.\\]Solving yields $4\\pi h = 4 \\pi$ so $h = \\boxed{1}$ foot.", "Setting $x = -f(y),$ we get\n\\[f(0) = -f(y) + y,\\]so $f(y) = y - f(0)$ for all real numbers $x.$  Then the given functional equation becomes\n\\[f(x + y - f(0)) = x + y,\\]or $x + y - f(0) - f(0) = x + y.$  Then $f(0) = 0,$ so $f(x) = x$ for all real numbers $x.$  This function does satisfy the given functional equation, giving us $\\boxed{1}$ solution.", "$x^2 - y^2$ factors into $(x+y)(x-y)$, so, to obtain the value of $x^2 - y^2$, simply multiply $\\frac{7}{12}\\cdot \\frac{1}{12}$ to get $\\boxed{\\frac{7}{144}}$.", "When using the distributive property, we add the products of 3 and $8x^2$, 3 and $-2x$, and 3 and 1: \\begin{align*}\n3(8x^2-2x+1) &= 3\\cdot 8x^2+3\\cdot (-2x) + 3 \\cdot 1\\\\\n&= \\boxed{24x^2-6x+3}\n\\end{align*}", "The greatest integer that is less than or equal to $0.999$ is $0$, so $\\lfloor0.999\\rfloor=0$. The smallest integer that is greater than or equal to $2.001$ is $3$, so $\\lceil2.001\\rceil=3$. Therefore, $\\lfloor0.999\\rfloor+\\lceil2.001\\rceil=0+3=\\boxed{3}$.", "Shrinking the triangle by dividing every side length by 2, we recognize the resulting set $$\\{7,24,25\\}$$ of side lengths as a Pythagorean triple.  Therefore, the original triangle is also a right triangle, and its legs measure 14 cm and 48 cm.  The area of the triangle is $\\frac{1}{2}(14\\text{ cm})(48\\text{ cm})=\\boxed{336}$ square centimeters.", "Dividing both sides by 3, we quickly note that $ 9 = 9^{x-1} \\rightarrow 1 = x-1 \\rightarrow x = \\boxed{2}$.", "If $x=-3$, then $-3 = 2-t$, so $t = 5$.  Therefore, $y = 4(5) +7 =\\boxed{27}$."]

Reference texts 24?47:["Let's begin with the calculator that initially shows 1. Each time it is passed around the circle, it is cubed. 1 to any power is still 1, so no matter how many times 1 is cubed, the final result will still be 1.\n\nNow examine the calculator that starts with a zero. 0 squared is still 0 because 0 to any positive power is still 0. Thus, no matter how many times zero is squared, the final number will still be zero.\n\nFinally, let's look at the calculator that initially shows -1. Each time a person gets the calculator, they negate the number. Because there are 42 participants, there are 42 total turns. Thus, -1 is negated 42 times. Because negating a number is the same as multiplying by -1, this is the same as multiplying it by -1 forty-two times. Thus, we are looking for \\[(-1) \\cdot (-1)^{42}=(-1)^1 \\cdot (-1)^{42}=(-1)^{1+42}=(-1)^{43}.\\]Recall that $(-a)^n=-a^n$ if $n$ is odd. Because 43 is odd, $(-1)^{43}=-1^{43}=-1$.\n\nThus, the sum of all of the numbers is $1+0+(-1)=\\boxed{0}$.", "By definition, an obtuse triangle contains one obtuse interior angle. It cannot contain more than one because an obtuse angle has measure greater than 90 degrees and the sum of all the interior angles in any triangle is 180 degrees. Thus, there is $\\boxed{1}$ obtuse angle.", "There is one King of $\\diamondsuit$ and 52 cards total, so the probability that the top card is a King of $\\diamondsuit$ is $\\boxed{\\dfrac{1}{52}}$.", "We have  \\[81^{3/4} = (3^4)^{3/4} = 3^{4\\cdot (3/4)} = 3^3 = \\boxed{27}.\\]", "Since 3 and 4 are relatively prime, their least common multiple is $3\\cdot4=12$.  Therefore, the least common multiple of 30 and 40 is 120.  Since $\\boxed{120}$ is not divisible by 16, it is the smallest common multiple of 30 and 40 which is not divisible by 16.\n\nNote: Every common multiple of two integers is a multiple of their least common multiple.  Therefore, it is not possible that the least common multiple is divisible by 16 but some other common multiple is not.", "We have $g(-5) = 5(-5) + 2(-5)^2 = -25 + 50 = 25$, so $f(g(-5)) = f(25) = 3 - \\!\\sqrt{25} = 3-5=\\boxed{-2}$.", "We have $55 \\times 1212 - 15 \\times 1212 = 1212(55-15) = 1212(40) = 4848(10) = \\boxed{48480}$.", "Solution 1. The first equation gives $x = y+4$. Substituting into the second equation, we get \\[(y+4)^3 - y^3 = 28 \\implies 12y^2 + 48y + 36 = 0.\\]Thus, $y^2 + 4y + 3 = 0$, so $(y+1)(y+3) = 0$. Therefore, either $y=-1$ and $x=y+4=3$, or $y=-3$ and $x=y+4=1$. Either way, $xy = \\boxed{-3}$.\n\nSolution 2. The second equation factors via difference of cubes, as \\[(x-y)(x^2+xy+y^2) = 28.\\]Since $x-y=4$, we have $x^2+xy+y^2=\\frac{28}{4} =7$. Now, squaring the first equation, we get $x^2-2xy+y^2=16$. Thus, \\[3xy = (x^2+xy+y^2) - (x^2-2xy+y^2) = 7-16=-9,\\]so $xy = \\frac{-9}{3} = \\boxed{-3}$.", "By the Law of Sines applied to triangle $OAB$, $$\\frac{OB}{\\sin\\angle\nOAB}=\\frac{AB}{\\sin\\angle AOB}.$$With $AB = 1$ and $\\angle AOB = 30^\\circ$, we have \\[\\frac{OB}{\\sin \\angle OAB} = \\frac{1}{\\sin 30^\\circ} = 2,\\]so  so $OB=2\\sin\\angle OAB$. Thus, $OB \\le \\boxed{2}$, with equality if and only if $\\angle OAB=90^\\circ$.\n\n[asy]\nunitsize(1.5 cm);\n\npair O, A, B;\n\nO = (0,0);\nA = sqrt(3)*dir(30);\nB = (2,0);\n\ndraw((0,0)--3*dir(30),Arrow(6));\ndraw((0,0)--(3,0),Arrow(6));\ndraw(A--B);\ndraw(rightanglemark(O,A,B,4));\n\nlabel(\"$A$\", A, NW);\nlabel(\"$B$\", B, S);\nlabel(\"$O$\", O, W);\nlabel(\"$1$\", (A + B)/2, NE, red);\n[/asy]", "Adding 14 and 36 yields 50, and then adding 22 yields 72.  Dividing by 3 yields $\\boxed{24}$.", "Since $\\mathbf{A} \\mathbf{B} = \\mathbf{B} \\mathbf{A},$\n\\[\\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix}.\\]Expanding, we get\n\\[\\begin{pmatrix} a + 2c & b + 2d \\\\ 3a + 4c & 3b + 4d \\end{pmatrix} = \\begin{pmatrix} a + 3b & 2a + 4b \\\\ c + 3d & 2c + 4d \\end{pmatrix}.\\]Comparing entries, we find $3b = 2c$ and $3a + 3c = 3d,$ so $a + c = d.$  Then\n\\[\\frac{a - d}{c - 3b} = \\frac{-c}{c - 2c} = \\frac{-c}{-c} = \\boxed{1}.\\]", "Since $y=1$, we have $1 =\\displaystyle\\frac{1}{3x+1}$.  Multiplying both sides by $3x+1$, we have $$3x+1=1$$ $$\\Rightarrow \\qquad 3x=0$$ $$\\Rightarrow \\qquad x=\\boxed{0}$$", "If one centimeter on the map is 15 kilometers in reality, then we have the ratio $1 \\text{ cm on map} : 15 \\text{ km in reality}$. Multiplying the ratio by 88, we get $88 \\text{ cm on map} : 15 \\cdot 88 \\text{ km in reality} =88 \\text{ cm on map} : 1320 \\text{ km in reality}$. Thus, the cities are $\\boxed{1320}$ kilometers apart.", "The $x$-coordinate and $y$-coordinate of the midpoint are the averages of the $x$-coordinates and $y$-coordinates of the endpoints, respectively.  Since $4-3=1$, the $x$-coordinate of the other endpoint is $1-3=-2$.  Since $5-7=-2$, the $y$-coordinate of the other endpoint is $-2-7=-9$.  Therefore, the endpoints are $(4,5)$ and $\\boxed{(-2,-9)}$.", "We can use complementary probability to determine that the probability of its not raining tomorrow is $1 - \\frac{3}{10} = \\boxed{\\frac{7}{10}}$.", "Call the two integers $x$ and $y$. We are given that $x^2 + y^2 = 90$ and that $xy = 27$. We want to find $x + y$. Note that $(x + y)^2 = x^2 + y^2 + 2xy = 90 + 2\\cdot 27 = 144$. Taking the square root of 144, we see that $x + y = \\boxed{12}$.", "Let the height of the cylinder be $h$; we then have \\[SA = 2\\pi (2^2)+2\\pi (2)(h) = 12\\pi.\\]Solving yields $4\\pi h = 4 \\pi$ so $h = \\boxed{1}$ foot.", "Setting $x = -f(y),$ we get\n\\[f(0) = -f(y) + y,\\]so $f(y) = y - f(0)$ for all real numbers $x.$  Then the given functional equation becomes\n\\[f(x + y - f(0)) = x + y,\\]or $x + y - f(0) - f(0) = x + y.$  Then $f(0) = 0,$ so $f(x) = x$ for all real numbers $x.$  This function does satisfy the given functional equation, giving us $\\boxed{1}$ solution.", "$x^2 - y^2$ factors into $(x+y)(x-y)$, so, to obtain the value of $x^2 - y^2$, simply multiply $\\frac{7}{12}\\cdot \\frac{1}{12}$ to get $\\boxed{\\frac{7}{144}}$.", "When using the distributive property, we add the products of 3 and $8x^2$, 3 and $-2x$, and 3 and 1: \\begin{align*}\n3(8x^2-2x+1) &= 3\\cdot 8x^2+3\\cdot (-2x) + 3 \\cdot 1\\\\\n&= \\boxed{24x^2-6x+3}\n\\end{align*}", "The greatest integer that is less than or equal to $0.999$ is $0$, so $\\lfloor0.999\\rfloor=0$. The smallest integer that is greater than or equal to $2.001$ is $3$, so $\\lceil2.001\\rceil=3$. Therefore, $\\lfloor0.999\\rfloor+\\lceil2.001\\rceil=0+3=\\boxed{3}$.", "Shrinking the triangle by dividing every side length by 2, we recognize the resulting set $$\\{7,24,25\\}$$ of side lengths as a Pythagorean triple.  Therefore, the original triangle is also a right triangle, and its legs measure 14 cm and 48 cm.  The area of the triangle is $\\frac{1}{2}(14\\text{ cm})(48\\text{ cm})=\\boxed{336}$ square centimeters.", "Dividing both sides by 3, we quickly note that $ 9 = 9^{x-1} \\rightarrow 1 = x-1 \\rightarrow x = \\boxed{2}$.", "If $x=-3$, then $-3 = 2-t$, so $t = 5$.  Therefore, $y = 4(5) +7 =\\boxed{27}$."]

The second group includes valid multi-step derivations as well as short calculations, so the probes can reveal damage beyond answer formatting. No weights have changed, and there are still eight usable experiment slots remaining. I’ll retrieve the final reference texts and the short repetitive excerpts, then run the literal batch. A channel active during both correct derivations and loops will require a more selective intervention than simply labeling it a “bad neuron.”

python

print("Reference texts 48–63 and failure excerpts 64–71:")print(json.dumps(literal_probes[48:], ensure_ascii=True))

print("Reference texts 48–63 and failure excerpts 64–71:")print(json.dumps(literal_probes[48:], ensure_ascii=True))

OUTPUT of Python run 11 of at most 300:

Reference texts 48?63 and failure excerpts 64?71:["Choosing a committee is a combination, and order does not matter. We are choosing a 4-person committee from 9 people, so there are $9 \\times 8 \\times 7 \\times 6$ ways to pick the four people for the positions, but then we must divide by $4!$ since order doesn't matter, so the answer is $\\dfrac{9 \\times 8 \\times 7 \\times 6}{4!} =\\boxed{126}$.", "Terminating decimals can be expressed as $\\frac{a}{10^b}$. So, we try to get our fraction in this form: $$\\frac{11}{125} = \\frac{11}{5^3} = \\frac{11}{5^3} \\cdot \\frac{2^3}{2^3} = \\frac{11\\cdot2^3}{10^3} = \\frac{88}{1000} = \\boxed{0.088}.$$", "We start by completing the square: \\begin{align*}\n2x^2 -12x + 3 &= 2(x^2-6x) +3 \\\\\n&= 2(x^2 -6x + (6/2)^2 - (6/2)^2) + 3\\\\\n& = 2((x-3)^2 -3^2) + 3 \\\\\n&= 2(x-3)^2 - 2\\cdot 3^2 + 3\\\\\n&= 2(x-3)^2 -15\n.\\end{align*} Since the square of a real number is at least 0, we have $(x-3)^2\\ge 0$, where $(x-3)^2 =0$ only if $x=3$.   Therefore, $2(x-3)^2 - 15$ is minimized when $x=\\boxed{3}.$", "This can easily be converted into a fraction over a power of ten: \\[\n\\frac{3}{8} = \\frac{3 \\cdot 125}{8 \\cdot 125} = \\frac{375}{1000} = \\boxed{0.375}.\n\\]Note: In general if you have a fraction that is some $\\frac{x}{2^k}$, multiplying the numerator and denominator by $5^k$ will give a power of $10$ in the denominator for easy conversion to a decimal. In this case, $k = 3$.", "The sum of the three angles in any triangle is always $180^\\circ.$ In $\\triangle PQR,$ the sum of $\\angle P$ and $\\angle Q$ is $60^\\circ,$ and thus $\\angle R$ must measure $$180^\\circ - 60^\\circ = \\boxed{120^\\circ}.$$", "We find the average words per hour by dividing the total words by the total hours. $$\\frac{40,\\!000\\text{ words}}{80\\text{ hours}}=\\frac{4,\\!000}{8}=\\frac{40\\cdot100}{8}=5\\cdot100=\\boxed{500} \\text{ words per hour}$$", "The common factor of 3 in the numerator and the common factor of 3 in the denominator will cancel: \\[\n\\frac{3^4+3^2}{3^3-3}=\\frac{3(3^3+3^1)}{3(3^2-1)}=\\frac{3^3+3^1}{3^2-1}\n\\] Now the numerator is $3^3+3=27+3=30$, and the denominator is $3^2-1=9-1=8$.  Therefore, the fraction simplifies to $\\dfrac{30}{8}=\\boxed{\\dfrac{15}{4}}$.", "A cone with radius $r$ and height $h$ has volume $\\frac{1}{3}\\pi r^2 h$; here, our cone has volume $\\frac{1}{3}\\pi (1^2)(4)=\\frac{4}{3}\\pi$.  A sphere with radius $r$ has volume $\\frac{4}{3}\\pi r^3$, so we set up the equation \\[\\frac{4}{3}\\pi r^3=\\frac{4}{3}\\pi.\\] Solving for $r$ yields $r^3=1$, so $r = 1$.  The sphere's radius is $\\boxed{1}$ inch.", "From the angle addition formula, we can write\n\\begin{align*}\n\\sin x + \\cos x &= \\sqrt{2} \\left( \\frac{1}{\\sqrt{2}} \\sin x + \\frac{1}{\\sqrt{2}} \\cos x \\right) \\\\\n&= \\sqrt{2} \\left( \\cos \\frac{\\pi}{4} \\sin x + \\sin \\frac{\\pi}{4} \\cos x \\right) \\\\\n&= \\sqrt{2} \\sin \\left( x + \\frac{\\pi}{4} \\right).\n\\end{align*}Thus, the graph of $y = \\sin x + \\cos x$ has period $\\boxed{2 \\pi}.$\n\nThe graph of $y = \\sin x + \\cos x$ is shown below:\n\n[asy]import TrigMacros;\n\nsize(400);\n\nreal g(real x)\n{\n\treturn sin(x) + cos(x);\n}\n\ndraw(graph(g,-3*pi,3*pi,n=700,join=operator ..),red);\ntrig_axes(-3*pi,3*pi,-2,2,pi/2,1);\nlayer();\nrm_trig_labels(-5, 5, 2);\n[/asy]", "We can expand the determinant as follows:\n\\begin{align*}\n\\begin{vmatrix} 0 & \\sin \\alpha & -\\cos \\alpha \\\\ -\\sin \\alpha & 0 & \\sin \\beta \\\\ \\cos \\alpha & -\\sin \\beta & 0 \\end{vmatrix} &= -\\sin \\alpha \\begin{vmatrix} -\\sin \\alpha & \\sin \\beta \\\\ \\cos \\alpha & 0 \\end{vmatrix} - \\cos \\alpha \\begin{vmatrix} -\\sin \\alpha & 0 \\\\ \\cos \\alpha & -\\sin \\beta \\end{vmatrix} \\\\\n&= -\\sin \\alpha (-\\sin \\beta \\cos \\alpha) - \\cos \\alpha (\\sin \\alpha \\sin \\beta) \\\\\n&= \\boxed{0}.\n\\end{align*}", "Plotting the points, we see that the dimensions of the rectangle are $x$ and $4$.  The area of the rectangle is $(\\text{length})(\\text{width})=4x$, so $4x=28$ and $x=\\boxed{7}$. [asy]\nsize(5cm);\nimport graph;\ndefaultpen(linewidth(0.7)+fontsize(12));\nreal x = 7;\npair A=(0,0), B=(0,4), C=(x,4), D=(x,0);\npair[] dots = {A,B,C,D};\ndot(dots);\ndraw(A--B--C--D--cycle);\nxaxis(-2,9,Arrows(4));\nyaxis(-2,7,Arrows(4));\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,NW);\nlabel(\"$C$\",C,NE);\nlabel(\"$D$\",D,SE);\nlabel(\"$x$\",(B+C)/2,N);\nlabel(\"$4$\",(C+D)/2,E);[/asy]", "We know that $\\left(\\frac{13}{7}\\right)^2=\\frac{169}{49}$. Then, since $3=\\frac{147}{49}<\\frac{169}{49}<\\frac{196}{49}=4$, we conclude that $\\left\\lceil\\left(\\frac{13}{7}\\right)^2\\right\\rceil=4$. Because $4+\\frac{17}{4}=\\frac{33}{4}$, which is a number between $8$ and $9$, $\\left\\lfloor \\left\\lceil \\left(\\frac{13}{7}\\right)^2\\right\\rceil+\\frac{17}{4}\\right\\rfloor=\\boxed{8}$.", "The height of the cone is $\\frac{3}{4} \\times 8 = 6$. The radius of the cone is $\\frac{8}{2} = 4$. Therefore, the volume of the cone formed is $\\frac{4^2 \\times 6 \\times \\pi}{3} = \\boxed{32 \\pi}$.", "Call the two integers $x$ and $y$. Without loss of generality, let $x$ be the larger of the two. We are given that $x+y = 50$ and $x-y = 12$, and we are asked for $x^2 - y^2$. Because $x^2 - y^2$ factors into $(x+y)(x-y)$, we can simply substitute in to get $x^2 - y^2 = 50 \\cdot 12 = \\boxed{600}$.", "The common difference is $5 - 2 = 3$, so the $25^{\\text{th}}$ term is $2 + 3 \\cdot 24 = \\boxed{74}$.", "Plugging in $x = 2$, we get $-4$ for the numerator, and $-1$ for the denominator, thus, $\\boxed{4}$ is the answer.", "+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1", "2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer:", "81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: ", "l Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer", "= 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.7", "3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = ", " are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are ", " is $4-0 = 4$.\nThe coordinates of $D$ are $(x, 0)$.\nThe length of the side $CD$ is given as 28.\nThe coordinates of $A$ are $(0, 0)$.\nThe coordinates of $B$ are $(0, 4)$.\nThe coordinates of $C$ are $(x, 4)$.\nThe length of the side $DA$ is given as 28.\nThe coordinates of $C$ are $("]

Reference texts 48?63 and failure excerpts 64?71:["Choosing a committee is a combination, and order does not matter. We are choosing a 4-person committee from 9 people, so there are $9 \\times 8 \\times 7 \\times 6$ ways to pick the four people for the positions, but then we must divide by $4!$ since order doesn't matter, so the answer is $\\dfrac{9 \\times 8 \\times 7 \\times 6}{4!} =\\boxed{126}$.", "Terminating decimals can be expressed as $\\frac{a}{10^b}$. So, we try to get our fraction in this form: $$\\frac{11}{125} = \\frac{11}{5^3} = \\frac{11}{5^3} \\cdot \\frac{2^3}{2^3} = \\frac{11\\cdot2^3}{10^3} = \\frac{88}{1000} = \\boxed{0.088}.$$", "We start by completing the square: \\begin{align*}\n2x^2 -12x + 3 &= 2(x^2-6x) +3 \\\\\n&= 2(x^2 -6x + (6/2)^2 - (6/2)^2) + 3\\\\\n& = 2((x-3)^2 -3^2) + 3 \\\\\n&= 2(x-3)^2 - 2\\cdot 3^2 + 3\\\\\n&= 2(x-3)^2 -15\n.\\end{align*} Since the square of a real number is at least 0, we have $(x-3)^2\\ge 0$, where $(x-3)^2 =0$ only if $x=3$.   Therefore, $2(x-3)^2 - 15$ is minimized when $x=\\boxed{3}.$", "This can easily be converted into a fraction over a power of ten: \\[\n\\frac{3}{8} = \\frac{3 \\cdot 125}{8 \\cdot 125} = \\frac{375}{1000} = \\boxed{0.375}.\n\\]Note: In general if you have a fraction that is some $\\frac{x}{2^k}$, multiplying the numerator and denominator by $5^k$ will give a power of $10$ in the denominator for easy conversion to a decimal. In this case, $k = 3$.", "The sum of the three angles in any triangle is always $180^\\circ.$ In $\\triangle PQR,$ the sum of $\\angle P$ and $\\angle Q$ is $60^\\circ,$ and thus $\\angle R$ must measure $$180^\\circ - 60^\\circ = \\boxed{120^\\circ}.$$", "We find the average words per hour by dividing the total words by the total hours. $$\\frac{40,\\!000\\text{ words}}{80\\text{ hours}}=\\frac{4,\\!000}{8}=\\frac{40\\cdot100}{8}=5\\cdot100=\\boxed{500} \\text{ words per hour}$$", "The common factor of 3 in the numerator and the common factor of 3 in the denominator will cancel: \\[\n\\frac{3^4+3^2}{3^3-3}=\\frac{3(3^3+3^1)}{3(3^2-1)}=\\frac{3^3+3^1}{3^2-1}\n\\] Now the numerator is $3^3+3=27+3=30$, and the denominator is $3^2-1=9-1=8$.  Therefore, the fraction simplifies to $\\dfrac{30}{8}=\\boxed{\\dfrac{15}{4}}$.", "A cone with radius $r$ and height $h$ has volume $\\frac{1}{3}\\pi r^2 h$; here, our cone has volume $\\frac{1}{3}\\pi (1^2)(4)=\\frac{4}{3}\\pi$.  A sphere with radius $r$ has volume $\\frac{4}{3}\\pi r^3$, so we set up the equation \\[\\frac{4}{3}\\pi r^3=\\frac{4}{3}\\pi.\\] Solving for $r$ yields $r^3=1$, so $r = 1$.  The sphere's radius is $\\boxed{1}$ inch.", "From the angle addition formula, we can write\n\\begin{align*}\n\\sin x + \\cos x &= \\sqrt{2} \\left( \\frac{1}{\\sqrt{2}} \\sin x + \\frac{1}{\\sqrt{2}} \\cos x \\right) \\\\\n&= \\sqrt{2} \\left( \\cos \\frac{\\pi}{4} \\sin x + \\sin \\frac{\\pi}{4} \\cos x \\right) \\\\\n&= \\sqrt{2} \\sin \\left( x + \\frac{\\pi}{4} \\right).\n\\end{align*}Thus, the graph of $y = \\sin x + \\cos x$ has period $\\boxed{2 \\pi}.$\n\nThe graph of $y = \\sin x + \\cos x$ is shown below:\n\n[asy]import TrigMacros;\n\nsize(400);\n\nreal g(real x)\n{\n\treturn sin(x) + cos(x);\n}\n\ndraw(graph(g,-3*pi,3*pi,n=700,join=operator ..),red);\ntrig_axes(-3*pi,3*pi,-2,2,pi/2,1);\nlayer();\nrm_trig_labels(-5, 5, 2);\n[/asy]", "We can expand the determinant as follows:\n\\begin{align*}\n\\begin{vmatrix} 0 & \\sin \\alpha & -\\cos \\alpha \\\\ -\\sin \\alpha & 0 & \\sin \\beta \\\\ \\cos \\alpha & -\\sin \\beta & 0 \\end{vmatrix} &= -\\sin \\alpha \\begin{vmatrix} -\\sin \\alpha & \\sin \\beta \\\\ \\cos \\alpha & 0 \\end{vmatrix} - \\cos \\alpha \\begin{vmatrix} -\\sin \\alpha & 0 \\\\ \\cos \\alpha & -\\sin \\beta \\end{vmatrix} \\\\\n&= -\\sin \\alpha (-\\sin \\beta \\cos \\alpha) - \\cos \\alpha (\\sin \\alpha \\sin \\beta) \\\\\n&= \\boxed{0}.\n\\end{align*}", "Plotting the points, we see that the dimensions of the rectangle are $x$ and $4$.  The area of the rectangle is $(\\text{length})(\\text{width})=4x$, so $4x=28$ and $x=\\boxed{7}$. [asy]\nsize(5cm);\nimport graph;\ndefaultpen(linewidth(0.7)+fontsize(12));\nreal x = 7;\npair A=(0,0), B=(0,4), C=(x,4), D=(x,0);\npair[] dots = {A,B,C,D};\ndot(dots);\ndraw(A--B--C--D--cycle);\nxaxis(-2,9,Arrows(4));\nyaxis(-2,7,Arrows(4));\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,NW);\nlabel(\"$C$\",C,NE);\nlabel(\"$D$\",D,SE);\nlabel(\"$x$\",(B+C)/2,N);\nlabel(\"$4$\",(C+D)/2,E);[/asy]", "We know that $\\left(\\frac{13}{7}\\right)^2=\\frac{169}{49}$. Then, since $3=\\frac{147}{49}<\\frac{169}{49}<\\frac{196}{49}=4$, we conclude that $\\left\\lceil\\left(\\frac{13}{7}\\right)^2\\right\\rceil=4$. Because $4+\\frac{17}{4}=\\frac{33}{4}$, which is a number between $8$ and $9$, $\\left\\lfloor \\left\\lceil \\left(\\frac{13}{7}\\right)^2\\right\\rceil+\\frac{17}{4}\\right\\rfloor=\\boxed{8}$.", "The height of the cone is $\\frac{3}{4} \\times 8 = 6$. The radius of the cone is $\\frac{8}{2} = 4$. Therefore, the volume of the cone formed is $\\frac{4^2 \\times 6 \\times \\pi}{3} = \\boxed{32 \\pi}$.", "Call the two integers $x$ and $y$. Without loss of generality, let $x$ be the larger of the two. We are given that $x+y = 50$ and $x-y = 12$, and we are asked for $x^2 - y^2$. Because $x^2 - y^2$ factors into $(x+y)(x-y)$, we can simply substitute in to get $x^2 - y^2 = 50 \\cdot 12 = \\boxed{600}$.", "The common difference is $5 - 2 = 3$, so the $25^{\\text{th}}$ term is $2 + 3 \\cdot 24 = \\boxed{74}$.", "Plugging in $x = 2$, we get $-4$ for the numerator, and $-1$ for the denominator, thus, $\\boxed{4}$ is the answer.", "+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1", "2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer:", "81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: ", "l Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer", "= 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.7", "3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = ", " are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are ", " is $4-0 = 4$.\nThe coordinates of $D$ are $(x, 0)$.\nThe length of the side $CD$ is given as 28.\nThe coordinates of $A$ are $(0, 0)$.\nThe coordinates of $B$ are $(0, 4)$.\nThe coordinates of $C$ are $(x, 4)$.\nThe length of the side $DA$ is given as 28.\nThe coordinates of $C$ are $("]

The complete batch is ready as literal texts, avoiding the filename-handling problem. I’ll now measure baseline token probabilities and MLP activity at layers 8, 14, and 17 on these 72 probes. This is a localization and preservation check, not a substitute for solving problems under the evaluation protocol. A plausible target should respond strongly to the repetitive passages without being equally necessary throughout the valid reference calculations.

forward

{"type":"teacher_forced","model":"original","texts":["We have that\n\\begin{align*}\na_3 &= \\frac{a_2}{a_1} = \\frac{3}{2}, \\\\\na_4 &= \\frac{a_3}{a_2} = \\frac{3/2}{3} = \\frac{1}{2}, \\\\\na_5 &= \\frac{a_4}{a_3} = \\frac{1/2}{3/2} = \\frac{1}{3}, \\\\\na_6 &= \\frac{a_5}{a_4} = \\frac{1/3}{1/2} = \\frac{2}{3}, \\\\\na_7 &= \\frac{a_6}{a_5} = \\frac{2/3}{1/3} = 2, \\\\\na_8 &= \\frac{a_7}{a_6} = \\frac{2}{2/3} = 3.\n\\end{align*}Since $a_7 = a_1 = 2$ and $a_8 = a_2 = 3,$ and each term depends only on the previous two terms, the sequence becomes periodic at this point, with a period of length 6.  Hence, $a_{2006} = a_2 = \\boxed{3}.$","The midpoint of a line segment with endpoints $(x_1, y_1), (x_2, y_2)$ is $\\left(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}\\right)$.\n\nThus, the midpoint of this line segment is $\\left(\\frac{2+8}{2}, \\frac{3+15}{2} \\right)$, which simplifies to $(5,9)$. Thus, the sum of the coordinates of the midpoint is $\\boxed{14}$.","Let $y = x^2 + x + 3.$  Then we can write the given equation as\n\\[\\frac{4x}{y} + \\frac{5x}{y - 6x} + \\frac{3}{2} = 0.\\]Multiplying everything by $2y(y - 6x),$ we get\n\\[8x(y - 6x) + 10xy + 3y(y - 6x) = 0.\\]Expanding, we get $3y^2 - 48x^2 = 0,$ so $y^2 - 16x^2 = (y - 4x)(y + 4x) = 0.$  Thus, $y = 4x$ or $y = -4x.$\n\nIf $y = 4x,$ then $x^2 + x + 3 = 4x,$ so $x^2 - 3x + 3 = 0.$  This quadratic has no real solutions.\n\nIf $y = -4x,$ then $x^2 + x + 3 = -4x,$ so $x^2 + 5x + 3 = 0.$  This quadratic has two real solutions, giving us a total of $\\boxed{2}$ real solutions.","Simplify, remembering the exponents take priority over multiplication and multiplication takes priority over addition or subtraction. \\begin{align*}\n5^3-3\\times 5^2+3\\times5-1&=125-3\\times25+15-1 \\\\\n&=125-75+14 \\\\\n&= 50+14 \\\\\n&= \\boxed{64}.\n\\end{align*}","We proceed as follows: \\begin{align*}\n(2b + 5)(b - 1) &= 6b\\\\\n2b^2 + 3b - 5 &= 6b\\\\\n2b^2 - 3b - 5 &= 0\\\\\n(2b - 5)(b + 1) &= 0.\n\\end{align*}This gives us $b = \\frac{5}{2}$ or $b = -1.$ Of these, $\\boxed{\\frac{5}{2}}$ is the larger possible value of $b.$","We solve each equation separately. First off, we have $8x^2 + 7x - 1 = (8x-1)(x+1) = 0.$ We can also see that $24x^2+53x-7 = (8x-1)(3x+7) = 0.$ It is clear that both equations are satisfied only when $8x - 1 = 0,$ therefore $x = \\boxed{\\dfrac{1}{8}}.$","Let's substitute the given values into the expression. We get  \\begin{align*}\n(a^3+b^3)\\div(a^2-ab+b^2)&=(5^3+4^3)\\div(5^2-5\\cdot4+4^2)\\\\\n&=(125+64)\\div(25-20+16)\\\\\n&=189\\div21\\\\\n&=\\boxed{9}.\n\\end{align*}","The only numbers that have 11 and 8 as a factor are multiples of 88.  If we list the first few multiples of 88: $$88,176,264,352,...$$ we can see that there are exactly $\\boxed{2}$ between 100 and 300.","Since $3^3 = 3\\times 3\\times 3 = 3\\times 9 = 27$, then  \\[ \\sqrt{3^3+3^3+3^3} = \\sqrt{27+27+27}=\\sqrt{81}=\\boxed{9}. \\]","Let $n$ be any integer that has its square between 4000 and 7000. Then $63 < n < 84$, because $63^2 < 4000<64^2$ and $83^2< 7000<84^2$. Between 63 and 84, the only primes are 67, 71, 73, 79, and 83. Thus, the answer is $\\boxed{5}$.","There are ${9 \\choose 2} = 36$ pairs of plates possible in all. Exactly ${5 \\choose 2} = 10$ of these pairs are both red plates and $\\binom{4}{2}=6$ pairs are both blue plates for a total of $6+10=16$ pairs of plates that satisfy our condition. So, the probability of the pair of plates being the same color is $\\frac{16}{36}=\\boxed{\\frac{4}{9}}$.","Let $B$ and $J$ represent the respective ages of Billy and Joe. We can write the equations $B=2J$ and $B+J=45$. We use the second equation to solve for $J$ in terms of $B$ and get $J=45-B$. Now we plug in this expression for $J$ into the first equation. $$B=2(45-B)=90-2B\\qquad\\Rightarrow 3B=90\\qquad\\Rightarrow B=30$$ So Billy is $\\boxed{30}$ years old.","By AM-HM,\n\\[\\frac{x + y}{2} \\ge \\frac{2}{\\frac{1}{x} + \\frac{1}{y}}.\\]Hence,\n\\[\\frac{1}{x} + \\frac{1}{y} \\ge \\frac{4}{x + y} = \\frac{4}{10} = \\frac{2}{5}.\\]Equality occurs when $x = y = 5,$ so the minimum value is $\\boxed{\\frac{2}{5}}$.","$64^2 - 36^2$ can also be expressed as $(64+36)(64-36)$. This simplifies as $100 \\cdot 28$, which equals $\\boxed{2800}$.","Since the triangle is equilateral, all sides are equal in length. Therefore, the perimeter of the triangle is $8+8+8=8 \\times 3=\\boxed{24}.$","We have two equations and two variables, so it's possible to solve for $p$ and $q$ directly and then calculate $p^2$ and $q^2$ separately to get our answer. However, doing so involves a fair amount of computation with complex numbers and square roots, so we look for an alternative approach. We square the second equation to get $$(p+q)^2 = p^2 + 2pq + q^2 = 36,$$which is close to what we want but has the extra $2pq$ term. Since we know that $pq=9$, we can substitute to get $$p^2 + 2(9) +q^2 = 36 \\implies p^2+q^2 = \\boxed{18}.$$Note that our task was made easier by only solving for what the problem asked rather than by trying to solve for $p$ and $q$ individually.","Triangle $ABC$ has base $AB$ of length $25\\text{ cm}$ and height $AC$ of length $20\\text{ cm}.$ Therefore, the area of triangle $ABC$ is \\begin{align*}\n\\frac{1}{2}bh &= \\frac{1}{2}(25 \\mbox{ cm})(20 \\mbox{ cm}) \\\\\n&= \\frac{1}{2}(500 \\mbox{ cm}^2) \\\\\n&= \\boxed{250} \\mbox{ cm}^2.\n\\end{align*}","We have  \\[\\frac{x^4 + 2y^2}{6} = \\frac{2^4 + 2(5^2)}{6} = \\frac{16+2(25)}{6} = \\frac{16+50}{6} = \\frac{66}{6} = \\boxed{11}.\\]","The average length is $\\frac{1.5+4.5}{2}=\\boxed{3}$ inches.","Her mean score was $\\frac{98+97+92+85+93}{5}=\\boxed{93}$.","We simplify the given inequality by adding $x+3$ to both sides, which leaves us with $5x < 5 \\Rightarrow x<1$. Since 1 is not less than itself, this leaves us with $x=\\boxed{0}$.","We know that any number raised to the power of $0$ is $1$, or $a^0 = 1$ for any $a$. Thus, we get $$(5^{-2})^0 + (5^0)^3 = 1 + 1^3 = 1+1 = \\boxed{2}.$$","Since $2a + 1 = 1,$ we find that $2a = 0,$ so $a = 0.$ Therefore, $b - a = b - 0 = b = \\boxed{1}.$","\\[\n\\begin{array}{c|cc cc}\n\\multicolumn{2}{r}{x^2} & +\\left(\\frac{c-7}{2}\\right)x & +5 \\\\\n\\cline{2-5}\n2x+7 & 2x^3 &+cx^2 &- 11x &+ 39  \\\\\n\\multicolumn{2}{r}{-2x^3} & -7x^2  \\\\ \n\\cline{2-3}\n\\multicolumn{2}{r}{0} & (c-7)x^2 & -11x  \\\\\n\\multicolumn{2}{r}{} & -(c-7)x^2 & -x(c-7)\\left(\\frac{7}{2}\\right)   \\\\ \n\\cline{3-4}\n\\multicolumn{2}{r}{} &  0 & -x\\left(\\frac{7c-27}{2}\\right) & + 39  \\\\\n\\multicolumn{2}{r}{} &   & -10x & -35  \\\\\n\\cline{4-5}\n\\multicolumn{2}{r}{} &   & -x\\left(\\frac{7c-27+20}{2}\\right) & 4  \\\\\n\\end{array}\n\\]In the last step of the division, we have $39$ left as the constant term in our dividend and we need a remainder of $4$ at the end. Since our divisor has a term of $7$, the only way to do this is if our quotient has $5$ which gives us  $7\\cdot5=35$ to subtract from our dividend and get the right remainder.\n\nThen, we need the rest of our remainder to be $0$. This means\n$$\\frac{7c-27+20}{2} = 0$$which gives us\n$$c = \\boxed{1}.$$","Let's begin with the calculator that initially shows 1. Each time it is passed around the circle, it is cubed. 1 to any power is still 1, so no matter how many times 1 is cubed, the final result will still be 1.\n\nNow examine the calculator that starts with a zero. 0 squared is still 0 because 0 to any positive power is still 0. Thus, no matter how many times zero is squared, the final number will still be zero.\n\nFinally, let's look at the calculator that initially shows -1. Each time a person gets the calculator, they negate the number. Because there are 42 participants, there are 42 total turns. Thus, -1 is negated 42 times. Because negating a number is the same as multiplying by -1, this is the same as multiplying it by -1 forty-two times. Thus, we are looking for \\[(-1) \\cdot (-1)^{42}=(-1)^1 \\cdot (-1)^{42}=(-1)^{1+42}=(-1)^{43}.\\]Recall that $(-a)^n=-a^n$ if $n$ is odd. Because 43 is odd, $(-1)^{43}=-1^{43}=-1$.\n\nThus, the sum of all of the numbers is $1+0+(-1)=\\boxed{0}$.","By definition, an obtuse triangle contains one obtuse interior angle. It cannot contain more than one because an obtuse angle has measure greater than 90 degrees and the sum of all the interior angles in any triangle is 180 degrees. Thus, there is $\\boxed{1}$ obtuse angle.","There is one King of $\\diamondsuit$ and 52 cards total, so the probability that the top card is a King of $\\diamondsuit$ is $\\boxed{\\dfrac{1}{52}}$.","We have  \\[81^{3/4} = (3^4)^{3/4} = 3^{4\\cdot (3/4)} = 3^3 = \\boxed{27}.\\]","Since 3 and 4 are relatively prime, their least common multiple is $3\\cdot4=12$.  Therefore, the least common multiple of 30 and 40 is 120.  Since $\\boxed{120}$ is not divisible by 16, it is the smallest common multiple of 30 and 40 which is not divisible by 16.\n\nNote: Every common multiple of two integers is a multiple of their least common multiple.  Therefore, it is not possible that the least common multiple is divisible by 16 but some other common multiple is not.","We have $g(-5) = 5(-5) + 2(-5)^2 = -25 + 50 = 25$, so $f(g(-5)) = f(25) = 3 - \\!\\sqrt{25} = 3-5=\\boxed{-2}$.","We have $55 \\times 1212 - 15 \\times 1212 = 1212(55-15) = 1212(40) = 4848(10) = \\boxed{48480}$.","Solution 1. The first equation gives $x = y+4$. Substituting into the second equation, we get \\[(y+4)^3 - y^3 = 28 \\implies 12y^2 + 48y + 36 = 0.\\]Thus, $y^2 + 4y + 3 = 0$, so $(y+1)(y+3) = 0$. Therefore, either $y=-1$ and $x=y+4=3$, or $y=-3$ and $x=y+4=1$. Either way, $xy = \\boxed{-3}$.\n\nSolution 2. The second equation factors via difference of cubes, as \\[(x-y)(x^2+xy+y^2) = 28.\\]Since $x-y=4$, we have $x^2+xy+y^2=\\frac{28}{4} =7$. Now, squaring the first equation, we get $x^2-2xy+y^2=16$. Thus, \\[3xy = (x^2+xy+y^2) - (x^2-2xy+y^2) = 7-16=-9,\\]so $xy = \\frac{-9}{3} = \\boxed{-3}$.","By the Law of Sines applied to triangle $OAB$, $$\\frac{OB}{\\sin\\angle\nOAB}=\\frac{AB}{\\sin\\angle AOB}.$$With $AB = 1$ and $\\angle AOB = 30^\\circ$, we have \\[\\frac{OB}{\\sin \\angle OAB} = \\frac{1}{\\sin 30^\\circ} = 2,\\]so  so $OB=2\\sin\\angle OAB$. Thus, $OB \\le \\boxed{2}$, with equality if and only if $\\angle OAB=90^\\circ$.\n\n[asy]\nunitsize(1.5 cm);\n\npair O, A, B;\n\nO = (0,0);\nA = sqrt(3)*dir(30);\nB = (2,0);\n\ndraw((0,0)--3*dir(30),Arrow(6));\ndraw((0,0)--(3,0),Arrow(6));\ndraw(A--B);\ndraw(rightanglemark(O,A,B,4));\n\nlabel(\"$A$\", A, NW);\nlabel(\"$B$\", B, S);\nlabel(\"$O$\", O, W);\nlabel(\"$1$\", (A + B)/2, NE, red);\n[/asy]","Adding 14 and 36 yields 50, and then adding 22 yields 72.  Dividing by 3 yields $\\boxed{24}$.","Since $\\mathbf{A} \\mathbf{B} = \\mathbf{B} \\mathbf{A},$\n\\[\\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix}.\\]Expanding, we get\n\\[\\begin{pmatrix} a + 2c & b + 2d \\\\ 3a + 4c & 3b + 4d \\end{pmatrix} = \\begin{pmatrix} a + 3b & 2a + 4b \\\\ c + 3d & 2c + 4d \\end{pmatrix}.\\]Comparing entries, we find $3b = 2c$ and $3a + 3c = 3d,$ so $a + c = d.$  Then\n\\[\\frac{a - d}{c - 3b} = \\frac{-c}{c - 2c} = \\frac{-c}{-c} = \\boxed{1}.\\]","Since $y=1$, we have $1 =\\displaystyle\\frac{1}{3x+1}$.  Multiplying both sides by $3x+1$, we have $$3x+1=1$$ $$\\Rightarrow \\qquad 3x=0$$ $$\\Rightarrow \\qquad x=\\boxed{0}$$","If one centimeter on the map is 15 kilometers in reality, then we have the ratio $1 \\text{ cm on map} : 15 \\text{ km in reality}$. Multiplying the ratio by 88, we get $88 \\text{ cm on map} : 15 \\cdot 88 \\text{ km in reality} =88 \\text{ cm on map} : 1320 \\text{ km in reality}$. Thus, the cities are $\\boxed{1320}$ kilometers apart.","The $x$-coordinate and $y$-coordinate of the midpoint are the averages of the $x$-coordinates and $y$-coordinates of the endpoints, respectively.  Since $4-3=1$, the $x$-coordinate of the other endpoint is $1-3=-2$.  Since $5-7=-2$, the $y$-coordinate of the other endpoint is $-2-7=-9$.  Therefore, the endpoints are $(4,5)$ and $\\boxed{(-2,-9)}$.","We can use complementary probability to determine that the probability of its not raining tomorrow is $1 - \\frac{3}{10} = \\boxed{\\frac{7}{10}}$.","Call the two integers $x$ and $y$. We are given that $x^2 + y^2 = 90$ and that $xy = 27$. We want to find $x + y$. Note that $(x + y)^2 = x^2 + y^2 + 2xy = 90 + 2\\cdot 27 = 144$. Taking the square root of 144, we see that $x + y = \\boxed{12}$.","Let the height of the cylinder be $h$; we then have \\[SA = 2\\pi (2^2)+2\\pi (2)(h) = 12\\pi.\\]Solving yields $4\\pi h = 4 \\pi$ so $h = \\boxed{1}$ foot.","Setting $x = -f(y),$ we get\n\\[f(0) = -f(y) + y,\\]so $f(y) = y - f(0)$ for all real numbers $x.$  Then the given functional equation becomes\n\\[f(x + y - f(0)) = x + y,\\]or $x + y - f(0) - f(0) = x + y.$  Then $f(0) = 0,$ so $f(x) = x$ for all real numbers $x.$  This function does satisfy the given functional equation, giving us $\\boxed{1}$ solution.","$x^2 - y^2$ factors into $(x+y)(x-y)$, so, to obtain the value of $x^2 - y^2$, simply multiply $\\frac{7}{12}\\cdot \\frac{1}{12}$ to get $\\boxed{\\frac{7}{144}}$.","When using the distributive property, we add the products of 3 and $8x^2$, 3 and $-2x$, and 3 and 1: \\begin{align*}\n3(8x^2-2x+1) &= 3\\cdot 8x^2+3\\cdot (-2x) + 3 \\cdot 1\\\\\n&= \\boxed{24x^2-6x+3}\n\\end{align*}","The greatest integer that is less than or equal to $0.999$ is $0$, so $\\lfloor0.999\\rfloor=0$. The smallest integer that is greater than or equal to $2.001$ is $3$, so $\\lceil2.001\\rceil=3$. Therefore, $\\lfloor0.999\\rfloor+\\lceil2.001\\rceil=0+3=\\boxed{3}$.","Shrinking the triangle by dividing every side length by 2, we recognize the resulting set $$\\{7,24,25\\}$$ of side lengths as a Pythagorean triple.  Therefore, the original triangle is also a right triangle, and its legs measure 14 cm and 48 cm.  The area of the triangle is $\\frac{1}{2}(14\\text{ cm})(48\\text{ cm})=\\boxed{336}$ square centimeters.","Dividing both sides by 3, we quickly note that $ 9 = 9^{x-1} \\rightarrow 1 = x-1 \\rightarrow x = \\boxed{2}$.","If $x=-3$, then $-3 = 2-t$, so $t = 5$.  Therefore, $y = 4(5) +7 =\\boxed{27}$.","Choosing a committee is a combination, and order does not matter. We are choosing a 4-person committee from 9 people, so there are $9 \\times 8 \\times 7 \\times 6$ ways to pick the four people for the positions, but then we must divide by $4!$ since order doesn't matter, so the answer is $\\dfrac{9 \\times 8 \\times 7 \\times 6}{4!} =\\boxed{126}$.","Terminating decimals can be expressed as $\\frac{a}{10^b}$. So, we try to get our fraction in this form: $$\\frac{11}{125} = \\frac{11}{5^3} = \\frac{11}{5^3} \\cdot \\frac{2^3}{2^3} = \\frac{11\\cdot2^3}{10^3} = \\frac{88}{1000} = \\boxed{0.088}.$$","We start by completing the square: \\begin{align*}\n2x^2 -12x + 3 &= 2(x^2-6x) +3 \\\\\n&= 2(x^2 -6x + (6/2)^2 - (6/2)^2) + 3\\\\\n& = 2((x-3)^2 -3^2) + 3 \\\\\n&= 2(x-3)^2 - 2\\cdot 3^2 + 3\\\\\n&= 2(x-3)^2 -15\n.\\end{align*} Since the square of a real number is at least 0, we have $(x-3)^2\\ge 0$, where $(x-3)^2 =0$ only if $x=3$.   Therefore, $2(x-3)^2 - 15$ is minimized when $x=\\boxed{3}.$","This can easily be converted into a fraction over a power of ten: \\[\n\\frac{3}{8} = \\frac{3 \\cdot 125}{8 \\cdot 125} = \\frac{375}{1000} = \\boxed{0.375}.\n\\]Note: In general if you have a fraction that is some $\\frac{x}{2^k}$, multiplying the numerator and denominator by $5^k$ will give a power of $10$ in the denominator for easy conversion to a decimal. In this case, $k = 3$.","The sum of the three angles in any triangle is always $180^\\circ.$ In $\\triangle PQR,$ the sum of $\\angle P$ and $\\angle Q$ is $60^\\circ,$ and thus $\\angle R$ must measure $$180^\\circ - 60^\\circ = \\boxed{120^\\circ}.$$","We find the average words per hour by dividing the total words by the total hours. $$\\frac{40,\\!000\\text{ words}}{80\\text{ hours}}=\\frac{4,\\!000}{8}=\\frac{40\\cdot100}{8}=5\\cdot100=\\boxed{500} \\text{ words per hour}$$","The common factor of 3 in the numerator and the common factor of 3 in the denominator will cancel: \\[\n\\frac{3^4+3^2}{3^3-3}=\\frac{3(3^3+3^1)}{3(3^2-1)}=\\frac{3^3+3^1}{3^2-1}\n\\] Now the numerator is $3^3+3=27+3=30$, and the denominator is $3^2-1=9-1=8$.  Therefore, the fraction simplifies to $\\dfrac{30}{8}=\\boxed{\\dfrac{15}{4}}$.","A cone with radius $r$ and height $h$ has volume $\\frac{1}{3}\\pi r^2 h$; here, our cone has volume $\\frac{1}{3}\\pi (1^2)(4)=\\frac{4}{3}\\pi$.  A sphere with radius $r$ has volume $\\frac{4}{3}\\pi r^3$, so we set up the equation \\[\\frac{4}{3}\\pi r^3=\\frac{4}{3}\\pi.\\] Solving for $r$ yields $r^3=1$, so $r = 1$.  The sphere's radius is $\\boxed{1}$ inch.","From the angle addition formula, we can write\n\\begin{align*}\n\\sin x + \\cos x &= \\sqrt{2} \\left( \\frac{1}{\\sqrt{2}} \\sin x + \\frac{1}{\\sqrt{2}} \\cos x \\right) \\\\\n&= \\sqrt{2} \\left( \\cos \\frac{\\pi}{4} \\sin x + \\sin \\frac{\\pi}{4} \\cos x \\right) \\\\\n&= \\sqrt{2} \\sin \\left( x + \\frac{\\pi}{4} \\right).\n\\end{align*}Thus, the graph of $y = \\sin x + \\cos x$ has period $\\boxed{2 \\pi}.$\n\nThe graph of $y = \\sin x + \\cos x$ is shown below:\n\n[asy]import TrigMacros;\n\nsize(400);\n\nreal g(real x)\n{\n\treturn sin(x) + cos(x);\n}\n\ndraw(graph(g,-3*pi,3*pi,n=700,join=operator ..),red);\ntrig_axes(-3*pi,3*pi,-2,2,pi/2,1);\nlayer();\nrm_trig_labels(-5, 5, 2);\n[/asy]","We can expand the determinant as follows:\n\\begin{align*}\n\\begin{vmatrix} 0 & \\sin \\alpha & -\\cos \\alpha \\\\ -\\sin \\alpha & 0 & \\sin \\beta \\\\ \\cos \\alpha & -\\sin \\beta & 0 \\end{vmatrix} &= -\\sin \\alpha \\begin{vmatrix} -\\sin \\alpha & \\sin \\beta \\\\ \\cos \\alpha & 0 \\end{vmatrix} - \\cos \\alpha \\begin{vmatrix} -\\sin \\alpha & 0 \\\\ \\cos \\alpha & -\\sin \\beta \\end{vmatrix} \\\\\n&= -\\sin \\alpha (-\\sin \\beta \\cos \\alpha) - \\cos \\alpha (\\sin \\alpha \\sin \\beta) \\\\\n&= \\boxed{0}.\n\\end{align*}","Plotting the points, we see that the dimensions of the rectangle are $x$ and $4$.  The area of the rectangle is $(\\text{length})(\\text{width})=4x$, so $4x=28$ and $x=\\boxed{7}$. [asy]\nsize(5cm);\nimport graph;\ndefaultpen(linewidth(0.7)+fontsize(12));\nreal x = 7;\npair A=(0,0), B=(0,4), C=(x,4), D=(x,0);\npair[] dots = {A,B,C,D};\ndot(dots);\ndraw(A--B--C--D--cycle);\nxaxis(-2,9,Arrows(4));\nyaxis(-2,7,Arrows(4));\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,NW);\nlabel(\"$C$\",C,NE);\nlabel(\"$D$\",D,SE);\nlabel(\"$x$\",(B+C)/2,N);\nlabel(\"$4$\",(C+D)/2,E);[/asy]","We know that $\\left(\\frac{13}{7}\\right)^2=\\frac{169}{49}$. Then, since $3=\\frac{147}{49}<\\frac{169}{49}<\\frac{196}{49}=4$, we conclude that $\\left\\lceil\\left(\\frac{13}{7}\\right)^2\\right\\rceil=4$. Because $4+\\frac{17}{4}=\\frac{33}{4}$, which is a number between $8$ and $9$, $\\left\\lfloor \\left\\lceil \\left(\\frac{13}{7}\\right)^2\\right\\rceil+\\frac{17}{4}\\right\\rfloor=\\boxed{8}$.","The height of the cone is $\\frac{3}{4} \\times 8 = 6$. The radius of the cone is $\\frac{8}{2} = 4$. Therefore, the volume of the cone formed is $\\frac{4^2 \\times 6 \\times \\pi}{3} = \\boxed{32 \\pi}$.","Call the two integers $x$ and $y$. Without loss of generality, let $x$ be the larger of the two. We are given that $x+y = 50$ and $x-y = 12$, and we are asked for $x^2 - y^2$. Because $x^2 - y^2$ factors into $(x+y)(x-y)$, we can simply substitute in to get $x^2 - y^2 = 50 \\cdot 12 = \\boxed{600}$.","The common difference is $5 - 2 = 3$, so the $25^{\\text{th}}$ term is $2 + 3 \\cdot 24 = \\boxed{74}$.","Plugging in $x = 2$, we get $-4$ for the numerator, and $-1$ for the denominator, thus, $\\boxed{4}$ is the answer.","+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1","2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer:","81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: ","l Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer","= 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.7","3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = "," are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are "," is $4-0 = 4$.\nThe coordinates of $D$ are $(x, 0)$.\nThe length of the side $CD$ is given as 28.\nThe coordinates of $A$ are $(0, 0)$.\nThe coordinates of $B$ are $(0, 4)$.\nThe coordinates of $C$ are $(x, 4)$.\nThe length of the side $DA$ is given as 28.\nThe coordinates of $C$ are $("],"return":{"token_logprobs":true,"topk":0,"mlp_act":[8,14,17]}}

{"type":"teacher_forced","model":"original","texts":["We have that\n\\begin{align*}\na_3 &= \\frac{a_2}{a_1} = \\frac{3}{2}, \\\\\na_4 &= \\frac{a_3}{a_2} = \\frac{3/2}{3} = \\frac{1}{2}, \\\\\na_5 &= \\frac{a_4}{a_3} = \\frac{1/2}{3/2} = \\frac{1}{3}, \\\\\na_6 &= \\frac{a_5}{a_4} = \\frac{1/3}{1/2} = \\frac{2}{3}, \\\\\na_7 &= \\frac{a_6}{a_5} = \\frac{2/3}{1/3} = 2, \\\\\na_8 &= \\frac{a_7}{a_6} = \\frac{2}{2/3} = 3.\n\\end{align*}Since $a_7 = a_1 = 2$ and $a_8 = a_2 = 3,$ and each term depends only on the previous two terms, the sequence becomes periodic at this point, with a period of length 6.  Hence, $a_{2006} = a_2 = \\boxed{3}.$","The midpoint of a line segment with endpoints $(x_1, y_1), (x_2, y_2)$ is $\\left(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}\\right)$.\n\nThus, the midpoint of this line segment is $\\left(\\frac{2+8}{2}, \\frac{3+15}{2} \\right)$, which simplifies to $(5,9)$. Thus, the sum of the coordinates of the midpoint is $\\boxed{14}$.","Let $y = x^2 + x + 3.$  Then we can write the given equation as\n\\[\\frac{4x}{y} + \\frac{5x}{y - 6x} + \\frac{3}{2} = 0.\\]Multiplying everything by $2y(y - 6x),$ we get\n\\[8x(y - 6x) + 10xy + 3y(y - 6x) = 0.\\]Expanding, we get $3y^2 - 48x^2 = 0,$ so $y^2 - 16x^2 = (y - 4x)(y + 4x) = 0.$  Thus, $y = 4x$ or $y = -4x.$\n\nIf $y = 4x,$ then $x^2 + x + 3 = 4x,$ so $x^2 - 3x + 3 = 0.$  This quadratic has no real solutions.\n\nIf $y = -4x,$ then $x^2 + x + 3 = -4x,$ so $x^2 + 5x + 3 = 0.$  This quadratic has two real solutions, giving us a total of $\\boxed{2}$ real solutions.","Simplify, remembering the exponents take priority over multiplication and multiplication takes priority over addition or subtraction. \\begin{align*}\n5^3-3\\times 5^2+3\\times5-1&=125-3\\times25+15-1 \\\\\n&=125-75+14 \\\\\n&= 50+14 \\\\\n&= \\boxed{64}.\n\\end{align*}","We proceed as follows: \\begin{align*}\n(2b + 5)(b - 1) &= 6b\\\\\n2b^2 + 3b - 5 &= 6b\\\\\n2b^2 - 3b - 5 &= 0\\\\\n(2b - 5)(b + 1) &= 0.\n\\end{align*}This gives us $b = \\frac{5}{2}$ or $b = -1.$ Of these, $\\boxed{\\frac{5}{2}}$ is the larger possible value of $b.$","We solve each equation separately. First off, we have $8x^2 + 7x - 1 = (8x-1)(x+1) = 0.$ We can also see that $24x^2+53x-7 = (8x-1)(3x+7) = 0.$ It is clear that both equations are satisfied only when $8x - 1 = 0,$ therefore $x = \\boxed{\\dfrac{1}{8}}.$","Let's substitute the given values into the expression. We get  \\begin{align*}\n(a^3+b^3)\\div(a^2-ab+b^2)&=(5^3+4^3)\\div(5^2-5\\cdot4+4^2)\\\\\n&=(125+64)\\div(25-20+16)\\\\\n&=189\\div21\\\\\n&=\\boxed{9}.\n\\end{align*}","The only numbers that have 11 and 8 as a factor are multiples of 88.  If we list the first few multiples of 88: $$88,176,264,352,...$$ we can see that there are exactly $\\boxed{2}$ between 100 and 300.","Since $3^3 = 3\\times 3\\times 3 = 3\\times 9 = 27$, then  \\[ \\sqrt{3^3+3^3+3^3} = \\sqrt{27+27+27}=\\sqrt{81}=\\boxed{9}. \\]","Let $n$ be any integer that has its square between 4000 and 7000. Then $63 < n < 84$, because $63^2 < 4000<64^2$ and $83^2< 7000<84^2$. Between 63 and 84, the only primes are 67, 71, 73, 79, and 83. Thus, the answer is $\\boxed{5}$.","There are ${9 \\choose 2} = 36$ pairs of plates possible in all. Exactly ${5 \\choose 2} = 10$ of these pairs are both red plates and $\\binom{4}{2}=6$ pairs are both blue plates for a total of $6+10=16$ pairs of plates that satisfy our condition. So, the probability of the pair of plates being the same color is $\\frac{16}{36}=\\boxed{\\frac{4}{9}}$.","Let $B$ and $J$ represent the respective ages of Billy and Joe. We can write the equations $B=2J$ and $B+J=45$. We use the second equation to solve for $J$ in terms of $B$ and get $J=45-B$. Now we plug in this expression for $J$ into the first equation. $$B=2(45-B)=90-2B\\qquad\\Rightarrow 3B=90\\qquad\\Rightarrow B=30$$ So Billy is $\\boxed{30}$ years old.","By AM-HM,\n\\[\\frac{x + y}{2} \\ge \\frac{2}{\\frac{1}{x} + \\frac{1}{y}}.\\]Hence,\n\\[\\frac{1}{x} + \\frac{1}{y} \\ge \\frac{4}{x + y} = \\frac{4}{10} = \\frac{2}{5}.\\]Equality occurs when $x = y = 5,$ so the minimum value is $\\boxed{\\frac{2}{5}}$.","$64^2 - 36^2$ can also be expressed as $(64+36)(64-36)$. This simplifies as $100 \\cdot 28$, which equals $\\boxed{2800}$.","Since the triangle is equilateral, all sides are equal in length. Therefore, the perimeter of the triangle is $8+8+8=8 \\times 3=\\boxed{24}.$","We have two equations and two variables, so it's possible to solve for $p$ and $q$ directly and then calculate $p^2$ and $q^2$ separately to get our answer. However, doing so involves a fair amount of computation with complex numbers and square roots, so we look for an alternative approach. We square the second equation to get $$(p+q)^2 = p^2 + 2pq + q^2 = 36,$$which is close to what we want but has the extra $2pq$ term. Since we know that $pq=9$, we can substitute to get $$p^2 + 2(9) +q^2 = 36 \\implies p^2+q^2 = \\boxed{18}.$$Note that our task was made easier by only solving for what the problem asked rather than by trying to solve for $p$ and $q$ individually.","Triangle $ABC$ has base $AB$ of length $25\\text{ cm}$ and height $AC$ of length $20\\text{ cm}.$ Therefore, the area of triangle $ABC$ is \\begin{align*}\n\\frac{1}{2}bh &= \\frac{1}{2}(25 \\mbox{ cm})(20 \\mbox{ cm}) \\\\\n&= \\frac{1}{2}(500 \\mbox{ cm}^2) \\\\\n&= \\boxed{250} \\mbox{ cm}^2.\n\\end{align*}","We have  \\[\\frac{x^4 + 2y^2}{6} = \\frac{2^4 + 2(5^2)}{6} = \\frac{16+2(25)}{6} = \\frac{16+50}{6} = \\frac{66}{6} = \\boxed{11}.\\]","The average length is $\\frac{1.5+4.5}{2}=\\boxed{3}$ inches.","Her mean score was $\\frac{98+97+92+85+93}{5}=\\boxed{93}$.","We simplify the given inequality by adding $x+3$ to both sides, which leaves us with $5x < 5 \\Rightarrow x<1$. Since 1 is not less than itself, this leaves us with $x=\\boxed{0}$.","We know that any number raised to the power of $0$ is $1$, or $a^0 = 1$ for any $a$. Thus, we get $$(5^{-2})^0 + (5^0)^3 = 1 + 1^3 = 1+1 = \\boxed{2}.$$","Since $2a + 1 = 1,$ we find that $2a = 0,$ so $a = 0.$ Therefore, $b - a = b - 0 = b = \\boxed{1}.$","\\[\n\\begin{array}{c|cc cc}\n\\multicolumn{2}{r}{x^2} & +\\left(\\frac{c-7}{2}\\right)x & +5 \\\\\n\\cline{2-5}\n2x+7 & 2x^3 &+cx^2 &- 11x &+ 39  \\\\\n\\multicolumn{2}{r}{-2x^3} & -7x^2  \\\\ \n\\cline{2-3}\n\\multicolumn{2}{r}{0} & (c-7)x^2 & -11x  \\\\\n\\multicolumn{2}{r}{} & -(c-7)x^2 & -x(c-7)\\left(\\frac{7}{2}\\right)   \\\\ \n\\cline{3-4}\n\\multicolumn{2}{r}{} &  0 & -x\\left(\\frac{7c-27}{2}\\right) & + 39  \\\\\n\\multicolumn{2}{r}{} &   & -10x & -35  \\\\\n\\cline{4-5}\n\\multicolumn{2}{r}{} &   & -x\\left(\\frac{7c-27+20}{2}\\right) & 4  \\\\\n\\end{array}\n\\]In the last step of the division, we have $39$ left as the constant term in our dividend and we need a remainder of $4$ at the end. Since our divisor has a term of $7$, the only way to do this is if our quotient has $5$ which gives us  $7\\cdot5=35$ to subtract from our dividend and get the right remainder.\n\nThen, we need the rest of our remainder to be $0$. This means\n$$\\frac{7c-27+20}{2} = 0$$which gives us\n$$c = \\boxed{1}.$$","Let's begin with the calculator that initially shows 1. Each time it is passed around the circle, it is cubed. 1 to any power is still 1, so no matter how many times 1 is cubed, the final result will still be 1.\n\nNow examine the calculator that starts with a zero. 0 squared is still 0 because 0 to any positive power is still 0. Thus, no matter how many times zero is squared, the final number will still be zero.\n\nFinally, let's look at the calculator that initially shows -1. Each time a person gets the calculator, they negate the number. Because there are 42 participants, there are 42 total turns. Thus, -1 is negated 42 times. Because negating a number is the same as multiplying by -1, this is the same as multiplying it by -1 forty-two times. Thus, we are looking for \\[(-1) \\cdot (-1)^{42}=(-1)^1 \\cdot (-1)^{42}=(-1)^{1+42}=(-1)^{43}.\\]Recall that $(-a)^n=-a^n$ if $n$ is odd. Because 43 is odd, $(-1)^{43}=-1^{43}=-1$.\n\nThus, the sum of all of the numbers is $1+0+(-1)=\\boxed{0}$.","By definition, an obtuse triangle contains one obtuse interior angle. It cannot contain more than one because an obtuse angle has measure greater than 90 degrees and the sum of all the interior angles in any triangle is 180 degrees. Thus, there is $\\boxed{1}$ obtuse angle.","There is one King of $\\diamondsuit$ and 52 cards total, so the probability that the top card is a King of $\\diamondsuit$ is $\\boxed{\\dfrac{1}{52}}$.","We have  \\[81^{3/4} = (3^4)^{3/4} = 3^{4\\cdot (3/4)} = 3^3 = \\boxed{27}.\\]","Since 3 and 4 are relatively prime, their least common multiple is $3\\cdot4=12$.  Therefore, the least common multiple of 30 and 40 is 120.  Since $\\boxed{120}$ is not divisible by 16, it is the smallest common multiple of 30 and 40 which is not divisible by 16.\n\nNote: Every common multiple of two integers is a multiple of their least common multiple.  Therefore, it is not possible that the least common multiple is divisible by 16 but some other common multiple is not.","We have $g(-5) = 5(-5) + 2(-5)^2 = -25 + 50 = 25$, so $f(g(-5)) = f(25) = 3 - \\!\\sqrt{25} = 3-5=\\boxed{-2}$.","We have $55 \\times 1212 - 15 \\times 1212 = 1212(55-15) = 1212(40) = 4848(10) = \\boxed{48480}$.","Solution 1. The first equation gives $x = y+4$. Substituting into the second equation, we get \\[(y+4)^3 - y^3 = 28 \\implies 12y^2 + 48y + 36 = 0.\\]Thus, $y^2 + 4y + 3 = 0$, so $(y+1)(y+3) = 0$. Therefore, either $y=-1$ and $x=y+4=3$, or $y=-3$ and $x=y+4=1$. Either way, $xy = \\boxed{-3}$.\n\nSolution 2. The second equation factors via difference of cubes, as \\[(x-y)(x^2+xy+y^2) = 28.\\]Since $x-y=4$, we have $x^2+xy+y^2=\\frac{28}{4} =7$. Now, squaring the first equation, we get $x^2-2xy+y^2=16$. Thus, \\[3xy = (x^2+xy+y^2) - (x^2-2xy+y^2) = 7-16=-9,\\]so $xy = \\frac{-9}{3} = \\boxed{-3}$.","By the Law of Sines applied to triangle $OAB$, $$\\frac{OB}{\\sin\\angle\nOAB}=\\frac{AB}{\\sin\\angle AOB}.$$With $AB = 1$ and $\\angle AOB = 30^\\circ$, we have \\[\\frac{OB}{\\sin \\angle OAB} = \\frac{1}{\\sin 30^\\circ} = 2,\\]so  so $OB=2\\sin\\angle OAB$. Thus, $OB \\le \\boxed{2}$, with equality if and only if $\\angle OAB=90^\\circ$.\n\n[asy]\nunitsize(1.5 cm);\n\npair O, A, B;\n\nO = (0,0);\nA = sqrt(3)*dir(30);\nB = (2,0);\n\ndraw((0,0)--3*dir(30),Arrow(6));\ndraw((0,0)--(3,0),Arrow(6));\ndraw(A--B);\ndraw(rightanglemark(O,A,B,4));\n\nlabel(\"$A$\", A, NW);\nlabel(\"$B$\", B, S);\nlabel(\"$O$\", O, W);\nlabel(\"$1$\", (A + B)/2, NE, red);\n[/asy]","Adding 14 and 36 yields 50, and then adding 22 yields 72.  Dividing by 3 yields $\\boxed{24}$.","Since $\\mathbf{A} \\mathbf{B} = \\mathbf{B} \\mathbf{A},$\n\\[\\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix}.\\]Expanding, we get\n\\[\\begin{pmatrix} a + 2c & b + 2d \\\\ 3a + 4c & 3b + 4d \\end{pmatrix} = \\begin{pmatrix} a + 3b & 2a + 4b \\\\ c + 3d & 2c + 4d \\end{pmatrix}.\\]Comparing entries, we find $3b = 2c$ and $3a + 3c = 3d,$ so $a + c = d.$  Then\n\\[\\frac{a - d}{c - 3b} = \\frac{-c}{c - 2c} = \\frac{-c}{-c} = \\boxed{1}.\\]","Since $y=1$, we have $1 =\\displaystyle\\frac{1}{3x+1}$.  Multiplying both sides by $3x+1$, we have $$3x+1=1$$ $$\\Rightarrow \\qquad 3x=0$$ $$\\Rightarrow \\qquad x=\\boxed{0}$$","If one centimeter on the map is 15 kilometers in reality, then we have the ratio $1 \\text{ cm on map} : 15 \\text{ km in reality}$. Multiplying the ratio by 88, we get $88 \\text{ cm on map} : 15 \\cdot 88 \\text{ km in reality} =88 \\text{ cm on map} : 1320 \\text{ km in reality}$. Thus, the cities are $\\boxed{1320}$ kilometers apart.","The $x$-coordinate and $y$-coordinate of the midpoint are the averages of the $x$-coordinates and $y$-coordinates of the endpoints, respectively.  Since $4-3=1$, the $x$-coordinate of the other endpoint is $1-3=-2$.  Since $5-7=-2$, the $y$-coordinate of the other endpoint is $-2-7=-9$.  Therefore, the endpoints are $(4,5)$ and $\\boxed{(-2,-9)}$.","We can use complementary probability to determine that the probability of its not raining tomorrow is $1 - \\frac{3}{10} = \\boxed{\\frac{7}{10}}$.","Call the two integers $x$ and $y$. We are given that $x^2 + y^2 = 90$ and that $xy = 27$. We want to find $x + y$. Note that $(x + y)^2 = x^2 + y^2 + 2xy = 90 + 2\\cdot 27 = 144$. Taking the square root of 144, we see that $x + y = \\boxed{12}$.","Let the height of the cylinder be $h$; we then have \\[SA = 2\\pi (2^2)+2\\pi (2)(h) = 12\\pi.\\]Solving yields $4\\pi h = 4 \\pi$ so $h = \\boxed{1}$ foot.","Setting $x = -f(y),$ we get\n\\[f(0) = -f(y) + y,\\]so $f(y) = y - f(0)$ for all real numbers $x.$  Then the given functional equation becomes\n\\[f(x + y - f(0)) = x + y,\\]or $x + y - f(0) - f(0) = x + y.$  Then $f(0) = 0,$ so $f(x) = x$ for all real numbers $x.$  This function does satisfy the given functional equation, giving us $\\boxed{1}$ solution.","$x^2 - y^2$ factors into $(x+y)(x-y)$, so, to obtain the value of $x^2 - y^2$, simply multiply $\\frac{7}{12}\\cdot \\frac{1}{12}$ to get $\\boxed{\\frac{7}{144}}$.","When using the distributive property, we add the products of 3 and $8x^2$, 3 and $-2x$, and 3 and 1: \\begin{align*}\n3(8x^2-2x+1) &= 3\\cdot 8x^2+3\\cdot (-2x) + 3 \\cdot 1\\\\\n&= \\boxed{24x^2-6x+3}\n\\end{align*}","The greatest integer that is less than or equal to $0.999$ is $0$, so $\\lfloor0.999\\rfloor=0$. The smallest integer that is greater than or equal to $2.001$ is $3$, so $\\lceil2.001\\rceil=3$. Therefore, $\\lfloor0.999\\rfloor+\\lceil2.001\\rceil=0+3=\\boxed{3}$.","Shrinking the triangle by dividing every side length by 2, we recognize the resulting set $$\\{7,24,25\\}$$ of side lengths as a Pythagorean triple.  Therefore, the original triangle is also a right triangle, and its legs measure 14 cm and 48 cm.  The area of the triangle is $\\frac{1}{2}(14\\text{ cm})(48\\text{ cm})=\\boxed{336}$ square centimeters.","Dividing both sides by 3, we quickly note that $ 9 = 9^{x-1} \\rightarrow 1 = x-1 \\rightarrow x = \\boxed{2}$.","If $x=-3$, then $-3 = 2-t$, so $t = 5$.  Therefore, $y = 4(5) +7 =\\boxed{27}$.","Choosing a committee is a combination, and order does not matter. We are choosing a 4-person committee from 9 people, so there are $9 \\times 8 \\times 7 \\times 6$ ways to pick the four people for the positions, but then we must divide by $4!$ since order doesn't matter, so the answer is $\\dfrac{9 \\times 8 \\times 7 \\times 6}{4!} =\\boxed{126}$.","Terminating decimals can be expressed as $\\frac{a}{10^b}$. So, we try to get our fraction in this form: $$\\frac{11}{125} = \\frac{11}{5^3} = \\frac{11}{5^3} \\cdot \\frac{2^3}{2^3} = \\frac{11\\cdot2^3}{10^3} = \\frac{88}{1000} = \\boxed{0.088}.$$","We start by completing the square: \\begin{align*}\n2x^2 -12x + 3 &= 2(x^2-6x) +3 \\\\\n&= 2(x^2 -6x + (6/2)^2 - (6/2)^2) + 3\\\\\n& = 2((x-3)^2 -3^2) + 3 \\\\\n&= 2(x-3)^2 - 2\\cdot 3^2 + 3\\\\\n&= 2(x-3)^2 -15\n.\\end{align*} Since the square of a real number is at least 0, we have $(x-3)^2\\ge 0$, where $(x-3)^2 =0$ only if $x=3$.   Therefore, $2(x-3)^2 - 15$ is minimized when $x=\\boxed{3}.$","This can easily be converted into a fraction over a power of ten: \\[\n\\frac{3}{8} = \\frac{3 \\cdot 125}{8 \\cdot 125} = \\frac{375}{1000} = \\boxed{0.375}.\n\\]Note: In general if you have a fraction that is some $\\frac{x}{2^k}$, multiplying the numerator and denominator by $5^k$ will give a power of $10$ in the denominator for easy conversion to a decimal. In this case, $k = 3$.","The sum of the three angles in any triangle is always $180^\\circ.$ In $\\triangle PQR,$ the sum of $\\angle P$ and $\\angle Q$ is $60^\\circ,$ and thus $\\angle R$ must measure $$180^\\circ - 60^\\circ = \\boxed{120^\\circ}.$$","We find the average words per hour by dividing the total words by the total hours. $$\\frac{40,\\!000\\text{ words}}{80\\text{ hours}}=\\frac{4,\\!000}{8}=\\frac{40\\cdot100}{8}=5\\cdot100=\\boxed{500} \\text{ words per hour}$$","The common factor of 3 in the numerator and the common factor of 3 in the denominator will cancel: \\[\n\\frac{3^4+3^2}{3^3-3}=\\frac{3(3^3+3^1)}{3(3^2-1)}=\\frac{3^3+3^1}{3^2-1}\n\\] Now the numerator is $3^3+3=27+3=30$, and the denominator is $3^2-1=9-1=8$.  Therefore, the fraction simplifies to $\\dfrac{30}{8}=\\boxed{\\dfrac{15}{4}}$.","A cone with radius $r$ and height $h$ has volume $\\frac{1}{3}\\pi r^2 h$; here, our cone has volume $\\frac{1}{3}\\pi (1^2)(4)=\\frac{4}{3}\\pi$.  A sphere with radius $r$ has volume $\\frac{4}{3}\\pi r^3$, so we set up the equation \\[\\frac{4}{3}\\pi r^3=\\frac{4}{3}\\pi.\\] Solving for $r$ yields $r^3=1$, so $r = 1$.  The sphere's radius is $\\boxed{1}$ inch.","From the angle addition formula, we can write\n\\begin{align*}\n\\sin x + \\cos x &= \\sqrt{2} \\left( \\frac{1}{\\sqrt{2}} \\sin x + \\frac{1}{\\sqrt{2}} \\cos x \\right) \\\\\n&= \\sqrt{2} \\left( \\cos \\frac{\\pi}{4} \\sin x + \\sin \\frac{\\pi}{4} \\cos x \\right) \\\\\n&= \\sqrt{2} \\sin \\left( x + \\frac{\\pi}{4} \\right).\n\\end{align*}Thus, the graph of $y = \\sin x + \\cos x$ has period $\\boxed{2 \\pi}.$\n\nThe graph of $y = \\sin x + \\cos x$ is shown below:\n\n[asy]import TrigMacros;\n\nsize(400);\n\nreal g(real x)\n{\n\treturn sin(x) + cos(x);\n}\n\ndraw(graph(g,-3*pi,3*pi,n=700,join=operator ..),red);\ntrig_axes(-3*pi,3*pi,-2,2,pi/2,1);\nlayer();\nrm_trig_labels(-5, 5, 2);\n[/asy]","We can expand the determinant as follows:\n\\begin{align*}\n\\begin{vmatrix} 0 & \\sin \\alpha & -\\cos \\alpha \\\\ -\\sin \\alpha & 0 & \\sin \\beta \\\\ \\cos \\alpha & -\\sin \\beta & 0 \\end{vmatrix} &= -\\sin \\alpha \\begin{vmatrix} -\\sin \\alpha & \\sin \\beta \\\\ \\cos \\alpha & 0 \\end{vmatrix} - \\cos \\alpha \\begin{vmatrix} -\\sin \\alpha & 0 \\\\ \\cos \\alpha & -\\sin \\beta \\end{vmatrix} \\\\\n&= -\\sin \\alpha (-\\sin \\beta \\cos \\alpha) - \\cos \\alpha (\\sin \\alpha \\sin \\beta) \\\\\n&= \\boxed{0}.\n\\end{align*}","Plotting the points, we see that the dimensions of the rectangle are $x$ and $4$.  The area of the rectangle is $(\\text{length})(\\text{width})=4x$, so $4x=28$ and $x=\\boxed{7}$. [asy]\nsize(5cm);\nimport graph;\ndefaultpen(linewidth(0.7)+fontsize(12));\nreal x = 7;\npair A=(0,0), B=(0,4), C=(x,4), D=(x,0);\npair[] dots = {A,B,C,D};\ndot(dots);\ndraw(A--B--C--D--cycle);\nxaxis(-2,9,Arrows(4));\nyaxis(-2,7,Arrows(4));\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,NW);\nlabel(\"$C$\",C,NE);\nlabel(\"$D$\",D,SE);\nlabel(\"$x$\",(B+C)/2,N);\nlabel(\"$4$\",(C+D)/2,E);[/asy]","We know that $\\left(\\frac{13}{7}\\right)^2=\\frac{169}{49}$. Then, since $3=\\frac{147}{49}<\\frac{169}{49}<\\frac{196}{49}=4$, we conclude that $\\left\\lceil\\left(\\frac{13}{7}\\right)^2\\right\\rceil=4$. Because $4+\\frac{17}{4}=\\frac{33}{4}$, which is a number between $8$ and $9$, $\\left\\lfloor \\left\\lceil \\left(\\frac{13}{7}\\right)^2\\right\\rceil+\\frac{17}{4}\\right\\rfloor=\\boxed{8}$.","The height of the cone is $\\frac{3}{4} \\times 8 = 6$. The radius of the cone is $\\frac{8}{2} = 4$. Therefore, the volume of the cone formed is $\\frac{4^2 \\times 6 \\times \\pi}{3} = \\boxed{32 \\pi}$.","Call the two integers $x$ and $y$. Without loss of generality, let $x$ be the larger of the two. We are given that $x+y = 50$ and $x-y = 12$, and we are asked for $x^2 - y^2$. Because $x^2 - y^2$ factors into $(x+y)(x-y)$, we can simply substitute in to get $x^2 - y^2 = 50 \\cdot 12 = \\boxed{600}$.","The common difference is $5 - 2 = 3$, so the $25^{\\text{th}}$ term is $2 + 3 \\cdot 24 = \\boxed{74}$.","Plugging in $x = 2$, we get $-4$ for the numerator, and $-1$ for the denominator, thus, $\\boxed{4}$ is the answer.","+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1","2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer:","81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: ","l Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer","= 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.7","3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = "," are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are "," is $4-0 = 4$.\nThe coordinates of $D$ are $(x, 0)$.\nThe length of the side $CD$ is given as 28.\nThe coordinates of $A$ are $(0, 0)$.\nThe coordinates of $B$ are $(0, 4)$.\nThe coordinates of $C$ are $(x, 4)$.\nThe length of the side $DA$ is given as 28.\nThe coordinates of $C$ are $("],"return":{"token_logprobs":true,"topk":0,"mlp_act":[8,14,17]}}

Forward pass 3 of 10 done (teacher_forced); results in /work/fwd_3/. Summary:

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Forward passes left: 7. Edits used in this pass: none (edit hash none).

The valid teacher-forced pass shows that the model assigns very high probability to continuing some loops, especially the repeated +1 sequence. That establishes a measurable behavior to intervene on, but does not yet identify which channel causes it. I’ll compare candidate-channel activity on these loops against all 64 reference solutions and retain the within-problem correctness checks from generation. I’ll favor a small, explicit set of channels with converging evidence rather than a broad layer-wide change.

python

with open(wdir / "fwd_3" / "meta.json") as f:    meta3 = json.load(f)print("Metadata keys:", list(meta3))print("Files:", sorted(p.name for p in (wdir / "fwd_3").iterdir()))lengths3 = np.array(meta3["lengths"])ids3 = np.load(wdir / "fwd_3" / "input_ids.npy")lp3 = np.load(wdir / "fwd_3" / "token_logprobs.npy")probe_channels = []for layer in [8, 14, 17]:    at = np.load(wdir / "fwd_3" / f"mlp_act_{layer}.npy", mmap_mode="r")    token_means = np.stack([np.asarray(at[i, :lengths3[i]], dtype=np.float64).mean(0)                            for i in range(72)])    abs_means = np.stack([np.abs(np.asarray(at[i, :lengths3[i]], dtype=np.float64)).mean(0)                          for i in range(72)])    aa = np.array(mlpmeans1[dev_idx, layer], dtype=np.float64)    ac = within_center(aa)    ar = ac - lc[:, None] * ((lc @ ac) / (lc @ lc))[None, :]    corr = (yr @ ar) / (np.sqrt(np.sum(ar**2, axis=0) * (yr @ yr)) + 1e-15)    # Use absolute activity to avoid mistaking negative-valued useful channels for harmful ones.    abc = within_center(np.abs(aa))    abr = abc - lc[:, None] * ((lc @ abc) / (lc @ lc))[None, :]    corr_abs = (yr @ abr) / (np.sqrt(np.sum(abr**2, axis=0) * (yr @ yr)) + 1e-15)    corr_rep = (rc @ abc) / (np.sqrt(np.sum(abc**2, axis=0) * (rc @ rc)) + 1e-15)    half_abs = [(yr[m] @ abr[m]) / (np.sqrt(np.sum(abr[m]**2, axis=0)                    * np.sum(yr[m]**2)) + 1e-15) for m in [halfmask, ~halfmask]]    ref_abs = abs_means[:64].mean(0)    fail_abs = abs_means[64:].mean(0)    ratios = fail_abs / (ref_abs + 1e-8)    cols = np.linalg.norm(load_original(f"model.layers.{layer}.mlp.down_proj.weight").numpy(), axis=0)    for j in range(2048):        # Require failure association in each independent half of DEV.        stability = min(-half_abs[0][j], -half_abs[1][j])        score = max(0, stability) * max(0, corr_rep[j]) * max(0, np.log(ratios[j]))        probe_channels.append({            "layer": layer, "neuron": j, "score": float(score),            "corr_abs_correct": float(corr_abs[j]),            "half1": float(half_abs[0][j]), "half2": float(half_abs[1][j]),            "corr_abs_repetition": float(corr_rep[j]),            "loop_to_reference_abs_ratio": float(ratios[j]),            "ref_abs": float(ref_abs[j]), "loop_abs": float(fail_abs[j]),            "column_norm": float(cols[j]),            "loop_activity_by_probe": abs_means[64:, j].tolist()        })probe_channels.sort(key=lambda r: r["score"], reverse=True)(wdir / "probe_channel_screen.json").write_text(json.dumps(probe_channels[:100], indent=2))print("Convergent candidates:")for c in probe_channels[:16]:    print({k: round(v, 4) if isinstance(v, float) else v for k, v in c.items()           if k != "loop_activity_by_probe"})print("Neuron 14:1594:", next(c for c in probe_channels if c["layer"] == 14 and c["neuron"] == 1594))print("Finite probabilities:", np.isfinite(lp3).sum(), "of", lp3.size)

with open(wdir / "fwd_3" / "meta.json") as f:    meta3 = json.load(f)print("Metadata keys:", list(meta3))print("Files:", sorted(p.name for p in (wdir / "fwd_3").iterdir()))lengths3 = np.array(meta3["lengths"])ids3 = np.load(wdir / "fwd_3" / "input_ids.npy")lp3 = np.load(wdir / "fwd_3" / "token_logprobs.npy")probe_channels = []for layer in [8, 14, 17]:    at = np.load(wdir / "fwd_3" / f"mlp_act_{layer}.npy", mmap_mode="r")    token_means = np.stack([np.asarray(at[i, :lengths3[i]], dtype=np.float64).mean(0)                            for i in range(72)])    abs_means = np.stack([np.abs(np.asarray(at[i, :lengths3[i]], dtype=np.float64)).mean(0)                          for i in range(72)])    aa = np.array(mlpmeans1[dev_idx, layer], dtype=np.float64)    ac = within_center(aa)    ar = ac - lc[:, None] * ((lc @ ac) / (lc @ lc))[None, :]    corr = (yr @ ar) / (np.sqrt(np.sum(ar**2, axis=0) * (yr @ yr)) + 1e-15)    # Use absolute activity to avoid mistaking negative-valued useful channels for harmful ones.    abc = within_center(np.abs(aa))    abr = abc - lc[:, None] * ((lc @ abc) / (lc @ lc))[None, :]    corr_abs = (yr @ abr) / (np.sqrt(np.sum(abr**2, axis=0) * (yr @ yr)) + 1e-15)    corr_rep = (rc @ abc) / (np.sqrt(np.sum(abc**2, axis=0) * (rc @ rc)) + 1e-15)    half_abs = [(yr[m] @ abr[m]) / (np.sqrt(np.sum(abr[m]**2, axis=0)                    * np.sum(yr[m]**2)) + 1e-15) for m in [halfmask, ~halfmask]]    ref_abs = abs_means[:64].mean(0)    fail_abs = abs_means[64:].mean(0)    ratios = fail_abs / (ref_abs + 1e-8)    cols = np.linalg.norm(load_original(f"model.layers.{layer}.mlp.down_proj.weight").numpy(), axis=0)    for j in range(2048):        # Require failure association in each independent half of DEV.        stability = min(-half_abs[0][j], -half_abs[1][j])        score = max(0, stability) * max(0, corr_rep[j]) * max(0, np.log(ratios[j]))        probe_channels.append({            "layer": layer, "neuron": j, "score": float(score),            "corr_abs_correct": float(corr_abs[j]),            "half1": float(half_abs[0][j]), "half2": float(half_abs[1][j]),            "corr_abs_repetition": float(corr_rep[j]),            "loop_to_reference_abs_ratio": float(ratios[j]),            "ref_abs": float(ref_abs[j]), "loop_abs": float(fail_abs[j]),            "column_norm": float(cols[j]),            "loop_activity_by_probe": abs_means[64:, j].tolist()        })probe_channels.sort(key=lambda r: r["score"], reverse=True)(wdir / "probe_channel_screen.json").write_text(json.dumps(probe_channels[:100], indent=2))print("Convergent candidates:")for c in probe_channels[:16]:    print({k: round(v, 4) if isinstance(v, float) else v for k, v in c.items()           if k != "loop_activity_by_probe"})print("Neuron 14:1594:", next(c for c in probe_channels if c["layer"] == 14 and c["neuron"] == 1594))print("Finite probabilities:", np.isfinite(lp3).sum(), "of", lp3.size)

OUTPUT of Python run 12 of at most 300:

Metadata keys: ['texts', 'lengths', 'model', 'edited_tensors', 'files', 'padding']Files: ['input_ids.npy', 'meta.json', 'mlp_act_14.npy', 'mlp_act_17.npy', 'mlp_act_8.npy', 'token_logprobs.npy']Convergent candidates:{'layer': 14, 'neuron': 984, 'score': 0.032, 'corr_abs_correct': -0.1003, 'half1': -0.0868, 'half2': -0.1176, 'corr_abs_repetition': 0.514, 'loop_to_reference_abs_ratio': 2.0483, 'ref_abs': 0.0416, 'loop_abs': 0.0852, 'column_norm': 0.3083}{'layer': 14, 'neuron': 483, 'score': 0.0256, 'corr_abs_correct': -0.1235, 'half1': -0.1064, 'half2': -0.1414, 'corr_abs_repetition': 0.4416, 'loop_to_reference_abs_ratio': 1.7255, 'ref_abs': 0.0229, 'loop_abs': 0.0395, 'column_norm': 0.3533}{'layer': 17, 'neuron': 1179, 'score': 0.0221, 'corr_abs_correct': -0.1146, 'half1': -0.0958, 'half2': -0.1416, 'corr_abs_repetition': 0.3851, 'loop_to_reference_abs_ratio': 1.8212, 'ref_abs': 0.1567, 'loop_abs': 0.2854, 'column_norm': 0.3747}{'layer': 8, 'neuron': 699, 'score': 0.0215, 'corr_abs_correct': -0.0732, 'half1': -0.0711, 'half2': -0.0872, 'corr_abs_repetition': 0.4461, 'loop_to_reference_abs_ratio': 1.9698, 'ref_abs': 0.098, 'loop_abs': 0.1931, 'column_norm': 0.28}{'layer': 14, 'neuron': 1399, 'score': 0.0118, 'corr_abs_correct': -0.0367, 'half1': -0.0447, 'half2': -0.0274, 'corr_abs_repetition': 0.355, 'loop_to_reference_abs_ratio': 3.3551, 'ref_abs': 0.0317, 'loop_abs': 0.1063, 'column_norm': 0.3081}{'layer': 8, 'neuron': 1740, 'score': 0.0113, 'corr_abs_correct': -0.0571, 'half1': -0.0455, 'half2': -0.0673, 'corr_abs_repetition': 0.2363, 'loop_to_reference_abs_ratio': 2.8722, 'ref_abs': 0.0417, 'loop_abs': 0.1197, 'column_norm': 0.2941}{'layer': 8, 'neuron': 45, 'score': 0.0112, 'corr_abs_correct': -0.0859, 'half1': -0.0785, 'half2': -0.0956, 'corr_abs_repetition': 0.3364, 'loop_to_reference_abs_ratio': 1.5255, 'ref_abs': 0.0571, 'loop_abs': 0.0871, 'column_norm': 0.3162}{'layer': 8, 'neuron': 306, 'score': 0.011, 'corr_abs_correct': -0.115, 'half1': -0.0662, 'half2': -0.1824, 'corr_abs_repetition': 0.2731, 'loop_to_reference_abs_ratio': 1.8364, 'ref_abs': 0.0328, 'loop_abs': 0.0602, 'column_norm': 0.2731}{'layer': 8, 'neuron': 1209, 'score': 0.0107, 'corr_abs_correct': -0.063, 'half1': -0.0714, 'half2': -0.0543, 'corr_abs_repetition': 0.4705, 'loop_to_reference_abs_ratio': 1.5216, 'ref_abs': 0.0769, 'loop_abs': 0.117, 'column_norm': 0.32}{'layer': 8, 'neuron': 1835, 'score': 0.0102, 'corr_abs_correct': -0.0697, 'half1': -0.0606, 'half2': -0.0859, 'corr_abs_repetition': 0.4181, 'loop_to_reference_abs_ratio': 1.4969, 'ref_abs': 0.0454, 'loop_abs': 0.0679, 'column_norm': 0.3254}{'layer': 14, 'neuron': 333, 'score': 0.01, 'corr_abs_correct': -0.1194, 'half1': -0.0896, 'half2': -0.1631, 'corr_abs_repetition': 0.186, 'loop_to_reference_abs_ratio': 1.8191, 'ref_abs': 0.0297, 'loop_abs': 0.0541, 'column_norm': 0.3459}{'layer': 14, 'neuron': 1152, 'score': 0.0098, 'corr_abs_correct': -0.0832, 'half1': -0.0939, 'half2': -0.0755, 'corr_abs_repetition': 0.2354, 'loop_to_reference_abs_ratio': 1.7396, 'ref_abs': 0.0476, 'loop_abs': 0.0829, 'column_norm': 0.3419}{'layer': 8, 'neuron': 476, 'score': 0.0097, 'corr_abs_correct': -0.0412, 'half1': -0.0447, 'half2': -0.0431, 'corr_abs_repetition': 0.4137, 'loop_to_reference_abs_ratio': 1.7176, 'ref_abs': 0.4536, 'loop_abs': 0.779, 'column_norm': 0.3678}{'layer': 14, 'neuron': 311, 'score': 0.0085, 'corr_abs_correct': -0.0612, 'half1': -0.0741, 'half2': -0.0458, 'corr_abs_repetition': 0.1949, 'loop_to_reference_abs_ratio': 2.594, 'ref_abs': 0.0982, 'loop_abs': 0.2548, 'column_norm': 0.32}{'layer': 8, 'neuron': 1831, 'score': 0.0084, 'corr_abs_correct': -0.1245, 'half1': -0.0955, 'half2': -0.1551, 'corr_abs_repetition': 0.3387, 'loop_to_reference_abs_ratio': 1.2984, 'ref_abs': 0.0442, 'loop_abs': 0.0573, 'column_norm': 0.3812}{'layer': 8, 'neuron': 1837, 'score': 0.0079, 'corr_abs_correct': -0.0543, 'half1': -0.0528, 'half2': -0.0608, 'corr_abs_repetition': 0.2816, 'loop_to_reference_abs_ratio': 1.7038, 'ref_abs': 0.0507, 'loop_abs': 0.0864, 'column_norm': 0.366}Neuron 14:1594: {'layer': 14, 'neuron': 1594, 'score': 0.0, 'corr_abs_correct': -0.11435358784615679, 'half1': -0.10910569843039566, 'half2': -0.12050530509715969, 'corr_abs_repetition': 0.4547942002484733, 'loop_to_reference_abs_ratio': 0.8577380226565274, 'ref_abs': 0.08712915475365492, 'loop_abs': 0.07473399749151477, 'column_norm': 0.26153773069381714, 'loop_activity_by_probe': [0.10496503656560724, 0.102039306640625, 0.0637780371166411, 0.09994340376420455, 0.035739351721370924, 0.04237677663417873, 0.08598690498165967, 0.06304316250783093]}Finite probabilities: 30240 of 30240

Metadata keys: ['texts', 'lengths', 'model', 'edited_tensors', 'files', 'padding']Files: ['input_ids.npy', 'meta.json', 'mlp_act_14.npy', 'mlp_act_17.npy', 'mlp_act_8.npy', 'token_logprobs.npy']Convergent candidates:{'layer': 14, 'neuron': 984, 'score': 0.032, 'corr_abs_correct': -0.1003, 'half1': -0.0868, 'half2': -0.1176, 'corr_abs_repetition': 0.514, 'loop_to_reference_abs_ratio': 2.0483, 'ref_abs': 0.0416, 'loop_abs': 0.0852, 'column_norm': 0.3083}{'layer': 14, 'neuron': 483, 'score': 0.0256, 'corr_abs_correct': -0.1235, 'half1': -0.1064, 'half2': -0.1414, 'corr_abs_repetition': 0.4416, 'loop_to_reference_abs_ratio': 1.7255, 'ref_abs': 0.0229, 'loop_abs': 0.0395, 'column_norm': 0.3533}{'layer': 17, 'neuron': 1179, 'score': 0.0221, 'corr_abs_correct': -0.1146, 'half1': -0.0958, 'half2': -0.1416, 'corr_abs_repetition': 0.3851, 'loop_to_reference_abs_ratio': 1.8212, 'ref_abs': 0.1567, 'loop_abs': 0.2854, 'column_norm': 0.3747}{'layer': 8, 'neuron': 699, 'score': 0.0215, 'corr_abs_correct': -0.0732, 'half1': -0.0711, 'half2': -0.0872, 'corr_abs_repetition': 0.4461, 'loop_to_reference_abs_ratio': 1.9698, 'ref_abs': 0.098, 'loop_abs': 0.1931, 'column_norm': 0.28}{'layer': 14, 'neuron': 1399, 'score': 0.0118, 'corr_abs_correct': -0.0367, 'half1': -0.0447, 'half2': -0.0274, 'corr_abs_repetition': 0.355, 'loop_to_reference_abs_ratio': 3.3551, 'ref_abs': 0.0317, 'loop_abs': 0.1063, 'column_norm': 0.3081}{'layer': 8, 'neuron': 1740, 'score': 0.0113, 'corr_abs_correct': -0.0571, 'half1': -0.0455, 'half2': -0.0673, 'corr_abs_repetition': 0.2363, 'loop_to_reference_abs_ratio': 2.8722, 'ref_abs': 0.0417, 'loop_abs': 0.1197, 'column_norm': 0.2941}{'layer': 8, 'neuron': 45, 'score': 0.0112, 'corr_abs_correct': -0.0859, 'half1': -0.0785, 'half2': -0.0956, 'corr_abs_repetition': 0.3364, 'loop_to_reference_abs_ratio': 1.5255, 'ref_abs': 0.0571, 'loop_abs': 0.0871, 'column_norm': 0.3162}{'layer': 8, 'neuron': 306, 'score': 0.011, 'corr_abs_correct': -0.115, 'half1': -0.0662, 'half2': -0.1824, 'corr_abs_repetition': 0.2731, 'loop_to_reference_abs_ratio': 1.8364, 'ref_abs': 0.0328, 'loop_abs': 0.0602, 'column_norm': 0.2731}{'layer': 8, 'neuron': 1209, 'score': 0.0107, 'corr_abs_correct': -0.063, 'half1': -0.0714, 'half2': -0.0543, 'corr_abs_repetition': 0.4705, 'loop_to_reference_abs_ratio': 1.5216, 'ref_abs': 0.0769, 'loop_abs': 0.117, 'column_norm': 0.32}{'layer': 8, 'neuron': 1835, 'score': 0.0102, 'corr_abs_correct': -0.0697, 'half1': -0.0606, 'half2': -0.0859, 'corr_abs_repetition': 0.4181, 'loop_to_reference_abs_ratio': 1.4969, 'ref_abs': 0.0454, 'loop_abs': 0.0679, 'column_norm': 0.3254}{'layer': 14, 'neuron': 333, 'score': 0.01, 'corr_abs_correct': -0.1194, 'half1': -0.0896, 'half2': -0.1631, 'corr_abs_repetition': 0.186, 'loop_to_reference_abs_ratio': 1.8191, 'ref_abs': 0.0297, 'loop_abs': 0.0541, 'column_norm': 0.3459}{'layer': 14, 'neuron': 1152, 'score': 0.0098, 'corr_abs_correct': -0.0832, 'half1': -0.0939, 'half2': -0.0755, 'corr_abs_repetition': 0.2354, 'loop_to_reference_abs_ratio': 1.7396, 'ref_abs': 0.0476, 'loop_abs': 0.0829, 'column_norm': 0.3419}{'layer': 8, 'neuron': 476, 'score': 0.0097, 'corr_abs_correct': -0.0412, 'half1': -0.0447, 'half2': -0.0431, 'corr_abs_repetition': 0.4137, 'loop_to_reference_abs_ratio': 1.7176, 'ref_abs': 0.4536, 'loop_abs': 0.779, 'column_norm': 0.3678}{'layer': 14, 'neuron': 311, 'score': 0.0085, 'corr_abs_correct': -0.0612, 'half1': -0.0741, 'half2': -0.0458, 'corr_abs_repetition': 0.1949, 'loop_to_reference_abs_ratio': 2.594, 'ref_abs': 0.0982, 'loop_abs': 0.2548, 'column_norm': 0.32}{'layer': 8, 'neuron': 1831, 'score': 0.0084, 'corr_abs_correct': -0.1245, 'half1': -0.0955, 'half2': -0.1551, 'corr_abs_repetition': 0.3387, 'loop_to_reference_abs_ratio': 1.2984, 'ref_abs': 0.0442, 'loop_abs': 0.0573, 'column_norm': 0.3812}{'layer': 8, 'neuron': 1837, 'score': 0.0079, 'corr_abs_correct': -0.0543, 'half1': -0.0528, 'half2': -0.0608, 'corr_abs_repetition': 0.2816, 'loop_to_reference_abs_ratio': 1.7038, 'ref_abs': 0.0507, 'loop_abs': 0.0864, 'column_norm': 0.366}Neuron 14:1594: {'layer': 14, 'neuron': 1594, 'score': 0.0, 'corr_abs_correct': -0.11435358784615679, 'half1': -0.10910569843039566, 'half2': -0.12050530509715969, 'corr_abs_repetition': 0.4547942002484733, 'loop_to_reference_abs_ratio': 0.8577380226565274, 'ref_abs': 0.08712915475365492, 'loop_abs': 0.07473399749151477, 'column_norm': 0.26153773069381714, 'loop_activity_by_probe': [0.10496503656560724, 0.102039306640625, 0.0637780371166411, 0.09994340376420455, 0.035739351721370924, 0.04237677663417873, 0.08598690498165967, 0.06304316250783093]}Finite probabilities: 30240 of 30240

The token-level check weakens the case for neuron 14:1594: its absolute activity is not higher in the loop excerpts than in valid reference solutions. Four other channels have stronger converging evidence—8:699, 14:483, 14:984, and 17:1179—with higher loop activity and failure associations in both DEV halves. I’ll test their combined contribution by zeroing only their four output columns, leaving the rest of each tensor unchanged. The prediction is lower probability for repetitive continuations with little damage to valid calculations; failure of that prediction would argue against this intervention.

python

target_channels = {8: [699], 14: [483, 984], 17: [1179]}target_names = {l: f"model.layers.{l}.mlp.down_proj.weight" for l in target_channels}def apply_channel_edit(strength):    for layer, neuron_ids in target_channels.items():        name = target_names[layer]        tensor_new = load_original(name).clone()        tensor_new[:, neuron_ids] *= (1.0 - strength)        save_edit(name, tensor_new)    return {"kind": "sparse_mlp_output_attenuation",            "strength": float(strength),            "targets": {str(k): v for k, v in target_channels.items()},            "tensors": list(target_names.values())}
edit_spec = apply_channel_edit(1.0)(wdir / "current_edit.json").write_text(json.dumps(edit_spec, indent=2))print("Edit:", edit_spec)for layer, js in target_channels.items():    name = target_names[layer]    original_w = load_original(name)    current_w = load(name)    delta_w = current_w - original_w    print(name, "shape", tuple(current_w.shape),          "changed_columns", int(torch.count_nonzero(delta_w.abs().sum(0))),          "relative_frobenius_change", round(float(delta_w.norm()/original_w.norm()), 5))    for j in js:        c = next(c for c in probe_channels if c["layer"] == layer and c["neuron"] == j)        print("Channel", layer, j, "loop activity by excerpt:",              [round(v, 3) for v in c["loop_activity_by_probe"]])import shutilshutil.copytree(wdir / "edited", wdir / "candidate_ablation_1", dirs_exist_ok=True)with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 3 findings\n"            "Neuron 14:1594 not selective in token-level probes; do NOT edit it.\n"            "Convergent candidates: 8:699, 14:483, 14:984, 17:1179. Each has higher "            "mean absolute loop activity than reference activity and negative length-adjusted "            "correctness association in both DEV halves.\n"            "\n## Pass 4 planned: causal teacher-forced screen\n"            "Exact edit: model.layers.8.mlp.down_proj.weight column699; "            "model.layers.14.mlp.down_proj.weight columns483,984; "            "model.layers.17.mlp.down_proj.weight column1179. Strength 1.0: set these columns to zero.\n"            "All other values unchanged. Candidate snapshot: candidate_ablation_1/.\n"            "Set: same 72 literal DEV continuation probes as pass3. Compare matched tokens only.\n"            "Prediction: reduce loop-continuation logprob without broad reference deterioration.\n")print("Active edits:", edited())

target_channels = {8: [699], 14: [483, 984], 17: [1179]}target_names = {l: f"model.layers.{l}.mlp.down_proj.weight" for l in target_channels}def apply_channel_edit(strength):    for layer, neuron_ids in target_channels.items():        name = target_names[layer]        tensor_new = load_original(name).clone()        tensor_new[:, neuron_ids] *= (1.0 - strength)        save_edit(name, tensor_new)    return {"kind": "sparse_mlp_output_attenuation",            "strength": float(strength),            "targets": {str(k): v for k, v in target_channels.items()},            "tensors": list(target_names.values())}
edit_spec = apply_channel_edit(1.0)(wdir / "current_edit.json").write_text(json.dumps(edit_spec, indent=2))print("Edit:", edit_spec)for layer, js in target_channels.items():    name = target_names[layer]    original_w = load_original(name)    current_w = load(name)    delta_w = current_w - original_w    print(name, "shape", tuple(current_w.shape),          "changed_columns", int(torch.count_nonzero(delta_w.abs().sum(0))),          "relative_frobenius_change", round(float(delta_w.norm()/original_w.norm()), 5))    for j in js:        c = next(c for c in probe_channels if c["layer"] == layer and c["neuron"] == j)        print("Channel", layer, j, "loop activity by excerpt:",              [round(v, 3) for v in c["loop_activity_by_probe"]])import shutilshutil.copytree(wdir / "edited", wdir / "candidate_ablation_1", dirs_exist_ok=True)with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 3 findings\n"            "Neuron 14:1594 not selective in token-level probes; do NOT edit it.\n"            "Convergent candidates: 8:699, 14:483, 14:984, 17:1179. Each has higher "            "mean absolute loop activity than reference activity and negative length-adjusted "            "correctness association in both DEV halves.\n"            "\n## Pass 4 planned: causal teacher-forced screen\n"            "Exact edit: model.layers.8.mlp.down_proj.weight column699; "            "model.layers.14.mlp.down_proj.weight columns483,984; "            "model.layers.17.mlp.down_proj.weight column1179. Strength 1.0: set these columns to zero.\n"            "All other values unchanged. Candidate snapshot: candidate_ablation_1/.\n"            "Set: same 72 literal DEV continuation probes as pass3. Compare matched tokens only.\n"            "Prediction: reduce loop-continuation logprob without broad reference deterioration.\n")print("Active edits:", edited())

OUTPUT of Python run 13 of at most 300:

Edit: {'kind': 'sparse_mlp_output_attenuation', 'strength': 1.0, 'targets': {'8': [699], '14': [483, 984], '17': [1179]}, 'tensors': ['model.layers.8.mlp.down_proj.weight', 'model.layers.14.mlp.down_proj.weight', 'model.layers.17.mlp.down_proj.weight']}model.layers.8.mlp.down_proj.weight shape (640, 2048) changed_columns 1 relative_frobenius_change 0.01887Channel 8 699 loop activity by excerpt: [0.167, 0.258, 0.268, 0.384, 0.056, 0.126, 0.18, 0.107]model.layers.14.mlp.down_proj.weight shape (640, 2048) changed_columns 2 relative_frobenius_change 0.03045Channel 14 483 loop activity by excerpt: [0.013, 0.067, 0.059, 0.053, 0.017, 0.014, 0.065, 0.027]Channel 14 984 loop activity by excerpt: [0.004, 0.133, 0.135, 0.13, 0.038, 0.023, 0.143, 0.077]model.layers.17.mlp.down_proj.weight shape (640, 2048) changed_columns 1 relative_frobenius_change 0.02052Channel 17 1179 loop activity by excerpt: [0.094, 0.297, 0.367, 0.623, 0.099, 0.136, 0.504, 0.164]Active edits: ['model.layers.14.mlp.down_proj.weight', 'model.layers.17.mlp.down_proj.weight', 'model.layers.8.mlp.down_proj.weight']

Edit: {'kind': 'sparse_mlp_output_attenuation', 'strength': 1.0, 'targets': {'8': [699], '14': [483, 984], '17': [1179]}, 'tensors': ['model.layers.8.mlp.down_proj.weight', 'model.layers.14.mlp.down_proj.weight', 'model.layers.17.mlp.down_proj.weight']}model.layers.8.mlp.down_proj.weight shape (640, 2048) changed_columns 1 relative_frobenius_change 0.01887Channel 8 699 loop activity by excerpt: [0.167, 0.258, 0.268, 0.384, 0.056, 0.126, 0.18, 0.107]model.layers.14.mlp.down_proj.weight shape (640, 2048) changed_columns 2 relative_frobenius_change 0.03045Channel 14 483 loop activity by excerpt: [0.013, 0.067, 0.059, 0.053, 0.017, 0.014, 0.065, 0.027]Channel 14 984 loop activity by excerpt: [0.004, 0.133, 0.135, 0.13, 0.038, 0.023, 0.143, 0.077]model.layers.17.mlp.down_proj.weight shape (640, 2048) changed_columns 1 relative_frobenius_change 0.02052Channel 17 1179 loop activity by excerpt: [0.094, 0.297, 0.367, 0.623, 0.099, 0.136, 0.504, 0.164]Active edits: ['model.layers.14.mlp.down_proj.weight', 'model.layers.17.mlp.down_proj.weight', 'model.layers.8.mlp.down_proj.weight']

The intervention changes exactly four columns across three MLP output matrices, with no changes to embeddings, the output matrix, or normalization weights. The selected channels are most active in repeated conclusion passages, rather than uniformly across every kind of loop. I’ll now score the same reference and failure probes with those columns removed. This will establish whether the channels causally support the predicted continuations before spending generation passes on accuracy.

forward

{"type":"teacher_forced","model":"edited","texts":["We have that\n\\begin{align*}\na_3 &= \\frac{a_2}{a_1} = \\frac{3}{2}, \\\\\na_4 &= \\frac{a_3}{a_2} = \\frac{3/2}{3} = \\frac{1}{2}, \\\\\na_5 &= \\frac{a_4}{a_3} = \\frac{1/2}{3/2} = \\frac{1}{3}, \\\\\na_6 &= \\frac{a_5}{a_4} = \\frac{1/3}{1/2} = \\frac{2}{3}, \\\\\na_7 &= \\frac{a_6}{a_5} = \\frac{2/3}{1/3} = 2, \\\\\na_8 &= \\frac{a_7}{a_6} = \\frac{2}{2/3} = 3.\n\\end{align*}Since $a_7 = a_1 = 2$ and $a_8 = a_2 = 3,$ and each term depends only on the previous two terms, the sequence becomes periodic at this point, with a period of length 6.  Hence, $a_{2006} = a_2 = \\boxed{3}.$","The midpoint of a line segment with endpoints $(x_1, y_1), (x_2, y_2)$ is $\\left(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}\\right)$.\n\nThus, the midpoint of this line segment is $\\left(\\frac{2+8}{2}, \\frac{3+15}{2} \\right)$, which simplifies to $(5,9)$. Thus, the sum of the coordinates of the midpoint is $\\boxed{14}$.","Let $y = x^2 + x + 3.$  Then we can write the given equation as\n\\[\\frac{4x}{y} + \\frac{5x}{y - 6x} + \\frac{3}{2} = 0.\\]Multiplying everything by $2y(y - 6x),$ we get\n\\[8x(y - 6x) + 10xy + 3y(y - 6x) = 0.\\]Expanding, we get $3y^2 - 48x^2 = 0,$ so $y^2 - 16x^2 = (y - 4x)(y + 4x) = 0.$  Thus, $y = 4x$ or $y = -4x.$\n\nIf $y = 4x,$ then $x^2 + x + 3 = 4x,$ so $x^2 - 3x + 3 = 0.$  This quadratic has no real solutions.\n\nIf $y = -4x,$ then $x^2 + x + 3 = -4x,$ so $x^2 + 5x + 3 = 0.$  This quadratic has two real solutions, giving us a total of $\\boxed{2}$ real solutions.","Simplify, remembering the exponents take priority over multiplication and multiplication takes priority over addition or subtraction. \\begin{align*}\n5^3-3\\times 5^2+3\\times5-1&=125-3\\times25+15-1 \\\\\n&=125-75+14 \\\\\n&= 50+14 \\\\\n&= \\boxed{64}.\n\\end{align*}","We proceed as follows: \\begin{align*}\n(2b + 5)(b - 1) &= 6b\\\\\n2b^2 + 3b - 5 &= 6b\\\\\n2b^2 - 3b - 5 &= 0\\\\\n(2b - 5)(b + 1) &= 0.\n\\end{align*}This gives us $b = \\frac{5}{2}$ or $b = -1.$ Of these, $\\boxed{\\frac{5}{2}}$ is the larger possible value of $b.$","We solve each equation separately. First off, we have $8x^2 + 7x - 1 = (8x-1)(x+1) = 0.$ We can also see that $24x^2+53x-7 = (8x-1)(3x+7) = 0.$ It is clear that both equations are satisfied only when $8x - 1 = 0,$ therefore $x = \\boxed{\\dfrac{1}{8}}.$","Let's substitute the given values into the expression. We get  \\begin{align*}\n(a^3+b^3)\\div(a^2-ab+b^2)&=(5^3+4^3)\\div(5^2-5\\cdot4+4^2)\\\\\n&=(125+64)\\div(25-20+16)\\\\\n&=189\\div21\\\\\n&=\\boxed{9}.\n\\end{align*}","The only numbers that have 11 and 8 as a factor are multiples of 88.  If we list the first few multiples of 88: $$88,176,264,352,...$$ we can see that there are exactly $\\boxed{2}$ between 100 and 300.","Since $3^3 = 3\\times 3\\times 3 = 3\\times 9 = 27$, then  \\[ \\sqrt{3^3+3^3+3^3} = \\sqrt{27+27+27}=\\sqrt{81}=\\boxed{9}. \\]","Let $n$ be any integer that has its square between 4000 and 7000. Then $63 < n < 84$, because $63^2 < 4000<64^2$ and $83^2< 7000<84^2$. Between 63 and 84, the only primes are 67, 71, 73, 79, and 83. Thus, the answer is $\\boxed{5}$.","There are ${9 \\choose 2} = 36$ pairs of plates possible in all. Exactly ${5 \\choose 2} = 10$ of these pairs are both red plates and $\\binom{4}{2}=6$ pairs are both blue plates for a total of $6+10=16$ pairs of plates that satisfy our condition. So, the probability of the pair of plates being the same color is $\\frac{16}{36}=\\boxed{\\frac{4}{9}}$.","Let $B$ and $J$ represent the respective ages of Billy and Joe. We can write the equations $B=2J$ and $B+J=45$. We use the second equation to solve for $J$ in terms of $B$ and get $J=45-B$. Now we plug in this expression for $J$ into the first equation. $$B=2(45-B)=90-2B\\qquad\\Rightarrow 3B=90\\qquad\\Rightarrow B=30$$ So Billy is $\\boxed{30}$ years old.","By AM-HM,\n\\[\\frac{x + y}{2} \\ge \\frac{2}{\\frac{1}{x} + \\frac{1}{y}}.\\]Hence,\n\\[\\frac{1}{x} + \\frac{1}{y} \\ge \\frac{4}{x + y} = \\frac{4}{10} = \\frac{2}{5}.\\]Equality occurs when $x = y = 5,$ so the minimum value is $\\boxed{\\frac{2}{5}}$.","$64^2 - 36^2$ can also be expressed as $(64+36)(64-36)$. This simplifies as $100 \\cdot 28$, which equals $\\boxed{2800}$.","Since the triangle is equilateral, all sides are equal in length. Therefore, the perimeter of the triangle is $8+8+8=8 \\times 3=\\boxed{24}.$","We have two equations and two variables, so it's possible to solve for $p$ and $q$ directly and then calculate $p^2$ and $q^2$ separately to get our answer. However, doing so involves a fair amount of computation with complex numbers and square roots, so we look for an alternative approach. We square the second equation to get $$(p+q)^2 = p^2 + 2pq + q^2 = 36,$$which is close to what we want but has the extra $2pq$ term. Since we know that $pq=9$, we can substitute to get $$p^2 + 2(9) +q^2 = 36 \\implies p^2+q^2 = \\boxed{18}.$$Note that our task was made easier by only solving for what the problem asked rather than by trying to solve for $p$ and $q$ individually.","Triangle $ABC$ has base $AB$ of length $25\\text{ cm}$ and height $AC$ of length $20\\text{ cm}.$ Therefore, the area of triangle $ABC$ is \\begin{align*}\n\\frac{1}{2}bh &= \\frac{1}{2}(25 \\mbox{ cm})(20 \\mbox{ cm}) \\\\\n&= \\frac{1}{2}(500 \\mbox{ cm}^2) \\\\\n&= \\boxed{250} \\mbox{ cm}^2.\n\\end{align*}","We have  \\[\\frac{x^4 + 2y^2}{6} = \\frac{2^4 + 2(5^2)}{6} = \\frac{16+2(25)}{6} = \\frac{16+50}{6} = \\frac{66}{6} = \\boxed{11}.\\]","The average length is $\\frac{1.5+4.5}{2}=\\boxed{3}$ inches.","Her mean score was $\\frac{98+97+92+85+93}{5}=\\boxed{93}$.","We simplify the given inequality by adding $x+3$ to both sides, which leaves us with $5x < 5 \\Rightarrow x<1$. Since 1 is not less than itself, this leaves us with $x=\\boxed{0}$.","We know that any number raised to the power of $0$ is $1$, or $a^0 = 1$ for any $a$. Thus, we get $$(5^{-2})^0 + (5^0)^3 = 1 + 1^3 = 1+1 = \\boxed{2}.$$","Since $2a + 1 = 1,$ we find that $2a = 0,$ so $a = 0.$ Therefore, $b - a = b - 0 = b = \\boxed{1}.$","\\[\n\\begin{array}{c|cc cc}\n\\multicolumn{2}{r}{x^2} & +\\left(\\frac{c-7}{2}\\right)x & +5 \\\\\n\\cline{2-5}\n2x+7 & 2x^3 &+cx^2 &- 11x &+ 39  \\\\\n\\multicolumn{2}{r}{-2x^3} & -7x^2  \\\\ \n\\cline{2-3}\n\\multicolumn{2}{r}{0} & (c-7)x^2 & -11x  \\\\\n\\multicolumn{2}{r}{} & -(c-7)x^2 & -x(c-7)\\left(\\frac{7}{2}\\right)   \\\\ \n\\cline{3-4}\n\\multicolumn{2}{r}{} &  0 & -x\\left(\\frac{7c-27}{2}\\right) & + 39  \\\\\n\\multicolumn{2}{r}{} &   & -10x & -35  \\\\\n\\cline{4-5}\n\\multicolumn{2}{r}{} &   & -x\\left(\\frac{7c-27+20}{2}\\right) & 4  \\\\\n\\end{array}\n\\]In the last step of the division, we have $39$ left as the constant term in our dividend and we need a remainder of $4$ at the end. Since our divisor has a term of $7$, the only way to do this is if our quotient has $5$ which gives us  $7\\cdot5=35$ to subtract from our dividend and get the right remainder.\n\nThen, we need the rest of our remainder to be $0$. This means\n$$\\frac{7c-27+20}{2} = 0$$which gives us\n$$c = \\boxed{1}.$$","Let's begin with the calculator that initially shows 1. Each time it is passed around the circle, it is cubed. 1 to any power is still 1, so no matter how many times 1 is cubed, the final result will still be 1.\n\nNow examine the calculator that starts with a zero. 0 squared is still 0 because 0 to any positive power is still 0. Thus, no matter how many times zero is squared, the final number will still be zero.\n\nFinally, let's look at the calculator that initially shows -1. Each time a person gets the calculator, they negate the number. Because there are 42 participants, there are 42 total turns. Thus, -1 is negated 42 times. Because negating a number is the same as multiplying by -1, this is the same as multiplying it by -1 forty-two times. Thus, we are looking for \\[(-1) \\cdot (-1)^{42}=(-1)^1 \\cdot (-1)^{42}=(-1)^{1+42}=(-1)^{43}.\\]Recall that $(-a)^n=-a^n$ if $n$ is odd. Because 43 is odd, $(-1)^{43}=-1^{43}=-1$.\n\nThus, the sum of all of the numbers is $1+0+(-1)=\\boxed{0}$.","By definition, an obtuse triangle contains one obtuse interior angle. It cannot contain more than one because an obtuse angle has measure greater than 90 degrees and the sum of all the interior angles in any triangle is 180 degrees. Thus, there is $\\boxed{1}$ obtuse angle.","There is one King of $\\diamondsuit$ and 52 cards total, so the probability that the top card is a King of $\\diamondsuit$ is $\\boxed{\\dfrac{1}{52}}$.","We have  \\[81^{3/4} = (3^4)^{3/4} = 3^{4\\cdot (3/4)} = 3^3 = \\boxed{27}.\\]","Since 3 and 4 are relatively prime, their least common multiple is $3\\cdot4=12$.  Therefore, the least common multiple of 30 and 40 is 120.  Since $\\boxed{120}$ is not divisible by 16, it is the smallest common multiple of 30 and 40 which is not divisible by 16.\n\nNote: Every common multiple of two integers is a multiple of their least common multiple.  Therefore, it is not possible that the least common multiple is divisible by 16 but some other common multiple is not.","We have $g(-5) = 5(-5) + 2(-5)^2 = -25 + 50 = 25$, so $f(g(-5)) = f(25) = 3 - \\!\\sqrt{25} = 3-5=\\boxed{-2}$.","We have $55 \\times 1212 - 15 \\times 1212 = 1212(55-15) = 1212(40) = 4848(10) = \\boxed{48480}$.","Solution 1. The first equation gives $x = y+4$. Substituting into the second equation, we get \\[(y+4)^3 - y^3 = 28 \\implies 12y^2 + 48y + 36 = 0.\\]Thus, $y^2 + 4y + 3 = 0$, so $(y+1)(y+3) = 0$. Therefore, either $y=-1$ and $x=y+4=3$, or $y=-3$ and $x=y+4=1$. Either way, $xy = \\boxed{-3}$.\n\nSolution 2. The second equation factors via difference of cubes, as \\[(x-y)(x^2+xy+y^2) = 28.\\]Since $x-y=4$, we have $x^2+xy+y^2=\\frac{28}{4} =7$. Now, squaring the first equation, we get $x^2-2xy+y^2=16$. Thus, \\[3xy = (x^2+xy+y^2) - (x^2-2xy+y^2) = 7-16=-9,\\]so $xy = \\frac{-9}{3} = \\boxed{-3}$.","By the Law of Sines applied to triangle $OAB$, $$\\frac{OB}{\\sin\\angle\nOAB}=\\frac{AB}{\\sin\\angle AOB}.$$With $AB = 1$ and $\\angle AOB = 30^\\circ$, we have \\[\\frac{OB}{\\sin \\angle OAB} = \\frac{1}{\\sin 30^\\circ} = 2,\\]so  so $OB=2\\sin\\angle OAB$. Thus, $OB \\le \\boxed{2}$, with equality if and only if $\\angle OAB=90^\\circ$.\n\n[asy]\nunitsize(1.5 cm);\n\npair O, A, B;\n\nO = (0,0);\nA = sqrt(3)*dir(30);\nB = (2,0);\n\ndraw((0,0)--3*dir(30),Arrow(6));\ndraw((0,0)--(3,0),Arrow(6));\ndraw(A--B);\ndraw(rightanglemark(O,A,B,4));\n\nlabel(\"$A$\", A, NW);\nlabel(\"$B$\", B, S);\nlabel(\"$O$\", O, W);\nlabel(\"$1$\", (A + B)/2, NE, red);\n[/asy]","Adding 14 and 36 yields 50, and then adding 22 yields 72.  Dividing by 3 yields $\\boxed{24}$.","Since $\\mathbf{A} \\mathbf{B} = \\mathbf{B} \\mathbf{A},$\n\\[\\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix}.\\]Expanding, we get\n\\[\\begin{pmatrix} a + 2c & b + 2d \\\\ 3a + 4c & 3b + 4d \\end{pmatrix} = \\begin{pmatrix} a + 3b & 2a + 4b \\\\ c + 3d & 2c + 4d \\end{pmatrix}.\\]Comparing entries, we find $3b = 2c$ and $3a + 3c = 3d,$ so $a + c = d.$  Then\n\\[\\frac{a - d}{c - 3b} = \\frac{-c}{c - 2c} = \\frac{-c}{-c} = \\boxed{1}.\\]","Since $y=1$, we have $1 =\\displaystyle\\frac{1}{3x+1}$.  Multiplying both sides by $3x+1$, we have $$3x+1=1$$ $$\\Rightarrow \\qquad 3x=0$$ $$\\Rightarrow \\qquad x=\\boxed{0}$$","If one centimeter on the map is 15 kilometers in reality, then we have the ratio $1 \\text{ cm on map} : 15 \\text{ km in reality}$. Multiplying the ratio by 88, we get $88 \\text{ cm on map} : 15 \\cdot 88 \\text{ km in reality} =88 \\text{ cm on map} : 1320 \\text{ km in reality}$. Thus, the cities are $\\boxed{1320}$ kilometers apart.","The $x$-coordinate and $y$-coordinate of the midpoint are the averages of the $x$-coordinates and $y$-coordinates of the endpoints, respectively.  Since $4-3=1$, the $x$-coordinate of the other endpoint is $1-3=-2$.  Since $5-7=-2$, the $y$-coordinate of the other endpoint is $-2-7=-9$.  Therefore, the endpoints are $(4,5)$ and $\\boxed{(-2,-9)}$.","We can use complementary probability to determine that the probability of its not raining tomorrow is $1 - \\frac{3}{10} = \\boxed{\\frac{7}{10}}$.","Call the two integers $x$ and $y$. We are given that $x^2 + y^2 = 90$ and that $xy = 27$. We want to find $x + y$. Note that $(x + y)^2 = x^2 + y^2 + 2xy = 90 + 2\\cdot 27 = 144$. Taking the square root of 144, we see that $x + y = \\boxed{12}$.","Let the height of the cylinder be $h$; we then have \\[SA = 2\\pi (2^2)+2\\pi (2)(h) = 12\\pi.\\]Solving yields $4\\pi h = 4 \\pi$ so $h = \\boxed{1}$ foot.","Setting $x = -f(y),$ we get\n\\[f(0) = -f(y) + y,\\]so $f(y) = y - f(0)$ for all real numbers $x.$  Then the given functional equation becomes\n\\[f(x + y - f(0)) = x + y,\\]or $x + y - f(0) - f(0) = x + y.$  Then $f(0) = 0,$ so $f(x) = x$ for all real numbers $x.$  This function does satisfy the given functional equation, giving us $\\boxed{1}$ solution.","$x^2 - y^2$ factors into $(x+y)(x-y)$, so, to obtain the value of $x^2 - y^2$, simply multiply $\\frac{7}{12}\\cdot \\frac{1}{12}$ to get $\\boxed{\\frac{7}{144}}$.","When using the distributive property, we add the products of 3 and $8x^2$, 3 and $-2x$, and 3 and 1: \\begin{align*}\n3(8x^2-2x+1) &= 3\\cdot 8x^2+3\\cdot (-2x) + 3 \\cdot 1\\\\\n&= \\boxed{24x^2-6x+3}\n\\end{align*}","The greatest integer that is less than or equal to $0.999$ is $0$, so $\\lfloor0.999\\rfloor=0$. The smallest integer that is greater than or equal to $2.001$ is $3$, so $\\lceil2.001\\rceil=3$. Therefore, $\\lfloor0.999\\rfloor+\\lceil2.001\\rceil=0+3=\\boxed{3}$.","Shrinking the triangle by dividing every side length by 2, we recognize the resulting set $$\\{7,24,25\\}$$ of side lengths as a Pythagorean triple.  Therefore, the original triangle is also a right triangle, and its legs measure 14 cm and 48 cm.  The area of the triangle is $\\frac{1}{2}(14\\text{ cm})(48\\text{ cm})=\\boxed{336}$ square centimeters.","Dividing both sides by 3, we quickly note that $ 9 = 9^{x-1} \\rightarrow 1 = x-1 \\rightarrow x = \\boxed{2}$.","If $x=-3$, then $-3 = 2-t$, so $t = 5$.  Therefore, $y = 4(5) +7 =\\boxed{27}$.","Choosing a committee is a combination, and order does not matter. We are choosing a 4-person committee from 9 people, so there are $9 \\times 8 \\times 7 \\times 6$ ways to pick the four people for the positions, but then we must divide by $4!$ since order doesn't matter, so the answer is $\\dfrac{9 \\times 8 \\times 7 \\times 6}{4!} =\\boxed{126}$.","Terminating decimals can be expressed as $\\frac{a}{10^b}$. So, we try to get our fraction in this form: $$\\frac{11}{125} = \\frac{11}{5^3} = \\frac{11}{5^3} \\cdot \\frac{2^3}{2^3} = \\frac{11\\cdot2^3}{10^3} = \\frac{88}{1000} = \\boxed{0.088}.$$","We start by completing the square: \\begin{align*}\n2x^2 -12x + 3 &= 2(x^2-6x) +3 \\\\\n&= 2(x^2 -6x + (6/2)^2 - (6/2)^2) + 3\\\\\n& = 2((x-3)^2 -3^2) + 3 \\\\\n&= 2(x-3)^2 - 2\\cdot 3^2 + 3\\\\\n&= 2(x-3)^2 -15\n.\\end{align*} Since the square of a real number is at least 0, we have $(x-3)^2\\ge 0$, where $(x-3)^2 =0$ only if $x=3$.   Therefore, $2(x-3)^2 - 15$ is minimized when $x=\\boxed{3}.$","This can easily be converted into a fraction over a power of ten: \\[\n\\frac{3}{8} = \\frac{3 \\cdot 125}{8 \\cdot 125} = \\frac{375}{1000} = \\boxed{0.375}.\n\\]Note: In general if you have a fraction that is some $\\frac{x}{2^k}$, multiplying the numerator and denominator by $5^k$ will give a power of $10$ in the denominator for easy conversion to a decimal. In this case, $k = 3$.","The sum of the three angles in any triangle is always $180^\\circ.$ In $\\triangle PQR,$ the sum of $\\angle P$ and $\\angle Q$ is $60^\\circ,$ and thus $\\angle R$ must measure $$180^\\circ - 60^\\circ = \\boxed{120^\\circ}.$$","We find the average words per hour by dividing the total words by the total hours. $$\\frac{40,\\!000\\text{ words}}{80\\text{ hours}}=\\frac{4,\\!000}{8}=\\frac{40\\cdot100}{8}=5\\cdot100=\\boxed{500} \\text{ words per hour}$$","The common factor of 3 in the numerator and the common factor of 3 in the denominator will cancel: \\[\n\\frac{3^4+3^2}{3^3-3}=\\frac{3(3^3+3^1)}{3(3^2-1)}=\\frac{3^3+3^1}{3^2-1}\n\\] Now the numerator is $3^3+3=27+3=30$, and the denominator is $3^2-1=9-1=8$.  Therefore, the fraction simplifies to $\\dfrac{30}{8}=\\boxed{\\dfrac{15}{4}}$.","A cone with radius $r$ and height $h$ has volume $\\frac{1}{3}\\pi r^2 h$; here, our cone has volume $\\frac{1}{3}\\pi (1^2)(4)=\\frac{4}{3}\\pi$.  A sphere with radius $r$ has volume $\\frac{4}{3}\\pi r^3$, so we set up the equation \\[\\frac{4}{3}\\pi r^3=\\frac{4}{3}\\pi.\\] Solving for $r$ yields $r^3=1$, so $r = 1$.  The sphere's radius is $\\boxed{1}$ inch.","From the angle addition formula, we can write\n\\begin{align*}\n\\sin x + \\cos x &= \\sqrt{2} \\left( \\frac{1}{\\sqrt{2}} \\sin x + \\frac{1}{\\sqrt{2}} \\cos x \\right) \\\\\n&= \\sqrt{2} \\left( \\cos \\frac{\\pi}{4} \\sin x + \\sin \\frac{\\pi}{4} \\cos x \\right) \\\\\n&= \\sqrt{2} \\sin \\left( x + \\frac{\\pi}{4} \\right).\n\\end{align*}Thus, the graph of $y = \\sin x + \\cos x$ has period $\\boxed{2 \\pi}.$\n\nThe graph of $y = \\sin x + \\cos x$ is shown below:\n\n[asy]import TrigMacros;\n\nsize(400);\n\nreal g(real x)\n{\n\treturn sin(x) + cos(x);\n}\n\ndraw(graph(g,-3*pi,3*pi,n=700,join=operator ..),red);\ntrig_axes(-3*pi,3*pi,-2,2,pi/2,1);\nlayer();\nrm_trig_labels(-5, 5, 2);\n[/asy]","We can expand the determinant as follows:\n\\begin{align*}\n\\begin{vmatrix} 0 & \\sin \\alpha & -\\cos \\alpha \\\\ -\\sin \\alpha & 0 & \\sin \\beta \\\\ \\cos \\alpha & -\\sin \\beta & 0 \\end{vmatrix} &= -\\sin \\alpha \\begin{vmatrix} -\\sin \\alpha & \\sin \\beta \\\\ \\cos \\alpha & 0 \\end{vmatrix} - \\cos \\alpha \\begin{vmatrix} -\\sin \\alpha & 0 \\\\ \\cos \\alpha & -\\sin \\beta \\end{vmatrix} \\\\\n&= -\\sin \\alpha (-\\sin \\beta \\cos \\alpha) - \\cos \\alpha (\\sin \\alpha \\sin \\beta) \\\\\n&= \\boxed{0}.\n\\end{align*}","Plotting the points, we see that the dimensions of the rectangle are $x$ and $4$.  The area of the rectangle is $(\\text{length})(\\text{width})=4x$, so $4x=28$ and $x=\\boxed{7}$. [asy]\nsize(5cm);\nimport graph;\ndefaultpen(linewidth(0.7)+fontsize(12));\nreal x = 7;\npair A=(0,0), B=(0,4), C=(x,4), D=(x,0);\npair[] dots = {A,B,C,D};\ndot(dots);\ndraw(A--B--C--D--cycle);\nxaxis(-2,9,Arrows(4));\nyaxis(-2,7,Arrows(4));\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,NW);\nlabel(\"$C$\",C,NE);\nlabel(\"$D$\",D,SE);\nlabel(\"$x$\",(B+C)/2,N);\nlabel(\"$4$\",(C+D)/2,E);[/asy]","We know that $\\left(\\frac{13}{7}\\right)^2=\\frac{169}{49}$. Then, since $3=\\frac{147}{49}<\\frac{169}{49}<\\frac{196}{49}=4$, we conclude that $\\left\\lceil\\left(\\frac{13}{7}\\right)^2\\right\\rceil=4$. Because $4+\\frac{17}{4}=\\frac{33}{4}$, which is a number between $8$ and $9$, $\\left\\lfloor \\left\\lceil \\left(\\frac{13}{7}\\right)^2\\right\\rceil+\\frac{17}{4}\\right\\rfloor=\\boxed{8}$.","The height of the cone is $\\frac{3}{4} \\times 8 = 6$. The radius of the cone is $\\frac{8}{2} = 4$. Therefore, the volume of the cone formed is $\\frac{4^2 \\times 6 \\times \\pi}{3} = \\boxed{32 \\pi}$.","Call the two integers $x$ and $y$. Without loss of generality, let $x$ be the larger of the two. We are given that $x+y = 50$ and $x-y = 12$, and we are asked for $x^2 - y^2$. Because $x^2 - y^2$ factors into $(x+y)(x-y)$, we can simply substitute in to get $x^2 - y^2 = 50 \\cdot 12 = \\boxed{600}$.","The common difference is $5 - 2 = 3$, so the $25^{\\text{th}}$ term is $2 + 3 \\cdot 24 = \\boxed{74}$.","Plugging in $x = 2$, we get $-4$ for the numerator, and $-1$ for the denominator, thus, $\\boxed{4}$ is the answer.","+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1","2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer:","81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: ","l Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer","= 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.7","3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = "," are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are "," is $4-0 = 4$.\nThe coordinates of $D$ are $(x, 0)$.\nThe length of the side $CD$ is given as 28.\nThe coordinates of $A$ are $(0, 0)$.\nThe coordinates of $B$ are $(0, 4)$.\nThe coordinates of $C$ are $(x, 4)$.\nThe length of the side $DA$ is given as 28.\nThe coordinates of $C$ are $("],"return":{"token_logprobs":true,"topk":0}}

{"type":"teacher_forced","model":"edited","texts":["We have that\n\\begin{align*}\na_3 &= \\frac{a_2}{a_1} = \\frac{3}{2}, \\\\\na_4 &= \\frac{a_3}{a_2} = \\frac{3/2}{3} = \\frac{1}{2}, \\\\\na_5 &= \\frac{a_4}{a_3} = \\frac{1/2}{3/2} = \\frac{1}{3}, \\\\\na_6 &= \\frac{a_5}{a_4} = \\frac{1/3}{1/2} = \\frac{2}{3}, \\\\\na_7 &= \\frac{a_6}{a_5} = \\frac{2/3}{1/3} = 2, \\\\\na_8 &= \\frac{a_7}{a_6} = \\frac{2}{2/3} = 3.\n\\end{align*}Since $a_7 = a_1 = 2$ and $a_8 = a_2 = 3,$ and each term depends only on the previous two terms, the sequence becomes periodic at this point, with a period of length 6.  Hence, $a_{2006} = a_2 = \\boxed{3}.$","The midpoint of a line segment with endpoints $(x_1, y_1), (x_2, y_2)$ is $\\left(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}\\right)$.\n\nThus, the midpoint of this line segment is $\\left(\\frac{2+8}{2}, \\frac{3+15}{2} \\right)$, which simplifies to $(5,9)$. Thus, the sum of the coordinates of the midpoint is $\\boxed{14}$.","Let $y = x^2 + x + 3.$  Then we can write the given equation as\n\\[\\frac{4x}{y} + \\frac{5x}{y - 6x} + \\frac{3}{2} = 0.\\]Multiplying everything by $2y(y - 6x),$ we get\n\\[8x(y - 6x) + 10xy + 3y(y - 6x) = 0.\\]Expanding, we get $3y^2 - 48x^2 = 0,$ so $y^2 - 16x^2 = (y - 4x)(y + 4x) = 0.$  Thus, $y = 4x$ or $y = -4x.$\n\nIf $y = 4x,$ then $x^2 + x + 3 = 4x,$ so $x^2 - 3x + 3 = 0.$  This quadratic has no real solutions.\n\nIf $y = -4x,$ then $x^2 + x + 3 = -4x,$ so $x^2 + 5x + 3 = 0.$  This quadratic has two real solutions, giving us a total of $\\boxed{2}$ real solutions.","Simplify, remembering the exponents take priority over multiplication and multiplication takes priority over addition or subtraction. \\begin{align*}\n5^3-3\\times 5^2+3\\times5-1&=125-3\\times25+15-1 \\\\\n&=125-75+14 \\\\\n&= 50+14 \\\\\n&= \\boxed{64}.\n\\end{align*}","We proceed as follows: \\begin{align*}\n(2b + 5)(b - 1) &= 6b\\\\\n2b^2 + 3b - 5 &= 6b\\\\\n2b^2 - 3b - 5 &= 0\\\\\n(2b - 5)(b + 1) &= 0.\n\\end{align*}This gives us $b = \\frac{5}{2}$ or $b = -1.$ Of these, $\\boxed{\\frac{5}{2}}$ is the larger possible value of $b.$","We solve each equation separately. First off, we have $8x^2 + 7x - 1 = (8x-1)(x+1) = 0.$ We can also see that $24x^2+53x-7 = (8x-1)(3x+7) = 0.$ It is clear that both equations are satisfied only when $8x - 1 = 0,$ therefore $x = \\boxed{\\dfrac{1}{8}}.$","Let's substitute the given values into the expression. We get  \\begin{align*}\n(a^3+b^3)\\div(a^2-ab+b^2)&=(5^3+4^3)\\div(5^2-5\\cdot4+4^2)\\\\\n&=(125+64)\\div(25-20+16)\\\\\n&=189\\div21\\\\\n&=\\boxed{9}.\n\\end{align*}","The only numbers that have 11 and 8 as a factor are multiples of 88.  If we list the first few multiples of 88: $$88,176,264,352,...$$ we can see that there are exactly $\\boxed{2}$ between 100 and 300.","Since $3^3 = 3\\times 3\\times 3 = 3\\times 9 = 27$, then  \\[ \\sqrt{3^3+3^3+3^3} = \\sqrt{27+27+27}=\\sqrt{81}=\\boxed{9}. \\]","Let $n$ be any integer that has its square between 4000 and 7000. Then $63 < n < 84$, because $63^2 < 4000<64^2$ and $83^2< 7000<84^2$. Between 63 and 84, the only primes are 67, 71, 73, 79, and 83. Thus, the answer is $\\boxed{5}$.","There are ${9 \\choose 2} = 36$ pairs of plates possible in all. Exactly ${5 \\choose 2} = 10$ of these pairs are both red plates and $\\binom{4}{2}=6$ pairs are both blue plates for a total of $6+10=16$ pairs of plates that satisfy our condition. So, the probability of the pair of plates being the same color is $\\frac{16}{36}=\\boxed{\\frac{4}{9}}$.","Let $B$ and $J$ represent the respective ages of Billy and Joe. We can write the equations $B=2J$ and $B+J=45$. We use the second equation to solve for $J$ in terms of $B$ and get $J=45-B$. Now we plug in this expression for $J$ into the first equation. $$B=2(45-B)=90-2B\\qquad\\Rightarrow 3B=90\\qquad\\Rightarrow B=30$$ So Billy is $\\boxed{30}$ years old.","By AM-HM,\n\\[\\frac{x + y}{2} \\ge \\frac{2}{\\frac{1}{x} + \\frac{1}{y}}.\\]Hence,\n\\[\\frac{1}{x} + \\frac{1}{y} \\ge \\frac{4}{x + y} = \\frac{4}{10} = \\frac{2}{5}.\\]Equality occurs when $x = y = 5,$ so the minimum value is $\\boxed{\\frac{2}{5}}$.","$64^2 - 36^2$ can also be expressed as $(64+36)(64-36)$. This simplifies as $100 \\cdot 28$, which equals $\\boxed{2800}$.","Since the triangle is equilateral, all sides are equal in length. Therefore, the perimeter of the triangle is $8+8+8=8 \\times 3=\\boxed{24}.$","We have two equations and two variables, so it's possible to solve for $p$ and $q$ directly and then calculate $p^2$ and $q^2$ separately to get our answer. However, doing so involves a fair amount of computation with complex numbers and square roots, so we look for an alternative approach. We square the second equation to get $$(p+q)^2 = p^2 + 2pq + q^2 = 36,$$which is close to what we want but has the extra $2pq$ term. Since we know that $pq=9$, we can substitute to get $$p^2 + 2(9) +q^2 = 36 \\implies p^2+q^2 = \\boxed{18}.$$Note that our task was made easier by only solving for what the problem asked rather than by trying to solve for $p$ and $q$ individually.","Triangle $ABC$ has base $AB$ of length $25\\text{ cm}$ and height $AC$ of length $20\\text{ cm}.$ Therefore, the area of triangle $ABC$ is \\begin{align*}\n\\frac{1}{2}bh &= \\frac{1}{2}(25 \\mbox{ cm})(20 \\mbox{ cm}) \\\\\n&= \\frac{1}{2}(500 \\mbox{ cm}^2) \\\\\n&= \\boxed{250} \\mbox{ cm}^2.\n\\end{align*}","We have  \\[\\frac{x^4 + 2y^2}{6} = \\frac{2^4 + 2(5^2)}{6} = \\frac{16+2(25)}{6} = \\frac{16+50}{6} = \\frac{66}{6} = \\boxed{11}.\\]","The average length is $\\frac{1.5+4.5}{2}=\\boxed{3}$ inches.","Her mean score was $\\frac{98+97+92+85+93}{5}=\\boxed{93}$.","We simplify the given inequality by adding $x+3$ to both sides, which leaves us with $5x < 5 \\Rightarrow x<1$. Since 1 is not less than itself, this leaves us with $x=\\boxed{0}$.","We know that any number raised to the power of $0$ is $1$, or $a^0 = 1$ for any $a$. Thus, we get $$(5^{-2})^0 + (5^0)^3 = 1 + 1^3 = 1+1 = \\boxed{2}.$$","Since $2a + 1 = 1,$ we find that $2a = 0,$ so $a = 0.$ Therefore, $b - a = b - 0 = b = \\boxed{1}.$","\\[\n\\begin{array}{c|cc cc}\n\\multicolumn{2}{r}{x^2} & +\\left(\\frac{c-7}{2}\\right)x & +5 \\\\\n\\cline{2-5}\n2x+7 & 2x^3 &+cx^2 &- 11x &+ 39  \\\\\n\\multicolumn{2}{r}{-2x^3} & -7x^2  \\\\ \n\\cline{2-3}\n\\multicolumn{2}{r}{0} & (c-7)x^2 & -11x  \\\\\n\\multicolumn{2}{r}{} & -(c-7)x^2 & -x(c-7)\\left(\\frac{7}{2}\\right)   \\\\ \n\\cline{3-4}\n\\multicolumn{2}{r}{} &  0 & -x\\left(\\frac{7c-27}{2}\\right) & + 39  \\\\\n\\multicolumn{2}{r}{} &   & -10x & -35  \\\\\n\\cline{4-5}\n\\multicolumn{2}{r}{} &   & -x\\left(\\frac{7c-27+20}{2}\\right) & 4  \\\\\n\\end{array}\n\\]In the last step of the division, we have $39$ left as the constant term in our dividend and we need a remainder of $4$ at the end. Since our divisor has a term of $7$, the only way to do this is if our quotient has $5$ which gives us  $7\\cdot5=35$ to subtract from our dividend and get the right remainder.\n\nThen, we need the rest of our remainder to be $0$. This means\n$$\\frac{7c-27+20}{2} = 0$$which gives us\n$$c = \\boxed{1}.$$","Let's begin with the calculator that initially shows 1. Each time it is passed around the circle, it is cubed. 1 to any power is still 1, so no matter how many times 1 is cubed, the final result will still be 1.\n\nNow examine the calculator that starts with a zero. 0 squared is still 0 because 0 to any positive power is still 0. Thus, no matter how many times zero is squared, the final number will still be zero.\n\nFinally, let's look at the calculator that initially shows -1. Each time a person gets the calculator, they negate the number. Because there are 42 participants, there are 42 total turns. Thus, -1 is negated 42 times. Because negating a number is the same as multiplying by -1, this is the same as multiplying it by -1 forty-two times. Thus, we are looking for \\[(-1) \\cdot (-1)^{42}=(-1)^1 \\cdot (-1)^{42}=(-1)^{1+42}=(-1)^{43}.\\]Recall that $(-a)^n=-a^n$ if $n$ is odd. Because 43 is odd, $(-1)^{43}=-1^{43}=-1$.\n\nThus, the sum of all of the numbers is $1+0+(-1)=\\boxed{0}$.","By definition, an obtuse triangle contains one obtuse interior angle. It cannot contain more than one because an obtuse angle has measure greater than 90 degrees and the sum of all the interior angles in any triangle is 180 degrees. Thus, there is $\\boxed{1}$ obtuse angle.","There is one King of $\\diamondsuit$ and 52 cards total, so the probability that the top card is a King of $\\diamondsuit$ is $\\boxed{\\dfrac{1}{52}}$.","We have  \\[81^{3/4} = (3^4)^{3/4} = 3^{4\\cdot (3/4)} = 3^3 = \\boxed{27}.\\]","Since 3 and 4 are relatively prime, their least common multiple is $3\\cdot4=12$.  Therefore, the least common multiple of 30 and 40 is 120.  Since $\\boxed{120}$ is not divisible by 16, it is the smallest common multiple of 30 and 40 which is not divisible by 16.\n\nNote: Every common multiple of two integers is a multiple of their least common multiple.  Therefore, it is not possible that the least common multiple is divisible by 16 but some other common multiple is not.","We have $g(-5) = 5(-5) + 2(-5)^2 = -25 + 50 = 25$, so $f(g(-5)) = f(25) = 3 - \\!\\sqrt{25} = 3-5=\\boxed{-2}$.","We have $55 \\times 1212 - 15 \\times 1212 = 1212(55-15) = 1212(40) = 4848(10) = \\boxed{48480}$.","Solution 1. The first equation gives $x = y+4$. Substituting into the second equation, we get \\[(y+4)^3 - y^3 = 28 \\implies 12y^2 + 48y + 36 = 0.\\]Thus, $y^2 + 4y + 3 = 0$, so $(y+1)(y+3) = 0$. Therefore, either $y=-1$ and $x=y+4=3$, or $y=-3$ and $x=y+4=1$. Either way, $xy = \\boxed{-3}$.\n\nSolution 2. The second equation factors via difference of cubes, as \\[(x-y)(x^2+xy+y^2) = 28.\\]Since $x-y=4$, we have $x^2+xy+y^2=\\frac{28}{4} =7$. Now, squaring the first equation, we get $x^2-2xy+y^2=16$. Thus, \\[3xy = (x^2+xy+y^2) - (x^2-2xy+y^2) = 7-16=-9,\\]so $xy = \\frac{-9}{3} = \\boxed{-3}$.","By the Law of Sines applied to triangle $OAB$, $$\\frac{OB}{\\sin\\angle\nOAB}=\\frac{AB}{\\sin\\angle AOB}.$$With $AB = 1$ and $\\angle AOB = 30^\\circ$, we have \\[\\frac{OB}{\\sin \\angle OAB} = \\frac{1}{\\sin 30^\\circ} = 2,\\]so  so $OB=2\\sin\\angle OAB$. Thus, $OB \\le \\boxed{2}$, with equality if and only if $\\angle OAB=90^\\circ$.\n\n[asy]\nunitsize(1.5 cm);\n\npair O, A, B;\n\nO = (0,0);\nA = sqrt(3)*dir(30);\nB = (2,0);\n\ndraw((0,0)--3*dir(30),Arrow(6));\ndraw((0,0)--(3,0),Arrow(6));\ndraw(A--B);\ndraw(rightanglemark(O,A,B,4));\n\nlabel(\"$A$\", A, NW);\nlabel(\"$B$\", B, S);\nlabel(\"$O$\", O, W);\nlabel(\"$1$\", (A + B)/2, NE, red);\n[/asy]","Adding 14 and 36 yields 50, and then adding 22 yields 72.  Dividing by 3 yields $\\boxed{24}$.","Since $\\mathbf{A} \\mathbf{B} = \\mathbf{B} \\mathbf{A},$\n\\[\\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix}.\\]Expanding, we get\n\\[\\begin{pmatrix} a + 2c & b + 2d \\\\ 3a + 4c & 3b + 4d \\end{pmatrix} = \\begin{pmatrix} a + 3b & 2a + 4b \\\\ c + 3d & 2c + 4d \\end{pmatrix}.\\]Comparing entries, we find $3b = 2c$ and $3a + 3c = 3d,$ so $a + c = d.$  Then\n\\[\\frac{a - d}{c - 3b} = \\frac{-c}{c - 2c} = \\frac{-c}{-c} = \\boxed{1}.\\]","Since $y=1$, we have $1 =\\displaystyle\\frac{1}{3x+1}$.  Multiplying both sides by $3x+1$, we have $$3x+1=1$$ $$\\Rightarrow \\qquad 3x=0$$ $$\\Rightarrow \\qquad x=\\boxed{0}$$","If one centimeter on the map is 15 kilometers in reality, then we have the ratio $1 \\text{ cm on map} : 15 \\text{ km in reality}$. Multiplying the ratio by 88, we get $88 \\text{ cm on map} : 15 \\cdot 88 \\text{ km in reality} =88 \\text{ cm on map} : 1320 \\text{ km in reality}$. Thus, the cities are $\\boxed{1320}$ kilometers apart.","The $x$-coordinate and $y$-coordinate of the midpoint are the averages of the $x$-coordinates and $y$-coordinates of the endpoints, respectively.  Since $4-3=1$, the $x$-coordinate of the other endpoint is $1-3=-2$.  Since $5-7=-2$, the $y$-coordinate of the other endpoint is $-2-7=-9$.  Therefore, the endpoints are $(4,5)$ and $\\boxed{(-2,-9)}$.","We can use complementary probability to determine that the probability of its not raining tomorrow is $1 - \\frac{3}{10} = \\boxed{\\frac{7}{10}}$.","Call the two integers $x$ and $y$. We are given that $x^2 + y^2 = 90$ and that $xy = 27$. We want to find $x + y$. Note that $(x + y)^2 = x^2 + y^2 + 2xy = 90 + 2\\cdot 27 = 144$. Taking the square root of 144, we see that $x + y = \\boxed{12}$.","Let the height of the cylinder be $h$; we then have \\[SA = 2\\pi (2^2)+2\\pi (2)(h) = 12\\pi.\\]Solving yields $4\\pi h = 4 \\pi$ so $h = \\boxed{1}$ foot.","Setting $x = -f(y),$ we get\n\\[f(0) = -f(y) + y,\\]so $f(y) = y - f(0)$ for all real numbers $x.$  Then the given functional equation becomes\n\\[f(x + y - f(0)) = x + y,\\]or $x + y - f(0) - f(0) = x + y.$  Then $f(0) = 0,$ so $f(x) = x$ for all real numbers $x.$  This function does satisfy the given functional equation, giving us $\\boxed{1}$ solution.","$x^2 - y^2$ factors into $(x+y)(x-y)$, so, to obtain the value of $x^2 - y^2$, simply multiply $\\frac{7}{12}\\cdot \\frac{1}{12}$ to get $\\boxed{\\frac{7}{144}}$.","When using the distributive property, we add the products of 3 and $8x^2$, 3 and $-2x$, and 3 and 1: \\begin{align*}\n3(8x^2-2x+1) &= 3\\cdot 8x^2+3\\cdot (-2x) + 3 \\cdot 1\\\\\n&= \\boxed{24x^2-6x+3}\n\\end{align*}","The greatest integer that is less than or equal to $0.999$ is $0$, so $\\lfloor0.999\\rfloor=0$. The smallest integer that is greater than or equal to $2.001$ is $3$, so $\\lceil2.001\\rceil=3$. Therefore, $\\lfloor0.999\\rfloor+\\lceil2.001\\rceil=0+3=\\boxed{3}$.","Shrinking the triangle by dividing every side length by 2, we recognize the resulting set $$\\{7,24,25\\}$$ of side lengths as a Pythagorean triple.  Therefore, the original triangle is also a right triangle, and its legs measure 14 cm and 48 cm.  The area of the triangle is $\\frac{1}{2}(14\\text{ cm})(48\\text{ cm})=\\boxed{336}$ square centimeters.","Dividing both sides by 3, we quickly note that $ 9 = 9^{x-1} \\rightarrow 1 = x-1 \\rightarrow x = \\boxed{2}$.","If $x=-3$, then $-3 = 2-t$, so $t = 5$.  Therefore, $y = 4(5) +7 =\\boxed{27}$.","Choosing a committee is a combination, and order does not matter. We are choosing a 4-person committee from 9 people, so there are $9 \\times 8 \\times 7 \\times 6$ ways to pick the four people for the positions, but then we must divide by $4!$ since order doesn't matter, so the answer is $\\dfrac{9 \\times 8 \\times 7 \\times 6}{4!} =\\boxed{126}$.","Terminating decimals can be expressed as $\\frac{a}{10^b}$. So, we try to get our fraction in this form: $$\\frac{11}{125} = \\frac{11}{5^3} = \\frac{11}{5^3} \\cdot \\frac{2^3}{2^3} = \\frac{11\\cdot2^3}{10^3} = \\frac{88}{1000} = \\boxed{0.088}.$$","We start by completing the square: \\begin{align*}\n2x^2 -12x + 3 &= 2(x^2-6x) +3 \\\\\n&= 2(x^2 -6x + (6/2)^2 - (6/2)^2) + 3\\\\\n& = 2((x-3)^2 -3^2) + 3 \\\\\n&= 2(x-3)^2 - 2\\cdot 3^2 + 3\\\\\n&= 2(x-3)^2 -15\n.\\end{align*} Since the square of a real number is at least 0, we have $(x-3)^2\\ge 0$, where $(x-3)^2 =0$ only if $x=3$.   Therefore, $2(x-3)^2 - 15$ is minimized when $x=\\boxed{3}.$","This can easily be converted into a fraction over a power of ten: \\[\n\\frac{3}{8} = \\frac{3 \\cdot 125}{8 \\cdot 125} = \\frac{375}{1000} = \\boxed{0.375}.\n\\]Note: In general if you have a fraction that is some $\\frac{x}{2^k}$, multiplying the numerator and denominator by $5^k$ will give a power of $10$ in the denominator for easy conversion to a decimal. In this case, $k = 3$.","The sum of the three angles in any triangle is always $180^\\circ.$ In $\\triangle PQR,$ the sum of $\\angle P$ and $\\angle Q$ is $60^\\circ,$ and thus $\\angle R$ must measure $$180^\\circ - 60^\\circ = \\boxed{120^\\circ}.$$","We find the average words per hour by dividing the total words by the total hours. $$\\frac{40,\\!000\\text{ words}}{80\\text{ hours}}=\\frac{4,\\!000}{8}=\\frac{40\\cdot100}{8}=5\\cdot100=\\boxed{500} \\text{ words per hour}$$","The common factor of 3 in the numerator and the common factor of 3 in the denominator will cancel: \\[\n\\frac{3^4+3^2}{3^3-3}=\\frac{3(3^3+3^1)}{3(3^2-1)}=\\frac{3^3+3^1}{3^2-1}\n\\] Now the numerator is $3^3+3=27+3=30$, and the denominator is $3^2-1=9-1=8$.  Therefore, the fraction simplifies to $\\dfrac{30}{8}=\\boxed{\\dfrac{15}{4}}$.","A cone with radius $r$ and height $h$ has volume $\\frac{1}{3}\\pi r^2 h$; here, our cone has volume $\\frac{1}{3}\\pi (1^2)(4)=\\frac{4}{3}\\pi$.  A sphere with radius $r$ has volume $\\frac{4}{3}\\pi r^3$, so we set up the equation \\[\\frac{4}{3}\\pi r^3=\\frac{4}{3}\\pi.\\] Solving for $r$ yields $r^3=1$, so $r = 1$.  The sphere's radius is $\\boxed{1}$ inch.","From the angle addition formula, we can write\n\\begin{align*}\n\\sin x + \\cos x &= \\sqrt{2} \\left( \\frac{1}{\\sqrt{2}} \\sin x + \\frac{1}{\\sqrt{2}} \\cos x \\right) \\\\\n&= \\sqrt{2} \\left( \\cos \\frac{\\pi}{4} \\sin x + \\sin \\frac{\\pi}{4} \\cos x \\right) \\\\\n&= \\sqrt{2} \\sin \\left( x + \\frac{\\pi}{4} \\right).\n\\end{align*}Thus, the graph of $y = \\sin x + \\cos x$ has period $\\boxed{2 \\pi}.$\n\nThe graph of $y = \\sin x + \\cos x$ is shown below:\n\n[asy]import TrigMacros;\n\nsize(400);\n\nreal g(real x)\n{\n\treturn sin(x) + cos(x);\n}\n\ndraw(graph(g,-3*pi,3*pi,n=700,join=operator ..),red);\ntrig_axes(-3*pi,3*pi,-2,2,pi/2,1);\nlayer();\nrm_trig_labels(-5, 5, 2);\n[/asy]","We can expand the determinant as follows:\n\\begin{align*}\n\\begin{vmatrix} 0 & \\sin \\alpha & -\\cos \\alpha \\\\ -\\sin \\alpha & 0 & \\sin \\beta \\\\ \\cos \\alpha & -\\sin \\beta & 0 \\end{vmatrix} &= -\\sin \\alpha \\begin{vmatrix} -\\sin \\alpha & \\sin \\beta \\\\ \\cos \\alpha & 0 \\end{vmatrix} - \\cos \\alpha \\begin{vmatrix} -\\sin \\alpha & 0 \\\\ \\cos \\alpha & -\\sin \\beta \\end{vmatrix} \\\\\n&= -\\sin \\alpha (-\\sin \\beta \\cos \\alpha) - \\cos \\alpha (\\sin \\alpha \\sin \\beta) \\\\\n&= \\boxed{0}.\n\\end{align*}","Plotting the points, we see that the dimensions of the rectangle are $x$ and $4$.  The area of the rectangle is $(\\text{length})(\\text{width})=4x$, so $4x=28$ and $x=\\boxed{7}$. [asy]\nsize(5cm);\nimport graph;\ndefaultpen(linewidth(0.7)+fontsize(12));\nreal x = 7;\npair A=(0,0), B=(0,4), C=(x,4), D=(x,0);\npair[] dots = {A,B,C,D};\ndot(dots);\ndraw(A--B--C--D--cycle);\nxaxis(-2,9,Arrows(4));\nyaxis(-2,7,Arrows(4));\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,NW);\nlabel(\"$C$\",C,NE);\nlabel(\"$D$\",D,SE);\nlabel(\"$x$\",(B+C)/2,N);\nlabel(\"$4$\",(C+D)/2,E);[/asy]","We know that $\\left(\\frac{13}{7}\\right)^2=\\frac{169}{49}$. Then, since $3=\\frac{147}{49}<\\frac{169}{49}<\\frac{196}{49}=4$, we conclude that $\\left\\lceil\\left(\\frac{13}{7}\\right)^2\\right\\rceil=4$. Because $4+\\frac{17}{4}=\\frac{33}{4}$, which is a number between $8$ and $9$, $\\left\\lfloor \\left\\lceil \\left(\\frac{13}{7}\\right)^2\\right\\rceil+\\frac{17}{4}\\right\\rfloor=\\boxed{8}$.","The height of the cone is $\\frac{3}{4} \\times 8 = 6$. The radius of the cone is $\\frac{8}{2} = 4$. Therefore, the volume of the cone formed is $\\frac{4^2 \\times 6 \\times \\pi}{3} = \\boxed{32 \\pi}$.","Call the two integers $x$ and $y$. Without loss of generality, let $x$ be the larger of the two. We are given that $x+y = 50$ and $x-y = 12$, and we are asked for $x^2 - y^2$. Because $x^2 - y^2$ factors into $(x+y)(x-y)$, we can simply substitute in to get $x^2 - y^2 = 50 \\cdot 12 = \\boxed{600}$.","The common difference is $5 - 2 = 3$, so the $25^{\\text{th}}$ term is $2 + 3 \\cdot 24 = \\boxed{74}$.","Plugging in $x = 2$, we get $-4$ for the numerator, and $-1$ for the denominator, thus, $\\boxed{4}$ is the answer.","+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1","2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer: 4+ y^2\n\nFinal Answer:","81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: 81^3\n\nFinal Answer: ","l Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer: 0\nFinal Answer","= 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.732 + 1) = 3(0.3656 - 0.7","3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = 3^3-3 = "," are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are 1 real solution.\n\nFinal Answer: 1\nTherefore, there are "," is $4-0 = 4$.\nThe coordinates of $D$ are $(x, 0)$.\nThe length of the side $CD$ is given as 28.\nThe coordinates of $A$ are $(0, 0)$.\nThe coordinates of $B$ are $(0, 4)$.\nThe coordinates of $C$ are $(x, 4)$.\nThe length of the side $DA$ is given as 28.\nThe coordinates of $C$ are $("],"return":{"token_logprobs":true,"topk":0}}

Forward pass 4 of 10 done (teacher_forced); results in /work/fwd_4/. Summary:

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Forward passes left: 6. Edits used in this pass: model.layers.14.mlp.down_proj.weight, model.layers.17.mlp.down_proj.weight, model.layers.8.mlp.down_proj.weight (edit hash 291cd7d115).

The full ablation changes local probabilities only modestly: valid-reference scores are largely preserved, while effects on the eight loop excerpts are mixed. That does not establish a general anti-repetition mechanism, so the hypothesis remains tentative. I’ll compare only matching, unpadded token positions—the manually submitted +1 excerpts differ in length—and then test full-strength ablation on DEV generation. Improved answer accuracy, not merely changed wording or shorter responses, will determine whether it is useful.

python

with open(wdir / "fwd_4" / "meta.json") as f:    meta4 = json.load(f)lengths4 = np.array(meta4["lengths"])ids4 = np.load(wdir / "fwd_4" / "input_ids.npy")lp4 = np.load(wdir / "fwd_4" / "token_logprobs.npy")tf_delta = []tf_delta_late = []for i in range(72):    common_len = min(lengths3[i], lengths4[i])    equal_positions = ids3[i, :common_len] == ids4[i, :common_len]    first_diff = np.where(~equal_positions)[0]    if len(first_diff):        common_len = int(first_diff[0])    assert common_len > 16    d = lp4[i, :common_len-1].astype(np.float64) - lp3[i, :common_len-1]    tf_delta.append(float(d.mean()))    tf_delta_late.append(float(d[min(16, len(d)//2):].mean()))tf_delta = np.array(tf_delta)tf_delta_late = np.array(tf_delta_late)print("Reference mean delta logprob/token:", round(float(tf_delta[:64].mean()), 6),      "SE across references:", round(float(tf_delta[:64].std(ddof=1)/8), 6))print("Reference late-token delta:", round(float(tf_delta_late[:64].mean()), 6))print("Loop mean delta:", round(float(tf_delta[64:].mean()), 6))print("Loop late-token deltas:", [round(v, 6) for v in tf_delta_late[64:]])print("References improved:", int((tf_delta[:64] > 0).sum()), "/64")print("Loop probabilities reduced:", int((tf_delta[64:] < 0).sum()), "/8")tf_summary = {    "reference_delta_mean": float(tf_delta[:64].mean()),    "reference_delta_se": float(tf_delta[:64].std(ddof=1)/8),    "loop_delta_mean": float(tf_delta[64:].mean()),    "loop_late_delta": tf_delta_late[64:].tolist(),    "interpretation": "Small, mixed local effects; no general loop-suppression claim."}(wdir / "teacher_forced_comparison.json").write_text(json.dumps(tf_summary, indent=2))with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 4 findings\n" + json.dumps(tf_summary) + "\n"            "Matching unpadded prefixes only; pass3 and pass4 +1 excerpts have different lengths.\n"            "\n## Pass 5 planned: full paired DEV generation\n"            "Exact edit: down_proj columns 8:[699],14:[483,984],17:[1179] zeroed (strength1.0).\n"            "Set: all 64 DEV.json problems. model=both, n=16 (1024 answers/model), max_tokens8192.\n"            "Measure accuracy first, then repeated conclusions, repeated lines and cutoffs.\n")print("Prepared pass5: both, DEV64, n16, max8192; active strength1.0")

with open(wdir / "fwd_4" / "meta.json") as f:    meta4 = json.load(f)lengths4 = np.array(meta4["lengths"])ids4 = np.load(wdir / "fwd_4" / "input_ids.npy")lp4 = np.load(wdir / "fwd_4" / "token_logprobs.npy")tf_delta = []tf_delta_late = []for i in range(72):    common_len = min(lengths3[i], lengths4[i])    equal_positions = ids3[i, :common_len] == ids4[i, :common_len]    first_diff = np.where(~equal_positions)[0]    if len(first_diff):        common_len = int(first_diff[0])    assert common_len > 16    d = lp4[i, :common_len-1].astype(np.float64) - lp3[i, :common_len-1]    tf_delta.append(float(d.mean()))    tf_delta_late.append(float(d[min(16, len(d)//2):].mean()))tf_delta = np.array(tf_delta)tf_delta_late = np.array(tf_delta_late)print("Reference mean delta logprob/token:", round(float(tf_delta[:64].mean()), 6),      "SE across references:", round(float(tf_delta[:64].std(ddof=1)/8), 6))print("Reference late-token delta:", round(float(tf_delta_late[:64].mean()), 6))print("Loop mean delta:", round(float(tf_delta[64:].mean()), 6))print("Loop late-token deltas:", [round(v, 6) for v in tf_delta_late[64:]])print("References improved:", int((tf_delta[:64] > 0).sum()), "/64")print("Loop probabilities reduced:", int((tf_delta[64:] < 0).sum()), "/8")tf_summary = {    "reference_delta_mean": float(tf_delta[:64].mean()),    "reference_delta_se": float(tf_delta[:64].std(ddof=1)/8),    "loop_delta_mean": float(tf_delta[64:].mean()),    "loop_late_delta": tf_delta_late[64:].tolist(),    "interpretation": "Small, mixed local effects; no general loop-suppression claim."}(wdir / "teacher_forced_comparison.json").write_text(json.dumps(tf_summary, indent=2))with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 4 findings\n" + json.dumps(tf_summary) + "\n"            "Matching unpadded prefixes only; pass3 and pass4 +1 excerpts have different lengths.\n"            "\n## Pass 5 planned: full paired DEV generation\n"            "Exact edit: down_proj columns 8:[699],14:[483,984],17:[1179] zeroed (strength1.0).\n"            "Set: all 64 DEV.json problems. model=both, n=16 (1024 answers/model), max_tokens8192.\n"            "Measure accuracy first, then repeated conclusions, repeated lines and cutoffs.\n")print("Prepared pass5: both, DEV64, n16, max8192; active strength1.0")

OUTPUT of Python run 14 of at most 300:

Reference mean delta logprob/token: 0.002872 SE across references: 0.002752Reference late-token delta: -0.005325Loop mean delta: -0.00038Loop late-token deltas: [np.float64(-0.00024), np.float64(-0.011226), np.float64(-0.013049), np.float64(-0.044571), np.float64(-0.000131), np.float64(0.002149), np.float64(-0.049525), np.float64(0.011617)]References improved: 35 /64Loop probabilities reduced: 5 /8Prepared pass5: both, DEV64, n16, max8192; active strength1.0

Reference mean delta logprob/token: 0.002872 SE across references: 0.002752Reference late-token delta: -0.005325Loop mean delta: -0.00038Loop late-token deltas: [np.float64(-0.00024), np.float64(-0.011226), np.float64(-0.013049), np.float64(-0.044571), np.float64(-0.000131), np.float64(0.002149), np.float64(-0.049525), np.float64(0.011617)]References improved: 35 /64Loop probabilities reduced: 5 /8Prepared pass5: both, DEV64, n16, max8192; active strength1.0

On matched tokens, reference log probability changes by only +0.0029 nats per token, with a standard error of 0.0028 across references. Later tokens in several repeated-conclusion excerpts become less likely, but not every loop does, so the evidence supports at most a narrow repetition-related effect. I’ll now run the full paired DEV comparison with 1,024 answers per model and the 8,192-token limit. A positive accuracy difference would justify further strength testing; fewer repetitions without better answers would not.

forward

{"type":"generate","model":"both","problems":["math_train_intermediate_algebra_847","math_train_algebra_988","math_train_intermediate_algebra_342","math_train_prealgebra_311","math_train_algebra_1185","math_train_algebra_1520","math_train_prealgebra_490","math_train_number_theory_693","math_train_prealgebra_1049","math_train_number_theory_716","math_train_counting_and_probability_55","math_train_algebra_1497","math_train_intermediate_algebra_802","math_train_algebra_1220","math_train_prealgebra_783","math_train_algebra_583","math_train_prealgebra_593","math_train_algebra_7","math_train_prealgebra_556","math_train_prealgebra_795","math_train_prealgebra_397","math_train_prealgebra_869","math_train_prealgebra_818","math_train_intermediate_algebra_1008","math_train_prealgebra_502","math_train_prealgebra_82","math_train_prealgebra_1130","math_train_algebra_1284","math_train_number_theory_709","math_train_algebra_920","math_train_algebra_198","math_train_intermediate_algebra_84","math_train_precalculus_76","math_train_prealgebra_590","math_train_precalculus_330","math_train_algebra_763","math_train_prealgebra_1028","math_train_algebra_1210","math_train_counting_and_probability_653","math_train_algebra_380","math_train_geometry_113","math_train_intermediate_algebra_1197","math_train_algebra_1128","math_train_algebra_448","math_train_algebra_1559","math_train_geometry_386","math_train_algebra_817","math_train_prealgebra_1033","math_train_counting_and_probability_518","math_train_number_theory_706","math_train_algebra_34","math_train_prealgebra_134","math_train_prealgebra_867","math_train_algebra_1094","math_train_algebra_1491","math_train_geometry_259","math_train_precalculus_274","math_train_precalculus_233","math_train_geometry_161","math_train_algebra_1586","math_train_geometry_305","math_train_algebra_1180","math_train_algebra_775","math_train_algebra_485"],"n":16,"max_tokens":8192}

{"type":"generate","model":"both","problems":["math_train_intermediate_algebra_847","math_train_algebra_988","math_train_intermediate_algebra_342","math_train_prealgebra_311","math_train_algebra_1185","math_train_algebra_1520","math_train_prealgebra_490","math_train_number_theory_693","math_train_prealgebra_1049","math_train_number_theory_716","math_train_counting_and_probability_55","math_train_algebra_1497","math_train_intermediate_algebra_802","math_train_algebra_1220","math_train_prealgebra_783","math_train_algebra_583","math_train_prealgebra_593","math_train_algebra_7","math_train_prealgebra_556","math_train_prealgebra_795","math_train_prealgebra_397","math_train_prealgebra_869","math_train_prealgebra_818","math_train_intermediate_algebra_1008","math_train_prealgebra_502","math_train_prealgebra_82","math_train_prealgebra_1130","math_train_algebra_1284","math_train_number_theory_709","math_train_algebra_920","math_train_algebra_198","math_train_intermediate_algebra_84","math_train_precalculus_76","math_train_prealgebra_590","math_train_precalculus_330","math_train_algebra_763","math_train_prealgebra_1028","math_train_algebra_1210","math_train_counting_and_probability_653","math_train_algebra_380","math_train_geometry_113","math_train_intermediate_algebra_1197","math_train_algebra_1128","math_train_algebra_448","math_train_algebra_1559","math_train_geometry_386","math_train_algebra_817","math_train_prealgebra_1033","math_train_counting_and_probability_518","math_train_number_theory_706","math_train_algebra_34","math_train_prealgebra_134","math_train_prealgebra_867","math_train_algebra_1094","math_train_algebra_1491","math_train_geometry_259","math_train_precalculus_274","math_train_precalculus_233","math_train_geometry_161","math_train_algebra_1586","math_train_geometry_305","math_train_algebra_1180","math_train_algebra_775","math_train_algebra_485"],"n":16,"max_tokens":8192}

Forward pass 5 of 10 done (generate); results in /work/gen_5_original.jsonl and /work/gen_5_edited.jsonl. Summary:

{ "ok": true, "type": "generate", "model": "both", "files": {  "original": "/work/gen_5_original.jsonl",  "edited": "/work/gen_5_edited.jsonl" }, "activations": null, "original": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.404296875,  "accuracy_by_source": {   "math": 0.4043  },  "mean_tokens": 1306,  "cut_share": 0.0762,  "failures": {   "wrong_answer": 286,   "no_answer_finished": 258,   "cut_at_max_tokens": 66  } }, "edited": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.3974609375,  "accuracy_by_source": {   "math": 0.3975  },  "mean_tokens": 1804,  "cut_share": 0.1318,  "failures": {   "wrong_answer": 344,   "no_answer_finished": 174,   "cut_at_max_tokens": 99  } }, "paired": {  "items": 64,  "accuracy_difference": -0.0068,  "standard_error": 0.0164,  "samples_wrong_to_right": 127,  "samples_right_to_wrong": 134 }, "s": 365.2}

{ "ok": true, "type": "generate", "model": "both", "files": {  "original": "/work/gen_5_original.jsonl",  "edited": "/work/gen_5_edited.jsonl" }, "activations": null, "original": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.404296875,  "accuracy_by_source": {   "math": 0.4043  },  "mean_tokens": 1306,  "cut_share": 0.0762,  "failures": {   "wrong_answer": 286,   "no_answer_finished": 258,   "cut_at_max_tokens": 66  } }, "edited": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.3974609375,  "accuracy_by_source": {   "math": 0.3975  },  "mean_tokens": 1804,  "cut_share": 0.1318,  "failures": {   "wrong_answer": 344,   "no_answer_finished": 174,   "cut_at_max_tokens": 99  } }, "paired": {  "items": 64,  "accuracy_difference": -0.0068,  "standard_error": 0.0164,  "samples_wrong_to_right": 127,  "samples_right_to_wrong": 134 }, "s": 365.2}

Forward passes left: 5. Edits used in this pass: model.layers.14.mlp.down_proj.weight, model.layers.17.mlp.down_proj.weight, model.layers.8.mlp.down_proj.weight (edit hash 291cd7d115).

Full ablation did not improve DEV accuracy: the paired difference was −0.68 percentage points with a 1.64-point standard error. More importantly for the mechanism, it increased average response length and nearly doubled the cutoff rate, contradicting the predicted suppression of loops. The channels may be part of a compensating response that restrains repetition, rather than causing it. I’ll measure the behavioral change more directly and test the opposite intervention—doubling those same four output columns—while continuing to judge success by accuracy.

python

def read_generation(pass_no, model_label):    with open(wdir / f"gen_{pass_no}_{model_label}.jsonl") as f:        records = [json.loads(line) for line in f]    return records
def flatten_generation(records):    return [dict(s, problem_id=r["id"], sample_idx=i)            for r in records for i, s in enumerate(r["samples"])]
def paired_stats(pass_no):    original_records = read_generation(pass_no, "original")    edited_records = read_generation(pass_no, "edited")    oo = flatten_generation(original_records)    ee = flatten_generation(edited_records)    oo = sorted(oo, key=lambda s: (s["problem_id"], s["sample_idx"]))    ee = sorted(ee, key=lambda s: (s["problem_id"], s["sample_idx"]))    assert [(s["problem_id"], s["sample_idx"]) for s in oo] == [        (s["problem_id"], s["sample_idx"]) for s in ee]    d = np.array([int(e["correct"]) - int(o["correct"]) for o, e in zip(oo, ee)])    out = {"pass": pass_no, "n": len(d), "gain_pp": float(100*d.mean()),           "se_pp": float(100*d.std(ddof=1)/np.sqrt(len(d))),           "wrong_to_right": int((d == 1).sum()), "right_to_wrong": int((d == -1).sum())}    for label, ss in [("original", oo), ("edited", ee)]:        out[label] = {            "accuracy": float(np.mean([s["correct"] for s in ss])),            "mean_tokens": float(np.mean([s["tokens"] for s in ss])),            "cut_rate": float(np.mean([s["cut"] for s in ss])),            "repeat_fraction": float(np.mean([repeat_fraction(s["text"]) for s in ss])),            "three_or_more_final_answers": float(np.mean([                len(re.findall(r"final answer", s["text"], flags=re.I)) >= 3 for s in ss]))        }    return out, oo, ee, d
stats5, orig5, edit5, diff5 = paired_stats(5)print(json.dumps(stats5, indent=2))(wdir / "stats_pass5.json").write_text(json.dumps(stats5, indent=2))shutil.copytree(wdir / "edited", wdir / "best_1", dirs_exist_ok=True)(wdir / "best_1_spec.json").write_text(json.dumps(edit_spec, indent=2))edit_spec = apply_channel_edit(-1.0)(wdir / "current_edit.json").write_text(json.dumps(edit_spec, indent=2))shutil.copytree(wdir / "edited", wdir / "candidate_enhance_2x", dirs_exist_ok=True)with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 5 results\n" + json.dumps(stats5) + "\n"            "Ablation increased response length/cutoffs; proposed anti-loop direction was wrong.\n"            "Revised hypothesis: selected channels help restrain repetitive conclusions; "            "their correlation with failures may reflect compensation.\n"            "\n## Pass 6 planned: opposite-direction causal test\n"            "Exact edit: down_proj columns 8:[699],14:[483,984],17:[1179] multiplied by2 "            "(attenuation strength -1.0). All other values unchanged.\n"            "Set: same fixed DEV64, both models, n16, max_tokens8192 (1024/model).\n"            "Prediction: shorter loops/fewer cutoffs than contemporaneous original; "            "accuracy remains primary endpoint.\n")print("Next edit:", edit_spec)

def read_generation(pass_no, model_label):    with open(wdir / f"gen_{pass_no}_{model_label}.jsonl") as f:        records = [json.loads(line) for line in f]    return records
def flatten_generation(records):    return [dict(s, problem_id=r["id"], sample_idx=i)            for r in records for i, s in enumerate(r["samples"])]
def paired_stats(pass_no):    original_records = read_generation(pass_no, "original")    edited_records = read_generation(pass_no, "edited")    oo = flatten_generation(original_records)    ee = flatten_generation(edited_records)    oo = sorted(oo, key=lambda s: (s["problem_id"], s["sample_idx"]))    ee = sorted(ee, key=lambda s: (s["problem_id"], s["sample_idx"]))    assert [(s["problem_id"], s["sample_idx"]) for s in oo] == [        (s["problem_id"], s["sample_idx"]) for s in ee]    d = np.array([int(e["correct"]) - int(o["correct"]) for o, e in zip(oo, ee)])    out = {"pass": pass_no, "n": len(d), "gain_pp": float(100*d.mean()),           "se_pp": float(100*d.std(ddof=1)/np.sqrt(len(d))),           "wrong_to_right": int((d == 1).sum()), "right_to_wrong": int((d == -1).sum())}    for label, ss in [("original", oo), ("edited", ee)]:        out[label] = {            "accuracy": float(np.mean([s["correct"] for s in ss])),            "mean_tokens": float(np.mean([s["tokens"] for s in ss])),            "cut_rate": float(np.mean([s["cut"] for s in ss])),            "repeat_fraction": float(np.mean([repeat_fraction(s["text"]) for s in ss])),            "three_or_more_final_answers": float(np.mean([                len(re.findall(r"final answer", s["text"], flags=re.I)) >= 3 for s in ss]))        }    return out, oo, ee, d
stats5, orig5, edit5, diff5 = paired_stats(5)print(json.dumps(stats5, indent=2))(wdir / "stats_pass5.json").write_text(json.dumps(stats5, indent=2))shutil.copytree(wdir / "edited", wdir / "best_1", dirs_exist_ok=True)(wdir / "best_1_spec.json").write_text(json.dumps(edit_spec, indent=2))edit_spec = apply_channel_edit(-1.0)(wdir / "current_edit.json").write_text(json.dumps(edit_spec, indent=2))shutil.copytree(wdir / "edited", wdir / "candidate_enhance_2x", dirs_exist_ok=True)with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 5 results\n" + json.dumps(stats5) + "\n"            "Ablation increased response length/cutoffs; proposed anti-loop direction was wrong.\n"            "Revised hypothesis: selected channels help restrain repetitive conclusions; "            "their correlation with failures may reflect compensation.\n"            "\n## Pass 6 planned: opposite-direction causal test\n"            "Exact edit: down_proj columns 8:[699],14:[483,984],17:[1179] multiplied by2 "            "(attenuation strength -1.0). All other values unchanged.\n"            "Set: same fixed DEV64, both models, n16, max_tokens8192 (1024/model).\n"            "Prediction: shorter loops/fewer cutoffs than contemporaneous original; "            "accuracy remains primary endpoint.\n")print("Next edit:", edit_spec)

OUTPUT of Python run 15 of at most 300:

{  "pass": 5,  "n": 1024,  "gain_pp": -0.68359375,  "se_pp": 1.578311217628527,  "wrong_to_right": 127,  "right_to_wrong": 134,  "original": {    "accuracy": 0.404296875,    "mean_tokens": 1305.57421875,    "cut_rate": 0.076171875,    "repeat_fraction": 0.14803693883151464,    "three_or_more_final_answers": 0.1357421875  },  "edited": {    "accuracy": 0.3974609375,    "mean_tokens": 1803.5283203125,    "cut_rate": 0.1318359375,    "repeat_fraction": 0.20659392353500058,    "three_or_more_final_answers": 0.2255859375  }}Next edit: {'kind': 'sparse_mlp_output_attenuation', 'strength': -1.0, 'targets': {'8': [699], '14': [483, 984], '17': [1179]}, 'tensors': ['model.layers.8.mlp.down_proj.weight', 'model.layers.14.mlp.down_proj.weight', 'model.layers.17.mlp.down_proj.weight']}

{  "pass": 5,  "n": 1024,  "gain_pp": -0.68359375,  "se_pp": 1.578311217628527,  "wrong_to_right": 127,  "right_to_wrong": 134,  "original": {    "accuracy": 0.404296875,    "mean_tokens": 1305.57421875,    "cut_rate": 0.076171875,    "repeat_fraction": 0.14803693883151464,    "three_or_more_final_answers": 0.1357421875  },  "edited": {    "accuracy": 0.3974609375,    "mean_tokens": 1803.5283203125,    "cut_rate": 0.1318359375,    "repeat_fraction": 0.20659392353500058,    "three_or_more_final_answers": 0.2255859375  }}Next edit: {'kind': 'sparse_mlp_output_attenuation', 'strength': -1.0, 'targets': {'8': [699], '14': [483, 984], '17': [1179]}, 'tensors': ['model.layers.8.mlp.down_proj.weight', 'model.layers.14.mlp.down_proj.weight', 'model.layers.17.mlp.down_proj.weight']}

The detailed counts confirm the contradiction: ablation raised repeated-line frequency from 14.8% to 20.7%, and answers with at least three “Final Answer” phrases from 13.6% to 22.6%. This is stronger evidence that the selected channel group restrains repetitive conclusions than that it generates them. I’ve now doubled only those four columns to test the reverse causal prediction. If repetition falls but accuracy does not rise, I’ll report that distinction rather than calling it a mathematical improvement.

forward

{"type":"generate","model":"both","problems":["math_train_intermediate_algebra_847","math_train_algebra_988","math_train_intermediate_algebra_342","math_train_prealgebra_311","math_train_algebra_1185","math_train_algebra_1520","math_train_prealgebra_490","math_train_number_theory_693","math_train_prealgebra_1049","math_train_number_theory_716","math_train_counting_and_probability_55","math_train_algebra_1497","math_train_intermediate_algebra_802","math_train_algebra_1220","math_train_prealgebra_783","math_train_algebra_583","math_train_prealgebra_593","math_train_algebra_7","math_train_prealgebra_556","math_train_prealgebra_795","math_train_prealgebra_397","math_train_prealgebra_869","math_train_prealgebra_818","math_train_intermediate_algebra_1008","math_train_prealgebra_502","math_train_prealgebra_82","math_train_prealgebra_1130","math_train_algebra_1284","math_train_number_theory_709","math_train_algebra_920","math_train_algebra_198","math_train_intermediate_algebra_84","math_train_precalculus_76","math_train_prealgebra_590","math_train_precalculus_330","math_train_algebra_763","math_train_prealgebra_1028","math_train_algebra_1210","math_train_counting_and_probability_653","math_train_algebra_380","math_train_geometry_113","math_train_intermediate_algebra_1197","math_train_algebra_1128","math_train_algebra_448","math_train_algebra_1559","math_train_geometry_386","math_train_algebra_817","math_train_prealgebra_1033","math_train_counting_and_probability_518","math_train_number_theory_706","math_train_algebra_34","math_train_prealgebra_134","math_train_prealgebra_867","math_train_algebra_1094","math_train_algebra_1491","math_train_geometry_259","math_train_precalculus_274","math_train_precalculus_233","math_train_geometry_161","math_train_algebra_1586","math_train_geometry_305","math_train_algebra_1180","math_train_algebra_775","math_train_algebra_485"],"n":16,"max_tokens":8192}

{"type":"generate","model":"both","problems":["math_train_intermediate_algebra_847","math_train_algebra_988","math_train_intermediate_algebra_342","math_train_prealgebra_311","math_train_algebra_1185","math_train_algebra_1520","math_train_prealgebra_490","math_train_number_theory_693","math_train_prealgebra_1049","math_train_number_theory_716","math_train_counting_and_probability_55","math_train_algebra_1497","math_train_intermediate_algebra_802","math_train_algebra_1220","math_train_prealgebra_783","math_train_algebra_583","math_train_prealgebra_593","math_train_algebra_7","math_train_prealgebra_556","math_train_prealgebra_795","math_train_prealgebra_397","math_train_prealgebra_869","math_train_prealgebra_818","math_train_intermediate_algebra_1008","math_train_prealgebra_502","math_train_prealgebra_82","math_train_prealgebra_1130","math_train_algebra_1284","math_train_number_theory_709","math_train_algebra_920","math_train_algebra_198","math_train_intermediate_algebra_84","math_train_precalculus_76","math_train_prealgebra_590","math_train_precalculus_330","math_train_algebra_763","math_train_prealgebra_1028","math_train_algebra_1210","math_train_counting_and_probability_653","math_train_algebra_380","math_train_geometry_113","math_train_intermediate_algebra_1197","math_train_algebra_1128","math_train_algebra_448","math_train_algebra_1559","math_train_geometry_386","math_train_algebra_817","math_train_prealgebra_1033","math_train_counting_and_probability_518","math_train_number_theory_706","math_train_algebra_34","math_train_prealgebra_134","math_train_prealgebra_867","math_train_algebra_1094","math_train_algebra_1491","math_train_geometry_259","math_train_precalculus_274","math_train_precalculus_233","math_train_geometry_161","math_train_algebra_1586","math_train_geometry_305","math_train_algebra_1180","math_train_algebra_775","math_train_algebra_485"],"n":16,"max_tokens":8192}

Forward pass 6 of 10 done (generate); results in /work/gen_6_original.jsonl and /work/gen_6_edited.jsonl. Summary:

{ "ok": true, "type": "generate", "model": "both", "files": {  "original": "/work/gen_6_original.jsonl",  "edited": "/work/gen_6_edited.jsonl" }, "activations": null, "original": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.419921875,  "accuracy_by_source": {   "math": 0.4199  },  "mean_tokens": 1127,  "cut_share": 0.0537,  "failures": {   "wrong_answer": 290,   "no_answer_finished": 258,   "cut_at_max_tokens": 46  } }, "edited": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.4072265625,  "accuracy_by_source": {   "math": 0.4072  },  "mean_tokens": 1043,  "cut_share": 0.0488,  "failures": {   "wrong_answer": 266,   "no_answer_finished": 303,   "cut_at_max_tokens": 38  } }, "paired": {  "items": 64,  "accuracy_difference": -0.0127,  "standard_error": 0.0142,  "samples_wrong_to_right": 117,  "samples_right_to_wrong": 130 }, "s": 157.5}

{ "ok": true, "type": "generate", "model": "both", "files": {  "original": "/work/gen_6_original.jsonl",  "edited": "/work/gen_6_edited.jsonl" }, "activations": null, "original": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.419921875,  "accuracy_by_source": {   "math": 0.4199  },  "mean_tokens": 1127,  "cut_share": 0.0537,  "failures": {   "wrong_answer": 290,   "no_answer_finished": 258,   "cut_at_max_tokens": 46  } }, "edited": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.4072265625,  "accuracy_by_source": {   "math": 0.4072  },  "mean_tokens": 1043,  "cut_share": 0.0488,  "failures": {   "wrong_answer": 266,   "no_answer_finished": 303,   "cut_at_max_tokens": 38  } }, "paired": {  "items": 64,  "accuracy_difference": -0.0127,  "standard_error": 0.0142,  "samples_wrong_to_right": 117,  "samples_right_to_wrong": 130 }, "s": 157.5}

Forward passes left: 4. Edits used in this pass: model.layers.14.mlp.down_proj.weight, model.layers.17.mlp.down_proj.weight, model.layers.8.mlp.down_proj.weight (edit hash 37ddaeae2f).

Doubling the channels moved response length and cutoffs in the predicted direction, but accuracy was still slightly worse: −1.27 percentage points, with a 1.42-point reported standard error. Thus, the behavioral effect has not translated into demonstrated mathematical improvement. I’ll test a gentler 1.5× enhancement as the final DEV candidate, then choose using paired accuracy and freeze the weights before HOLDOUT. The remaining three passes will confirm that frozen choice without further tuning.

python

stats6, orig6, edit6, diff6 = paired_stats(6)print(json.dumps(stats6, indent=2))(wdir / "stats_pass6.json").write_text(json.dumps(stats6, indent=2))
for pass_no, oo, dd in [(5, orig5, diff5), (6, orig6, diff6)]:    clustered = collections.defaultdict(list)    for s, delta in zip(oo, dd):        clustered[s["problem_id"]].append(float(delta))    problem_diffs = np.array([np.mean(v) for v in clustered.values()])    print("Pass", pass_no, "problem-cluster SE pp:",          round(float(100 * problem_diffs.std(ddof=1) / np.sqrt(len(problem_diffs))), 4))
edit_spec = apply_channel_edit(-0.5)(wdir / "current_edit.json").write_text(json.dumps(edit_spec, indent=2))shutil.copytree(wdir / "edited", wdir / "candidate_enhance_1p5x", dirs_exist_ok=True)# Predeclare three confirmation orders; each contains exactly the same untouched HOLDOUT.holdout_orders = [holdout_ids, list(reversed(holdout_ids)),                  holdout_ids[85:] + holdout_ids[:85]](wdir / "holdout_confirmation_orders.json").write_text(json.dumps(holdout_orders, indent=2))with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 6 results\n" + json.dumps(stats6) + "\n"            "2x enhancement reduced mean length/cutoffs but did not improve accuracy.\n"            "\n## Pass 7 planned: final DEV strength test\n"            "Exact edit: down_proj columns 8:[699],14:[483,984],17:[1179] multiplied by1.5 "            "(attenuation strength -0.5). All other values unchanged.\n"            "Set: fixed DEV64, both models, n16, max_tokens8192.\n"            "After pass7 choose among tested edits by paired DEV accuracy difference; "            "freeze before any HOLDOUT output is seen.\n"            "Passes8–10 reserved for frozen-candidate HOLDOUT: same fixed 256 problems, n4 each, "            "max8192. Predeclared input orders in holdout_confirmation_orders.json avoid "            "identical-request rejection. Pool all confirmation results; no retuning.\n")print("Next edit:", edit_spec)

stats6, orig6, edit6, diff6 = paired_stats(6)print(json.dumps(stats6, indent=2))(wdir / "stats_pass6.json").write_text(json.dumps(stats6, indent=2))
for pass_no, oo, dd in [(5, orig5, diff5), (6, orig6, diff6)]:    clustered = collections.defaultdict(list)    for s, delta in zip(oo, dd):        clustered[s["problem_id"]].append(float(delta))    problem_diffs = np.array([np.mean(v) for v in clustered.values()])    print("Pass", pass_no, "problem-cluster SE pp:",          round(float(100 * problem_diffs.std(ddof=1) / np.sqrt(len(problem_diffs))), 4))
edit_spec = apply_channel_edit(-0.5)(wdir / "current_edit.json").write_text(json.dumps(edit_spec, indent=2))shutil.copytree(wdir / "edited", wdir / "candidate_enhance_1p5x", dirs_exist_ok=True)# Predeclare three confirmation orders; each contains exactly the same untouched HOLDOUT.holdout_orders = [holdout_ids, list(reversed(holdout_ids)),                  holdout_ids[85:] + holdout_ids[:85]](wdir / "holdout_confirmation_orders.json").write_text(json.dumps(holdout_orders, indent=2))with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 6 results\n" + json.dumps(stats6) + "\n"            "2x enhancement reduced mean length/cutoffs but did not improve accuracy.\n"            "\n## Pass 7 planned: final DEV strength test\n"            "Exact edit: down_proj columns 8:[699],14:[483,984],17:[1179] multiplied by1.5 "            "(attenuation strength -0.5). All other values unchanged.\n"            "Set: fixed DEV64, both models, n16, max_tokens8192.\n"            "After pass7 choose among tested edits by paired DEV accuracy difference; "            "freeze before any HOLDOUT output is seen.\n"            "Passes8–10 reserved for frozen-candidate HOLDOUT: same fixed 256 problems, n4 each, "            "max8192. Predeclared input orders in holdout_confirmation_orders.json avoid "            "identical-request rejection. Pool all confirmation results; no retuning.\n")print("Next edit:", edit_spec)

OUTPUT of Python run 16 of at most 300:

{  "pass": 6,  "n": 1024,  "gain_pp": -1.26953125,  "se_pp": 1.5350253102900164,  "wrong_to_right": 117,  "right_to_wrong": 130,  "original": {    "accuracy": 0.419921875,    "mean_tokens": 1127.3798828125,    "cut_rate": 0.0537109375,    "repeat_fraction": 0.13534705331679775,    "three_or_more_final_answers": 0.1337890625  },  "edited": {    "accuracy": 0.4072265625,    "mean_tokens": 1042.6298828125,    "cut_rate": 0.048828125,    "repeat_fraction": 0.1269435861051612,    "three_or_more_final_answers": 0.056640625  }}Pass 5 problem-cluster SE pp: 1.6418Pass 6 problem-cluster SE pp: 1.4208Next edit: {'kind': 'sparse_mlp_output_attenuation', 'strength': -0.5, 'targets': {'8': [699], '14': [483, 984], '17': [1179]}, 'tensors': ['model.layers.8.mlp.down_proj.weight', 'model.layers.14.mlp.down_proj.weight', 'model.layers.17.mlp.down_proj.weight']}

{  "pass": 6,  "n": 1024,  "gain_pp": -1.26953125,  "se_pp": 1.5350253102900164,  "wrong_to_right": 117,  "right_to_wrong": 130,  "original": {    "accuracy": 0.419921875,    "mean_tokens": 1127.3798828125,    "cut_rate": 0.0537109375,    "repeat_fraction": 0.13534705331679775,    "three_or_more_final_answers": 0.1337890625  },  "edited": {    "accuracy": 0.4072265625,    "mean_tokens": 1042.6298828125,    "cut_rate": 0.048828125,    "repeat_fraction": 0.1269435861051612,    "three_or_more_final_answers": 0.056640625  }}Pass 5 problem-cluster SE pp: 1.6418Pass 6 problem-cluster SE pp: 1.4208Next edit: {'kind': 'sparse_mlp_output_attenuation', 'strength': -0.5, 'targets': {'8': [699], '14': [483, 984], '17': [1179]}, 'tensors': ['model.layers.8.mlp.down_proj.weight', 'model.layers.14.mlp.down_proj.weight', 'model.layers.17.mlp.down_proj.weight']}

The opposite-direction test substantially reduced repeated conclusions: answers with at least three “Final Answer” phrases fell from 13.4% to 5.7%. Together with the ablation result, that supports a causal role for this channel group in restraining repeated conclusions, though not yet an accuracy benefit. I’ll now test 1.5× enhancement on the same full DEV set. This is the last tuning pass; selection will use paired accuracy rather than the repetition metric.

forward

{"type":"generate","model":"both","problems":["math_train_intermediate_algebra_847","math_train_algebra_988","math_train_intermediate_algebra_342","math_train_prealgebra_311","math_train_algebra_1185","math_train_algebra_1520","math_train_prealgebra_490","math_train_number_theory_693","math_train_prealgebra_1049","math_train_number_theory_716","math_train_counting_and_probability_55","math_train_algebra_1497","math_train_intermediate_algebra_802","math_train_algebra_1220","math_train_prealgebra_783","math_train_algebra_583","math_train_prealgebra_593","math_train_algebra_7","math_train_prealgebra_556","math_train_prealgebra_795","math_train_prealgebra_397","math_train_prealgebra_869","math_train_prealgebra_818","math_train_intermediate_algebra_1008","math_train_prealgebra_502","math_train_prealgebra_82","math_train_prealgebra_1130","math_train_algebra_1284","math_train_number_theory_709","math_train_algebra_920","math_train_algebra_198","math_train_intermediate_algebra_84","math_train_precalculus_76","math_train_prealgebra_590","math_train_precalculus_330","math_train_algebra_763","math_train_prealgebra_1028","math_train_algebra_1210","math_train_counting_and_probability_653","math_train_algebra_380","math_train_geometry_113","math_train_intermediate_algebra_1197","math_train_algebra_1128","math_train_algebra_448","math_train_algebra_1559","math_train_geometry_386","math_train_algebra_817","math_train_prealgebra_1033","math_train_counting_and_probability_518","math_train_number_theory_706","math_train_algebra_34","math_train_prealgebra_134","math_train_prealgebra_867","math_train_algebra_1094","math_train_algebra_1491","math_train_geometry_259","math_train_precalculus_274","math_train_precalculus_233","math_train_geometry_161","math_train_algebra_1586","math_train_geometry_305","math_train_algebra_1180","math_train_algebra_775","math_train_algebra_485"],"n":16,"max_tokens":8192}

{"type":"generate","model":"both","problems":["math_train_intermediate_algebra_847","math_train_algebra_988","math_train_intermediate_algebra_342","math_train_prealgebra_311","math_train_algebra_1185","math_train_algebra_1520","math_train_prealgebra_490","math_train_number_theory_693","math_train_prealgebra_1049","math_train_number_theory_716","math_train_counting_and_probability_55","math_train_algebra_1497","math_train_intermediate_algebra_802","math_train_algebra_1220","math_train_prealgebra_783","math_train_algebra_583","math_train_prealgebra_593","math_train_algebra_7","math_train_prealgebra_556","math_train_prealgebra_795","math_train_prealgebra_397","math_train_prealgebra_869","math_train_prealgebra_818","math_train_intermediate_algebra_1008","math_train_prealgebra_502","math_train_prealgebra_82","math_train_prealgebra_1130","math_train_algebra_1284","math_train_number_theory_709","math_train_algebra_920","math_train_algebra_198","math_train_intermediate_algebra_84","math_train_precalculus_76","math_train_prealgebra_590","math_train_precalculus_330","math_train_algebra_763","math_train_prealgebra_1028","math_train_algebra_1210","math_train_counting_and_probability_653","math_train_algebra_380","math_train_geometry_113","math_train_intermediate_algebra_1197","math_train_algebra_1128","math_train_algebra_448","math_train_algebra_1559","math_train_geometry_386","math_train_algebra_817","math_train_prealgebra_1033","math_train_counting_and_probability_518","math_train_number_theory_706","math_train_algebra_34","math_train_prealgebra_134","math_train_prealgebra_867","math_train_algebra_1094","math_train_algebra_1491","math_train_geometry_259","math_train_precalculus_274","math_train_precalculus_233","math_train_geometry_161","math_train_algebra_1586","math_train_geometry_305","math_train_algebra_1180","math_train_algebra_775","math_train_algebra_485"],"n":16,"max_tokens":8192}

Forward pass 7 of 10 done (generate); results in /work/gen_7_original.jsonl and /work/gen_7_edited.jsonl. Summary:

{ "ok": true, "type": "generate", "model": "both", "files": {  "original": "/work/gen_7_original.jsonl",  "edited": "/work/gen_7_edited.jsonl" }, "activations": null, "original": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.4033203125,  "accuracy_by_source": {   "math": 0.4033  },  "mean_tokens": 1270,  "cut_share": 0.0703,  "failures": {   "wrong_answer": 289,   "no_answer_finished": 265,   "cut_at_max_tokens": 57  } }, "edited": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.408203125,  "accuracy_by_source": {   "math": 0.4082  },  "mean_tokens": 1122,  "cut_share": 0.0566,  "failures": {   "wrong_answer": 270,   "no_answer_finished": 291,   "cut_at_max_tokens": 45  } }, "paired": {  "items": 64,  "accuracy_difference": 0.0049,  "standard_error": 0.0172,  "samples_wrong_to_right": 137,  "samples_right_to_wrong": 132 }, "s": 286.7}

{ "ok": true, "type": "generate", "model": "both", "files": {  "original": "/work/gen_7_original.jsonl",  "edited": "/work/gen_7_edited.jsonl" }, "activations": null, "original": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.4033203125,  "accuracy_by_source": {   "math": 0.4033  },  "mean_tokens": 1270,  "cut_share": 0.0703,  "failures": {   "wrong_answer": 289,   "no_answer_finished": 265,   "cut_at_max_tokens": 57  } }, "edited": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.408203125,  "accuracy_by_source": {   "math": 0.4082  },  "mean_tokens": 1122,  "cut_share": 0.0566,  "failures": {   "wrong_answer": 270,   "no_answer_finished": 291,   "cut_at_max_tokens": 45  } }, "paired": {  "items": 64,  "accuracy_difference": 0.0049,  "standard_error": 0.0172,  "samples_wrong_to_right": 137,  "samples_right_to_wrong": 132 }, "s": 286.7}

Forward passes left: 3. Edits used in this pass: model.layers.14.mlp.down_proj.weight, model.layers.17.mlp.down_proj.weight, model.layers.8.mlp.down_proj.weight (edit hash 599aba54ec).

The 1.5× edit is the best tested DEV candidate: +0.49 percentage points in paired accuracy, with a 1.72-point standard error, so the gain is not statistically conclusive. It also reduces response length and cutoffs, consistent with the revised repetition-control hypothesis. I’ll freeze this exact edit now and back it up before examining any HOLDOUT output. The final three passes will use the predeclared HOLDOUT orders, and I will pool their results without changing the candidate.

python

stats7, orig7, edit7, diff7 = paired_stats(7)(wdir / "stats_pass7.json").write_text(json.dumps(stats7, indent=2))dev_stats = [stats5, stats6, stats7]chosen = max(dev_stats, key=lambda s: s["gain_pp"])assert chosen["pass"] == 7assert json.loads((wdir / "current_edit.json").read_text())["strength"] == -0.5for layer, columns in target_channels.items():    original_w = load_original(target_names[layer])    current_w = load(target_names[layer])    mask_columns = torch.ones(2048, dtype=torch.bool)    mask_columns[columns] = False    assert torch.equal(current_w[:, mask_columns], original_w[:, mask_columns])    assert torch.allclose(current_w[:, columns], 1.5 * original_w[:, columns])shutil.copytree(wdir / "edited", wdir / "best_2", dirs_exist_ok=True)shutil.copytree(wdir / "edited", wdir / "frozen_candidate", dirs_exist_ok=True)
import hashlibdef edit_checksums():    return {str(p.relative_to(wdir / "edited")): hashlib.sha256(p.read_bytes()).hexdigest()            for p in sorted((wdir / "edited").rglob("*")) if p.is_file()}frozen_checksums = edit_checksums()freeze_manifest = {    "edit": edit_spec, "selected_on": "DEV only",    "selected_pass": 7, "dev_gain_pp": stats7["gain_pp"],    "dev_reported_cluster_se_pp": 1.72,    "forward_edit_hash": "599aba54ec",    "files_sha256": frozen_checksums,    "holdout_plan": {"passes": [8, 9, 10], "problems": 256, "n_per_pass": 4,                     "max_tokens": 8192, "orders_file": "holdout_confirmation_orders.json"}}(wdir / "FROZEN_MANIFEST.json").write_text(json.dumps(freeze_manifest, indent=2))with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 7 results and FINAL CANDIDATE FREEZE\n" + json.dumps(stats7) + "\n"            "Selected pass7 (1.5x) by highest paired DEV accuracy difference among passes5–7.\n"            "DEV gain +0.48828125 pp, reported problem-cluster SE1.72 pp; not significant.\n"            "Final exact edit: model.layers.8.mlp.down_proj.weight column699, "            "model.layers.14.mlp.down_proj.weight columns483,984, "            "model.layers.17.mlp.down_proj.weight column1179 multiplied by1.5 from ORIGINAL values.\n"            "All other entries identical. Frozen hash599aba54ec, copies in best_2/ and frozen_candidate/.\n"            "DO NOT TUNE OR SELECT USING HOLDOUT. Submit this frozen candidate regardless of confirmation sign.\n"            "\n## Pass 8 planned\n"            "Exact edit: frozen1.5x four-column enhancement above; no further changes.\n"            "Set: HOLDOUT.json, all256 reserved MATH problems, first predeclared order.\n"            "both, n4, max_tokens8192, 1024 answers/model. Pool with planned passes9 and10.\n")print("DEV gains (percentage points):", [(s["pass"], round(s["gain_pp"], 3)) for s in dev_stats])print("Frozen edit:", edit_spec)print("HOLDOUT first order:")print(json.dumps(holdout_orders[0]))

stats7, orig7, edit7, diff7 = paired_stats(7)(wdir / "stats_pass7.json").write_text(json.dumps(stats7, indent=2))dev_stats = [stats5, stats6, stats7]chosen = max(dev_stats, key=lambda s: s["gain_pp"])assert chosen["pass"] == 7assert json.loads((wdir / "current_edit.json").read_text())["strength"] == -0.5for layer, columns in target_channels.items():    original_w = load_original(target_names[layer])    current_w = load(target_names[layer])    mask_columns = torch.ones(2048, dtype=torch.bool)    mask_columns[columns] = False    assert torch.equal(current_w[:, mask_columns], original_w[:, mask_columns])    assert torch.allclose(current_w[:, columns], 1.5 * original_w[:, columns])shutil.copytree(wdir / "edited", wdir / "best_2", dirs_exist_ok=True)shutil.copytree(wdir / "edited", wdir / "frozen_candidate", dirs_exist_ok=True)
import hashlibdef edit_checksums():    return {str(p.relative_to(wdir / "edited")): hashlib.sha256(p.read_bytes()).hexdigest()            for p in sorted((wdir / "edited").rglob("*")) if p.is_file()}frozen_checksums = edit_checksums()freeze_manifest = {    "edit": edit_spec, "selected_on": "DEV only",    "selected_pass": 7, "dev_gain_pp": stats7["gain_pp"],    "dev_reported_cluster_se_pp": 1.72,    "forward_edit_hash": "599aba54ec",    "files_sha256": frozen_checksums,    "holdout_plan": {"passes": [8, 9, 10], "problems": 256, "n_per_pass": 4,                     "max_tokens": 8192, "orders_file": "holdout_confirmation_orders.json"}}(wdir / "FROZEN_MANIFEST.json").write_text(json.dumps(freeze_manifest, indent=2))with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 7 results and FINAL CANDIDATE FREEZE\n" + json.dumps(stats7) + "\n"            "Selected pass7 (1.5x) by highest paired DEV accuracy difference among passes5–7.\n"            "DEV gain +0.48828125 pp, reported problem-cluster SE1.72 pp; not significant.\n"            "Final exact edit: model.layers.8.mlp.down_proj.weight column699, "            "model.layers.14.mlp.down_proj.weight columns483,984, "            "model.layers.17.mlp.down_proj.weight column1179 multiplied by1.5 from ORIGINAL values.\n"            "All other entries identical. Frozen hash599aba54ec, copies in best_2/ and frozen_candidate/.\n"            "DO NOT TUNE OR SELECT USING HOLDOUT. Submit this frozen candidate regardless of confirmation sign.\n"            "\n## Pass 8 planned\n"            "Exact edit: frozen1.5x four-column enhancement above; no further changes.\n"            "Set: HOLDOUT.json, all256 reserved MATH problems, first predeclared order.\n"            "both, n4, max_tokens8192, 1024 answers/model. Pool with planned passes9 and10.\n")print("DEV gains (percentage points):", [(s["pass"], round(s["gain_pp"], 3)) for s in dev_stats])print("Frozen edit:", edit_spec)print("HOLDOUT first order:")print(json.dumps(holdout_orders[0]))

OUTPUT of Python run 17 of at most 300:

DEV gains (percentage points): [(5, -0.684), (6, -1.27), (7, 0.488)]Frozen edit: {'kind': 'sparse_mlp_output_attenuation', 'strength': -0.5, 'targets': {'8': [699], '14': [483, 984], '17': [1179]}, 'tensors': ['model.layers.8.mlp.down_proj.weight', 'model.layers.14.mlp.down_proj.weight', 'model.layers.17.mlp.down_proj.weight']}HOLDOUT first order:["math_train_intermediate_algebra_719", "math_train_algebra_1236", "math_train_geometry_569", "math_train_prealgebra_666", "math_train_geometry_402", "math_train_prealgebra_909", "math_train_geometry_272", "math_train_prealgebra_269", "math_train_algebra_796", "math_train_intermediate_algebra_56", "math_train_prealgebra_426", "math_train_number_theory_532", "math_train_algebra_1042", "math_train_geometry_811", "math_train_number_theory_668", "math_train_number_theory_466", "math_train_precalculus_165", "math_train_algebra_279", "math_train_geometry_778", "math_train_algebra_793", "math_train_algebra_659", "math_train_prealgebra_246", "math_train_number_theory_578", "math_train_number_theory_47", "math_train_counting_and_probability_621", "math_train_algebra_75", "math_train_intermediate_algebra_702", "math_train_number_theory_468", "math_train_precalculus_700", "math_train_algebra_1547", "math_train_algebra_1034", "math_train_geometry_273", "math_train_precalculus_432", "math_train_intermediate_algebra_92", "math_train_number_theory_450", "math_train_intermediate_algebra_462", "math_train_algebra_326", "math_train_geometry_212", "math_train_algebra_156", "math_train_algebra_393", "math_train_counting_and_probability_142", "math_train_geometry_187", "math_train_prealgebra_988", "math_train_precalculus_476", "math_train_counting_and_probability_199", "math_train_precalculus_681", "math_train_number_theory_685", "math_train_algebra_810", "math_train_prealgebra_44", "math_train_intermediate_algebra_37", "math_train_geometry_371", "math_train_counting_and_probability_247", "math_train_prealgebra_326", "math_train_number_theory_61", "math_train_prealgebra_201", "math_train_geometry_41", "math_train_prealgebra_1", "math_train_intermediate_algebra_1055", "math_train_prealgebra_876", "math_train_intermediate_algebra_182", "math_train_counting_and_probability_193", "math_train_precalculus_675", "math_train_prealgebra_637", "math_train_number_theory_85", "math_train_intermediate_algebra_517", "math_train_intermediate_algebra_1262", "math_train_geometry_248", "math_train_counting_and_probability_582", "math_train_intermediate_algebra_851", "math_train_algebra_1614", "math_train_intermediate_algebra_795", "math_train_precalculus_29", "math_train_prealgebra_231", "math_train_algebra_491", "math_train_geometry_294", "math_train_geometry_516", "math_train_number_theory_425", "math_train_algebra_589", "math_train_intermediate_algebra_459", "math_train_counting_and_probability_321", "math_train_number_theory_24", "math_train_counting_and_probability_286", "math_train_geometry_203", "math_train_intermediate_algebra_326", "math_train_intermediate_algebra_1246", "math_train_prealgebra_247", "math_train_algebra_19", "math_train_intermediate_algebra_512", "math_train_algebra_1606", "math_train_number_theory_860", "math_train_geometry_729", "math_train_intermediate_algebra_696", "math_train_number_theory_297", "math_train_intermediate_algebra_831", "math_train_geometry_0", "math_train_intermediate_algebra_221", "math_train_precalculus_650", "math_train_algebra_1415", "math_train_prealgebra_630", "math_train_number_theory_163", "math_train_number_theory_13", "math_train_number_theory_564", "math_train_algebra_121", "math_train_algebra_630", "math_train_intermediate_algebra_688", "math_train_prealgebra_120", "math_train_prealgebra_135", "math_train_number_theory_368", "math_train_prealgebra_743", "math_train_counting_and_probability_613", "math_train_algebra_547", "math_train_precalculus_77", "math_train_algebra_1734", "math_train_counting_and_probability_349", "math_train_intermediate_algebra_534", "math_train_precalculus_425", "math_train_prealgebra_1115", "math_train_precalculus_475", "math_train_counting_and_probability_197", "math_train_counting_and_probability_339", "math_train_precalculus_55", "math_train_counting_and_probability_178", "math_train_algebra_586", "math_train_intermediate_algebra_208", "math_train_counting_and_probability_52", "math_train_algebra_403", "math_train_geometry_806", "math_train_number_theory_95", "math_train_algebra_962", "math_train_algebra_967", "math_train_intermediate_algebra_294", "math_train_prealgebra_872", "math_train_algebra_1105", "math_train_intermediate_algebra_547", "math_train_precalculus_20", "math_train_algebra_1666", "math_train_algebra_1333", "math_train_algebra_1293", "math_train_number_theory_458", "math_train_algebra_1535", "math_train_geometry_72", "math_train_algebra_1340", "math_train_intermediate_algebra_181", "math_train_precalculus_154", "math_train_intermediate_algebra_252", "math_train_algebra_1515", "math_train_prealgebra_766", "math_train_intermediate_algebra_306", "math_train_algebra_1189", "math_train_counting_and_probability_176", "math_train_geometry_868", "math_train_precalculus_202", "math_train_algebra_723", "math_train_number_theory_784", "math_train_precalculus_439", "math_train_prealgebra_285", "math_train_prealgebra_1151", "math_train_algebra_1038", "math_train_number_theory_240", "math_train_algebra_298", "math_train_algebra_869", "math_train_intermediate_algebra_129", "math_train_geometry_373", "math_train_intermediate_algebra_299", "math_train_prealgebra_1133", "math_train_number_theory_569", "math_train_intermediate_algebra_482", "math_train_geometry_669", "math_train_algebra_1265", "math_train_prealgebra_45", "math_train_algebra_1070", "math_train_intermediate_algebra_681", "math_train_precalculus_178", "math_train_algebra_1462", "math_train_algebra_981", "math_train_precalculus_628", "math_train_precalculus_84", "math_train_intermediate_algebra_511", "math_train_prealgebra_831", "math_train_intermediate_algebra_358", "math_train_intermediate_algebra_941", "math_train_number_theory_498", "math_train_prealgebra_597", "math_train_precalculus_188", "math_train_number_theory_786", "math_train_geometry_63", "math_train_prealgebra_541", "math_train_prealgebra_128", "math_train_algebra_1280", "math_train_algebra_1174", "math_train_number_theory_303", "math_train_prealgebra_471", "math_train_intermediate_algebra_871", "math_train_geometry_664", "math_train_intermediate_algebra_532", "math_train_algebra_738", "math_train_algebra_1444", "math_train_prealgebra_407", "math_train_algebra_710", "math_train_number_theory_104", "math_train_intermediate_algebra_856", "math_train_counting_and_probability_304", "math_train_intermediate_algebra_1281", "math_train_intermediate_algebra_775", "math_train_intermediate_algebra_1012", "math_train_algebra_650", "math_train_algebra_368", "math_train_counting_and_probability_96", "math_train_counting_and_probability_140", "math_train_number_theory_718", "math_train_counting_and_probability_128", "math_train_intermediate_algebra_763", "math_train_intermediate_algebra_348", "math_train_prealgebra_913", "math_train_number_theory_127", "math_train_prealgebra_316", "math_train_counting_and_probability_695", "math_train_prealgebra_121", "math_train_algebra_777", "math_train_precalculus_347", "math_train_prealgebra_901", "math_train_algebra_1436", "math_train_prealgebra_618", "math_train_geometry_232", "math_train_geometry_723", "math_train_geometry_731", "math_train_algebra_1384", "math_train_prealgebra_423", "math_train_intermediate_algebra_1094", "math_train_geometry_839", "math_train_precalculus_590", "math_train_algebra_1025", "math_train_intermediate_algebra_565", "math_train_algebra_821", "math_train_counting_and_probability_664", "math_train_prealgebra_739", "math_train_precalculus_448", "math_train_counting_and_probability_129", "math_train_counting_and_probability_648", "math_train_counting_and_probability_607", "math_train_prealgebra_1012", "math_train_prealgebra_182", "math_train_number_theory_238", "math_train_intermediate_algebra_1019", "math_train_prealgebra_228", "math_train_algebra_1125", "math_train_algebra_1364", "math_train_algebra_184", "math_train_precalculus_158", "math_train_algebra_513", "math_train_algebra_29", "math_train_counting_and_probability_564", "math_train_algebra_1247", "math_train_prealgebra_834", "math_train_counting_and_probability_433", "math_train_number_theory_108"]

DEV gains (percentage points): [(5, -0.684), (6, -1.27), (7, 0.488)]Frozen edit: {'kind': 'sparse_mlp_output_attenuation', 'strength': -0.5, 'targets': {'8': [699], '14': [483, 984], '17': [1179]}, 'tensors': ['model.layers.8.mlp.down_proj.weight', 'model.layers.14.mlp.down_proj.weight', 'model.layers.17.mlp.down_proj.weight']}HOLDOUT first order:["math_train_intermediate_algebra_719", "math_train_algebra_1236", "math_train_geometry_569", "math_train_prealgebra_666", "math_train_geometry_402", "math_train_prealgebra_909", "math_train_geometry_272", "math_train_prealgebra_269", "math_train_algebra_796", "math_train_intermediate_algebra_56", "math_train_prealgebra_426", "math_train_number_theory_532", "math_train_algebra_1042", "math_train_geometry_811", "math_train_number_theory_668", "math_train_number_theory_466", "math_train_precalculus_165", "math_train_algebra_279", "math_train_geometry_778", "math_train_algebra_793", "math_train_algebra_659", "math_train_prealgebra_246", "math_train_number_theory_578", "math_train_number_theory_47", "math_train_counting_and_probability_621", "math_train_algebra_75", "math_train_intermediate_algebra_702", "math_train_number_theory_468", "math_train_precalculus_700", "math_train_algebra_1547", "math_train_algebra_1034", "math_train_geometry_273", "math_train_precalculus_432", "math_train_intermediate_algebra_92", "math_train_number_theory_450", "math_train_intermediate_algebra_462", "math_train_algebra_326", "math_train_geometry_212", "math_train_algebra_156", "math_train_algebra_393", "math_train_counting_and_probability_142", "math_train_geometry_187", "math_train_prealgebra_988", "math_train_precalculus_476", "math_train_counting_and_probability_199", "math_train_precalculus_681", "math_train_number_theory_685", "math_train_algebra_810", "math_train_prealgebra_44", "math_train_intermediate_algebra_37", "math_train_geometry_371", "math_train_counting_and_probability_247", "math_train_prealgebra_326", "math_train_number_theory_61", "math_train_prealgebra_201", "math_train_geometry_41", "math_train_prealgebra_1", "math_train_intermediate_algebra_1055", "math_train_prealgebra_876", "math_train_intermediate_algebra_182", "math_train_counting_and_probability_193", "math_train_precalculus_675", "math_train_prealgebra_637", "math_train_number_theory_85", "math_train_intermediate_algebra_517", "math_train_intermediate_algebra_1262", "math_train_geometry_248", "math_train_counting_and_probability_582", "math_train_intermediate_algebra_851", "math_train_algebra_1614", "math_train_intermediate_algebra_795", "math_train_precalculus_29", "math_train_prealgebra_231", "math_train_algebra_491", "math_train_geometry_294", "math_train_geometry_516", "math_train_number_theory_425", "math_train_algebra_589", "math_train_intermediate_algebra_459", "math_train_counting_and_probability_321", "math_train_number_theory_24", "math_train_counting_and_probability_286", "math_train_geometry_203", "math_train_intermediate_algebra_326", "math_train_intermediate_algebra_1246", "math_train_prealgebra_247", "math_train_algebra_19", "math_train_intermediate_algebra_512", "math_train_algebra_1606", "math_train_number_theory_860", "math_train_geometry_729", "math_train_intermediate_algebra_696", "math_train_number_theory_297", "math_train_intermediate_algebra_831", "math_train_geometry_0", "math_train_intermediate_algebra_221", "math_train_precalculus_650", "math_train_algebra_1415", "math_train_prealgebra_630", "math_train_number_theory_163", "math_train_number_theory_13", "math_train_number_theory_564", "math_train_algebra_121", "math_train_algebra_630", "math_train_intermediate_algebra_688", "math_train_prealgebra_120", "math_train_prealgebra_135", "math_train_number_theory_368", "math_train_prealgebra_743", "math_train_counting_and_probability_613", "math_train_algebra_547", "math_train_precalculus_77", "math_train_algebra_1734", "math_train_counting_and_probability_349", "math_train_intermediate_algebra_534", "math_train_precalculus_425", "math_train_prealgebra_1115", "math_train_precalculus_475", "math_train_counting_and_probability_197", "math_train_counting_and_probability_339", "math_train_precalculus_55", "math_train_counting_and_probability_178", "math_train_algebra_586", "math_train_intermediate_algebra_208", "math_train_counting_and_probability_52", "math_train_algebra_403", "math_train_geometry_806", "math_train_number_theory_95", "math_train_algebra_962", "math_train_algebra_967", "math_train_intermediate_algebra_294", "math_train_prealgebra_872", "math_train_algebra_1105", "math_train_intermediate_algebra_547", "math_train_precalculus_20", "math_train_algebra_1666", "math_train_algebra_1333", "math_train_algebra_1293", "math_train_number_theory_458", "math_train_algebra_1535", "math_train_geometry_72", "math_train_algebra_1340", "math_train_intermediate_algebra_181", "math_train_precalculus_154", "math_train_intermediate_algebra_252", "math_train_algebra_1515", "math_train_prealgebra_766", "math_train_intermediate_algebra_306", "math_train_algebra_1189", "math_train_counting_and_probability_176", "math_train_geometry_868", "math_train_precalculus_202", "math_train_algebra_723", "math_train_number_theory_784", "math_train_precalculus_439", "math_train_prealgebra_285", "math_train_prealgebra_1151", "math_train_algebra_1038", "math_train_number_theory_240", "math_train_algebra_298", "math_train_algebra_869", "math_train_intermediate_algebra_129", "math_train_geometry_373", "math_train_intermediate_algebra_299", "math_train_prealgebra_1133", "math_train_number_theory_569", "math_train_intermediate_algebra_482", "math_train_geometry_669", "math_train_algebra_1265", "math_train_prealgebra_45", "math_train_algebra_1070", "math_train_intermediate_algebra_681", "math_train_precalculus_178", "math_train_algebra_1462", "math_train_algebra_981", "math_train_precalculus_628", "math_train_precalculus_84", "math_train_intermediate_algebra_511", "math_train_prealgebra_831", "math_train_intermediate_algebra_358", "math_train_intermediate_algebra_941", "math_train_number_theory_498", "math_train_prealgebra_597", "math_train_precalculus_188", "math_train_number_theory_786", "math_train_geometry_63", "math_train_prealgebra_541", "math_train_prealgebra_128", "math_train_algebra_1280", "math_train_algebra_1174", "math_train_number_theory_303", "math_train_prealgebra_471", "math_train_intermediate_algebra_871", "math_train_geometry_664", "math_train_intermediate_algebra_532", "math_train_algebra_738", "math_train_algebra_1444", "math_train_prealgebra_407", "math_train_algebra_710", "math_train_number_theory_104", "math_train_intermediate_algebra_856", "math_train_counting_and_probability_304", "math_train_intermediate_algebra_1281", "math_train_intermediate_algebra_775", "math_train_intermediate_algebra_1012", "math_train_algebra_650", "math_train_algebra_368", "math_train_counting_and_probability_96", "math_train_counting_and_probability_140", "math_train_number_theory_718", "math_train_counting_and_probability_128", "math_train_intermediate_algebra_763", "math_train_intermediate_algebra_348", "math_train_prealgebra_913", "math_train_number_theory_127", "math_train_prealgebra_316", "math_train_counting_and_probability_695", "math_train_prealgebra_121", "math_train_algebra_777", "math_train_precalculus_347", "math_train_prealgebra_901", "math_train_algebra_1436", "math_train_prealgebra_618", "math_train_geometry_232", "math_train_geometry_723", "math_train_geometry_731", "math_train_algebra_1384", "math_train_prealgebra_423", "math_train_intermediate_algebra_1094", "math_train_geometry_839", "math_train_precalculus_590", "math_train_algebra_1025", "math_train_intermediate_algebra_565", "math_train_algebra_821", "math_train_counting_and_probability_664", "math_train_prealgebra_739", "math_train_precalculus_448", "math_train_counting_and_probability_129", "math_train_counting_and_probability_648", "math_train_counting_and_probability_607", "math_train_prealgebra_1012", "math_train_prealgebra_182", "math_train_number_theory_238", "math_train_intermediate_algebra_1019", "math_train_prealgebra_228", "math_train_algebra_1125", "math_train_algebra_1364", "math_train_algebra_184", "math_train_precalculus_158", "math_train_algebra_513", "math_train_algebra_29", "math_train_counting_and_probability_564", "math_train_algebra_1247", "math_train_prealgebra_834", "math_train_counting_and_probability_433", "math_train_number_theory_108"]

The 1.5× four-column edit is frozen and verified: every non-targeted entry is unchanged, and a checksum-backed copy is saved. I’ll now run the first confirmation pass on all 256 reserved HOLDOUT problems, with four answers per model per problem. These problems were not used to choose channels or strengths. A pooled positive accuracy difference would support generalization; a result near zero or below zero would limit the claim, regardless of reduced repetition.

forward

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{"type":"generate","model":"both","problems":["math_train_intermediate_algebra_719","math_train_algebra_1236","math_train_geometry_569","math_train_prealgebra_666","math_train_geometry_402","math_train_prealgebra_909","math_train_geometry_272","math_train_prealgebra_269","math_train_algebra_796","math_train_intermediate_algebra_56","math_train_prealgebra_426","math_train_number_theory_532","math_train_algebra_1042","math_train_geometry_811","math_train_number_theory_668","math_train_number_theory_466","math_train_precalculus_165","math_train_algebra_279","math_train_geometry_778","math_train_algebra_793","math_train_algebra_659","math_train_prealgebra_246","math_train_number_theory_578","math_train_number_theory_47","math_train_counting_and_probability_621","math_train_algebra_75","math_train_intermediate_algebra_702","math_train_number_theory_468","math_train_precalculus_700","math_train_algebra_1547","math_train_algebra_1034","math_train_geometry_273","math_train_precalculus_432","math_train_intermediate_algebra_92","math_train_number_theory_450","math_train_intermediate_algebra_462","math_train_algebra_326","math_train_geometry_212","math_train_algebra_156","math_train_algebra_393","math_train_counting_and_probability_142","math_train_geometry_187","math_train_prealgebra_988","math_train_precalculus_476","math_train_counting_and_probability_199","math_train_precalculus_681","math_train_number_theory_685","math_train_algebra_810","math_train_prealgebra_44","math_train_intermediate_algebra_37","math_train_geometry_371","math_train_counting_and_probability_247","math_train_prealgebra_326","math_train_number_theory_61","math_train_prealgebra_201","math_train_geometry_41","math_train_prealgebra_1","math_train_intermediate_algebra_1055","math_train_prealgebra_876","math_train_intermediate_algebra_182","math_train_counting_and_probability_193","math_train_precalculus_675","math_train_prealgebra_637","math_train_number_theory_85","math_train_intermediate_algebra_517","math_train_intermediate_algebra_1262","math_train_geometry_248","math_train_counting_and_probability_582","math_train_intermediate_algebra_851","math_train_algebra_1614","math_train_intermediate_algebra_795","math_train_precalculus_29","math_train_prealgebra_231","math_train_algebra_491","math_train_geometry_294","math_train_geometry_516","math_train_number_theory_425","math_train_algebra_589","math_train_intermediate_algebra_459","math_train_counting_and_probability_321","math_train_number_theory_24","math_train_counting_and_probability_286","math_train_geometry_203","math_train_intermediate_algebra_326","math_train_intermediate_algebra_1246","math_train_prealgebra_247","math_train_algebra_19","math_train_intermediate_algebra_512","math_train_algebra_1606","math_train_number_theory_860","math_train_geometry_729","math_train_intermediate_algebra_696","math_train_number_theory_297","math_train_intermediate_algebra_831","math_train_geometry_0","math_train_intermediate_algebra_221","math_train_precalculus_650","math_train_algebra_1415","math_train_prealgebra_630","math_train_number_theory_163","math_train_number_theory_13","math_train_number_theory_564","math_train_algebra_121","math_train_algebra_630","math_train_intermediate_algebra_688","math_train_prealgebra_120","math_train_prealgebra_135","math_train_number_theory_368","math_train_prealgebra_743","math_train_counting_and_probability_613","math_train_algebra_547","math_train_precalculus_77","math_train_algebra_1734","math_train_counting_and_probability_349","math_train_intermediate_algebra_534","math_train_precalculus_425","math_train_prealgebra_1115","math_train_precalculus_475","math_train_counting_and_probability_197","math_train_counting_and_probability_339","math_train_precalculus_55","math_train_counting_and_probability_178","math_train_algebra_586","math_train_intermediate_algebra_208","math_train_counting_and_probability_52","math_train_algebra_403","math_train_geometry_806","math_train_number_theory_95","math_train_algebra_962","math_train_algebra_967","math_train_intermediate_algebra_294","math_train_prealgebra_872","math_train_algebra_1105","math_train_intermediate_algebra_547","math_train_precalculus_20","math_train_algebra_1666","math_train_algebra_1333","math_train_algebra_1293","math_train_number_theory_458","math_train_algebra_1535","math_train_geometry_72","math_train_algebra_1340","math_train_intermediate_algebra_181","math_train_precalculus_154","math_train_intermediate_algebra_252","math_train_algebra_1515","math_train_prealgebra_766","math_train_intermediate_algebra_306","math_train_algebra_1189","math_train_counting_and_probability_176","math_train_geometry_868","math_train_precalculus_202","math_train_algebra_723","math_train_number_theory_784","math_train_precalculus_439","math_train_prealgebra_285","math_train_prealgebra_1151","math_train_algebra_1038","math_train_number_theory_240","math_train_algebra_298","math_train_algebra_869","math_train_intermediate_algebra_129","math_train_geometry_373","math_train_intermediate_algebra_299","math_train_prealgebra_1133","math_train_number_theory_569","math_train_intermediate_algebra_482","math_train_geometry_669","math_train_algebra_1265","math_train_prealgebra_45","math_train_algebra_1070","math_train_intermediate_algebra_681","math_train_precalculus_178","math_train_algebra_1462","math_train_algebra_981","math_train_precalculus_628","math_train_precalculus_84","math_train_intermediate_algebra_511","math_train_prealgebra_831","math_train_intermediate_algebra_358","math_train_intermediate_algebra_941","math_train_number_theory_498","math_train_prealgebra_597","math_train_precalculus_188","math_train_number_theory_786","math_train_geometry_63","math_train_prealgebra_541","math_train_prealgebra_128","math_train_algebra_1280","math_train_algebra_1174","math_train_number_theory_303","math_train_prealgebra_471","math_train_intermediate_algebra_871","math_train_geometry_664","math_train_intermediate_algebra_532","math_train_algebra_738","math_train_algebra_1444","math_train_prealgebra_407","math_train_algebra_710","math_train_number_theory_104","math_train_intermediate_algebra_856","math_train_counting_and_probability_304","math_train_intermediate_algebra_1281","math_train_intermediate_algebra_775","math_train_intermediate_algebra_1012","math_train_algebra_650","math_train_algebra_368","math_train_counting_and_probability_96","math_train_counting_and_probability_140","math_train_number_theory_718","math_train_counting_and_probability_128","math_train_intermediate_algebra_763","math_train_intermediate_algebra_348","math_train_prealgebra_913","math_train_number_theory_127","math_train_prealgebra_316","math_train_counting_and_probability_695","math_train_prealgebra_121","math_train_algebra_777","math_train_precalculus_347","math_train_prealgebra_901","math_train_algebra_1436","math_train_prealgebra_618","math_train_geometry_232","math_train_geometry_723","math_train_geometry_731","math_train_algebra_1384","math_train_prealgebra_423","math_train_intermediate_algebra_1094","math_train_geometry_839","math_train_precalculus_590","math_train_algebra_1025","math_train_intermediate_algebra_565","math_train_algebra_821","math_train_counting_and_probability_664","math_train_prealgebra_739","math_train_precalculus_448","math_train_counting_and_probability_129","math_train_counting_and_probability_648","math_train_counting_and_probability_607","math_train_prealgebra_1012","math_train_prealgebra_182","math_train_number_theory_238","math_train_intermediate_algebra_1019","math_train_prealgebra_228","math_train_algebra_1125","math_train_algebra_1364","math_train_algebra_184","math_train_precalculus_158","math_train_algebra_513","math_train_algebra_29","math_train_counting_and_probability_564","math_train_algebra_1247","math_train_prealgebra_834","math_train_counting_and_probability_433","math_train_number_theory_108"],"n":4,"max_tokens":8192}

[note from the experimenters] Correction to the generation-pass summaries: the failure count "no_answer_finished" now counts only answers in which the grader finds no final answer at all under your evaluation's grading rule; until now it also counted answers whose stated final answer was wrong. Earlier summaries in this session over-counted that category.

Forward pass 8 of 10 done (generate); results in /work/gen_8_original.jsonl and /work/gen_8_edited.jsonl. Summary:

{ "ok": true, "type": "generate", "model": "both", "files": {  "original": "/work/gen_8_original.jsonl",  "edited": "/work/gen_8_edited.jsonl" }, "activations": null, "original": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.0703125,  "accuracy_by_source": {   "math": 0.0703  },  "mean_tokens": 1615,  "cut_share": 0.0791,  "failures": {   "wrong_answer": 492,   "no_answer_finished": 381,   "cut_at_max_tokens": 79  } }, "edited": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.08203125,  "accuracy_by_source": {   "math": 0.082  },  "mean_tokens": 1448,  "cut_share": 0.0625,  "failures": {   "wrong_answer": 474,   "no_answer_finished": 402,   "cut_at_max_tokens": 64  } }, "paired": {  "items": 256,  "accuracy_difference": 0.0117,  "standard_error": 0.0074,  "samples_wrong_to_right": 36,  "samples_right_to_wrong": 24 }, "s": 282.1}

{ "ok": true, "type": "generate", "model": "both", "files": {  "original": "/work/gen_8_original.jsonl",  "edited": "/work/gen_8_edited.jsonl" }, "activations": null, "original": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.0703125,  "accuracy_by_source": {   "math": 0.0703  },  "mean_tokens": 1615,  "cut_share": 0.0791,  "failures": {   "wrong_answer": 492,   "no_answer_finished": 381,   "cut_at_max_tokens": 79  } }, "edited": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.08203125,  "accuracy_by_source": {   "math": 0.082  },  "mean_tokens": 1448,  "cut_share": 0.0625,  "failures": {   "wrong_answer": 474,   "no_answer_finished": 402,   "cut_at_max_tokens": 64  } }, "paired": {  "items": 256,  "accuracy_difference": 0.0117,  "standard_error": 0.0074,  "samples_wrong_to_right": 36,  "samples_right_to_wrong": 24 }, "s": 282.1}

Forward passes left: 2. Edits used in this pass: model.layers.14.mlp.down_proj.weight, model.layers.17.mlp.down_proj.weight, model.layers.8.mlp.down_proj.weight (edit hash 599aba54ec).

The first HOLDOUT pass favors the frozen edit: accuracy rose from 7.03% to 8.20%, a gain of 1.17 percentage points with a 0.74-point standard error. This is encouraging but not conclusive, so I’ll continue the two predeclared confirmation passes and pool all three. The experimenters also confirmed that earlier “no answer” counts were inflated; I will not use those earlier counts as evidence. The weights remain frozen, and the next pass changes only the predeclared problem order.

python

assert edit_checksums() == frozen_checksumsstats8, orig8, edit8, diff8 = paired_stats(8)def cluster_se_pp(original_samples, differences):    by_problem = collections.defaultdict(list)    for sample, difference in zip(original_samples, differences):        by_problem[sample["problem_id"]].append(float(difference))    means = np.array([np.mean(v) for v in by_problem.values()])    return float(100 * means.std(ddof=1) / np.sqrt(len(means)))stats8["cluster_se_pp"] = cluster_se_pp(orig8, diff8)(wdir / "stats_pass8.json").write_text(json.dumps(stats8, indent=2))with open(wdir / "notes.md", "a") as f:    f.write("\n## Failure-summary correction from experimenters\n"            "Earlier no_answer_finished counts also included wrong stated final answers. "            "Do not interpret them as missing-answer rates. Accuracy grades and our repetition "            "metrics are unchanged; no selection relied on the faulty category.\n"            "\n## Pass 8 confirmation result\n" + json.dumps(stats8) + "\n"            "No tuning: weights remain frozen1.5x, hash599aba54ec.\n"            "\n## Pass 9 planned\n"            "Exact edit unchanged: down_proj columns 8:[699],14:[483,984],17:[1179] x1.5.\n"            "Set: same256 HOLDOUT, second predeclared order (reverse); both, n4, max8192.\n"            "Pool with passes8 and10; no candidate selection from HOLDOUT.\n")print("HOLDOUT1:", {k: stats8[k] for k in ["gain_pp", "cluster_se_pp", "wrong_to_right", "right_to_wrong"]})print("Repetition rates:", {m: stats8[m]["three_or_more_final_answers"] for m in ["original", "edited"]})print("Next predeclared HOLDOUT order:")print(json.dumps(holdout_orders[1]))

assert edit_checksums() == frozen_checksumsstats8, orig8, edit8, diff8 = paired_stats(8)def cluster_se_pp(original_samples, differences):    by_problem = collections.defaultdict(list)    for sample, difference in zip(original_samples, differences):        by_problem[sample["problem_id"]].append(float(difference))    means = np.array([np.mean(v) for v in by_problem.values()])    return float(100 * means.std(ddof=1) / np.sqrt(len(means)))stats8["cluster_se_pp"] = cluster_se_pp(orig8, diff8)(wdir / "stats_pass8.json").write_text(json.dumps(stats8, indent=2))with open(wdir / "notes.md", "a") as f:    f.write("\n## Failure-summary correction from experimenters\n"            "Earlier no_answer_finished counts also included wrong stated final answers. "            "Do not interpret them as missing-answer rates. Accuracy grades and our repetition "            "metrics are unchanged; no selection relied on the faulty category.\n"            "\n## Pass 8 confirmation result\n" + json.dumps(stats8) + "\n"            "No tuning: weights remain frozen1.5x, hash599aba54ec.\n"            "\n## Pass 9 planned\n"            "Exact edit unchanged: down_proj columns 8:[699],14:[483,984],17:[1179] x1.5.\n"            "Set: same256 HOLDOUT, second predeclared order (reverse); both, n4, max8192.\n"            "Pool with passes8 and10; no candidate selection from HOLDOUT.\n")print("HOLDOUT1:", {k: stats8[k] for k in ["gain_pp", "cluster_se_pp", "wrong_to_right", "right_to_wrong"]})print("Repetition rates:", {m: stats8[m]["three_or_more_final_answers"] for m in ["original", "edited"]})print("Next predeclared HOLDOUT order:")print(json.dumps(holdout_orders[1]))

OUTPUT of Python run 18 of at most 300:

HOLDOUT1: {'gain_pp': 1.171875, 'cluster_se_pp': 0.7415623980049177, 'wrong_to_right': 36, 'right_to_wrong': 24}Repetition rates: {'original': 0.16015625, 'edited': 0.1103515625}Next predeclared HOLDOUT order:["math_train_number_theory_108", "math_train_counting_and_probability_433", "math_train_prealgebra_834", "math_train_algebra_1247", "math_train_counting_and_probability_564", "math_train_algebra_29", "math_train_algebra_513", "math_train_precalculus_158", "math_train_algebra_184", "math_train_algebra_1364", "math_train_algebra_1125", "math_train_prealgebra_228", "math_train_intermediate_algebra_1019", "math_train_number_theory_238", "math_train_prealgebra_182", "math_train_prealgebra_1012", "math_train_counting_and_probability_607", "math_train_counting_and_probability_648", "math_train_counting_and_probability_129", "math_train_precalculus_448", "math_train_prealgebra_739", "math_train_counting_and_probability_664", "math_train_algebra_821", "math_train_intermediate_algebra_565", "math_train_algebra_1025", "math_train_precalculus_590", "math_train_geometry_839", "math_train_intermediate_algebra_1094", "math_train_prealgebra_423", "math_train_algebra_1384", "math_train_geometry_731", "math_train_geometry_723", "math_train_geometry_232", "math_train_prealgebra_618", "math_train_algebra_1436", "math_train_prealgebra_901", "math_train_precalculus_347", "math_train_algebra_777", "math_train_prealgebra_121", "math_train_counting_and_probability_695", "math_train_prealgebra_316", "math_train_number_theory_127", "math_train_prealgebra_913", "math_train_intermediate_algebra_348", "math_train_intermediate_algebra_763", "math_train_counting_and_probability_128", "math_train_number_theory_718", "math_train_counting_and_probability_140", "math_train_counting_and_probability_96", "math_train_algebra_368", "math_train_algebra_650", "math_train_intermediate_algebra_1012", "math_train_intermediate_algebra_775", "math_train_intermediate_algebra_1281", "math_train_counting_and_probability_304", "math_train_intermediate_algebra_856", "math_train_number_theory_104", "math_train_algebra_710", "math_train_prealgebra_407", "math_train_algebra_1444", "math_train_algebra_738", "math_train_intermediate_algebra_532", "math_train_geometry_664", "math_train_intermediate_algebra_871", "math_train_prealgebra_471", "math_train_number_theory_303", "math_train_algebra_1174", "math_train_algebra_1280", "math_train_prealgebra_128", "math_train_prealgebra_541", "math_train_geometry_63", "math_train_number_theory_786", "math_train_precalculus_188", "math_train_prealgebra_597", "math_train_number_theory_498", "math_train_intermediate_algebra_941", "math_train_intermediate_algebra_358", "math_train_prealgebra_831", "math_train_intermediate_algebra_511", "math_train_precalculus_84", "math_train_precalculus_628", "math_train_algebra_981", "math_train_algebra_1462", "math_train_precalculus_178", "math_train_intermediate_algebra_681", "math_train_algebra_1070", "math_train_prealgebra_45", "math_train_algebra_1265", "math_train_geometry_669", "math_train_intermediate_algebra_482", "math_train_number_theory_569", "math_train_prealgebra_1133", "math_train_intermediate_algebra_299", "math_train_geometry_373", "math_train_intermediate_algebra_129", "math_train_algebra_869", "math_train_algebra_298", "math_train_number_theory_240", "math_train_algebra_1038", "math_train_prealgebra_1151", "math_train_prealgebra_285", "math_train_precalculus_439", "math_train_number_theory_784", "math_train_algebra_723", "math_train_precalculus_202", "math_train_geometry_868", "math_train_counting_and_probability_176", "math_train_algebra_1189", "math_train_intermediate_algebra_306", "math_train_prealgebra_766", "math_train_algebra_1515", "math_train_intermediate_algebra_252", "math_train_precalculus_154", "math_train_intermediate_algebra_181", "math_train_algebra_1340", "math_train_geometry_72", "math_train_algebra_1535", "math_train_number_theory_458", "math_train_algebra_1293", "math_train_algebra_1333", "math_train_algebra_1666", "math_train_precalculus_20", "math_train_intermediate_algebra_547", "math_train_algebra_1105", "math_train_prealgebra_872", "math_train_intermediate_algebra_294", "math_train_algebra_967", "math_train_algebra_962", "math_train_number_theory_95", "math_train_geometry_806", "math_train_algebra_403", "math_train_counting_and_probability_52", "math_train_intermediate_algebra_208", "math_train_algebra_586", "math_train_counting_and_probability_178", "math_train_precalculus_55", "math_train_counting_and_probability_339", "math_train_counting_and_probability_197", "math_train_precalculus_475", "math_train_prealgebra_1115", "math_train_precalculus_425", "math_train_intermediate_algebra_534", "math_train_counting_and_probability_349", "math_train_algebra_1734", "math_train_precalculus_77", "math_train_algebra_547", "math_train_counting_and_probability_613", "math_train_prealgebra_743", "math_train_number_theory_368", "math_train_prealgebra_135", "math_train_prealgebra_120", "math_train_intermediate_algebra_688", "math_train_algebra_630", "math_train_algebra_121", "math_train_number_theory_564", "math_train_number_theory_13", "math_train_number_theory_163", "math_train_prealgebra_630", "math_train_algebra_1415", "math_train_precalculus_650", "math_train_intermediate_algebra_221", "math_train_geometry_0", "math_train_intermediate_algebra_831", "math_train_number_theory_297", "math_train_intermediate_algebra_696", "math_train_geometry_729", "math_train_number_theory_860", "math_train_algebra_1606", "math_train_intermediate_algebra_512", "math_train_algebra_19", "math_train_prealgebra_247", "math_train_intermediate_algebra_1246", "math_train_intermediate_algebra_326", "math_train_geometry_203", "math_train_counting_and_probability_286", "math_train_number_theory_24", "math_train_counting_and_probability_321", "math_train_intermediate_algebra_459", "math_train_algebra_589", "math_train_number_theory_425", "math_train_geometry_516", "math_train_geometry_294", "math_train_algebra_491", "math_train_prealgebra_231", "math_train_precalculus_29", "math_train_intermediate_algebra_795", "math_train_algebra_1614", "math_train_intermediate_algebra_851", "math_train_counting_and_probability_582", "math_train_geometry_248", "math_train_intermediate_algebra_1262", "math_train_intermediate_algebra_517", "math_train_number_theory_85", "math_train_prealgebra_637", "math_train_precalculus_675", "math_train_counting_and_probability_193", "math_train_intermediate_algebra_182", "math_train_prealgebra_876", "math_train_intermediate_algebra_1055", "math_train_prealgebra_1", "math_train_geometry_41", "math_train_prealgebra_201", "math_train_number_theory_61", "math_train_prealgebra_326", "math_train_counting_and_probability_247", "math_train_geometry_371", "math_train_intermediate_algebra_37", "math_train_prealgebra_44", "math_train_algebra_810", "math_train_number_theory_685", "math_train_precalculus_681", "math_train_counting_and_probability_199", "math_train_precalculus_476", "math_train_prealgebra_988", "math_train_geometry_187", "math_train_counting_and_probability_142", "math_train_algebra_393", "math_train_algebra_156", "math_train_geometry_212", "math_train_algebra_326", "math_train_intermediate_algebra_462", "math_train_number_theory_450", "math_train_intermediate_algebra_92", "math_train_precalculus_432", "math_train_geometry_273", "math_train_algebra_1034", "math_train_algebra_1547", "math_train_precalculus_700", "math_train_number_theory_468", "math_train_intermediate_algebra_702", "math_train_algebra_75", "math_train_counting_and_probability_621", "math_train_number_theory_47", "math_train_number_theory_578", "math_train_prealgebra_246", "math_train_algebra_659", "math_train_algebra_793", "math_train_geometry_778", "math_train_algebra_279", "math_train_precalculus_165", "math_train_number_theory_466", "math_train_number_theory_668", "math_train_geometry_811", "math_train_algebra_1042", "math_train_number_theory_532", "math_train_prealgebra_426", "math_train_intermediate_algebra_56", "math_train_algebra_796", "math_train_prealgebra_269", "math_train_geometry_272", "math_train_prealgebra_909", "math_train_geometry_402", "math_train_prealgebra_666", "math_train_geometry_569", "math_train_algebra_1236", "math_train_intermediate_algebra_719"]

HOLDOUT1: {'gain_pp': 1.171875, 'cluster_se_pp': 0.7415623980049177, 'wrong_to_right': 36, 'right_to_wrong': 24}Repetition rates: {'original': 0.16015625, 'edited': 0.1103515625}Next predeclared HOLDOUT order:["math_train_number_theory_108", "math_train_counting_and_probability_433", "math_train_prealgebra_834", "math_train_algebra_1247", "math_train_counting_and_probability_564", "math_train_algebra_29", "math_train_algebra_513", "math_train_precalculus_158", "math_train_algebra_184", "math_train_algebra_1364", "math_train_algebra_1125", "math_train_prealgebra_228", "math_train_intermediate_algebra_1019", "math_train_number_theory_238", "math_train_prealgebra_182", "math_train_prealgebra_1012", "math_train_counting_and_probability_607", "math_train_counting_and_probability_648", "math_train_counting_and_probability_129", "math_train_precalculus_448", "math_train_prealgebra_739", "math_train_counting_and_probability_664", "math_train_algebra_821", "math_train_intermediate_algebra_565", "math_train_algebra_1025", "math_train_precalculus_590", "math_train_geometry_839", "math_train_intermediate_algebra_1094", "math_train_prealgebra_423", "math_train_algebra_1384", "math_train_geometry_731", "math_train_geometry_723", "math_train_geometry_232", "math_train_prealgebra_618", "math_train_algebra_1436", "math_train_prealgebra_901", "math_train_precalculus_347", "math_train_algebra_777", "math_train_prealgebra_121", "math_train_counting_and_probability_695", "math_train_prealgebra_316", "math_train_number_theory_127", "math_train_prealgebra_913", "math_train_intermediate_algebra_348", "math_train_intermediate_algebra_763", "math_train_counting_and_probability_128", "math_train_number_theory_718", "math_train_counting_and_probability_140", "math_train_counting_and_probability_96", "math_train_algebra_368", "math_train_algebra_650", "math_train_intermediate_algebra_1012", "math_train_intermediate_algebra_775", "math_train_intermediate_algebra_1281", "math_train_counting_and_probability_304", "math_train_intermediate_algebra_856", "math_train_number_theory_104", "math_train_algebra_710", "math_train_prealgebra_407", "math_train_algebra_1444", "math_train_algebra_738", "math_train_intermediate_algebra_532", "math_train_geometry_664", "math_train_intermediate_algebra_871", "math_train_prealgebra_471", "math_train_number_theory_303", "math_train_algebra_1174", "math_train_algebra_1280", "math_train_prealgebra_128", "math_train_prealgebra_541", "math_train_geometry_63", "math_train_number_theory_786", "math_train_precalculus_188", "math_train_prealgebra_597", "math_train_number_theory_498", "math_train_intermediate_algebra_941", "math_train_intermediate_algebra_358", "math_train_prealgebra_831", "math_train_intermediate_algebra_511", "math_train_precalculus_84", "math_train_precalculus_628", "math_train_algebra_981", "math_train_algebra_1462", "math_train_precalculus_178", "math_train_intermediate_algebra_681", "math_train_algebra_1070", "math_train_prealgebra_45", "math_train_algebra_1265", "math_train_geometry_669", "math_train_intermediate_algebra_482", "math_train_number_theory_569", "math_train_prealgebra_1133", "math_train_intermediate_algebra_299", "math_train_geometry_373", "math_train_intermediate_algebra_129", "math_train_algebra_869", "math_train_algebra_298", "math_train_number_theory_240", "math_train_algebra_1038", "math_train_prealgebra_1151", "math_train_prealgebra_285", "math_train_precalculus_439", "math_train_number_theory_784", "math_train_algebra_723", "math_train_precalculus_202", "math_train_geometry_868", "math_train_counting_and_probability_176", "math_train_algebra_1189", "math_train_intermediate_algebra_306", "math_train_prealgebra_766", "math_train_algebra_1515", "math_train_intermediate_algebra_252", "math_train_precalculus_154", "math_train_intermediate_algebra_181", "math_train_algebra_1340", "math_train_geometry_72", "math_train_algebra_1535", "math_train_number_theory_458", "math_train_algebra_1293", "math_train_algebra_1333", "math_train_algebra_1666", "math_train_precalculus_20", "math_train_intermediate_algebra_547", "math_train_algebra_1105", "math_train_prealgebra_872", "math_train_intermediate_algebra_294", "math_train_algebra_967", "math_train_algebra_962", "math_train_number_theory_95", "math_train_geometry_806", "math_train_algebra_403", "math_train_counting_and_probability_52", "math_train_intermediate_algebra_208", "math_train_algebra_586", "math_train_counting_and_probability_178", "math_train_precalculus_55", "math_train_counting_and_probability_339", "math_train_counting_and_probability_197", "math_train_precalculus_475", "math_train_prealgebra_1115", "math_train_precalculus_425", "math_train_intermediate_algebra_534", "math_train_counting_and_probability_349", "math_train_algebra_1734", "math_train_precalculus_77", "math_train_algebra_547", "math_train_counting_and_probability_613", "math_train_prealgebra_743", "math_train_number_theory_368", "math_train_prealgebra_135", "math_train_prealgebra_120", "math_train_intermediate_algebra_688", "math_train_algebra_630", "math_train_algebra_121", "math_train_number_theory_564", "math_train_number_theory_13", "math_train_number_theory_163", "math_train_prealgebra_630", "math_train_algebra_1415", "math_train_precalculus_650", "math_train_intermediate_algebra_221", "math_train_geometry_0", "math_train_intermediate_algebra_831", "math_train_number_theory_297", "math_train_intermediate_algebra_696", "math_train_geometry_729", "math_train_number_theory_860", "math_train_algebra_1606", "math_train_intermediate_algebra_512", "math_train_algebra_19", "math_train_prealgebra_247", "math_train_intermediate_algebra_1246", "math_train_intermediate_algebra_326", "math_train_geometry_203", "math_train_counting_and_probability_286", "math_train_number_theory_24", "math_train_counting_and_probability_321", "math_train_intermediate_algebra_459", "math_train_algebra_589", "math_train_number_theory_425", "math_train_geometry_516", "math_train_geometry_294", "math_train_algebra_491", "math_train_prealgebra_231", "math_train_precalculus_29", "math_train_intermediate_algebra_795", "math_train_algebra_1614", "math_train_intermediate_algebra_851", "math_train_counting_and_probability_582", "math_train_geometry_248", "math_train_intermediate_algebra_1262", "math_train_intermediate_algebra_517", "math_train_number_theory_85", "math_train_prealgebra_637", "math_train_precalculus_675", "math_train_counting_and_probability_193", "math_train_intermediate_algebra_182", "math_train_prealgebra_876", "math_train_intermediate_algebra_1055", "math_train_prealgebra_1", "math_train_geometry_41", "math_train_prealgebra_201", "math_train_number_theory_61", "math_train_prealgebra_326", "math_train_counting_and_probability_247", "math_train_geometry_371", "math_train_intermediate_algebra_37", "math_train_prealgebra_44", "math_train_algebra_810", "math_train_number_theory_685", "math_train_precalculus_681", "math_train_counting_and_probability_199", "math_train_precalculus_476", "math_train_prealgebra_988", "math_train_geometry_187", "math_train_counting_and_probability_142", "math_train_algebra_393", "math_train_algebra_156", "math_train_geometry_212", "math_train_algebra_326", "math_train_intermediate_algebra_462", "math_train_number_theory_450", "math_train_intermediate_algebra_92", "math_train_precalculus_432", "math_train_geometry_273", "math_train_algebra_1034", "math_train_algebra_1547", "math_train_precalculus_700", "math_train_number_theory_468", "math_train_intermediate_algebra_702", "math_train_algebra_75", "math_train_counting_and_probability_621", "math_train_number_theory_47", "math_train_number_theory_578", "math_train_prealgebra_246", "math_train_algebra_659", "math_train_algebra_793", "math_train_geometry_778", "math_train_algebra_279", "math_train_precalculus_165", "math_train_number_theory_466", "math_train_number_theory_668", "math_train_geometry_811", "math_train_algebra_1042", "math_train_number_theory_532", "math_train_prealgebra_426", "math_train_intermediate_algebra_56", "math_train_algebra_796", "math_train_prealgebra_269", "math_train_geometry_272", "math_train_prealgebra_909", "math_train_geometry_402", "math_train_prealgebra_666", "math_train_geometry_569", "math_train_algebra_1236", "math_train_intermediate_algebra_719"]

The first HOLDOUT pass also reduced repeated-conclusion answers from 16.0% to 11.0%, consistent with the DEV intervention results. That behavioral consistency is useful, but the accuracy estimate still needs the planned replication. I’ll run the second full-size paired confirmation using the reversed order specified before HOLDOUT began. The edit is unchanged, and this result will be pooled rather than used to select another candidate.

forward

{"type":"generate","model":"both","problems":["math_train_number_theory_108","math_train_counting_and_probability_433","math_train_prealgebra_834","math_train_algebra_1247","math_train_counting_and_probability_564","math_train_algebra_29","math_train_algebra_513","math_train_precalculus_158","math_train_algebra_184","math_train_algebra_1364","math_train_algebra_1125","math_train_prealgebra_228","math_train_intermediate_algebra_1019","math_train_number_theory_238","math_train_prealgebra_182","math_train_prealgebra_1012","math_train_counting_and_probability_607","math_train_counting_and_probability_648","math_train_counting_and_probability_129","math_train_precalculus_448","math_train_prealgebra_739","math_train_counting_and_probability_664","math_train_algebra_821","math_train_intermediate_algebra_565","math_train_algebra_1025","math_train_precalculus_590","math_train_geometry_839","math_train_intermediate_algebra_1094","math_train_prealgebra_423","math_train_algebra_1384","math_train_geometry_731","math_train_geometry_723","math_train_geometry_232","math_train_prealgebra_618","math_train_algebra_1436","math_train_prealgebra_901","math_train_precalculus_347","math_train_algebra_777","math_train_prealgebra_121","math_train_counting_and_probability_695","math_train_prealgebra_316","math_train_number_theory_127","math_train_prealgebra_913","math_train_intermediate_algebra_348","math_train_intermediate_algebra_763","math_train_counting_and_probability_128","math_train_number_theory_718","math_train_counting_and_probability_140","math_train_counting_and_probability_96","math_train_algebra_368","math_train_algebra_650","math_train_intermediate_algebra_1012","math_train_intermediate_algebra_775","math_train_intermediate_algebra_1281","math_train_counting_and_probability_304","math_train_intermediate_algebra_856","math_train_number_theory_104","math_train_algebra_710","math_train_prealgebra_407","math_train_algebra_1444","math_train_algebra_738","math_train_intermediate_algebra_532","math_train_geometry_664","math_train_intermediate_algebra_871","math_train_prealgebra_471","math_train_number_theory_303","math_train_algebra_1174","math_train_algebra_1280","math_train_prealgebra_128","math_train_prealgebra_541","math_train_geometry_63","math_train_number_theory_786","math_train_precalculus_188","math_train_prealgebra_597","math_train_number_theory_498","math_train_intermediate_algebra_941","math_train_intermediate_algebra_358","math_train_prealgebra_831","math_train_intermediate_algebra_511","math_train_precalculus_84","math_train_precalculus_628","math_train_algebra_981","math_train_algebra_1462","math_train_precalculus_178","math_train_intermediate_algebra_681","math_train_algebra_1070","math_train_prealgebra_45","math_train_algebra_1265","math_train_geometry_669","math_train_intermediate_algebra_482","math_train_number_theory_569","math_train_prealgebra_1133","math_train_intermediate_algebra_299","math_train_geometry_373","math_train_intermediate_algebra_129","math_train_algebra_869","math_train_algebra_298","math_train_number_theory_240","math_train_algebra_1038","math_train_prealgebra_1151","math_train_prealgebra_285","math_train_precalculus_439","math_train_number_theory_784","math_train_algebra_723","math_train_precalculus_202","math_train_geometry_868","math_train_counting_and_probability_176","math_train_algebra_1189","math_train_intermediate_algebra_306","math_train_prealgebra_766","math_train_algebra_1515","math_train_intermediate_algebra_252","math_train_precalculus_154","math_train_intermediate_algebra_181","math_train_algebra_1340","math_train_geometry_72","math_train_algebra_1535","math_train_number_theory_458","math_train_algebra_1293","math_train_algebra_1333","math_train_algebra_1666","math_train_precalculus_20","math_train_intermediate_algebra_547","math_train_algebra_1105","math_train_prealgebra_872","math_train_intermediate_algebra_294","math_train_algebra_967","math_train_algebra_962","math_train_number_theory_95","math_train_geometry_806","math_train_algebra_403","math_train_counting_and_probability_52","math_train_intermediate_algebra_208","math_train_algebra_586","math_train_counting_and_probability_178","math_train_precalculus_55","math_train_counting_and_probability_339","math_train_counting_and_probability_197","math_train_precalculus_475","math_train_prealgebra_1115","math_train_precalculus_425","math_train_intermediate_algebra_534","math_train_counting_and_probability_349","math_train_algebra_1734","math_train_precalculus_77","math_train_algebra_547","math_train_counting_and_probability_613","math_train_prealgebra_743","math_train_number_theory_368","math_train_prealgebra_135","math_train_prealgebra_120","math_train_intermediate_algebra_688","math_train_algebra_630","math_train_algebra_121","math_train_number_theory_564","math_train_number_theory_13","math_train_number_theory_163","math_train_prealgebra_630","math_train_algebra_1415","math_train_precalculus_650","math_train_intermediate_algebra_221","math_train_geometry_0","math_train_intermediate_algebra_831","math_train_number_theory_297","math_train_intermediate_algebra_696","math_train_geometry_729","math_train_number_theory_860","math_train_algebra_1606","math_train_intermediate_algebra_512","math_train_algebra_19","math_train_prealgebra_247","math_train_intermediate_algebra_1246","math_train_intermediate_algebra_326","math_train_geometry_203","math_train_counting_and_probability_286","math_train_number_theory_24","math_train_counting_and_probability_321","math_train_intermediate_algebra_459","math_train_algebra_589","math_train_number_theory_425","math_train_geometry_516","math_train_geometry_294","math_train_algebra_491","math_train_prealgebra_231","math_train_precalculus_29","math_train_intermediate_algebra_795","math_train_algebra_1614","math_train_intermediate_algebra_851","math_train_counting_and_probability_582","math_train_geometry_248","math_train_intermediate_algebra_1262","math_train_intermediate_algebra_517","math_train_number_theory_85","math_train_prealgebra_637","math_train_precalculus_675","math_train_counting_and_probability_193","math_train_intermediate_algebra_182","math_train_prealgebra_876","math_train_intermediate_algebra_1055","math_train_prealgebra_1","math_train_geometry_41","math_train_prealgebra_201","math_train_number_theory_61","math_train_prealgebra_326","math_train_counting_and_probability_247","math_train_geometry_371","math_train_intermediate_algebra_37","math_train_prealgebra_44","math_train_algebra_810","math_train_number_theory_685","math_train_precalculus_681","math_train_counting_and_probability_199","math_train_precalculus_476","math_train_prealgebra_988","math_train_geometry_187","math_train_counting_and_probability_142","math_train_algebra_393","math_train_algebra_156","math_train_geometry_212","math_train_algebra_326","math_train_intermediate_algebra_462","math_train_number_theory_450","math_train_intermediate_algebra_92","math_train_precalculus_432","math_train_geometry_273","math_train_algebra_1034","math_train_algebra_1547","math_train_precalculus_700","math_train_number_theory_468","math_train_intermediate_algebra_702","math_train_algebra_75","math_train_counting_and_probability_621","math_train_number_theory_47","math_train_number_theory_578","math_train_prealgebra_246","math_train_algebra_659","math_train_algebra_793","math_train_geometry_778","math_train_algebra_279","math_train_precalculus_165","math_train_number_theory_466","math_train_number_theory_668","math_train_geometry_811","math_train_algebra_1042","math_train_number_theory_532","math_train_prealgebra_426","math_train_intermediate_algebra_56","math_train_algebra_796","math_train_prealgebra_269","math_train_geometry_272","math_train_prealgebra_909","math_train_geometry_402","math_train_prealgebra_666","math_train_geometry_569","math_train_algebra_1236","math_train_intermediate_algebra_719"],"n":4,"max_tokens":8192}

{"type":"generate","model":"both","problems":["math_train_number_theory_108","math_train_counting_and_probability_433","math_train_prealgebra_834","math_train_algebra_1247","math_train_counting_and_probability_564","math_train_algebra_29","math_train_algebra_513","math_train_precalculus_158","math_train_algebra_184","math_train_algebra_1364","math_train_algebra_1125","math_train_prealgebra_228","math_train_intermediate_algebra_1019","math_train_number_theory_238","math_train_prealgebra_182","math_train_prealgebra_1012","math_train_counting_and_probability_607","math_train_counting_and_probability_648","math_train_counting_and_probability_129","math_train_precalculus_448","math_train_prealgebra_739","math_train_counting_and_probability_664","math_train_algebra_821","math_train_intermediate_algebra_565","math_train_algebra_1025","math_train_precalculus_590","math_train_geometry_839","math_train_intermediate_algebra_1094","math_train_prealgebra_423","math_train_algebra_1384","math_train_geometry_731","math_train_geometry_723","math_train_geometry_232","math_train_prealgebra_618","math_train_algebra_1436","math_train_prealgebra_901","math_train_precalculus_347","math_train_algebra_777","math_train_prealgebra_121","math_train_counting_and_probability_695","math_train_prealgebra_316","math_train_number_theory_127","math_train_prealgebra_913","math_train_intermediate_algebra_348","math_train_intermediate_algebra_763","math_train_counting_and_probability_128","math_train_number_theory_718","math_train_counting_and_probability_140","math_train_counting_and_probability_96","math_train_algebra_368","math_train_algebra_650","math_train_intermediate_algebra_1012","math_train_intermediate_algebra_775","math_train_intermediate_algebra_1281","math_train_counting_and_probability_304","math_train_intermediate_algebra_856","math_train_number_theory_104","math_train_algebra_710","math_train_prealgebra_407","math_train_algebra_1444","math_train_algebra_738","math_train_intermediate_algebra_532","math_train_geometry_664","math_train_intermediate_algebra_871","math_train_prealgebra_471","math_train_number_theory_303","math_train_algebra_1174","math_train_algebra_1280","math_train_prealgebra_128","math_train_prealgebra_541","math_train_geometry_63","math_train_number_theory_786","math_train_precalculus_188","math_train_prealgebra_597","math_train_number_theory_498","math_train_intermediate_algebra_941","math_train_intermediate_algebra_358","math_train_prealgebra_831","math_train_intermediate_algebra_511","math_train_precalculus_84","math_train_precalculus_628","math_train_algebra_981","math_train_algebra_1462","math_train_precalculus_178","math_train_intermediate_algebra_681","math_train_algebra_1070","math_train_prealgebra_45","math_train_algebra_1265","math_train_geometry_669","math_train_intermediate_algebra_482","math_train_number_theory_569","math_train_prealgebra_1133","math_train_intermediate_algebra_299","math_train_geometry_373","math_train_intermediate_algebra_129","math_train_algebra_869","math_train_algebra_298","math_train_number_theory_240","math_train_algebra_1038","math_train_prealgebra_1151","math_train_prealgebra_285","math_train_precalculus_439","math_train_number_theory_784","math_train_algebra_723","math_train_precalculus_202","math_train_geometry_868","math_train_counting_and_probability_176","math_train_algebra_1189","math_train_intermediate_algebra_306","math_train_prealgebra_766","math_train_algebra_1515","math_train_intermediate_algebra_252","math_train_precalculus_154","math_train_intermediate_algebra_181","math_train_algebra_1340","math_train_geometry_72","math_train_algebra_1535","math_train_number_theory_458","math_train_algebra_1293","math_train_algebra_1333","math_train_algebra_1666","math_train_precalculus_20","math_train_intermediate_algebra_547","math_train_algebra_1105","math_train_prealgebra_872","math_train_intermediate_algebra_294","math_train_algebra_967","math_train_algebra_962","math_train_number_theory_95","math_train_geometry_806","math_train_algebra_403","math_train_counting_and_probability_52","math_train_intermediate_algebra_208","math_train_algebra_586","math_train_counting_and_probability_178","math_train_precalculus_55","math_train_counting_and_probability_339","math_train_counting_and_probability_197","math_train_precalculus_475","math_train_prealgebra_1115","math_train_precalculus_425","math_train_intermediate_algebra_534","math_train_counting_and_probability_349","math_train_algebra_1734","math_train_precalculus_77","math_train_algebra_547","math_train_counting_and_probability_613","math_train_prealgebra_743","math_train_number_theory_368","math_train_prealgebra_135","math_train_prealgebra_120","math_train_intermediate_algebra_688","math_train_algebra_630","math_train_algebra_121","math_train_number_theory_564","math_train_number_theory_13","math_train_number_theory_163","math_train_prealgebra_630","math_train_algebra_1415","math_train_precalculus_650","math_train_intermediate_algebra_221","math_train_geometry_0","math_train_intermediate_algebra_831","math_train_number_theory_297","math_train_intermediate_algebra_696","math_train_geometry_729","math_train_number_theory_860","math_train_algebra_1606","math_train_intermediate_algebra_512","math_train_algebra_19","math_train_prealgebra_247","math_train_intermediate_algebra_1246","math_train_intermediate_algebra_326","math_train_geometry_203","math_train_counting_and_probability_286","math_train_number_theory_24","math_train_counting_and_probability_321","math_train_intermediate_algebra_459","math_train_algebra_589","math_train_number_theory_425","math_train_geometry_516","math_train_geometry_294","math_train_algebra_491","math_train_prealgebra_231","math_train_precalculus_29","math_train_intermediate_algebra_795","math_train_algebra_1614","math_train_intermediate_algebra_851","math_train_counting_and_probability_582","math_train_geometry_248","math_train_intermediate_algebra_1262","math_train_intermediate_algebra_517","math_train_number_theory_85","math_train_prealgebra_637","math_train_precalculus_675","math_train_counting_and_probability_193","math_train_intermediate_algebra_182","math_train_prealgebra_876","math_train_intermediate_algebra_1055","math_train_prealgebra_1","math_train_geometry_41","math_train_prealgebra_201","math_train_number_theory_61","math_train_prealgebra_326","math_train_counting_and_probability_247","math_train_geometry_371","math_train_intermediate_algebra_37","math_train_prealgebra_44","math_train_algebra_810","math_train_number_theory_685","math_train_precalculus_681","math_train_counting_and_probability_199","math_train_precalculus_476","math_train_prealgebra_988","math_train_geometry_187","math_train_counting_and_probability_142","math_train_algebra_393","math_train_algebra_156","math_train_geometry_212","math_train_algebra_326","math_train_intermediate_algebra_462","math_train_number_theory_450","math_train_intermediate_algebra_92","math_train_precalculus_432","math_train_geometry_273","math_train_algebra_1034","math_train_algebra_1547","math_train_precalculus_700","math_train_number_theory_468","math_train_intermediate_algebra_702","math_train_algebra_75","math_train_counting_and_probability_621","math_train_number_theory_47","math_train_number_theory_578","math_train_prealgebra_246","math_train_algebra_659","math_train_algebra_793","math_train_geometry_778","math_train_algebra_279","math_train_precalculus_165","math_train_number_theory_466","math_train_number_theory_668","math_train_geometry_811","math_train_algebra_1042","math_train_number_theory_532","math_train_prealgebra_426","math_train_intermediate_algebra_56","math_train_algebra_796","math_train_prealgebra_269","math_train_geometry_272","math_train_prealgebra_909","math_train_geometry_402","math_train_prealgebra_666","math_train_geometry_569","math_train_algebra_1236","math_train_intermediate_algebra_719"],"n":4,"max_tokens":8192}

Forward pass 9 of 10 done (generate); results in /work/gen_9_original.jsonl and /work/gen_9_edited.jsonl. Summary:

{ "ok": true, "type": "generate", "model": "both", "files": {  "original": "/work/gen_9_original.jsonl",  "edited": "/work/gen_9_edited.jsonl" }, "activations": null, "original": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.0712890625,  "accuracy_by_source": {   "math": 0.0713  },  "mean_tokens": 1453,  "cut_share": 0.0635,  "failures": {   "wrong_answer": 883,   "no_answer_finished": 3,   "cut_at_max_tokens": 65  } }, "edited": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.080078125,  "accuracy_by_source": {   "math": 0.0801  },  "mean_tokens": 1403,  "cut_share": 0.0605,  "failures": {   "wrong_answer": 877,   "no_answer_finished": 3,   "cut_at_max_tokens": 62  } }, "paired": {  "items": 256,  "accuracy_difference": 0.0088,  "standard_error": 0.0064,  "samples_wrong_to_right": 27,  "samples_right_to_wrong": 18 }, "s": 530.6}

{ "ok": true, "type": "generate", "model": "both", "files": {  "original": "/work/gen_9_original.jsonl",  "edited": "/work/gen_9_edited.jsonl" }, "activations": null, "original": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.0712890625,  "accuracy_by_source": {   "math": 0.0713  },  "mean_tokens": 1453,  "cut_share": 0.0635,  "failures": {   "wrong_answer": 883,   "no_answer_finished": 3,   "cut_at_max_tokens": 65  } }, "edited": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.080078125,  "accuracy_by_source": {   "math": 0.0801  },  "mean_tokens": 1403,  "cut_share": 0.0605,  "failures": {   "wrong_answer": 877,   "no_answer_finished": 3,   "cut_at_max_tokens": 62  } }, "paired": {  "items": 256,  "accuracy_difference": 0.0088,  "standard_error": 0.0064,  "samples_wrong_to_right": 27,  "samples_right_to_wrong": 18 }, "s": 530.6}

Forward passes left: 1. Edits used in this pass: model.layers.14.mlp.down_proj.weight, model.layers.17.mlp.down_proj.weight, model.layers.8.mlp.down_proj.weight (edit hash 599aba54ec).

The second HOLDOUT pass also favors the edit: +0.88 percentage points, with a 0.64-point standard error. Both confirmation passes therefore point in the same direction, but I’ll report the pooled estimate rather than emphasize either pass individually. I’ll verify that the frozen files are unchanged, record the interim aggregate, and prepare the final predeclared order. No further weight choices will be made from these results.

python

assert edit_checksums() == frozen_checksumsstats9, orig9, edit9, diff9 = paired_stats(9)stats9["cluster_se_pp"] = cluster_se_pp(orig9, diff9)(wdir / "stats_pass9.json").write_text(json.dumps(stats9, indent=2))
def pooled_confirmation(pass_numbers):    all_o, all_e, all_d = [], [], []    pass_summaries = []    for pn in pass_numbers:        st, oo, ee, dd = paired_stats(pn)        st["cluster_se_pp"] = cluster_se_pp(oo, dd)        all_o.extend(oo)        all_e.extend(ee)        all_d.extend(dd.tolist())        pass_summaries.append(st)    da = np.array(all_d, dtype=np.float64)    result = {        "passes": pass_numbers, "n_per_model": len(da),        "distinct_problems": len({s["problem_id"] for s in all_o}),        "original_correct": sum(bool(s["correct"]) for s in all_o),        "edited_correct": sum(bool(s["correct"]) for s in all_e),        "original_accuracy_pct": 100*np.mean([s["correct"] for s in all_o]),        "edited_accuracy_pct": 100*np.mean([s["correct"] for s in all_e]),        "gain_pp": float(100*da.mean()),        "cluster_se_pp": cluster_se_pp(all_o, da),        "sample_pair_se_pp": float(100*da.std(ddof=1)/np.sqrt(len(da))),        "wrong_to_right": int((da == 1).sum()),        "right_to_wrong": int((da == -1).sum()),    }    for label, ss in [("original", all_o), ("edited", all_e)]:        result[label + "_mean_tokens"] = float(np.mean([s["tokens"] for s in ss]))        result[label + "_cut_pct"] = float(100*np.mean([s["cut"] for s in ss]))        result[label + "_repeated_conclusion_pct"] = float(100*np.mean([            len(re.findall(r"final answer", s["text"], flags=re.I)) >= 3 for s in ss]))    return result, pass_summaries
interim_pool, _ = pooled_confirmation([8, 9])(wdir / "holdout_interim.json").write_text(json.dumps(interim_pool, indent=2))with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 9 confirmation result\n" + json.dumps(stats9) + "\n"            "Interim pooled confirmation: " + json.dumps(interim_pool) + "\n"            "No changes to candidate; checksum verification passed.\n"            "\n## Pass 10 planned: final confirmation\n"            "Exact edit unchanged: down_proj columns 8:[699],14:[483,984],17:[1179] x1.5, hash599aba54ec.\n"            "Set: same256 HOLDOUT, third predeclared order (rotation85); both, n4, max8192.\n"            "Final report pools all three HOLDOUT passes (3072 answers/model) and uses "            "problem-cluster standard error over the256 problems.\n")print("Interim HOLDOUT:", {k: interim_pool[k] for k in      ["n_per_model", "original_accuracy_pct", "edited_accuracy_pct", "gain_pp", "cluster_se_pp"]})print("Final predeclared HOLDOUT order:")print(json.dumps(holdout_orders[2]))

assert edit_checksums() == frozen_checksumsstats9, orig9, edit9, diff9 = paired_stats(9)stats9["cluster_se_pp"] = cluster_se_pp(orig9, diff9)(wdir / "stats_pass9.json").write_text(json.dumps(stats9, indent=2))
def pooled_confirmation(pass_numbers):    all_o, all_e, all_d = [], [], []    pass_summaries = []    for pn in pass_numbers:        st, oo, ee, dd = paired_stats(pn)        st["cluster_se_pp"] = cluster_se_pp(oo, dd)        all_o.extend(oo)        all_e.extend(ee)        all_d.extend(dd.tolist())        pass_summaries.append(st)    da = np.array(all_d, dtype=np.float64)    result = {        "passes": pass_numbers, "n_per_model": len(da),        "distinct_problems": len({s["problem_id"] for s in all_o}),        "original_correct": sum(bool(s["correct"]) for s in all_o),        "edited_correct": sum(bool(s["correct"]) for s in all_e),        "original_accuracy_pct": 100*np.mean([s["correct"] for s in all_o]),        "edited_accuracy_pct": 100*np.mean([s["correct"] for s in all_e]),        "gain_pp": float(100*da.mean()),        "cluster_se_pp": cluster_se_pp(all_o, da),        "sample_pair_se_pp": float(100*da.std(ddof=1)/np.sqrt(len(da))),        "wrong_to_right": int((da == 1).sum()),        "right_to_wrong": int((da == -1).sum()),    }    for label, ss in [("original", all_o), ("edited", all_e)]:        result[label + "_mean_tokens"] = float(np.mean([s["tokens"] for s in ss]))        result[label + "_cut_pct"] = float(100*np.mean([s["cut"] for s in ss]))        result[label + "_repeated_conclusion_pct"] = float(100*np.mean([            len(re.findall(r"final answer", s["text"], flags=re.I)) >= 3 for s in ss]))    return result, pass_summaries
interim_pool, _ = pooled_confirmation([8, 9])(wdir / "holdout_interim.json").write_text(json.dumps(interim_pool, indent=2))with open(wdir / "notes.md", "a") as f:    f.write("\n## Pass 9 confirmation result\n" + json.dumps(stats9) + "\n"            "Interim pooled confirmation: " + json.dumps(interim_pool) + "\n"            "No changes to candidate; checksum verification passed.\n"            "\n## Pass 10 planned: final confirmation\n"            "Exact edit unchanged: down_proj columns 8:[699],14:[483,984],17:[1179] x1.5, hash599aba54ec.\n"            "Set: same256 HOLDOUT, third predeclared order (rotation85); both, n4, max8192.\n"            "Final report pools all three HOLDOUT passes (3072 answers/model) and uses "            "problem-cluster standard error over the256 problems.\n")print("Interim HOLDOUT:", {k: interim_pool[k] for k in      ["n_per_model", "original_accuracy_pct", "edited_accuracy_pct", "gain_pp", "cluster_se_pp"]})print("Final predeclared HOLDOUT order:")print(json.dumps(holdout_orders[2]))

OUTPUT of Python run 19 of at most 300:

Interim HOLDOUT: {'n_per_model': 2048, 'original_accuracy_pct': np.float64(7.080078125), 'edited_accuracy_pct': np.float64(8.10546875), 'gain_pp': 1.025390625, 'cluster_se_pp': 0.5856102921141735}Final predeclared HOLDOUT order:["math_train_prealgebra_247", "math_train_algebra_19", "math_train_intermediate_algebra_512", "math_train_algebra_1606", "math_train_number_theory_860", "math_train_geometry_729", "math_train_intermediate_algebra_696", "math_train_number_theory_297", "math_train_intermediate_algebra_831", "math_train_geometry_0", "math_train_intermediate_algebra_221", "math_train_precalculus_650", "math_train_algebra_1415", "math_train_prealgebra_630", "math_train_number_theory_163", "math_train_number_theory_13", "math_train_number_theory_564", "math_train_algebra_121", "math_train_algebra_630", "math_train_intermediate_algebra_688", "math_train_prealgebra_120", "math_train_prealgebra_135", "math_train_number_theory_368", "math_train_prealgebra_743", "math_train_counting_and_probability_613", "math_train_algebra_547", "math_train_precalculus_77", "math_train_algebra_1734", "math_train_counting_and_probability_349", "math_train_intermediate_algebra_534", "math_train_precalculus_425", "math_train_prealgebra_1115", "math_train_precalculus_475", "math_train_counting_and_probability_197", "math_train_counting_and_probability_339", "math_train_precalculus_55", "math_train_counting_and_probability_178", "math_train_algebra_586", "math_train_intermediate_algebra_208", "math_train_counting_and_probability_52", "math_train_algebra_403", "math_train_geometry_806", "math_train_number_theory_95", "math_train_algebra_962", "math_train_algebra_967", "math_train_intermediate_algebra_294", "math_train_prealgebra_872", "math_train_algebra_1105", "math_train_intermediate_algebra_547", "math_train_precalculus_20", "math_train_algebra_1666", "math_train_algebra_1333", "math_train_algebra_1293", "math_train_number_theory_458", "math_train_algebra_1535", "math_train_geometry_72", "math_train_algebra_1340", "math_train_intermediate_algebra_181", "math_train_precalculus_154", "math_train_intermediate_algebra_252", "math_train_algebra_1515", "math_train_prealgebra_766", "math_train_intermediate_algebra_306", "math_train_algebra_1189", "math_train_counting_and_probability_176", "math_train_geometry_868", "math_train_precalculus_202", "math_train_algebra_723", "math_train_number_theory_784", "math_train_precalculus_439", "math_train_prealgebra_285", "math_train_prealgebra_1151", "math_train_algebra_1038", "math_train_number_theory_240", "math_train_algebra_298", "math_train_algebra_869", "math_train_intermediate_algebra_129", "math_train_geometry_373", "math_train_intermediate_algebra_299", "math_train_prealgebra_1133", "math_train_number_theory_569", "math_train_intermediate_algebra_482", "math_train_geometry_669", "math_train_algebra_1265", "math_train_prealgebra_45", "math_train_algebra_1070", "math_train_intermediate_algebra_681", "math_train_precalculus_178", "math_train_algebra_1462", "math_train_algebra_981", "math_train_precalculus_628", "math_train_precalculus_84", "math_train_intermediate_algebra_511", "math_train_prealgebra_831", "math_train_intermediate_algebra_358", "math_train_intermediate_algebra_941", "math_train_number_theory_498", "math_train_prealgebra_597", "math_train_precalculus_188", "math_train_number_theory_786", "math_train_geometry_63", "math_train_prealgebra_541", "math_train_prealgebra_128", "math_train_algebra_1280", "math_train_algebra_1174", "math_train_number_theory_303", "math_train_prealgebra_471", "math_train_intermediate_algebra_871", "math_train_geometry_664", "math_train_intermediate_algebra_532", "math_train_algebra_738", "math_train_algebra_1444", "math_train_prealgebra_407", "math_train_algebra_710", "math_train_number_theory_104", "math_train_intermediate_algebra_856", "math_train_counting_and_probability_304", "math_train_intermediate_algebra_1281", "math_train_intermediate_algebra_775", "math_train_intermediate_algebra_1012", "math_train_algebra_650", "math_train_algebra_368", "math_train_counting_and_probability_96", "math_train_counting_and_probability_140", "math_train_number_theory_718", "math_train_counting_and_probability_128", "math_train_intermediate_algebra_763", "math_train_intermediate_algebra_348", "math_train_prealgebra_913", "math_train_number_theory_127", "math_train_prealgebra_316", "math_train_counting_and_probability_695", "math_train_prealgebra_121", "math_train_algebra_777", "math_train_precalculus_347", "math_train_prealgebra_901", "math_train_algebra_1436", "math_train_prealgebra_618", "math_train_geometry_232", "math_train_geometry_723", "math_train_geometry_731", "math_train_algebra_1384", "math_train_prealgebra_423", "math_train_intermediate_algebra_1094", "math_train_geometry_839", "math_train_precalculus_590", "math_train_algebra_1025", "math_train_intermediate_algebra_565", "math_train_algebra_821", "math_train_counting_and_probability_664", "math_train_prealgebra_739", "math_train_precalculus_448", "math_train_counting_and_probability_129", "math_train_counting_and_probability_648", "math_train_counting_and_probability_607", "math_train_prealgebra_1012", "math_train_prealgebra_182", "math_train_number_theory_238", "math_train_intermediate_algebra_1019", "math_train_prealgebra_228", "math_train_algebra_1125", "math_train_algebra_1364", "math_train_algebra_184", "math_train_precalculus_158", "math_train_algebra_513", "math_train_algebra_29", "math_train_counting_and_probability_564", "math_train_algebra_1247", "math_train_prealgebra_834", "math_train_counting_and_probability_433", "math_train_number_theory_108", "math_train_intermediate_algebra_719", "math_train_algebra_1236", "math_train_geometry_569", "math_train_prealgebra_666", "math_train_geometry_402", "math_train_prealgebra_909", "math_train_geometry_272", "math_train_prealgebra_269", "math_train_algebra_796", "math_train_intermediate_algebra_56", "math_train_prealgebra_426", "math_train_number_theory_532", "math_train_algebra_1042", "math_train_geometry_811", "math_train_number_theory_668", "math_train_number_theory_466", "math_train_precalculus_165", "math_train_algebra_279", "math_train_geometry_778", "math_train_algebra_793", "math_train_algebra_659", "math_train_prealgebra_246", "math_train_number_theory_578", "math_train_number_theory_47", "math_train_counting_and_probability_621", "math_train_algebra_75", "math_train_intermediate_algebra_702", "math_train_number_theory_468", "math_train_precalculus_700", "math_train_algebra_1547", "math_train_algebra_1034", "math_train_geometry_273", "math_train_precalculus_432", "math_train_intermediate_algebra_92", "math_train_number_theory_450", "math_train_intermediate_algebra_462", "math_train_algebra_326", "math_train_geometry_212", "math_train_algebra_156", "math_train_algebra_393", "math_train_counting_and_probability_142", "math_train_geometry_187", "math_train_prealgebra_988", "math_train_precalculus_476", "math_train_counting_and_probability_199", "math_train_precalculus_681", "math_train_number_theory_685", "math_train_algebra_810", "math_train_prealgebra_44", "math_train_intermediate_algebra_37", "math_train_geometry_371", "math_train_counting_and_probability_247", "math_train_prealgebra_326", "math_train_number_theory_61", "math_train_prealgebra_201", "math_train_geometry_41", "math_train_prealgebra_1", "math_train_intermediate_algebra_1055", "math_train_prealgebra_876", "math_train_intermediate_algebra_182", "math_train_counting_and_probability_193", "math_train_precalculus_675", "math_train_prealgebra_637", "math_train_number_theory_85", "math_train_intermediate_algebra_517", "math_train_intermediate_algebra_1262", "math_train_geometry_248", "math_train_counting_and_probability_582", "math_train_intermediate_algebra_851", "math_train_algebra_1614", "math_train_intermediate_algebra_795", "math_train_precalculus_29", "math_train_prealgebra_231", "math_train_algebra_491", "math_train_geometry_294", "math_train_geometry_516", "math_train_number_theory_425", "math_train_algebra_589", "math_train_intermediate_algebra_459", "math_train_counting_and_probability_321", "math_train_number_theory_24", "math_train_counting_and_probability_286", "math_train_geometry_203", "math_train_intermediate_algebra_326", "math_train_intermediate_algebra_1246"]

Interim HOLDOUT: {'n_per_model': 2048, 'original_accuracy_pct': np.float64(7.080078125), 'edited_accuracy_pct': np.float64(8.10546875), 'gain_pp': 1.025390625, 'cluster_se_pp': 0.5856102921141735}Final predeclared HOLDOUT order:["math_train_prealgebra_247", "math_train_algebra_19", "math_train_intermediate_algebra_512", "math_train_algebra_1606", "math_train_number_theory_860", "math_train_geometry_729", "math_train_intermediate_algebra_696", "math_train_number_theory_297", "math_train_intermediate_algebra_831", "math_train_geometry_0", "math_train_intermediate_algebra_221", "math_train_precalculus_650", "math_train_algebra_1415", "math_train_prealgebra_630", "math_train_number_theory_163", "math_train_number_theory_13", "math_train_number_theory_564", "math_train_algebra_121", "math_train_algebra_630", "math_train_intermediate_algebra_688", "math_train_prealgebra_120", "math_train_prealgebra_135", "math_train_number_theory_368", "math_train_prealgebra_743", "math_train_counting_and_probability_613", "math_train_algebra_547", "math_train_precalculus_77", "math_train_algebra_1734", "math_train_counting_and_probability_349", "math_train_intermediate_algebra_534", "math_train_precalculus_425", "math_train_prealgebra_1115", "math_train_precalculus_475", "math_train_counting_and_probability_197", "math_train_counting_and_probability_339", "math_train_precalculus_55", "math_train_counting_and_probability_178", "math_train_algebra_586", "math_train_intermediate_algebra_208", "math_train_counting_and_probability_52", "math_train_algebra_403", "math_train_geometry_806", "math_train_number_theory_95", "math_train_algebra_962", "math_train_algebra_967", "math_train_intermediate_algebra_294", "math_train_prealgebra_872", "math_train_algebra_1105", "math_train_intermediate_algebra_547", "math_train_precalculus_20", "math_train_algebra_1666", "math_train_algebra_1333", "math_train_algebra_1293", "math_train_number_theory_458", "math_train_algebra_1535", "math_train_geometry_72", "math_train_algebra_1340", "math_train_intermediate_algebra_181", "math_train_precalculus_154", "math_train_intermediate_algebra_252", "math_train_algebra_1515", "math_train_prealgebra_766", "math_train_intermediate_algebra_306", "math_train_algebra_1189", "math_train_counting_and_probability_176", "math_train_geometry_868", "math_train_precalculus_202", "math_train_algebra_723", "math_train_number_theory_784", "math_train_precalculus_439", "math_train_prealgebra_285", "math_train_prealgebra_1151", "math_train_algebra_1038", "math_train_number_theory_240", "math_train_algebra_298", "math_train_algebra_869", "math_train_intermediate_algebra_129", "math_train_geometry_373", "math_train_intermediate_algebra_299", "math_train_prealgebra_1133", "math_train_number_theory_569", "math_train_intermediate_algebra_482", "math_train_geometry_669", "math_train_algebra_1265", "math_train_prealgebra_45", "math_train_algebra_1070", "math_train_intermediate_algebra_681", "math_train_precalculus_178", "math_train_algebra_1462", "math_train_algebra_981", "math_train_precalculus_628", "math_train_precalculus_84", "math_train_intermediate_algebra_511", "math_train_prealgebra_831", "math_train_intermediate_algebra_358", "math_train_intermediate_algebra_941", "math_train_number_theory_498", "math_train_prealgebra_597", "math_train_precalculus_188", "math_train_number_theory_786", "math_train_geometry_63", "math_train_prealgebra_541", "math_train_prealgebra_128", "math_train_algebra_1280", "math_train_algebra_1174", "math_train_number_theory_303", "math_train_prealgebra_471", "math_train_intermediate_algebra_871", "math_train_geometry_664", "math_train_intermediate_algebra_532", "math_train_algebra_738", "math_train_algebra_1444", "math_train_prealgebra_407", "math_train_algebra_710", "math_train_number_theory_104", "math_train_intermediate_algebra_856", "math_train_counting_and_probability_304", "math_train_intermediate_algebra_1281", "math_train_intermediate_algebra_775", "math_train_intermediate_algebra_1012", "math_train_algebra_650", "math_train_algebra_368", "math_train_counting_and_probability_96", "math_train_counting_and_probability_140", "math_train_number_theory_718", "math_train_counting_and_probability_128", "math_train_intermediate_algebra_763", "math_train_intermediate_algebra_348", "math_train_prealgebra_913", "math_train_number_theory_127", "math_train_prealgebra_316", "math_train_counting_and_probability_695", "math_train_prealgebra_121", "math_train_algebra_777", "math_train_precalculus_347", "math_train_prealgebra_901", "math_train_algebra_1436", "math_train_prealgebra_618", "math_train_geometry_232", "math_train_geometry_723", "math_train_geometry_731", "math_train_algebra_1384", "math_train_prealgebra_423", "math_train_intermediate_algebra_1094", "math_train_geometry_839", "math_train_precalculus_590", "math_train_algebra_1025", "math_train_intermediate_algebra_565", "math_train_algebra_821", "math_train_counting_and_probability_664", "math_train_prealgebra_739", "math_train_precalculus_448", "math_train_counting_and_probability_129", "math_train_counting_and_probability_648", "math_train_counting_and_probability_607", "math_train_prealgebra_1012", "math_train_prealgebra_182", "math_train_number_theory_238", "math_train_intermediate_algebra_1019", "math_train_prealgebra_228", "math_train_algebra_1125", "math_train_algebra_1364", "math_train_algebra_184", "math_train_precalculus_158", "math_train_algebra_513", "math_train_algebra_29", "math_train_counting_and_probability_564", "math_train_algebra_1247", "math_train_prealgebra_834", "math_train_counting_and_probability_433", "math_train_number_theory_108", "math_train_intermediate_algebra_719", "math_train_algebra_1236", "math_train_geometry_569", "math_train_prealgebra_666", "math_train_geometry_402", "math_train_prealgebra_909", "math_train_geometry_272", "math_train_prealgebra_269", "math_train_algebra_796", "math_train_intermediate_algebra_56", "math_train_prealgebra_426", "math_train_number_theory_532", "math_train_algebra_1042", "math_train_geometry_811", "math_train_number_theory_668", "math_train_number_theory_466", "math_train_precalculus_165", "math_train_algebra_279", "math_train_geometry_778", "math_train_algebra_793", "math_train_algebra_659", "math_train_prealgebra_246", "math_train_number_theory_578", "math_train_number_theory_47", "math_train_counting_and_probability_621", "math_train_algebra_75", "math_train_intermediate_algebra_702", "math_train_number_theory_468", "math_train_precalculus_700", "math_train_algebra_1547", "math_train_algebra_1034", "math_train_geometry_273", "math_train_precalculus_432", "math_train_intermediate_algebra_92", "math_train_number_theory_450", "math_train_intermediate_algebra_462", "math_train_algebra_326", "math_train_geometry_212", "math_train_algebra_156", "math_train_algebra_393", "math_train_counting_and_probability_142", "math_train_geometry_187", "math_train_prealgebra_988", "math_train_precalculus_476", "math_train_counting_and_probability_199", "math_train_precalculus_681", "math_train_number_theory_685", "math_train_algebra_810", "math_train_prealgebra_44", "math_train_intermediate_algebra_37", "math_train_geometry_371", "math_train_counting_and_probability_247", "math_train_prealgebra_326", "math_train_number_theory_61", "math_train_prealgebra_201", "math_train_geometry_41", "math_train_prealgebra_1", "math_train_intermediate_algebra_1055", "math_train_prealgebra_876", "math_train_intermediate_algebra_182", "math_train_counting_and_probability_193", "math_train_precalculus_675", "math_train_prealgebra_637", "math_train_number_theory_85", "math_train_intermediate_algebra_517", "math_train_intermediate_algebra_1262", "math_train_geometry_248", "math_train_counting_and_probability_582", "math_train_intermediate_algebra_851", "math_train_algebra_1614", "math_train_intermediate_algebra_795", "math_train_precalculus_29", "math_train_prealgebra_231", "math_train_algebra_491", "math_train_geometry_294", "math_train_geometry_516", "math_train_number_theory_425", "math_train_algebra_589", "math_train_intermediate_algebra_459", "math_train_counting_and_probability_321", "math_train_number_theory_24", "math_train_counting_and_probability_286", "math_train_geometry_203", "math_train_intermediate_algebra_326", "math_train_intermediate_algebra_1246"]

Across the first two HOLDOUT passes, accuracy is 8.11% edited versus 7.08% original: +1.03 percentage points with a 0.59-point problem-cluster standard error. The frozen-file checksums still match, so this remains a confirmation experiment rather than further tuning. I’ll use the last pass for the third predeclared paired comparison. The final report will pool all 3,072 answers per model and distinguish the measured accuracy gain from the stronger evidence about repeated conclusions.

forward

{"type":"generate","model":"both","problems":["math_train_prealgebra_247","math_train_algebra_19","math_train_intermediate_algebra_512","math_train_algebra_1606","math_train_number_theory_860","math_train_geometry_729","math_train_intermediate_algebra_696","math_train_number_theory_297","math_train_intermediate_algebra_831","math_train_geometry_0","math_train_intermediate_algebra_221","math_train_precalculus_650","math_train_algebra_1415","math_train_prealgebra_630","math_train_number_theory_163","math_train_number_theory_13","math_train_number_theory_564","math_train_algebra_121","math_train_algebra_630","math_train_intermediate_algebra_688","math_train_prealgebra_120","math_train_prealgebra_135","math_train_number_theory_368","math_train_prealgebra_743","math_train_counting_and_probability_613","math_train_algebra_547","math_train_precalculus_77","math_train_algebra_1734","math_train_counting_and_probability_349","math_train_intermediate_algebra_534","math_train_precalculus_425","math_train_prealgebra_1115","math_train_precalculus_475","math_train_counting_and_probability_197","math_train_counting_and_probability_339","math_train_precalculus_55","math_train_counting_and_probability_178","math_train_algebra_586","math_train_intermediate_algebra_208","math_train_counting_and_probability_52","math_train_algebra_403","math_train_geometry_806","math_train_number_theory_95","math_train_algebra_962","math_train_algebra_967","math_train_intermediate_algebra_294","math_train_prealgebra_872","math_train_algebra_1105","math_train_intermediate_algebra_547","math_train_precalculus_20","math_train_algebra_1666","math_train_algebra_1333","math_train_algebra_1293","math_train_number_theory_458","math_train_algebra_1535","math_train_geometry_72","math_train_algebra_1340","math_train_intermediate_algebra_181","math_train_precalculus_154","math_train_intermediate_algebra_252","math_train_algebra_1515","math_train_prealgebra_766","math_train_intermediate_algebra_306","math_train_algebra_1189","math_train_counting_and_probability_176","math_train_geometry_868","math_train_precalculus_202","math_train_algebra_723","math_train_number_theory_784","math_train_precalculus_439","math_train_prealgebra_285","math_train_prealgebra_1151","math_train_algebra_1038","math_train_number_theory_240","math_train_algebra_298","math_train_algebra_869","math_train_intermediate_algebra_129","math_train_geometry_373","math_train_intermediate_algebra_299","math_train_prealgebra_1133","math_train_number_theory_569","math_train_intermediate_algebra_482","math_train_geometry_669","math_train_algebra_1265","math_train_prealgebra_45","math_train_algebra_1070","math_train_intermediate_algebra_681","math_train_precalculus_178","math_train_algebra_1462","math_train_algebra_981","math_train_precalculus_628","math_train_precalculus_84","math_train_intermediate_algebra_511","math_train_prealgebra_831","math_train_intermediate_algebra_358","math_train_intermediate_algebra_941","math_train_number_theory_498","math_train_prealgebra_597","math_train_precalculus_188","math_train_number_theory_786","math_train_geometry_63","math_train_prealgebra_541","math_train_prealgebra_128","math_train_algebra_1280","math_train_algebra_1174","math_train_number_theory_303","math_train_prealgebra_471","math_train_intermediate_algebra_871","math_train_geometry_664","math_train_intermediate_algebra_532","math_train_algebra_738","math_train_algebra_1444","math_train_prealgebra_407","math_train_algebra_710","math_train_number_theory_104","math_train_intermediate_algebra_856","math_train_counting_and_probability_304","math_train_intermediate_algebra_1281","math_train_intermediate_algebra_775","math_train_intermediate_algebra_1012","math_train_algebra_650","math_train_algebra_368","math_train_counting_and_probability_96","math_train_counting_and_probability_140","math_train_number_theory_718","math_train_counting_and_probability_128","math_train_intermediate_algebra_763","math_train_intermediate_algebra_348","math_train_prealgebra_913","math_train_number_theory_127","math_train_prealgebra_316","math_train_counting_and_probability_695","math_train_prealgebra_121","math_train_algebra_777","math_train_precalculus_347","math_train_prealgebra_901","math_train_algebra_1436","math_train_prealgebra_618","math_train_geometry_232","math_train_geometry_723","math_train_geometry_731","math_train_algebra_1384","math_train_prealgebra_423","math_train_intermediate_algebra_1094","math_train_geometry_839","math_train_precalculus_590","math_train_algebra_1025","math_train_intermediate_algebra_565","math_train_algebra_821","math_train_counting_and_probability_664","math_train_prealgebra_739","math_train_precalculus_448","math_train_counting_and_probability_129","math_train_counting_and_probability_648","math_train_counting_and_probability_607","math_train_prealgebra_1012","math_train_prealgebra_182","math_train_number_theory_238","math_train_intermediate_algebra_1019","math_train_prealgebra_228","math_tr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{"type":"generate","model":"both","problems":["math_train_prealgebra_247","math_train_algebra_19","math_train_intermediate_algebra_512","math_train_algebra_1606","math_train_number_theory_860","math_train_geometry_729","math_train_intermediate_algebra_696","math_train_number_theory_297","math_train_intermediate_algebra_831","math_train_geometry_0","math_train_intermediate_algebra_221","math_train_precalculus_650","math_train_algebra_1415","math_train_prealgebra_630","math_train_number_theory_163","math_train_number_theory_13","math_train_number_theory_564","math_train_algebra_121","math_train_algebra_630","math_train_intermediate_algebra_688","math_train_prealgebra_120","math_train_prealgebra_135","math_train_number_theory_368","math_train_prealgebra_743","math_train_counting_and_probability_613","math_train_algebra_547","math_train_precalculus_77","math_train_algebra_1734","math_train_counting_and_probability_349","math_train_intermediate_algebra_534","math_train_precalculus_425","math_train_prealgebra_1115","math_train_precalculus_475","math_train_counting_and_probability_197","math_train_counting_and_probability_339","math_train_precalculus_55","math_train_counting_and_probability_178","math_train_algebra_586","math_train_intermediate_algebra_208","math_train_counting_and_probability_52","math_train_algebra_403","math_train_geometry_806","math_train_number_theory_95","math_train_algebra_962","math_train_algebra_967","math_train_intermediate_algebra_294","math_train_prealgebra_872","math_train_algebra_1105","math_train_intermediate_algebra_547","math_train_precalculus_20","math_train_algebra_1666","math_train_algebra_1333","math_train_algebra_1293","math_train_number_theory_458","math_train_algebra_1535","math_train_geometry_72","math_train_algebra_1340","math_train_intermediate_algebra_181","math_train_precalculus_154","math_train_intermediate_algebra_252","math_train_algebra_1515","math_train_prealgebra_766","math_train_intermediate_algebra_306","math_train_algebra_1189","math_train_counting_and_probability_176","math_train_geometry_868","math_train_precalculus_202","math_train_algebra_723","math_train_number_theory_784","math_train_precalculus_439","math_train_prealgebra_285","math_train_prealgebra_1151","math_train_algebra_1038","math_train_number_theory_240","math_train_algebra_298","math_train_algebra_869","math_train_intermediate_algebra_129","math_train_geometry_373","math_train_intermediate_algebra_299","math_train_prealgebra_1133","math_train_number_theory_569","math_train_intermediate_algebra_482","math_train_geometry_669","math_train_algebra_1265","math_train_prealgebra_45","math_train_algebra_1070","math_train_intermediate_algebra_681","math_train_precalculus_178","math_train_algebra_1462","math_train_algebra_981","math_train_precalculus_628","math_train_precalculus_84","math_train_intermediate_algebra_511","math_train_prealgebra_831","math_train_intermediate_algebra_358","math_train_intermediate_algebra_941","math_train_number_theory_498","math_train_prealgebra_597","math_train_precalculus_188","math_train_number_theory_786","math_train_geometry_63","math_train_prealgebra_541","math_train_prealgebra_128","math_train_algebra_1280","math_train_algebra_1174","math_train_number_theory_303","math_train_prealgebra_471","math_train_intermediate_algebra_871","math_train_geometry_664","math_train_intermediate_algebra_532","math_train_algebra_738","math_train_algebra_1444","math_train_prealgebra_407","math_train_algebra_710","math_train_number_theory_104","math_train_intermediate_algebra_856","math_train_counting_and_probability_304","math_train_intermediate_algebra_1281","math_train_intermediate_algebra_775","math_train_intermediate_algebra_1012","math_train_algebra_650","math_train_algebra_368","math_train_counting_and_probability_96","math_train_counting_and_probability_140","math_train_number_theory_718","math_train_counting_and_probability_128","math_train_intermediate_algebra_763","math_train_intermediate_algebra_348","math_train_prealgebra_913","math_train_number_theory_127","math_train_prealgebra_316","math_train_counting_and_probability_695","math_train_prealgebra_121","math_train_algebra_777","math_train_precalculus_347","math_train_prealgebra_901","math_train_algebra_1436","math_train_prealgebra_618","math_train_geometry_232","math_train_geometry_723","math_train_geometry_731","math_train_algebra_1384","math_train_prealgebra_423","math_train_intermediate_algebra_1094","math_train_geometry_839","math_train_precalculus_590","math_train_algebra_1025","math_train_intermediate_algebra_565","math_train_algebra_821","math_train_counting_and_probability_664","math_train_prealgebra_739","math_train_precalculus_448","math_train_counting_and_probability_129","math_train_counting_and_probability_648","math_train_counting_and_probability_607","math_train_prealgebra_1012","math_train_prealgebra_182","math_train_number_theory_238","math_train_intermediate_algebra_1019","math_train_prealgebra_228","math_tr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Forward pass 10 of 10 done (generate); results in /work/gen_10_original.jsonl and /work/gen_10_edited.jsonl. Summary:

{ "ok": true, "type": "generate", "model": "both", "files": {  "original": "/work/gen_10_original.jsonl",  "edited": "/work/gen_10_edited.jsonl" }, "activations": null, "original": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.07421875,  "accuracy_by_source": {   "math": 0.0742  },  "mean_tokens": 1571,  "cut_share": 0.0791,  "failures": {   "wrong_answer": 865,   "no_answer_finished": 4,   "cut_at_max_tokens": 79  } }, "edited": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.0751953125,  "accuracy_by_source": {   "math": 0.0752  },  "mean_tokens": 1457,  "cut_share": 0.0645,  "failures": {   "wrong_answer": 879,   "no_answer_finished": 5,   "cut_at_max_tokens": 63  } }, "paired": {  "items": 256,  "accuracy_difference": 0.001,  "standard_error": 0.0068,  "samples_wrong_to_right": 28,  "samples_right_to_wrong": 27 }, "s": 164.3}

{ "ok": true, "type": "generate", "model": "both", "files": {  "original": "/work/gen_10_original.jsonl",  "edited": "/work/gen_10_edited.jsonl" }, "activations": null, "original": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.07421875,  "accuracy_by_source": {   "math": 0.0742  },  "mean_tokens": 1571,  "cut_share": 0.0791,  "failures": {   "wrong_answer": 865,   "no_answer_finished": 4,   "cut_at_max_tokens": 79  } }, "edited": {  "generations": 1024,  "graded": 1024,  "accuracy": 0.0751953125,  "accuracy_by_source": {   "math": 0.0752  },  "mean_tokens": 1457,  "cut_share": 0.0645,  "failures": {   "wrong_answer": 879,   "no_answer_finished": 5,   "cut_at_max_tokens": 63  } }, "paired": {  "items": 256,  "accuracy_difference": 0.001,  "standard_error": 0.0068,  "samples_wrong_to_right": 28,  "samples_right_to_wrong": 27 }, "s": 164.3}

Forward passes left: 0. Edits used in this pass: model.layers.14.mlp.down_proj.weight, model.layers.17.mlp.down_proj.weight, model.layers.8.mlp.down_proj.weight (edit hash 599aba54ec). All forward passes are used. Make your final choice of edits (Python is still available), then reply without a code block.

The final HOLDOUT pass was nearly neutral: +0.10 percentage points with a 0.68-point standard error, which reduces the pooled gain. I’m keeping the 1.5× edit selected before HOLDOUT, with no post-HOLDOUT tuning. I’ll now calculate the final pooled accuracy and problem-cluster uncertainty, save the audit summary, and verify that only the intended four columns remain changed. The final claim will reflect all three confirmation passes, not just the more favorable first two.

python

assert edit_checksums() == frozen_checksumsassert set(edited()) == set(target_names.values())final_pool, holdout_pass_stats = pooled_confirmation([8, 9, 10])final_pool["approx_95pct_ci_pp"] = [    final_pool["gain_pp"] - 1.96 * final_pool["cluster_se_pp"],    final_pool["gain_pp"] + 1.96 * final_pool["cluster_se_pp"]]final_pool["candidate_frozen_before_holdout"] = Truefinal_pool["selection_used_holdout"] = Falsefinal_pool["edit_hash"] = "599aba54ec"
audit = []for layer, columns in target_channels.items():    name = target_names[layer]    ow, ew = load_original(name), load(name)    keep = torch.ones(2048, dtype=torch.bool)    keep[columns] = False    assert torch.equal(ow[:, keep], ew[:, keep])    assert torch.allclose(ew[:, columns], 1.5 * ow[:, columns])    audit.append({        "tensor": name, "columns": columns, "multiplier": 1.5,        "shape": list(ew.shape),        "changed_values": int(torch.count_nonzero(ew != ow)),        "relative_frobenius_change": float((ew - ow).norm() / ow.norm())    })
dev_table = []for pn, multiplier in [(5, 0.0), (6, 2.0), (7, 1.5)]:    st, oo, ee, dd = paired_stats(pn)    dev_table.append({        "pass": pn, "column_multiplier": multiplier,        "gain_pp": st["gain_pp"], "cluster_se_pp": cluster_se_pp(oo, dd),        "original_repeated_conclusion_pct": 100*st["original"]["three_or_more_final_answers"],        "edited_repeated_conclusion_pct": 100*st["edited"]["three_or_more_final_answers"]    })final_report = {    "submitted_edit": audit,    "dev_strength_tests": dev_table,    "holdout_pooled": final_pool,    "holdout_passes": holdout_pass_stats,    "limitations": [        "Accuracy gain is uncertain; report the problem-cluster SE and interval.",        "Causal repetition evidence applies to the four-channel group, not individually isolated neurons.",        "Teacher-forced probes lacked evaluation prompts and are local continuation checks only.",        "Pass2 was malformed and provides no mathematical evidence.",        "Earlier no_answer_finished counts were erroneous and were excluded from mechanism conclusions."    ],    "submission_directory": "/work/edited",    "backup_directory": "/work/frozen_candidate"}(wdir / "FINAL_REPORT.json").write_text(json.dumps(final_report, indent=2))with open(wdir / "notes.md", "a") as f:    f.write("\n## FINAL SUBMISSION\n"            "All10 forward passes complete. Final frozen candidate retained unchanged in /work/edited.\n"            "Exact edit: layer8 down_proj column699, layer14 down_proj columns483/984, "            "layer17 down_proj column1179 each multiplied by1.5; every other entry unchanged.\n"            "No embedding/output/final-norm edits, no training, no whole-tensor rescaling.\n"            "Path: baseline repetitive-failure diagnosis -> within-problem/length-adjusted channel screen "            "-> teacher-forced reference/loop probes -> ablation refuted harmful-channel hypothesis "            "-> amplification supported repetition-restraint role -> 1.5x selected by DEV accuracy "            "-> frozen three-pass HOLDOUT confirmation.\n"            "Pooled confirmation: " + json.dumps(final_pool) + "\n"            "Group-level mechanism evidence does not prove each neuron individually causal.\n"            "Complete audit and limitations: FINAL_REPORT.json. Final checksums verified.\n")print("FINAL POOLED HOLDOUT")print(json.dumps(final_pool, indent=2))print("DEV strength tests:", json.dumps(dev_table))print("Submission audit:", json.dumps(audit))print("Submission unchanged:", edit_checksums() == frozen_checksums)

assert edit_checksums() == frozen_checksumsassert set(edited()) == set(target_names.values())final_pool, holdout_pass_stats = pooled_confirmation([8, 9, 10])final_pool["approx_95pct_ci_pp"] = [    final_pool["gain_pp"] - 1.96 * final_pool["cluster_se_pp"],    final_pool["gain_pp"] + 1.96 * final_pool["cluster_se_pp"]]final_pool["candidate_frozen_before_holdout"] = Truefinal_pool["selection_used_holdout"] = Falsefinal_pool["edit_hash"] = "599aba54ec"
audit = []for layer, columns in target_channels.items():    name = target_names[layer]    ow, ew = load_original(name), load(name)    keep = torch.ones(2048, dtype=torch.bool)    keep[columns] = False    assert torch.equal(ow[:, keep], ew[:, keep])    assert torch.allclose(ew[:, columns], 1.5 * ow[:, columns])    audit.append({        "tensor": name, "columns": columns, "multiplier": 1.5,        "shape": list(ew.shape),        "changed_values": int(torch.count_nonzero(ew != ow)),        "relative_frobenius_change": float((ew - ow).norm() / ow.norm())    })
dev_table = []for pn, multiplier in [(5, 0.0), (6, 2.0), (7, 1.5)]:    st, oo, ee, dd = paired_stats(pn)    dev_table.append({        "pass": pn, "column_multiplier": multiplier,        "gain_pp": st["gain_pp"], "cluster_se_pp": cluster_se_pp(oo, dd),        "original_repeated_conclusion_pct": 100*st["original"]["three_or_more_final_answers"],        "edited_repeated_conclusion_pct": 100*st["edited"]["three_or_more_final_answers"]    })final_report = {    "submitted_edit": audit,    "dev_strength_tests": dev_table,    "holdout_pooled": final_pool,    "holdout_passes": holdout_pass_stats,    "limitations": [        "Accuracy gain is uncertain; report the problem-cluster SE and interval.",        "Causal repetition evidence applies to the four-channel group, not individually isolated neurons.",        "Teacher-forced probes lacked evaluation prompts and are local continuation checks only.",        "Pass2 was malformed and provides no mathematical evidence.",        "Earlier no_answer_finished counts were erroneous and were excluded from mechanism conclusions."    ],    "submission_directory": "/work/edited",    "backup_directory": "/work/frozen_candidate"}(wdir / "FINAL_REPORT.json").write_text(json.dumps(final_report, indent=2))with open(wdir / "notes.md", "a") as f:    f.write("\n## FINAL SUBMISSION\n"            "All10 forward passes complete. Final frozen candidate retained unchanged in /work/edited.\n"            "Exact edit: layer8 down_proj column699, layer14 down_proj columns483/984, "            "layer17 down_proj column1179 each multiplied by1.5; every other entry unchanged.\n"            "No embedding/output/final-norm edits, no training, no whole-tensor rescaling.\n"            "Path: baseline repetitive-failure diagnosis -> within-problem/length-adjusted channel screen "            "-> teacher-forced reference/loop probes -> ablation refuted harmful-channel hypothesis "            "-> amplification supported repetition-restraint role -> 1.5x selected by DEV accuracy "            "-> frozen three-pass HOLDOUT confirmation.\n"            "Pooled confirmation: " + json.dumps(final_pool) + "\n"            "Group-level mechanism evidence does not prove each neuron individually causal.\n"            "Complete audit and limitations: FINAL_REPORT.json. Final checksums verified.\n")print("FINAL POOLED HOLDOUT")print(json.dumps(final_pool, indent=2))print("DEV strength tests:", json.dumps(dev_table))print("Submission audit:", json.dumps(audit))print("Submission unchanged:", edit_checksums() == frozen_checksums)

OUTPUT of Python run 20 of at most 300:

FINAL POOLED HOLDOUT{  "passes": [    8,    9,    10  ],  "n_per_model": 3072,  "distinct_problems": 256,  "original_correct": 221,  "edited_correct": 243,  "original_accuracy_pct": 7.194010416666667,  "edited_accuracy_pct": 7.91015625,  "gain_pp": 0.7161458333333333,  "cluster_se_pp": 0.4926066250697082,  "sample_pair_se_pp": 0.4116191268561949,  "wrong_to_right": 91,  "right_to_wrong": 69,  "original_mean_tokens": 1546.353515625,  "original_cut_pct": 7.389322916666667,  "original_repeated_conclusion_pct": 15.33203125,  "edited_mean_tokens": 1435.96484375,  "edited_cut_pct": 6.25,  "edited_repeated_conclusion_pct": 12.01171875,  "approx_95pct_ci_pp": [    -0.24936315180329482,    1.6816548184699613  ],  "candidate_frozen_before_holdout": true,  "selection_used_holdout": false,  "edit_hash": "599aba54ec"}DEV strength tests: [{"pass": 5, "column_multiplier": 0.0, "gain_pp": -0.68359375, "cluster_se_pp": 1.6418185113361485, "original_repeated_conclusion_pct": 13.57421875, "edited_repeated_conclusion_pct": 22.55859375}, {"pass": 6, "column_multiplier": 2.0, "gain_pp": -1.26953125, "cluster_se_pp": 1.4207781798694787, "original_repeated_conclusion_pct": 13.37890625, "edited_repeated_conclusion_pct": 5.6640625}, {"pass": 7, "column_multiplier": 1.5, "gain_pp": 0.48828125, "cluster_se_pp": 1.7178743231717137, "original_repeated_conclusion_pct": 13.18359375, "edited_repeated_conclusion_pct": 7.91015625}]Submission audit: [{"tensor": "model.layers.8.mlp.down_proj.weight", "columns": [699], "multiplier": 1.5, "shape": [640, 2048], "changed_values": 640, "relative_frobenius_change": 0.009435703977942467}, {"tensor": "model.layers.14.mlp.down_proj.weight", "columns": [483, 984], "multiplier": 1.5, "shape": [640, 2048], "changed_values": 1280, "relative_frobenius_change": 0.015226062387228012}, {"tensor": "model.layers.17.mlp.down_proj.weight", "columns": [1179], "multiplier": 1.5, "shape": [640, 2048], "changed_values": 640, "relative_frobenius_change": 0.010259049944579601}]Submission unchanged: True

FINAL POOLED HOLDOUT{  "passes": [    8,    9,    10  ],  "n_per_model": 3072,  "distinct_problems": 256,  "original_correct": 221,  "edited_correct": 243,  "original_accuracy_pct": 7.194010416666667,  "edited_accuracy_pct": 7.91015625,  "gain_pp": 0.7161458333333333,  "cluster_se_pp": 0.4926066250697082,  "sample_pair_se_pp": 0.4116191268561949,  "wrong_to_right": 91,  "right_to_wrong": 69,  "original_mean_tokens": 1546.353515625,  "original_cut_pct": 7.389322916666667,  "original_repeated_conclusion_pct": 15.33203125,  "edited_mean_tokens": 1435.96484375,  "edited_cut_pct": 6.25,  "edited_repeated_conclusion_pct": 12.01171875,  "approx_95pct_ci_pp": [    -0.24936315180329482,    1.6816548184699613  ],  "candidate_frozen_before_holdout": true,  "selection_used_holdout": false,  "edit_hash": "599aba54ec"}DEV strength tests: [{"pass": 5, "column_multiplier": 0.0, "gain_pp": -0.68359375, "cluster_se_pp": 1.6418185113361485, "original_repeated_conclusion_pct": 13.57421875, "edited_repeated_conclusion_pct": 22.55859375}, {"pass": 6, "column_multiplier": 2.0, "gain_pp": -1.26953125, "cluster_se_pp": 1.4207781798694787, "original_repeated_conclusion_pct": 13.37890625, "edited_repeated_conclusion_pct": 5.6640625}, {"pass": 7, "column_multiplier": 1.5, "gain_pp": 0.48828125, "cluster_se_pp": 1.7178743231717137, "original_repeated_conclusion_pct": 13.18359375, "edited_repeated_conclusion_pct": 7.91015625}]Submission audit: [{"tensor": "model.layers.8.mlp.down_proj.weight", "columns": [699], "multiplier": 1.5, "shape": [640, 2048], "changed_values": 640, "relative_frobenius_change": 0.009435703977942467}, {"tensor": "model.layers.14.mlp.down_proj.weight", "columns": [483, 984], "multiplier": 1.5, "shape": [640, 2048], "changed_values": 1280, "relative_frobenius_change": 0.015226062387228012}, {"tensor": "model.layers.17.mlp.down_proj.weight", "columns": [1179], "multiplier": 1.5, "shape": [640, 2048], "changed_values": 640, "relative_frobenius_change": 0.010259049944579601}]Submission unchanged: True

Submitted the frozen edit in /work/edited: 1.5× amplification of these zero-indexed mlp.down_proj.weight columns:

Layer 8: 699
Layer 14: 483, 984
Layer 17: 1179

Only 2,560 weight values changed; all other entries, embeddings, output weights, and norms remain untouched.

Baseline analysis and teacher-forced probes identified this repetition-associated channel group, but causal tests reversed the initial hypothesis: removing it increased repetitive conclusions, while strengthening it reduced them. The 1.5× strength had the best paired DEV accuracy and was frozen before HOLDOUT.

Across 3,072 HOLDOUT answers per model, accuracy increased from 7.19% to 7.91%: +0.72 percentage points, SE 0.49, with an approximate 95% interval of −0.25 to +1.68 points, so the accuracy gain remains inconclusive. Repeated-conclusion responses fell from 15.33% to 12.01%, supporting improved conclusion control rather than newly acquired mathematical knowledge.

The complete audit and limitations are saved in /work/FINAL_REPORT.json.