tarunsachdeva shared in

Public

Analyzed middle blocks of transformer model for learnedPublic

Agent

Codex

Model

GPT 6 Astra

Started

Nov 3, 3:08 AM

Messages15

Downloads0

Summary

In Progress

Understand
Reviewed model architecture and weight matrices.
Understand
Build
Performed linear algebra analyses on weight matrices.
Build
Understand
Computed singular values and clustering of weight components.
Understand
Understand
Analyzed matrix alignments and eigenvalue spectra.
Understand
Verify
Identified candidate positional-selector head and shift-like eigenpairs.
Verify

Below is a small character-level language model: a 4-layer transformer with 6 attention heads per layer, which reads up to 128 characters and outputs scores (logits) for the next character. This is exactly how it computes (numpy):

python

import numpy as npVOCAB = "\n " + "abcdefghijklmnopqrstuvwxyz"   # index 0 = newline, 1 = space, 2..27 = a..zHEADS, LAYERS = 6, 4
def layer_norm(x, w, b):                     # over the last axis, eps 1e-5    mu = x.mean(-1, keepdims=True)    var = ((x - mu) ** 2).mean(-1, keepdims=True)    return (x - mu) / np.sqrt(var + 1e-5) * w + b
def next_char_logits(text, p):    """text: up to 128 characters (only the last 128 are used); p: dict of the weight arrays.    Returns the 28 logits for the character that comes next."""    ids = [VOCAB.index(c) for c in text[-128:]]    n, d = len(ids), p["tok_emb"].shape[1]    hd = d // HEADS    x = p["tok_emb"][ids] + p["pos_emb"][:n]                  # (n, d); positions 0..n-1 from the start of the text    for l in range(LAYERS):                                    # layer l uses the tensors named <name>.<l>        h = layer_norm(x, p[f"ln1_w.{l}"], p[f"ln1_b.{l}"])        q, k, v = h @ p[f"W_q.{l}"].T, h @ p[f"W_k.{l}"].T, h @ p[f"W_v.{l}"].T        out = np.zeros((n, d))        for j in range(HEADS):                                 # head j uses dimensions j*hd .. (j+1)*hd - 1            s = slice(j * hd, (j + 1) * hd)            att = q[:, s] @ k[:, s].T / np.sqrt(hd)            # (n, n)            att = att + np.triu(np.full((n, n), -np.inf), 1)   # causal: position i sees positions 0..i            att = np.exp(att - att.max(-1, keepdims=True))            att = att / att.sum(-1, keepdims=True)            out[:, s] = att @ v[:, s]        x = x + out @ p[f"W_o.{l}"].T        h = layer_norm(x, p[f"ln2_w.{l}"], p[f"ln2_b.{l}"])        x = x + np.maximum(0.0, h @ p[f"W_fc.{l}"].T) @ p[f"W_proj.{l}"].T   # ReLU MLP, hidden 4 d    h = layer_norm(x, p["lnf_w"], p["lnf_b"])    return h[-1] @ p["W_out"].T + p["b_out"]                   # logits of the next character (28)

Training: from random initialization (every matrix entry drawn from a normal distribution with standard deviation 0.02, or 0.007071 for W_o and W_proj; LayerNorm weights 1 and biases 0; b_out 0), next-character prediction with AdamW (30,517 steps of 256 random 128-character windows, 1,000,000,000 characters in total; learning-rate warm-up then cosine decay; weight decay 0.01 on the matrices).

The training text was ONE of these three corpora:

python: Python source code (about 36,600 files of Python libraries, docstrings and comments removed; 320 million characters)
romaji: Japanese literature (the Aozora Bunko collection, about 17,000 works) written in romaji (Latin letters, Hepburn-style romanization; 564 million characters)
english: English text (English Wikipedia articles; 500 million characters) and it was used EITHER as is OR with the characters of every line randomly shuffled (each line keeps its letters and its length, the order of the characters inside the line is random, the order of the lines is kept). Normalization (the same for all three corpora, before anything else): lowercased; in each line the leading spaces (indentation) were kept, every other character that is not a letter a-z became a space, runs of spaces were collapsed to one and trailing spaces removed. So the text contains only newlines, spaces and the letters a-z (the 28 characters of VOCAB).

WHAT YOU CAN USE. The weights inside the four transformer blocks, exactly as trained (float32): for layer l = 0, 1, 2 and 3, ln1_w.l, ln1_b.l, W_q.l, W_k.l, W_v.l, W_o.l (attention) and ln2_w.l, ln2_b.l, W_fc.l, W_proj.l (MLP). The input layer (tok_emb, the 28 x d character embeddings, and pos_emb, the 128 x d position embeddings) and the output layer (lnf_w, lnf_b, W_out, b_out) are NOT available. The d = 384 dimensions of the residual stream carry no labels. Head j of layer l uses rows 64j .. 64j+63 of W_q.l, W_k.l and W_v.l and columns 64j .. 64j+63 of W_o.l.

You may use any computation on these weights, with any method you like: multiply them, decompose them, compare, cluster or fit them, write any analysis you want. Two things are NOT allowed:

A forward pass: do not run the network or any of its layers (LayerNorm, attention with its softmax, the MLP with its ReLU) on any input, whether real, random or constructed, in code, in your head or on paper. Treating the weight matrices as linear-algebra objects (products, eigenvalues, singular vectors, applying one weight matrix to directions taken from other weights) is fine; simulating what the network does to an input is not.
Search: do not look anything up (there is no internet access in any case).

HOW YOU COMPUTE. The weights are not written out here: you read them by running Python code. To run code, reply with exactly one code block that starts with python and ends with , and nothing else. We run it and reply with its printed output. Available: numpy, scipy, scikit-learn, PyTorch (CPU), pandas, sympy, networkx. Before your code runs, numpy is imported as np and W is a dict of the weight arrays by name: W['W_q.0'] is 384 x 384, W['W_fc.0'] is 1536 x 384, W['W_proj.0'] is 384 x 1536, W['ln1_w.0'] has 384 entries, and so on; matrices are (out, in), as used in the code above. There is nothing else to read (no other files, no network). Each run is a fresh process (nothing is kept between runs), may take up to 600 seconds, and its output is cut at 20,000 characters. You may run code up to 100 times. When you are done, reply WITHOUT a code block, with your final answer in the order given in the question.

QUESTION. What was this network trained on, and how does it use the context? Answer in this order: Observations: <what you see in the shown weights and what it suggests; quote the weights you rely on> Heads: <one line per head, L0H0 .. L0H5, L1H0 .. L1H5, L2H0 .. L2H5, L3H0 .. L3H5: what it most likely does, as one of: 'previous' (attends to the previous character), 'local' (a few characters back), 'broad' (spread far back), 'copy' (induction: attends to the character right after an earlier occurrence of the current character or context, so the network can repeat earlier text), or 'unclear'; then your evidence> Reasoning: <which training text fits these observations, and why> corpus: <python | romaji | english> order: <as is | shuffled> probabilities: python=

romaji=

english=

; as is=

shuffled=

Equal probabilities are not allowed. Even if the evidence is weak, give your best judgement.

python

import numpy as npimport scipy.linalg as lafrom threadpoolctl import threadpool_limitsthreadpool_limits(4)
def norm(x):    return float(la.norm(x))
print("All calculations below are weight-only linear algebra; no network evaluations.")for l in range(4):    print("\nLAYER", l)    for name in ["ln1_w", "ln1_b", "ln2_w", "ln2_b"]:        a = W[f"{name}.{l}"].astype(np.float64)        print(name, "min/mean/max/std/norm",              np.round([a.min(), a.mean(), a.max(), a.std(), norm(a)], 5))    for name in ["W_q", "W_k", "W_v", "W_o", "W_fc", "W_proj"]:        a = W[f"{name}.{l}"].astype(np.float64)        s = la.svdvals(a)        print(name, "Frob", round(norm(a), 4),              "top8_sv", np.round(s[:8], 4),              "stable_rank", round(np.sum(s*s)/(s[0]*s[0]), 3))    print("HEAD: QK_frob QK_top4_sv QK_top1_energy QK_sym_fraction QK_trace; "          "OV_frob OV_top4_sv OV_trace OV_eig_posmax OV_eig_negmin")    for j in range(6):        sl = slice(64*j, 64*(j+1))        q = W[f"W_q.{l}"][sl].astype(np.float64)        k = W[f"W_k.{l}"][sl].astype(np.float64)        v = W[f"W_v.{l}"][sl].astype(np.float64)        o = W[f"W_o.{l}"][:, sl].astype(np.float64)        a = q.T @ k        b = o @ v        sa, sb = la.svdvals(a)[:64], la.svdvals(b)[:64]        eb = la.eigvals(v @ o)        print(f"L{l}H{j}",              round(norm(a), 4), np.round(sa[:4], 4),              round(sa[0]**2/np.sum(sa**2), 4),              round(norm((a+a.T)/2)/norm(a), 4), round(np.trace(a), 4),              ";", round(norm(b), 4), np.round(sb[:4], 4),              round(np.trace(b), 4), round(eb.real.max(), 4),              round(eb.real.min(), 4))

OUTPUT of run 1 of at most 100:

All calculations below are weight-only linear algebra; no network evaluations.
LAYER 0ln1_w min/mean/max/std/norm [-0.001   0.2403  1.0341  0.2436  6.7051]ln1_b min/mean/max/std/norm [-0.0959  0.0008  0.091   0.0287  0.5635]ln2_w min/mean/max/std/norm [ 0.0044  0.8006  1.5072  0.3295 16.9648]ln2_b min/mean/max/std/norm [-0.4986  0.0022  0.3331  0.0683  1.3392]W_q Frob 32.2912 top8_sv [8.2519 8.2232 7.963  7.819  7.68   7.6416 7.5194 7.2054] stable_rank 15.313W_k Frob 30.3909 top8_sv [7.6464 7.4363 7.3304 6.9118 6.8285 6.6716 6.4724 6.4464] stable_rank 15.797W_v Frob 14.836 top8_sv [4.339  3.5292 3.2699 3.1656 2.9824 2.6465 2.589  2.5519] stable_rank 11.691W_o Frob 16.3713 top8_sv [7.1033 6.6074 4.2683 2.7702 2.5025 2.285  2.2638 2.2184] stable_rank 5.312W_fc Frob 46.2486 top8_sv [18.1759 11.0178  8.7366  7.4517  7.2123  7.0621  7.0112  6.6204] stable_rank 6.474W_proj Frob 39.3075 top8_sv [9.8257 7.3508 6.2651 5.8693 5.7965 5.6574 5.5163 5.4501] stable_rank 16.004HEAD: QK_frob QK_top4_sv QK_top1_energy QK_sym_fraction QK_trace; OV_frob OV_top4_sv OV_trace OV_eig_posmax OV_eig_negminL0H0 3.0479 [2.5097 0.8949 0.6966 0.4393] 0.678 0.7267 -1.7245 ; 12.9041 [10.2524  5.6035  3.4749  2.8184] -0.1807 1.1732 -0.4325L0H1 1.8376 [1.0971 0.7651 0.4518 0.343 ] 0.3564 0.7472 -1.2358 ; 10.1485 [7.662  5.1724 2.5819 2.0305] -0.7493 0.9812 -0.9241L0H2 182.4351 [59.8786 57.2786 55.5815 54.3474] 0.1077 0.6963 32.7753 ; 13.223 [4.0009 3.853  3.7432 3.5437] -4.6421 0.4428 -0.6979L0H3 11.3515 [6.568  5.9931 3.2141 2.6459] 0.3348 0.7629 -7.9874 ; 7.3079 [4.305  4.0456 2.4933 2.078 ] -2.0478 0.262 -0.6907L0H4 6.6069 [6.0218 2.0421 1.078  0.5566] 0.8307 0.7629 -4.7252 ; 7.3117 [4.8786 3.2357 2.5484 2.0826] -0.9671 0.3599 -0.4665L0H5 2.5787 [2.1893 0.543  0.3941 0.328 ] 0.7208 0.7288 -0.9664 ; 13.0384 [10.3142  5.4678  3.1482  2.9348] -0.0858 0.827 -0.41
LAYER 1ln1_w min/mean/max/std/norm [0.1361 0.4575 1.1239 0.1022 9.1868]ln1_b min/mean/max/std/norm [-1.4712 -0.005   1.1544  0.1672  3.2771]ln2_w min/mean/max/std/norm [-0.0019  1.0429  1.4856  0.2003 20.8093]ln2_b min/mean/max/std/norm [-0.3597  0.0005  0.3686  0.1364  2.6732]W_q Frob 32.3126 top8_sv [9.9032 7.7974 6.7042 5.9516 5.6721 5.6025 5.4021 5.3125] stable_rank 10.646W_k Frob 29.2506 top8_sv [13.3072  9.9327  5.8989  5.677   5.0458  4.622   4.4903  4.4044] stable_rank 4.832W_v Frob 27.3529 top8_sv [5.2845 4.9661 4.87   4.7328 4.6743 4.5853 4.4113 4.3216] stable_rank 26.792W_o Frob 25.9233 top8_sv [4.9657 3.2412 3.157  3.1119 3.0615 3.0531 2.9942 2.9545] stable_rank 27.253W_fc Frob 79.1265 top8_sv [21.4881 13.1349 12.8042 12.6247 12.3279 12.1184 11.8504 11.6908] stable_rank 13.56W_proj Frob 75.7103 top8_sv [8.1935 7.8941 7.1992 6.9972 6.671  6.4329 6.2745 6.1333] stable_rank 85.384HEAD: QK_frob QK_top4_sv QK_top1_energy QK_sym_fraction QK_trace; OV_frob OV_top4_sv OV_trace OV_eig_posmax OV_eig_negminL1H0 23.6654 [12.6145 11.6953  8.2823  7.0164] 0.2841 0.7351 -26.1053 ; 21.7066 [8.0989 7.5796 6.3982 6.2019] -21.6005 1.6279 -2.7755L1H1 35.5126 [19.649  17.056  10.7527  9.5417] 0.3061 0.7424 -44.9928 ; 22.226 [6.5833 6.0091 5.7392 5.681 ] -28.4652 0.5693 -2.0517L1H2 41.7812 [23.3819 18.3356 12.0604 11.2132] 0.3132 0.7448 -60.5649 ; 19.5165 [6.1346 5.7881 5.2517 4.9869] -21.8959 0.2894 -1.708L1H3 36.8369 [21.1075 17.3804 11.6312  9.1371] 0.3283 0.7402 -44.4839 ; 21.4133 [7.7675 6.8644 6.5273 6.0604] -20.2784 0.2159 -1.8799L1H4 37.3819 [22.9586 17.425  11.442   8.6811] 0.3772 0.7366 -46.5798 ; 22.0658 [6.4984 6.2533 5.8966 5.7835] -26.577 0.1462 -2.0645L1H5 36.2112 [20.9422 18.3428 10.3483  8.4275] 0.3345 0.7295 -39.7087 ; 22.4638 [6.2081 6.0193 5.803  5.6451] -29.2909 0.2127 -2.1146
LAYER 2ln1_w min/mean/max/std/norm [ 0.2078  0.7052  1.0144  0.0749 13.8969]ln1_b min/mean/max/std/norm [-1.3819 -0.001   1.0527  0.1525  2.9893]ln2_w min/mean/max/std/norm [ 0.0011  1.3123  1.4655  0.1255 25.8329]ln2_b min/mean/max/std/norm [-0.3761  0.005   0.3886  0.1537  3.0128]W_q Frob 39.3455 top8_sv [5.7962 5.1962 4.8092 4.6159 4.6003 4.5145 4.4419 4.432 ] stable_rank 46.08W_k Frob 37.7688 top8_sv [10.6627  8.6244  4.9887  4.3937  4.241   4.0878  3.9564  3.8475] stable_rank 12.547W_v Frob 33.3048 top8_sv [3.7062 3.6245 3.5788 3.5334 3.5012 3.4679 3.4285 3.4153] stable_rank 80.753W_o Frob 32.8397 top8_sv [4.1025 3.9708 3.558  3.3813 3.3611 3.2925 3.1918 3.1686] stable_rank 64.077W_fc Frob 78.9668 top8_sv [15.3634  8.0212  7.8946  7.7247  7.5968  7.4905  7.2574  7.2158] stable_rank 26.419W_proj Frob 76.8311 top8_sv [9.4036 9.1867 8.7234 8.4703 8.0398 7.7955 7.6145 7.4501] stable_rank 66.756HEAD: QK_frob QK_top4_sv QK_top1_energy QK_sym_fraction QK_trace; OV_frob OV_top4_sv OV_trace OV_eig_posmax OV_eig_negminL2H0 38.5746 [11.6585  9.9934  9.1292  8.7027] 0.0913 0.702 1.9543 ; 22.8142 [4.8981 4.6278 4.5529 4.3995] -54.2674 0.056 -2.1124L2H1 33.1082 [9.2369 7.1417 7.0468 6.9344] 0.0778 0.7128 9.4557 ; 24.8446 [5.4684 5.1567 4.9683 4.7548] -50.1368 0.2665 -2.1816L2H2 25.2291 [8.8799 7.5578 6.1827 5.    ] 0.1239 0.7114 -16.2093 ; 28.7074 [7.8628 7.1802 6.9013 5.9893] -46.7717 0.8422 -2.9627L2H3 32.8218 [8.5292 7.9451 7.259  7.0144] 0.0675 0.7059 -11.8331 ; 24.5055 [6.1803 5.7997 5.5394 4.9316] -49.7953 0.2369 -2.7194L2H4 36.8275 [9.5029 8.8386 8.6143 8.1133] 0.0666 0.7045 -7.386 ; 23.2786 [5.0699 4.7745 4.6597 4.505 ] -45.5437 0.2805 -1.9452L2H5 35.3545 [9.6053 8.5044 7.7528 7.5417] 0.0738 0.7043 -10.2817 ; 23.3416 [4.8404 4.6932 4.4898 4.4832] -50.7511 0.2453 -2.3041
LAYER 3ln1_w min/mean/max/std/norm [ 0.2944  0.7546  0.8701  0.0542 14.8247]ln1_b min/mean/max/std/norm [-0.6069  0.0121  0.6502  0.0849  1.6801]ln2_w min/mean/max/std/norm [ 0.4364  1.2778  1.4306  0.0808 25.0891]ln2_b min/mean/max/std/norm [-0.6728  0.0043  0.3068  0.1276  2.5016]W_q Frob 32.9482 top8_sv [4.5598 4.2388 4.0054 3.9241 3.8529 3.7087 3.7044 3.6911] stable_rank 52.212W_k Frob 32.2094 top8_sv [6.0903 4.8989 4.1973 3.4559 3.3531 3.3051 3.2455 3.1608] stable_rank 27.97W_v Frob 39.7941 top8_sv [5.0598 4.9045 4.8481 4.769  4.6738 4.6053 4.5722 4.5185] stable_rank 61.854W_o Frob 40.455 top8_sv [9.5668 8.7592 5.3386 5.139  4.9162 4.8063 4.7743 4.7359] stable_rank 17.882W_fc Frob 77.7758 top8_sv [14.1624  8.1249  7.9152  7.3262  7.1484  6.8556  6.7628  6.6911] stable_rank 30.159W_proj Frob 82.3952 top8_sv [26.3094 20.6845 19.6986 19.2612 18.5937 18.1973 17.8617 17.6671] stable_rank 9.808HEAD: QK_frob QK_top4_sv QK_top1_energy QK_sym_fraction QK_trace; OV_frob OV_top4_sv OV_trace OV_eig_posmax OV_eig_negminL3H0 31.2433 [6.3237 5.9038 5.873  5.6567] 0.041 0.725 47.8937 ; 29.6831 [8.3838 7.5828 6.4196 6.2466] -72.8707 0.1992 -3.1664L3H1 25.789 [4.5716 4.43   4.4197 4.2853] 0.0314 0.7783 89.2162 ; 53.3976 [13.8537 13.6123 13.3779 12.3595] -70.0842 0.999 -5.3422L3H2 21.0416 [6.4644 5.7939 5.0121 4.6657] 0.0944 0.7065 3.1927 ; 35.2432 [11.8908  9.927   8.462   8.2846] -45.1262 0.6127 -3.171L3H3 22.3492 [6.8507 5.6549 4.6934 4.3132] 0.094 0.7018 1.5275 ; 35.9947 [12.4344 10.3968  9.0091  8.0613] -52.1078 0.6827 -3.1082L3H4 25.9947 [7.3521 5.9014 5.5575 5.2874] 0.08 0.7074 12.8867 ; 31.7775 [9.5258 8.963  7.0112 6.9513] -44.1288 0.5408 -2.9315L3H5 11.8506 [3.8648 3.3538 2.9899 2.7941] 0.1064 0.7157 1.3911 ; 59.2935 [21.9791 19.2007 17.4205 16.747 ] -60.0223 1.0804 -8.5079

python

import numpy as npimport scipy.linalg as lafrom threadpoolctl import threadpool_limitsthreadpool_limits(4)
A, B = {}, {}for l in range(4):    w = W[f"ln1_w.{l}"].astype(float)    bias = W[f"ln1_b.{l}"].astype(float)    print("\nLAYER", l, "(gain-weighted matrices; no LayerNorm evaluation)")    for j in range(6):        s = slice(64*j, 64*(j+1))        q0 = W[f"W_q.{l}"][s].astype(float)        k0 = W[f"W_k.{l}"][s].astype(float)        v0 = W[f"W_v.{l}"][s].astype(float)        o = W[f"W_o.{l}"][:,s].astype(float)        q, k, v = q0*w, k0*w, v0*w        a, b = q.T@k, o@v        A[l,j], B[l,j] = a, b        sv = la.svdvals(a)[:64]        qb, kb = q0@bias, k0@bias        key_linear = k.T@qb        query_linear = q.T@kb        print(f"L{l}H{j}",              "QKnorm", round(la.norm(a),4),              "QKsv", np.round(sv[:6],4),              "QKtrace", round(np.trace(a),4),              "OVnorm", round(la.norm(b),4),              "bias_key/query_norm", np.round([la.norm(key_linear),la.norm(query_linear)],4))
print("\nNormalized trace(A_lh @ B_0j), columns L0H0..L0H5")for l in range(1,4):    for j in range(6):        a = A[l,j]        vals = [np.trace(a@B[0,k])/(la.norm(a)*la.norm(B[0,k])) for k in range(6)]        print(f"L{l}H{j}", np.round(vals,4))
print("\nLargest earlier-head trace compositions (signed normalized trace)")for l in range(2,4):    for j in range(6):        a = A[l,j]        vals = []        for m in range(l):            for k in range(6):                b = B[m,k]                vals.append((float(np.trace(a@b)/(la.norm(a)*la.norm(b))), f"L{m}H{k}"))        print(f"L{l}H{j}", sorted(vals, reverse=True)[:5])
print("\nL0H2 effective QK spectrum")a = A[0,2]u,s,vt = la.svd(a, full_matrices=False)print("singular values", np.round(s[:40],5))ev = la.eigvals(a)ix = np.argsort(abs(ev))[::-1][:24]print("largest eigenvalues", np.round(ev[ix],4))print("left/right top-20 subspace overlap", np.round(la.svdvals(u[:,:20].T@vt[:20].T),4))
print("\nL0 query/key and value subspace overlap with L0H2 dominant QK subspace")p = la.orth(np.column_stack([u[:,:20],vt[:20].T]))for j in range(6):    s0 = slice(64*j,64*(j+1))    vals=[]    for name in ["W_q","W_k","W_v"]:        m = W[f"{name}.0"][s0].astype(float)*W["ln1_w.0"]        vals.append(la.norm(m@p)**2/la.norm(m)**2)    print(f"L0H{j}", np.round(vals,4))

OUTPUT of run 2 of at most 100:


LAYER 0 (gain-weighted matrices; no LayerNorm evaluation)L0H0 QKnorm 0.4367 QKsv [0.4114 0.0916 0.0567 0.0368 0.0343 0.0289] QKtrace -0.0379 OVnorm 9.0429 bias_key/query_norm [0.1091 0.1694]L0H1 QKnorm 0.2362 QKsv [0.1997 0.0761 0.0422 0.0353 0.0279 0.0247] QKtrace -0.1115 OVnorm 6.9661 bias_key/query_norm [0.0267 0.0107]L0H2 QKnorm 32.6269 QKsv [11.3199 10.1995  9.7194  9.6486  9.272   8.7533] QKtrace 9.5999 OVnorm 2.54 bias_key/query_norm [1.1448 1.5108]L0H3 QKnorm 3.1569 QKsv [2.8706 0.9529 0.621  0.3398 0.261  0.2181] QKtrace 1.5753 OVnorm 1.7662 bias_key/query_norm [0.4283 0.3216]L0H4 QKnorm 0.8508 QKsv [0.7136 0.4284 0.1319 0.0522 0.0419 0.0328] QKtrace -0.5674 OVnorm 4.0336 bias_key/query_norm [0.0901 0.0772]L0H5 QKnorm 0.6206 QKsv [0.6073 0.0861 0.0369 0.0299 0.0271 0.023 ] QKtrace -0.2776 OVnorm 9.3107 bias_key/query_norm [0.1299 0.2023]
LAYER 1 (gain-weighted matrices; no LayerNorm evaluation)L1H0 QKnorm 7.0989 QKsv [4.7823 3.8586 1.9815 1.5518 1.0526 0.8946] QKtrace -5.1542 OVnorm 10.1223 bias_key/query_norm [10.8591 13.1828]L1H1 QKnorm 10.5253 QKsv [7.0546 5.3302 3.2915 2.1355 1.6238 1.4283] QKtrace -9.2943 OVnorm 10.3335 bias_key/query_norm [18.2783 18.644 ]L1H2 QKnorm 12.41 QKsv [7.9573 6.8828 3.0682 2.603  2.115  1.8171] QKtrace -12.955 OVnorm 9.1806 bias_key/query_norm [20.7031 23.8558]L1H3 QKnorm 11.0075 QKsv [7.3565 5.9091 3.2626 2.029  1.764  1.3744] QKtrace -9.2582 OVnorm 10.0512 bias_key/query_norm [18.809  18.7776]L1H4 QKnorm 11.6052 QKsv [7.5816 6.9475 2.8775 2.0717 1.5464 1.3804] QKtrace -9.9582 OVnorm 10.401 bias_key/query_norm [18.3219 23.5867]L1H5 QKnorm 10.7065 QKsv [7.146  5.914  2.9776 1.8365 1.6321 1.4794] QKtrace -8.1163 OVnorm 10.6202 bias_key/query_norm [14.0211 19.6845]
LAYER 2 (gain-weighted matrices; no LayerNorm evaluation)L2H0 QKnorm 19.5056 QKsv [5.5592 5.0376 4.8195 4.5062 4.2407 4.0516] QKtrace -0.173 OVnorm 16.3068 bias_key/query_norm [13.5928 13.0054]L2H1 QKnorm 16.5448 QKsv [4.2136 3.5951 3.4598 3.3372 3.2779 3.161 ] QKtrace 4.237 OVnorm 17.7221 bias_key/query_norm [5.9578 7.4963]L2H2 QKnorm 12.9062 QKsv [4.8575 4.024  3.2103 2.5372 2.3832 2.2932] QKtrace -8.8427 OVnorm 20.5752 bias_key/query_norm [ 6.0447 12.5128]L2H3 QKnorm 16.6966 QKsv [4.4958 4.2052 3.9324 3.5228 3.2925 3.1557] QKtrace -5.9051 OVnorm 17.5062 bias_key/query_norm [11.702  11.0247]L2H4 QKnorm 18.7551 QKsv [5.0228 4.8342 4.2578 4.0652 3.7358 3.6109] QKtrace -3.8054 OVnorm 16.624 bias_key/query_norm [12.6211 13.1065]L2H5 QKnorm 18.0092 QKsv [5.1073 4.3177 3.95   3.8098 3.7436 3.615 ] QKtrace -5.578 OVnorm 16.6842 bias_key/query_norm [12.0719 11.6318]
LAYER 3 (gain-weighted matrices; no LayerNorm evaluation)L3H0 QKnorm 17.8612 QKsv [3.6121 3.3594 3.345  3.2422 3.2067 3.1565] QKtrace 27.2957 OVnorm 22.5984 bias_key/query_norm [2.0652 2.089 ]L3H1 QKnorm 14.6476 QKsv [2.6208 2.5393 2.4737 2.4472 2.4061 2.3836] QKtrace 50.449 OVnorm 40.8686 bias_key/query_norm [1.9323 1.0931]L3H2 QKnorm 12.0463 QKsv [3.7667 3.2887 2.8452 2.6572 2.4694 2.3229] QKtrace 1.7643 OVnorm 26.8815 bias_key/query_norm [2.3165 2.2163]L3H3 QKnorm 12.7936 QKsv [3.9788 3.2199 2.6861 2.4313 2.3645 2.3594] QKtrace 0.8934 OVnorm 27.4312 bias_key/query_norm [2.8828 2.4488]L3H4 QKnorm 14.8484 QKsv [4.2301 3.3    3.1849 2.9889 2.8823 2.6888] QKtrace 7.2288 OVnorm 24.1979 bias_key/query_norm [2.4576 2.0855]L3H5 QKnorm 6.679 QKsv [2.075  1.8792 1.6976 1.57   1.5251 1.2757] QKtrace 0.7931 OVnorm 45.4008 bias_key/query_norm [1.3167 0.7099]
Normalized trace(A_lh @ B_0j), columns L0H0..L0H5L1H0 [ 0.0388  0.0163 -0.0054 -0.0187  0.032   0.0041]L1H1 [ 0.0457  0.0329 -0.0001 -0.0143  0.0496  0.0187]L1H2 [ 0.0497  0.0415  0.0145 -0.0059  0.054   0.0222]L1H3 [ 0.0584  0.0467  0.0147 -0.0099  0.0601  0.0281]L1H4 [ 0.0796  0.0707  0.0133 -0.0046  0.0839  0.0459]L1H5 [ 0.0386  0.0174  0.0059 -0.0139  0.0363 -0.0111]L2H0 [ 0.0156  0.0124 -0.005  -0.0182  0.0183  0.0159]L2H1 [ 0.0098  0.0065 -0.0056 -0.0102  0.0104  0.0116]L2H2 [ 0.0295  0.0238  0.001  -0.0173  0.0355  0.01  ]L2H3 [ 0.0227  0.0162  0.0029 -0.012   0.023   0.0216]L2H4 [ 0.0293  0.0219 -0.008  -0.0169  0.0299  0.021 ]L2H5 [ 0.0222  0.0146  0.0003 -0.0172  0.0214  0.0168]L3H0 [ 0.0046  0.0048 -0.0022 -0.0031  0.007   0.0069]L3H1 [-0.0023 -0.0035  0.0949  0.001  -0.0052 -0.0028]L3H2 [ 0.016   0.0102  0.003  -0.0059  0.0141  0.0221]L3H3 [ 0.0104  0.0049  0.0012 -0.0018  0.0084  0.0144]L3H4 [ 0.0196  0.0205 -0.0048 -0.0013  0.0227  0.0282]L3H5 [-0.005  -0.0047  0.0025 -0.004  -0.0044 -0.003 ]
Largest earlier-head trace compositions (signed normalized trace)L2H0 [(0.01831488054770177, 'L0H4'), (0.015886681026769052, 'L0H5'), (0.015608857494623197, 'L0H0'), (0.012367579270624375, 'L0H1'), (-0.004385075845198281, 'L1H1')]L2H1 [(0.011626730747948765, 'L0H5'), (0.010425051990288375, 'L0H4'), (0.009803403048760628, 'L0H0'), (0.006525725144364843, 'L0H1'), (0.0030746091424649824, 'L1H3')]L2H2 [(0.03548392181702849, 'L0H4'), (0.029485073254451613, 'L0H0'), (0.023778341440254248, 'L0H1'), (0.009965517460207657, 'L0H5'), (0.007742027119735287, 'L1H4')]L2H3 [(0.023009692499583036, 'L0H4'), (0.022655398470314534, 'L0H0'), (0.02159495944098225, 'L0H5'), (0.016205740027523712, 'L0H1'), (0.002867347983306011, 'L0H2')]L2H4 [(0.02986057570010887, 'L0H4'), (0.02925880246372659, 'L0H0'), (0.021926972741687572, 'L0H1'), (0.02098021741375207, 'L0H5'), (-0.002996000199168058, 'L1H3')]L2H5 [(0.022163566772676495, 'L0H0'), (0.02137927166317656, 'L0H4'), (0.01679521007828187, 'L0H5'), (0.014566356586938216, 'L0H1'), (0.00035304827650425267, 'L1H2')]L3H0 [(0.00703240691007632, 'L0H4'), (0.006910922145819949, 'L0H5'), (0.004752035137208596, 'L0H1'), (0.004570716642140448, 'L0H0'), (-0.002247146526228783, 'L0H2')]L3H1 [(0.09486606986058785, 'L0H2'), (0.0811171357829096, 'L2H1'), (0.056764532759469445, 'L1H3'), (0.05549793736058776, 'L1H2'), (0.03612116030288871, 'L1H1')]L3H2 [(0.02205580262173361, 'L0H5'), (0.016017133729234214, 'L0H0'), (0.014117851397887326, 'L0H4'), (0.010234078605252283, 'L0H1'), (0.0029527680839085435, 'L0H2')]L3H3 [(0.014426395753234358, 'L0H5'), (0.010428059825454958, 'L0H0'), (0.008445143231364794, 'L0H4'), (0.004921340717805306, 'L0H1'), (0.0011732199754640239, 'L0H2')]L3H4 [(0.028195028938726028, 'L0H5'), (0.022746228327328106, 'L0H4'), (0.020535542760585734, 'L0H1'), (0.019626760961869532, 'L0H0'), (0.0007146672776497601, 'L2H0')]L3H5 [(0.01675224101706262, 'L2H3'), (0.01151727112046607, 'L2H1'), (0.0111474435163409, 'L1H0'), (0.011067087616306114, 'L2H4'), (0.01010903817591309, 'L2H2')]
L0H2 effective QK spectrumsingular values [11.3199 10.1995  9.7194  9.6486  9.272   8.7533  8.6907  8.1753  7.6461  7.3268  6.89    6.2532  5.8234  5.7713  4.1168  3.9356  3.4309  3.0088  2.5751  2.3197  1.5122  1.3247  1.0256  0.9255  0.8764  0.7851  0.601   0.4484  0.394   0.3479  0.2969  0.2686  0.2412  0.2179  0.1933  0.1571  0.156   0.1255  0.1161  0.1043]largest eigenvalues [ 5.0575+6.5134j  5.0575-6.5134j -1.8401+7.4813j -1.8401-7.4813j  2.6658+7.0019j  2.6658-7.0019j -6.2304+2.6725j -6.2304-2.6725j -6.5315+0.j  5.2072-3.2762j  5.2072+3.2762j  2.7023-4.0824j  2.7023+4.0824j  4.0955-0.1975j  4.0955+0.1975j  3.3297+0.j     -1.8666+1.9902j -1.8666-1.9902j -2.3481+1.0233j -2.3481-1.0233j  0.1685+1.0747j  0.1685-1.0747j -0.946 +0.j     -0.7171-0.0125j]left/right top-20 subspace overlap [0.9877 0.9845 0.9797 0.975  0.9689 0.9672 0.9649 0.9539 0.9482 0.9435 0.9412 0.9304 0.9103 0.8837 0.8478 0.8088 0.7843 0.6577 0.541  0.3834]
L0 query/key and value subspace overlap with L0H2 dominant QK subspaceL0H0 [0.6828 0.3702 0.8338]L0H1 [0.4909 0.3968 0.8247]L0H2 [0.9601 0.971  0.3177]L0H3 [0.6627 0.7298 0.3891]L0H4 [0.5381 0.4254 0.7063]L0H5 [0.6866 0.4371 0.8073]

python

import numpy as npimport scipy.linalg as lafrom threadpoolctl import threadpool_limitsthreadpool_limits(4)
for l in range(4):    print("\nLAYER",l)    for name in ["W_fc","W_proj"]:        m = W[f"{name}.{l}"].astype(float)        s = la.svdvals(m)        print(name,"first 40 SV:",np.round(s[:40],3))        print("SV indices 40,60,80,100,150,200,300,383:",              np.round(s[[40,60,80,100,150,200,300,383]],3))
def head(l,j):    sl=slice(j*64,(j+1)*64)    w=W[f"ln1_w.{l}"].astype(float)    q=W[f"W_q.{l}"][sl].astype(float)*w    k=W[f"W_k.{l}"][sl].astype(float)*w    v=W[f"W_v.{l}"][sl].astype(float)*w    o=W[f"W_o.{l}"][:,sl].astype(float)    return q.T@k,o@v
aa=[head(0,j)[0] for j in range(6)]bb=[head(0,j)[1] for j in range(6)]print("\nL0 OV matrix cosine similarities:")print(np.round([[np.sum(x*y)/(la.norm(x)*la.norm(y)) for y in bb] for x in bb],3))print("L0 QK matrix cosine similarities:")print(np.round([[np.sum(x*y)/(la.norm(x)*la.norm(y)) for y in aa] for x in aa],3))
print("\nL0H2 unweighted QK singular values:")q=W["W_q.0"][128:192].astype(float)k=W["W_k.0"][128:192].astype(float)u,s,vt=la.svd(q.T@k,full_matrices=False)print(np.round(s[:48],3))p=la.orth(np.column_stack((u[:,:20],vt[:20].T)))print("L0 raw Q/K/V energy in this subspace:")for j in range(6):    print(j, np.round([la.norm(W[f"{name}.0"][64*j:64*(j+1)]@p)**2/                       la.norm(W[f"{name}.0"][64*j:64*(j+1)])**2                       for name in ["W_q","W_k","W_v"]],4))
print("\nL1 key-bias weight directions and L0 OV outputs")for j in range(6):    sl=slice(j*64,(j+1)*64)    q=W["W_q.1"][sl].astype(float)    k=W["W_k.1"][sl].astype(float)    r=(k*W["ln1_w.1"]).T@(q@W["ln1_b.1"])    vals=[]    for b in bb:        vals.append(la.norm(r@b)/(la.norm(r)*la.norm(b)))    print(j,np.round(vals,4))
print("\nLinear-algebra-only MLP products: normalized trace with later QK")# These are matrix products, not MLP evaluations: there is no ReLU or input.mats={}for l in range(3):    f=W[f"W_fc.{l}"].astype(float)*W[f"ln2_w.{l}"]    mats[l]=W[f"W_proj.{l}"].astype(float)@ffor l in range(1,4):    for j in range(6):        a,b=head(l,j)        print(f"L{l}H{j}",              np.round([np.trace(a@mats[m])/(la.norm(a)*la.norm(mats[m]))                        for m in range(l)],4))

OUTPUT of run 3 of at most 100:


LAYER 0W_fc first 40 SV: [18.176 11.018  8.737  7.452  7.212  7.062  7.011  6.62   6.57   6.42   6.343  6.29   6.137  6.043  5.974  5.918  5.807  5.779  5.583  5.513  5.5  5.36   5.16   5.12   5.086  5.032  4.899  4.879  4.819  4.732  4.601  4.533  4.479  4.441  4.268  4.198  4.031  3.974  3.869  3.824]SV indices 40,60,80,100,150,200,300,383: [3.72  2.377 1.616 1.25  0.963 0.814 0.596 0.4  ]W_proj first 40 SV: [9.826 7.351 6.265 5.869 5.797 5.657 5.516 5.45  5.333 5.242 5.197 5.094 4.98  4.904 4.863 4.819 4.793 4.689 4.627 4.595 4.557 4.523 4.411 4.378 4.283 4.216 4.143 4.125 4.048 4.025 3.894 3.847 3.808 3.792 3.764 3.711 3.685 3.636 3.595 3.525]SV indices 40,60,80,100,150,200,300,383: [3.499 2.962 2.34  1.767 0.996 0.607 0.343 0.213]
LAYER 1W_fc first 40 SV: [21.488 13.135 12.804 12.625 12.328 12.118 11.85  11.691 11.43  11.269 11.049 10.846 10.646 10.62  10.324  9.928  9.758  9.699  9.592  9.353  9.219  9.169  9.074  8.921  8.816  8.729  8.516  8.442  8.433  8.247  8.106  7.853  7.745  7.696  7.617  7.54   7.422  7.349  7.124  7.084]SV indices 40,60,80,100,150,200,300,383: [6.851 5.28  4.27  3.346 1.842 1.274 0.846 0.411]W_proj first 40 SV: [8.193 7.894 7.199 6.997 6.671 6.433 6.274 6.133 6.099 6.021 5.942 5.865 5.829 5.789 5.698 5.675 5.646 5.629 5.581 5.564 5.553 5.538 5.528 5.477 5.456 5.447 5.438 5.403 5.399 5.38  5.367 5.351 5.332 5.327 5.314 5.296 5.252 5.235 5.219 5.208]SV indices 40,60,80,100,150,200,300,383: [5.197 4.925 4.704 4.479 3.985 3.509 2.629 0.319]
LAYER 2W_fc first 40 SV: [15.363  8.021  7.895  7.725  7.597  7.49   7.257  7.216  7.141  7.053  6.995  6.974  6.881  6.828  6.8    6.773  6.692  6.634  6.589  6.554  6.512  6.466  6.411  6.364  6.328  6.312  6.293  6.225  6.17   6.154  6.11   6.101  6.049  6.032  5.963  5.934  5.917  5.91   5.902  5.854]SV indices 40,60,80,100,150,200,300,383: [5.822 5.301 4.876 4.56  3.917 3.354 2.38  0.645]W_proj first 40 SV: [9.404 9.187 8.723 8.47  8.04  7.795 7.614 7.45  7.392 7.253 7.223 7.148 7.082 6.972 6.829 6.733 6.637 6.547 6.46  6.404 6.374 6.307 6.209 6.181 6.12  6.091 5.989 5.9   5.88  5.833 5.758 5.703 5.685 5.645 5.588 5.559 5.491 5.478 5.467 5.429]SV indices 40,60,80,100,150,200,300,383: [5.406 4.97  4.657 4.389 3.809 3.329 2.449 0.362]
LAYER 3W_fc first 40 SV: [14.162  8.125  7.915  7.326  7.148  6.856  6.763  6.691  6.489  6.436  6.387  6.31   6.26   6.177  6.144  6.088  6.066  6.017  5.962  5.934  5.883  5.855  5.843  5.781  5.752  5.72   5.697  5.657  5.632  5.617  5.593  5.568  5.519  5.517  5.495  5.45   5.431  5.389  5.379  5.361]SV indices 40,60,80,100,150,200,300,383: [5.341 4.986 4.708 4.465 3.94  3.5   2.678 0.683]W_proj first 40 SV: [26.309 20.685 19.699 19.261 18.594 18.197 17.862 17.667 17.044 16.749 16.039 15.151 14.756 14.326 14.05  13.5   13.293 12.531 12.401 11.958 11.74 11.594 10.367  8.351  8.064  7.914  7.461  7.166  5.833  4.072  3.221  3.139  2.842  2.8    2.687  2.571  2.403  2.328  2.244  2.185]SV indices 40,60,80,100,150,200,300,383: [2.181 1.645 1.422 1.244 0.961 0.765 0.509 0.318]
L0 OV matrix cosine similarities:[[1.    0.885 0.008 0.163 0.903 0.724] [0.885 1.    0.006 0.195 0.889 0.802] [0.008 0.006 1.    0.012 0.006 0.017] [0.163 0.195 0.012 1.    0.195 0.156] [0.903 0.889 0.006 0.195 1.    0.728] [0.724 0.802 0.017 0.156 0.728 1.   ]]L0 QK matrix cosine similarities:[[ 1.     0.209 -0.016 -0.014  0.115  0.018] [ 0.209  1.    -0.004 -0.036  0.209 -0.008] [-0.016 -0.004  1.     0.239 -0.006 -0.026] [-0.014 -0.036  0.239  1.    -0.054 -0.058] [ 0.115  0.209 -0.006 -0.054  1.     0.172] [ 0.018 -0.008 -0.026 -0.058  0.172  1.   ]]
L0H2 unweighted QK singular values:[59.879 57.279 55.581 54.347 51.234 49.452 47.184 44.678 43.21  41.651 39.871 35.376 31.577 26.606 23.538 21.633 19.958 18.271 16.837 14.93  13.224 10.856  9.22   8.099  8.066  6.534  4.851  4.339  3.972  3.207  3.015  2.925  2.44   2.351  1.993  1.798  1.513  1.451  1.349  1.297  1.132  1.074  0.976  0.837  0.819  0.709  0.665  0.625]L0 raw Q/K/V energy in this subspace:0 [0.2455 0.1965 0.368 ]1 [0.1627 0.197  0.3512]2 [0.9298 0.9329 0.1354]3 [0.3366 0.3524 0.1739]4 [0.1829 0.1872 0.2346]5 [0.2132 0.2109 0.3322]
L1 key-bias weight directions and L0 OV outputs0 [0.4286 0.5531 0.0093 0.0966 0.4738 0.7892]1 [0.4773 0.5822 0.0093 0.0781 0.5123 0.8106]2 [0.4795 0.582  0.0179 0.0758 0.5122 0.8098]3 [0.4589 0.5676 0.0232 0.0876 0.4939 0.799 ]4 [0.5101 0.5937 0.0332 0.0676 0.5308 0.8157]5 [0.4521 0.5668 0.0285 0.0787 0.4904 0.7964]
Linear-algebra-only MLP products: normalized trace with later QKL1H0 [-0.0073]L1H1 [-0.041]L1H2 [-0.0388]L1H3 [-0.0265]L1H4 [-0.0369]L1H5 [-0.0265]L2H0 [0.0028 0.0035]L2H1 [-0.0083  0.0235]L2H2 [0.0014 0.0149]L2H3 [-0.0047  0.0137]L2H4 [-0.0058  0.0046]L2H5 [ 0.001  -0.0012]L3H0 [-0.005  -0.0103  0.0057]L3H1 [-0.0464 -0.0133 -0.0229]L3H2 [-0.0034  0.0021  0.0054]L3H3 [-0.0012 -0.0023  0.012 ]L3H4 [-0.0011  0.0005  0.0093]L3H5 [-0.0067 -0.0014  0.0039]

python

import numpy as npimport scipy.linalg as lafrom threadpoolctl import threadpool_limitsthreadpool_limits(4)
A,B,Rkey = {},{},{}for l in range(4):    d=W[f"ln1_w.{l}"].astype(float)    z=W[f"ln1_b.{l}"].astype(float)    for j in range(6):        sl=slice(64*j,64*(j+1))        q=W[f"W_q.{l}"][sl].astype(float)        k=W[f"W_k.{l}"][sl].astype(float)        v=W[f"W_v.{l}"][sl].astype(float)        o=W[f"W_o.{l}"][:,sl].astype(float)        A[l,j]=(q*d).T@(k*d)        B[l,j]=o@(v*d)        Rkey[l,j]=(k*d).T@(q@z)
# Common write subspace of the four similar first-layer OV matrices.common=np.concatenate([B[0,j] for j in [0,1,4,5]],axis=1)u,s,vt=la.svd(common,full_matrices=False)print("Shared L0H0/H1/H4/H5 OV singular values:",np.round(s[:20],4))clock=u[:,:2]print("Fraction of each L0 OV's squared norm in shared top-2 write subspace:")for j in range(6):    print(j,round(la.norm(clock.T@B[0,j])**2/la.norm(B[0,j])**2,4))
print("\nLater QK energy on shared top-2 write subspace and bias-key alignment")print("head: query_fraction key_fraction key_bias_fraction key_bias_coordinates")for l in range(1,4):    for j in range(6):        a=A[l,j]        r=Rkey[l,j]        print(f"L{l}H{j}",              np.round([la.norm(clock.T@a)**2/la.norm(a)**2,                        la.norm(a@clock)**2/la.norm(a)**2,                        la.norm(clock.T@r)**2/la.norm(r)**2],4),              np.round(clock.T@r,3))
# A character-channel candidate: right singular subspace of L0H2 OV.u2,s2,vt2=la.svd(B[0,2],full_matrices=False)print("\nL0H2 gain-weighted OV singular values:",np.round(s2[:40],4))T=vt2[:27].Tprint("Projected compositions on L0H2's top-27 read subspace:")print("head: trace(A Bprev) projected_trace symmetric_fraction positive/negative_sym_eigenmass")for l in [1,2,3]:    for j in range(6):        a=A[l,j]        m=T.T@a@B[0,2]@T        sym=(m+m.T)/2        es=la.eigvalsh(sym)        print(f"L{l}H{j}",              np.round([np.trace(a@B[0,2]),np.trace(m),                        la.norm(sym)/la.norm(m),                        es[es>0].sum(),-es[es<0].sum()],4))
print("\nUnweighted L0H2 V singular values:")print(np.round(la.svdvals(W["W_v.0"][128:192].astype(float)),4))

OUTPUT of run 4 of at most 100:

Shared L0H0/H1/H4/H5 OV singular values: [11.996   8.4336  3.3624  1.312   0.9992  0.8615  0.7301  0.6058  0.5661  0.5402  0.4827  0.436   0.3932  0.3667  0.347   0.3209  0.306   0.2954  0.2622  0.246 ]Fraction of each L0 OV's squared norm in shared top-2 write subspace:0 0.91231 0.93852 0.00283 0.24854 0.85285 0.9345
Later QK energy on shared top-2 write subspace and bias-key alignmenthead: query_fraction key_fraction key_bias_fraction key_bias_coordinatesL1H0 [0.0614 0.5748 0.7729] [7.137 6.341]L1H1 [0.0648 0.5364 0.8146] [13.096 10.032]L1H2 [0.0606 0.5491 0.8131] [14.859 11.302]L1H3 [0.0591 0.5547 0.7933] [12.977 10.594]L1H4 [0.0569 0.6026 0.8268] [13.741  9.42 ]L1H5 [0.0677 0.5488 0.7836] [9.62  7.843]L2H0 [0.0275 0.0919 0.8536] [10.353  7.109]L2H1 [0.0082 0.0444 0.7929] [4.547 2.734]L2H2 [0.0203 0.1766 0.8019] [4.247 3.356]L2H3 [0.0381 0.092  0.8678] [9.315 5.662]L2H4 [0.0304 0.0922 0.8624] [10.055  6.023]L2H5 [0.0306 0.0882 0.8535] [9.473 5.885]L3H0 [0.0014 0.0127 0.1481] [0.676 0.418]L3H1 [0.001  0.0023 0.0316] [-0.327  0.106]L3H2 [0.0118 0.0693 0.5917] [1.495 0.97 ]L3H3 [0.0162 0.0655 0.666 ] [2.074 1.111]L3H4 [0.0054 0.044  0.5194] [1.545 0.865]L3H5 [0.0029 0.0084 0.0312] [ 0.232 -0.016]
L0H2 gain-weighted OV singular values: [0.7873 0.718  0.6827 0.6713 0.6378 0.6013 0.5692 0.5583 0.5504 0.5222 0.5151 0.4872 0.4805 0.4696 0.4391 0.4211 0.4006 0.3931 0.3471 0.3044 0.2772 0.2754 0.2565 0.2286 0.2181 0.205  0.1814 0.1707 0.1628 0.1593 0.1314 0.1184 0.1108 0.1014 0.0949 0.0807 0.0726 0.0649 0.0508 0.0469]Projected compositions on L0H2's top-27 read subspace:head: trace(A Bprev) projected_trace symmetric_fraction positive/negative_sym_eigenmassL1H0 [-0.0978 -0.109   0.7328  0.2629  0.372 ]L1H1 [-0.0033 -0.0118  0.7879  0.7768  0.7886]L1H2 [0.458  0.4426 0.7303 1.138  0.6953]L1H3 [0.4099 0.4019 0.7498 0.9661 0.5643]L1H4 [0.3922 0.3805 0.7224 0.9042 0.5237]L1H5 [0.1614 0.1521 0.7166 0.9226 0.7705]L2H0 [-0.248  -0.2914  0.707   1.0645  1.3559]L2H1 [-0.2372 -0.2557  0.7393  0.8213  1.077 ]L2H2 [0.0327 0.0371 0.7256 0.635  0.5978]L2H3 [0.1216 0.1512 0.7026 0.9162 0.765 ]L2H4 [-0.3803 -0.3931  0.705   0.8113  1.2044]L2H5 [0.0158 0.0345 0.7101 1.0638 1.0293]L3H0 [-0.1019 -0.0942  0.7048  0.9675  1.0617]L3H1 [3.5295 3.5096 0.9062 3.7617 0.252 ]L3H2 [0.0903 0.0916 0.7098 0.6001 0.5085]L3H3 [0.0381 0.0281 0.6743 0.5445 0.5164]L3H4 [-0.1812 -0.1594  0.7495  0.6723  0.8317]L3H5 [0.0419 0.0467 0.7255 0.3519 0.3052]
Unweighted L0H2 V singular values:[1.9539 1.8791 1.8467 1.8044 1.7716 1.7334 1.7213 1.6695 1.6478 1.5709 1.5539 1.5198 1.5167 1.4806 1.4422 1.331  1.2633 1.2307 1.0507 0.9934 0.9376 0.9084 0.843  0.8111 0.7783 0.7104 0.6947 0.6454 0.6367 0.599  0.5752 0.5655 0.5477 0.5344 0.5259 0.5052 0.4878 0.4767 0.4555 0.4349 0.427  0.4089 0.3946 0.3823 0.3712 0.3616 0.3501 0.3437 0.3341 0.323  0.3187 0.3139 0.302  0.2976 0.2964 0.2836 0.281  0.2722 0.2652 0.2598 0.2594 0.2497 0.2364 0.2305]

python

import numpy as npimport scipy.linalg as lafrom sklearn.cluster import KMeansfrom sklearn.metrics import silhouette_scorefrom threadpoolctl import threadpool_limitsthreadpool_limits(4)
d=W["ln1_w.0"].astype(float)v=W["W_v.0"][128:192].astype(float)*do=W["W_o.0"][:,128:192].astype(float)B=o@vF=W["W_fc.0"].astype(float)*W["ln2_w.0"]G=F@Bu,s,vt=la.svd(G-G.mean(axis=0),full_matrices=False)print("Singular values of centered W_fc.0 D_ln2.0 B_L0H2:")print(np.round(s[:50],4))
# Clustering weight rows, not activations or model inputs.for r in [20,24,27,32]:    X=(G-G.mean(axis=0))@vt[:r].T    X/=np.sqrt(s[:r])[None,:]    X/=np.maximum(la.norm(X,axis=1,keepdims=True),1e-12)    km=KMeans(n_clusters=r+1,n_init=10,random_state=0).fit(X)    sizes=np.bincount(km.labels_,minlength=r+1)    coh=np.array([la.norm(X[km.labels_==i].mean(axis=0)) for i in range(r+1)])    sil=silhouette_score(X,km.labels_,sample_size=1000,random_state=0)    print("rank",r,"clusters",r+1,"silhouette",round(sil,4),          "mean_cohesion",round(np.average(coh,weights=sizes),4))    print("sizes descending",np.sort(sizes)[::-1])
print("\nEffective QK symmetric eigenvalue masses, no input evaluation:")for l in range(4):    d=W[f"ln1_w.{l}"].astype(float)    for j in range(6):        sl=slice(64*j,64*(j+1))        q=W[f"W_q.{l}"][sl].astype(float)*d        k=W[f"W_k.{l}"][sl].astype(float)*d        a=q.T@k        ev=la.eigvalsh((a+a.T)/2)        pos=ev[ev>0].sum()        neg=-ev[ev<0].sum()        print(f"L{l}H{j}", "positive",round(pos,3),"negative",round(neg,3),              "signed_mass_fraction",round((pos-neg)/(pos+neg),4))

OUTPUT of run 5 of at most 100:

Singular values of centered W_fc.0 D_ln2.0 B_L0H2:[3.6485 2.7127 2.5966 2.3846 2.3094 1.9764 1.9523 1.9044 1.7751 1.6206 1.6077 1.4051 1.3241 1.2869 1.1878 1.1569 1.0534 0.8885 0.6879 0.4863 0.406 0.3707 0.3187 0.2846 0.2552 0.2371 0.2254 0.1925 0.1892 0.1811 0.1509 0.1421 0.1287 0.113  0.1058 0.0921 0.0829 0.0712 0.0631 0.0553 0.0482 0.0476 0.04   0.034  0.0309 0.0276 0.0267 0.0212 0.0188 0.0177]rank 20 clusters 21 silhouette 0.0421 mean_cohesion 0.5708sizes descending [108  92  91  89  88  87  83  80  79  78  77  75  74  73  71  70  59  50  42  38  32]rank 24 clusters 25 silhouette 0.0385 mean_cohesion 0.5578sizes descending [87 82 82 79 79 77 76 69 67 67 66 63 59 58 56 55 55 53 50 48 46 45 45 37 35]rank 27 clusters 28 silhouette 0.038 mean_cohesion 0.5501sizes descending [82 80 77 77 75 69 68 68 67 60 60 58 58 57 55 53 53 50 44 42 42 41 39 39 39 35 24 24]rank 32 clusters 33 silhouette 0.0362 mean_cohesion 0.5417sizes descending [69 63 62 60 59 59 58 58 54 53 51 50 50 48 48 47 46 46 46 45 45 42 41 40 38 38 37 37 33 31 30 27 25]
Effective QK symmetric eigenvalue masses, no input evaluation:L0H0 positive 0.526 negative 0.563 signed_mass_fraction -0.0348L0H1 positive 0.35 negative 0.462 signed_mass_fraction -0.1372L0H2 positive 57.4 negative 47.8 signed_mass_fraction 0.0913L0H3 positive 4.184 negative 2.609 signed_mass_fraction 0.2319L0H4 positive 0.591 negative 1.158 signed_mass_fraction -0.3243L0H5 positive 0.456 negative 0.733 signed_mass_fraction -0.2334L1H0 positive 8.996 negative 14.15 signed_mass_fraction -0.2227L1H1 positive 12.839 negative 22.133 signed_mass_fraction -0.2658L1H2 positive 14.061 negative 27.016 signed_mass_fraction -0.3154L1H3 positive 13.096 negative 22.354 signed_mass_fraction -0.2612L1H4 positive 13.29 negative 23.248 signed_mass_fraction -0.2725L1H5 positive 13.028 negative 21.144 signed_mass_fraction -0.2375L2H0 positive 62.04 negative 62.213 signed_mass_fraction -0.0014L2H1 positive 58.967 negative 54.73 signed_mass_fraction 0.0373L2H2 positive 38.345 negative 47.188 signed_mass_fraction -0.1034L2H3 positive 53.32 negative 59.226 signed_mass_fraction -0.0525L2H4 positive 60.195 negative 64.0 signed_mass_fraction -0.0306L2H5 positive 56.408 negative 61.986 signed_mass_fraction -0.0471L3H0 positive 78.679 negative 51.383 signed_mass_fraction 0.2099L3H1 positive 78.364 negative 27.915 signed_mass_fraction 0.4747L3H2 positive 42.154 negative 40.39 signed_mass_fraction 0.0214L3H3 positive 45.205 negative 44.312 signed_mass_fraction 0.01L3H4 positive 56.44 negative 49.211 signed_mass_fraction 0.0684L3H5 positive 22.97 negative 22.177 signed_mass_fraction 0.0176

python

import numpy as npimport scipy.linalg as lafrom threadpoolctl import threadpool_limitsthreadpool_limits(4)
d=W["ln1_w.0"].astype(float)A,B={},{}for j in range(6):    sl=slice(64*j,64*(j+1))    q=W["W_q.0"][sl].astype(float)*d    k=W["W_k.0"][sl].astype(float)*d    v=W["W_v.0"][sl].astype(float)*d    o=W["W_o.0"][:,sl].astype(float)    A[j]=q.T@k    B[j]=o@v
# Compare weight matrices with powers of the orthogonal polar factor# of L0H2's QK matrix. These are matrix alignments, not attention scores.q=W["W_q.0"][128:192].astype(float)*dk=W["W_k.0"][128:192].astype(float)*dc=q.T@q+k.T@kev,U=la.eigh(c)ix=np.argsort(ev)[::-1]ev,U=ev[ix],U[:,ix]for r in [16,20,24,32]:    Z=U[:,:r]/np.sqrt(ev[:r])[None,:]    t=Z.T@A[2]@Z    rot,_=la.polar(t)    print("\nrank",r,"whitened L0H2 singular values",np.round(la.svdvals(t),3))    print("Alignment with polar-factor powers, columns -4..4:")    for j in [0,1,3,4,5]:        m=Z.T@A[j]@Z        vals=[np.sum(m*np.linalg.matrix_power(rot,p))/              (la.norm(m)*np.sqrt(r)) for p in range(-4,5)]        print(f"L0H{j}",np.round(vals,4))
print("\nShared first-layer read subspace and positional-circuit overlap")stack=np.concatenate([B[j] for j in [0,1,4,5]],axis=0)_,s,vt=la.svd(stack,full_matrices=False)print("Shared-read singular values",np.round(s[:10],4))a_u,a_s,a_vt=la.svd(A[2],full_matrices=False)P=la.orth(np.column_stack([a_u[:,:20],a_vt[:20].T]))for i in range(4):    v=vt[i]    print("read direction",i,"energy in L0H2 QK subspace",          round(la.norm(P.T@v)**2,4))
# Check the induction-like composition without choosing one particular rank.a=(W["W_q.3"][64:128].astype(float)*W["ln1_w.3"]).T @ (    W["W_k.3"][64:128].astype(float)*W["ln1_w.3"])_,_,vt=la.svd(B[2],full_matrices=False)print("\nL3H1 x L0H2 induction-like composition, subspace-rank sensitivity")for r in [16,20,24,27,32,40,64]:    T=vt[:r].T    m=T.T@a@B[2]@T    e=la.eigvalsh((m+m.T)/2)    print(r,"trace",round(np.trace(m),4),          "positive_mass",round(e[e>0].sum(),4),          "negative_mass",round(-e[e<0].sum(),4),          "symmetry_fraction",round(la.norm((m+m.T)/2)/la.norm(m),4))

OUTPUT of run 6 of at most 100:


rank 16 whitened L0H2 singular values [0.617 0.575 0.566 0.53  0.521 0.507 0.493 0.472 0.468 0.443 0.419 0.405 0.391 0.37  0.083 0.017]Alignment with polar-factor powers, columns -4..4:L0H0 [-0.026   0.076   0.1024 -0.018  -0.0802 -0.1375 -0.0964  0.0035  0.1072]L0H1 [ 0.042   0.0077  0.007  -0.0733 -0.0518 -0.0235  0.0194 -0.0014  0.0007]L0H3 [-0.2319 -0.2995 -0.1674  0.1027  0.2253  0.2039 -0.0107 -0.2293 -0.2839]L0H4 [ 0.1195  0.0721 -0.047  -0.1282 -0.1541  0.0002  0.1201  0.0769 -0.0293]L0H5 [ 0.1541  0.1988  0.0798 -0.0938 -0.2027 -0.1472  0.0305  0.1617  0.1551]
rank 20 whitened L0H2 singular values [0.646 0.605 0.596 0.588 0.558 0.541 0.527 0.525 0.51  0.501 0.461 0.436 0.429 0.427 0.414 0.394 0.38  0.291 0.243 0.235]Alignment with polar-factor powers, columns -4..4:L0H0 [-0.0006  0.0326  0.0802  0.0299 -0.0569 -0.0544 -0.0776 -0.0236  0.0441]L0H1 [ 0.0427  0.0052 -0.0097 -0.0298 -0.0768  0.0005  0.0062  0.005  -0.0008]L0H3 [-0.1689 -0.2718 -0.164   0.0997  0.1326  0.199  -0.0295 -0.2152 -0.2294]L0H4 [ 0.0984  0.0222 -0.0847 -0.1366 -0.1324 -0.0288  0.0773  0.0386 -0.0272]L0H5 [ 0.1218  0.1262  0.0674 -0.0784 -0.1501 -0.1069  0.0388  0.1069  0.1294]
rank 24 whitened L0H2 singular values [0.695 0.655 0.624 0.617 0.606 0.591 0.571 0.565 0.543 0.54  0.521 0.504 0.495 0.488 0.454 0.428 0.412 0.39  0.351 0.338 0.283 0.201 0.095 0.025]Alignment with polar-factor powers, columns -4..4:L0H0 [-0.009   0.0091  0.0543  0.0603  0.051  -0.012   0.0038 -0.035   0.0292]L0H1 [ 0.0074  0.0155  0.0067  0.0354 -0.0988  0.0211  0.0051 -0.0227 -0.0332]L0H3 [-0.1302 -0.1833 -0.077   0.0664  0.0764  0.1546  0.0565 -0.0982 -0.1215]L0H4 [ 0.0085  0.0169 -0.0877  0.059  -0.1798  0.055  -0.082   0.0621 -0.077 ]L0H5 [ 0.027   0.0892  0.0264  0.0462 -0.1206  0.0031 -0.0609  0.0293  0.0147]
rank 32 whitened L0H2 singular values [0.89  0.764 0.721 0.711 0.684 0.67  0.651 0.634 0.623 0.612 0.593 0.586 0.57  0.562 0.539 0.526 0.514 0.474 0.457 0.417 0.385 0.356 0.345 0.311 0.282 0.256 0.188 0.121 0.049 0.03  0.013 0.   ]Alignment with polar-factor powers, columns -4..4:L0H0 [-0.0058 -0.0268 -0.0259  0.0236  0.0453 -0.0054 -0.0143  0.0132  0.0138]L0H1 [-0.0239 -0.0035 -0.0019 -0.0149 -0.0328 -0.023   0.0047  0.0314  0.0027]L0H3 [-0.0339  0.0386  0.0535 -0.0087 -0.0414  0.1335  0.013  -0.0334  0.0008]L0H4 [-0.0246 -0.0487  0.047   0.0855 -0.1421 -0.0777  0.0692 -0.0124 -0.0669]L0H5 [-0.0346 -0.0258  0.0829  0.0006 -0.1251  0.0222  0.092  -0.0212 -0.0558]
Shared first-layer read subspace and positional-circuit overlapShared-read singular values [12.4276  7.5882  3.5107  1.4158  1.1895  0.935   0.8361  0.731   0.6597  0.558 ]read direction 0 energy in L0H2 QK subspace 0.9829read direction 1 energy in L0H2 QK subspace 0.9488read direction 2 energy in L0H2 QK subspace 0.8122read direction 3 energy in L0H2 QK subspace 0.4707
L3H1 x L0H2 induction-like composition, subspace-rank sensitivity16 trace 3.2581 positive_mass 3.2621 negative_mass 0.0039 symmetry_fraction 0.933820 trace 3.4768 positive_mass 3.5445 negative_mass 0.0677 symmetry_fraction 0.924324 trace 3.4982 positive_mass 3.6684 negative_mass 0.1702 symmetry_fraction 0.914327 trace 3.5096 positive_mass 3.7617 negative_mass 0.252 symmetry_fraction 0.906232 trace 3.5278 positive_mass 3.8829 negative_mass 0.3551 symmetry_fraction 0.896140 trace 3.5302 positive_mass 4.081 negative_mass 0.5508 symmetry_fraction 0.877264 trace 3.5295 positive_mass 4.643 negative_mass 1.1134 symmetry_fraction 0.8402

Observations

These are weight-only inferences, not measured attention patterns. Exact distances and character identities remain uncertain.

Define (A_{lh}=D_lW_{q,lh}^{T}W_{k,lh}D_l) and (B_{lh}=W_{o,lh}W_{v,lh}D_l), where (D_l=\operatorname{diag}(\text{ln1_w.l})).

One exceptional positional-selector candidate: L0H2 has (|A|_F=32.627), versus 0.236–3.157 for the other first-layer heads. Its substantial complex eigenpairs, including (5.058\pm6.513i), suggest a shift-like positional circuit. “Previous character” is plausible, but the missing position embeddings prevent identifying its lag conclusively.
Four shared positional-summary heads: L0H0/H1/H4/H5 put respectively 91.2%, 93.9%, 85.3%, 93.5% of their OV energy into the same two-dimensional write subspace, called (C) below. Their two dominant shared read directions overlap the L0H2 QK subspace by 98.3% and 94.9%. This suggests positional summaries subsequently used for routing.
A particularly strong induction-like connection: L3H1 uniquely aligns with L0H2’s OV map: normalized (\operatorname{tr}(A_{31}B_{02})=0.0949). Restricted to the strongest 16 read directions, the symmetric composition has positive eigenvalue mass 3.2621, versus negative mass 0.0039. Conditional on L0H2 transporting the previous character, this is compelling successor-matching evidence.
Corpus identification is much weaker. W_proj.3 has singular values (s_1=26.309,\ s_{23}=10.367,\ s_{28}=7.166): substantial, diverse learned output directions. This does not recover letters or uniquely identify a language. Weight-row clustering also did not reveal clean character identities.

Heads

Roles below are provisional. “Clock-key energy” means QK energy reading the shared subspace (C); bias alignment refers to the key-linear weight direction generated by ln1_b.

L0H0: broad — small QK norm 0.437; 91.2% shared positional-summary write energy.
L0H1: broad — smallest QK norm, 0.236; 93.9% shared write energy.
L0H2: previous — exceptional, shift-like QK structure and the downstream induction-like connection; exact lag unverified.
L0H3: local — QK matrix cosine 0.239 with L0H2; weaker, lower-rank positional selection.
L0H4: broad — QK norm 0.851; 85.3% shared write energy.
L0H5: broad — QK norm 0.621; 93.5% shared write energy.

L1H0: local — clock-key energy 57.5%; bias alignment 77.3%.
L1H1: local — clock-key energy 53.6%; bias alignment 81.5%.
L1H2: local — clock-key energy 54.9%; strong clock-aligned key bias, norm 20.70.
L1H3: local — clock-key energy 55.5%; bias alignment 79.3%.
L1H4: local — highest L1 clock-key energy, 60.3%; bias alignment 82.7%.
L1H5: local — clock-key energy 54.9%; bias alignment 78.4%.

L2H0: local — strong key bias, norm 13.59, with 85.4% clock alignment.
L2H1: local — weaker positional bias, norm 5.96; 79.3% clock alignment. Lower-confidence assignment.
L2H2: local — largest L2 clock-key energy, 17.7%; bias alignment 80.2%.
L2H3: local — key-bias norm 11.70; clock alignment 86.8%.
L2H4: local — key-bias norm 12.62; clock alignment 86.2%.
L2H5: local — key-bias norm 12.07; clock alignment 85.4%.

L3H0: unclear — positive QK trace 27.30 suggests content matching, but no clear successor-matching composition.
L3H1: copy — uniquely strong, predominantly positive composition with L0H2; strongest induction candidate.
L3H2: unclear — modest clock-key energy, 6.9%; neither distance nor induction is established.
L3H3: unclear — modest clock-key energy, 6.5%; weak successor-composition evidence.
L3H4: unclear — clock-key energy 4.4%; content-dependent routing likely, but its range is unresolved.
L3H5: broad — weakest final-layer QK norm, 6.679, and only 0.84% clock-key energy; strong OV map suggests aggregation.

Reasoning

The strongest interpretation is positional summaries → nearby-character composition → later content matching and copying. Such an organized successor-matching circuit is substantially more useful for intact text than for independently shuffled lines.

English is a weak corpus preference, not a decoded identification. The diverse feature structure is compatible with English spelling and repeated words; Python remains a substantial alternative. The spectrum provides, at most, weak evidence against romaji’s more restricted orthography. The ordered-versus-shuffled conclusion is much stronger than the corpus choice.

corpus: english
order: as is
probabilities: python=0.35 romaji=0.17 english=0.48; as is=0.98 shuffled=0.02