Continued math visualizations development

{
  "type": "tool_reference",
  "tool_name": "mcp__traces__traces_search"
}
{
  "type": "tool_reference",
  "tool_name": "mcp__traces__traces_read"
}
{
  "type": "tool_reference",
  "tool_name": "mcp__traces__traces_lookup"
}
{
  "type": "tool_reference",
  "tool_name": "mcp__traces__traces_search_tools"
}
{
  "type": "tool_reference",
  "tool_name": "mcp__traces__traces_execute_tool"
}
{
  "type": "tool_reference",
  "tool_name": "mcp__traces__surface_build_instructions"
}

Found 2 trace(s) of 2.

## People

| User | Display name | User ID |
|---|---|---|
| @goldfishandswedishfish | Kara Melton | kd7ekkrfd435k459fw0hnpgq3187e6sg |

## Traces

| Started | User | Project | Agent | Status | Messages | Title | URL |
|---|---|---|---|---|---:|---|---|
| 2026-10-12 01:30 | @goldfishandswedishfish | karamelton | claude-code | In Progress | 18 | Reviewed recursive self-improvement paper | https://traces.com/s/52797cc3-0c4f-4505-b3fa-3bb171d53396 |
| 2026-10-12 01:26 | @goldfishandswedishfish | karamelton | claude-code | In Progress | 143 | Started building math proof visualizations | https://traces.com/s/189903e3-2f4c-44dd-8020-b4552bf40364 |

Canonical source: https://traces.com/surfaces.md
Content SHA-256: 86b2fc1482a8081bed50bfccac1049c9973eab53ae3b54dd83d2a86505d324ef
Latest trace in this namespace (use it to try each uploaded version): https://traces.com/s/52797cc3-0c4f-4505-b3fa-3bb171d53396
Workflow: write the HTML locally, then traces_surfaces_prepare_upload -> upload -> traces_surfaces_complete_upload (release defaults to false). Share the returned previewUrl with the user; it renders that exact version on their latest trace without changing what others see. Call traces_surfaces_release_version only when the user is happy.

---
name: create-surfaces
description: Create, revise, debug, or validate Traces surfaces: single-file HTML visualizations that run in the sandboxed trace-page iframe through window.traces.getTrace(). Use when building a new surface, changing its layout or interaction, diagnosing loading/rendering, or checking an artifact before publication.
---

# Prompt: build a Traces surface

You are writing a **Traces surface**: a single self-contained HTML file that renders a view of one AI agent trace (a recorded coding-agent session: messages, tool calls, thinking, token usage).

Traces stores your file as an immutable version, serves it from its own storage, and runs it inside a locked-down iframe on the trace page. The user opens it as a tab next to "Highlights" and "Full Trace".

## Output

One HTML file. Nothing else — no build step, no bundler, no package.json.

## Hard constraints (these are enforced; violating them means it won't publish or won't run)

- **Single file, ≤ 16 MiB.** All CSS and JS inline in the document.
- **Must contain a `<head>` element.**
- **No external resources at all.** No `<link>`, `<iframe>`, `<object>`, `<embed>`, `<base>` tags. No `@import`. No `src`/`action`/`formaction`/`poster` attribute pointing at `http:`, `https:` or `//`. That means no CDN scripts, no Google Fonts, no external stylesheets, no remote images.
- **Do not make outbound network requests.** CSP is `default-src 'none'; connect-src 'none'`, and the iframe is `sandbox="allow-scripts"` without `allow-same-origin`. `fetch`, `XMLHttpRequest`, WebSockets, and form submission all fail. You have no origin, no cookies, no `localStorage`, no access to the host page.
- **Upload screening rejects recognizable WebRTC access and scripted navigation**, including `RTCPeerConnection`, writes to `location` (except `hash`), `location.assign`/`replace`, `window.open`, meta refresh, and `javascript:` URLs. Use in-page controls rather than navigation. Scripts and inline event handlers must parse as JavaScript; each script is limited to 500,000 tokens. Comments, JSON data blocks, ordinary anchors and local variables are not treated as browser API use. Screening is best-effort, not a guarantee that arbitrary code cannot send data out.
- **Allowed:** inline `<script>` and inline `<style>`, `data:` URIs for images and fonts, inline SVG, canvas, and everything else that runs purely in-page.
- **Vendoring a library is allowed** but it must be inlined in the file and it counts against the 16 MiB. Prefer writing it yourself — a big bundle costs parse/compile time on every open, and the iframe shares nothing with the host page.

## The SDK

The host injects a global before your code runs:

```js
window.traces.version    // "surface-sdk.v1"
await window.traces.getTrace()   // Promise<SurfaceTrace>
```

`getTrace()` is the entire data API. There is exactly one call, it resolves once with everything you get, and there is no way to ask for more data. It rejects if the host doesn't deliver within 10 seconds — handle that and render a readable message rather than a blank page.

Data is pushed to you over a private channel; you never request it.

### Classifying (beta)

Classification is a beta feature and needs a signed-in viewer. For anyone signed out it fails with `errorCode: "unauthenticated"`, so every surface that uses it needs a fallback.

`window.traces.classify` answers typed questions about any JSON state you build from the trace:

```js
const result = await window.traces.classify({
  state: { command: call.args.command },
  questions: {
    destructive: { type: "bool", instructions: "Does `command` delete data?",
                   criteria: { true: "Removes files or history", false: "Leaves data intact" } },
    area: { type: "choice", instructions: "What does `command` touch?",
            criteria: { git: "Version control", files: "The filesystem", network: "Remote hosts" } },
    risk: { type: "score", instructions: "How consequential is `command`?",
            criteria: ["Read-only", "Local change", "Leaves the machine"] }
  }
});
// result.answers.destructive.probability · .area.choice / .probabilities · .risk.score (0 = first level)
```

It never rejects. Check `result.stopReason === "stop"`; otherwise `result.errorMessage` says why, and `result.errorCode` says what to do:
- `"unauthenticated"`, `"forbidden"` or `"quota_exceeded"`: this viewer can't classify. Stop and say so.
- `"rate_limited"`: the viewer is calling too fast. Back off briefly and retry.
- no code: this call failed. Render the fallback; the platform has already retried what a retry can fix.

Getting good answers:
- **Decide what each answer is about**, such as a tool call, a user turn or the whole trace. The `state` is that item plus the context needed to judge it. Every question sees all of the state, so unrelated content can sway an answer, and missing context makes it guess.
- **Ask everything about an item in one call.** Questions are answered independently against the state, so they don't affect each other, and the state is paid for once. Use a separate call only when a question needs different evidence or depends on another answer.
- **Point questions at the state.** Name fields with backticked paths, such as `` `action.output` ``, rather than "this" or "the output".
- **Choose how many items share a call.** One item per call gives each answer the cleanest evidence but makes the most calls, which is slower on long traces and repeats a little overhead per item. Several items per call is faster and lighter on limits, but every question then sees the other items, and content unrelated to a question can cost accuracy. Group items when speed matters more than the last bit of accuracy, when items are short, or when the question compares them. Give single-item checks that must be right (did it fail, did it touch secrets) their own call. When you group, give each item its own field and name it in its questions.
- **Send calls in parallel.** You don't need your own retries or concurrency tuning; the platform retries the model's own failures. Handle `"rate_limited"`, and classify what's on screen first if the trace is long. Oversized requests are refused, so trim `state` to the relevant excerpt.

## The data contract

```ts
type SurfaceTrace = {
  trace: {
    id: string;
    title?: string;
    agentId: string;                       // which coding agent produced it
    namespace: { id: string; slug: string };
    author?: { id: string; displayName?: string; avatarUrl?: string };
    model?: string;
    createdAt: number;                     // epoch ms
    startedAt?: number;
    messageCount?: number;                 // total in the trace, may exceed messages.length
    tokenUsage?: TokenUsage;
    project?: { name?: string; path?: string; gitRemoteUrl?: string; gitBranch?: string };
    analysis?: TraceAiAnalysis;            // optional AI summary, may be absent
  };
  messages: SurfaceTraceMessage[];
  truncated: boolean;
};

type SurfaceTraceMessage = {
  id: string;
  role: string;                            // e.g. user / assistant / system — do not assume a closed set
  text?: string;                           // flattened text, convenient but lossy
  model?: string;
  order?: number;
  timestamp?: number;                      // epoch ms
  parts: SurfaceTracePart[];
  tokenUsage?: TokenUsage;
};

type SurfaceTracePart = {
  type: "text" | "thinking" | "tool_call" | "tool_result" | "error" | "system_event";
  order: number;
  content: Record<string, unknown>;        // shape depends on type
  tokenUsage?: TokenUsage;
};
```

Part content shapes:

- `text` / `thinking` → `{ text }`
- `tool_call` → `{ callId, toolName, args }`
- `tool_result` → `{ callId, toolName, output, status: "success" | "error" }`
- `error` → `{ message }`
- `system_event` → `{ subtype, ... }` (open-ended; extra fields ride along verbatim)

Any content may also carry `startedAt`, `completedAt`, `durationMs`. Do **not** sum `durationMs` across parts — some agents report per-part timing and others stamp a whole turn's wall-clock onto every part in it, so summing double-counts.

Pair `tool_call` with `tool_result` on `callId`, not on adjacency.

## Rules for the code you write

- **Every field except `trace.id`, `trace.agentId`, `trace.namespace`, `trace.createdAt`, `messages[].id`, `messages[].role`, `messages[].parts` and `parts[].type/order/content` is optional.** Real traces are missing things — no title, no model, no token usage, no author. Render a sensible placeholder instead of `undefined` or `NaN`, and never crash on a missing field.
- **Handle `truncated: true` honestly.** You may be given a prefix of a long trace. If `truncated` is true, or `messages.length < trace.messageCount`, say so in the UI — "over the first N of M messages" — rather than presenting a partial aggregate as if it covered the whole trace.
- **Tolerate unknown values.** New part types, new `system_event` subtypes and new roles will appear. Skip or generically render what you don't recognise; never throw.
- **Escape all trace content.** Message text and tool output are untrusted arbitrary strings, frequently containing HTML, code and markup. Use `textContent`, never `innerHTML`, for anything derived from the data.
- **Don't sort by array position.** Use `order` / `timestamp` when you need sequence, and cope with them being absent.
- **Assume it can be big.** Up to a few thousand messages. Don't render thousands of DOM nodes eagerly if you can summarise, virtualise or collapse.
- **Render something immediately**, then fill in when `getTrace()` resolves: a loading state, an error state for the rejection case, and an empty state for a trace with no messages.
- **Keep states mutually exclusive.** If you use the HTML `hidden` attribute, include `[hidden] { display: none !important; }`. Author CSS such as `.state { display: grid }` can otherwise override the browser default and expose loading, error, empty, and content states simultaneously.
- **Define the surface's semantic units from the current request.** Do not assume what a “turn,” step, token, duration, category, or aggregate means based on another surface. Label estimated or derived values as such.
- Look reasonable on both a narrow and a wide viewport; the iframe is the width of the trace page.
- Keep it self-explanatory. No settings, no persistence — there is nowhere to save anything.

## Aesthetics — match traces.com

The surface renders inside the Traces trace page, so it should look like it belongs there rather than like an embedded third-party widget. Traces is a restrained, near-monochrome, information-dense UI: flat surfaces, hairline borders, one accent colour used sparingly, no drop shadows, no gradients, no rounded-pill decoration.

**The surface is a section of the trace page, not a page of its own.** It renders in a tab directly under the trace's own `<h1>` (the trace title) and the tab bar, so it must not restate that chrome:

- **Never paint a background on the whole surface.** Leave `html` and `body` transparent — no `background: var(--background)` on the root. `--background` is still fine for a *recessed* fill inside a card, just never on the root.
- **Never open with a title, an `<h1>`, or eyebrow text, and don't add a heading just because it's a section.** It's part of a layout, so start with the content — the tab name already acts as the surface's header. Add an `<h2>` only where it earns its place: a later section the tab name doesn't cover, or a sibling that needs telling apart.
- **Never add outer padding or margin on `html`/`body`.** The frontend already applies the page's outer gutter around the iframe; a root padding doubles it up as dead space. Put spacing on your own inner elements instead — `body { margin: 0 }`, no `padding` on `html` or `body`.

```html
<!-- Don't: a page header the trace page already gives you -->
<body>
  <p class="eyebrow">SKILL ACTIVITY</p>
  <h1>Skills loaded in this run</h1>
  <p class="lede">Open a skill to read its instructions and see every recorded load.</p>
  <section class="stats">…</section>
</body>

<!-- Do: open on the content; section headings from h2 down -->
<body>
  <section class="stats">…</section>
  <h2>Loaded skills</h2>
  <ul class="skill-list">…</ul>
</body>
```

**Always style both light and dark — never ship just one.** The host gives you no theme signal. Read `@media (prefers-color-scheme: dark)` and define both, for every color you introduce, not just the tokens below. Declare `color-scheme: light dark` so form controls and scrollbars follow.

Copy these tokens verbatim — they're the real values from the app:

```css
:root {
  color-scheme: light dark;
  --background: #f5f5f5;
  --foreground: #141414;
  --card: #ffffff;
  --muted: #f2f2f2;             /* recessed fills */
  --muted-foreground: #6b6b6b;  /* labels, secondary text */
  --faint-foreground: #808080;  /* timestamps, least important text */
  --border: #e3e3e3;
  --border-strong: #dbdbdb;
  --hairline: rgb(0 0 0 / 0.07);
  --accent: rgb(0 0 0 / 0.03);  /* hover fill */
  --primary: oklch(0.50 0.15 278);   /* indigo-violet; links, selection, focus */
  --destructive: #cf4415;       /* errors, deletions */
  --success: #4b7c0b;           /* success, additions */
  --radius: 0.625rem;           /* 10px cards; 5px for small controls */
}

@media (prefers-color-scheme: dark) {
  :root {
    --background: #141414;
    --foreground: #e2e2e2;
    --card: #262626;
    --muted: #111111;
    --muted-foreground: #a0a0a0;
    --faint-foreground: #737373;
    --border: #202020;
    --border-strong: #242424;
    --hairline: rgb(226 226 226 / 0.06);
    --accent: rgb(255 255 255 / 0.03);
  }
}
```

**Type.** Body is Inter in the app; you can't load it, so use `font-family: Inter, ui-sans-serif, system-ui, sans-serif` and let it fall back. Code, tool names, file paths, hashes, IDs and numeric columns are monospace: `ui-monospace, "Berkeley Mono", SFMono-Regular, Menlo, monospace`, ~13px. Body text 13–14px, section labels ~12px in `--muted-foreground`, occasionally uppercase with slight letter-spacing. **Inter is the only typeface** — no display, serif, or decorative face for titles or emphasis — unless the user's request explicitly names another one. Monospace, as above, is the only other family, reserved for code/numeric content.

**Weight.** Only two font weights: `font-weight: 400` (normal) and `font-weight: 500` (medium). Never 600/700/bold/semibold. Titles — including `<h2>`, the largest text you have — are `font-weight: 400`; size and tight tracking (`letter-spacing: -0.01em` to `-0.02em`) carry the hierarchy, not weight. Reserve `500` for small text that needs to read as distinct from adjacent body text at that *same* size — a list-item's title next to its description, a label next to its value. Everything else is 400.

**Restraint.** Use as few distinct type styles and spacing values as possible (ex: one font size, one text colour, one weight, one spacing unit — vary just one property when a genuinely different element needs a distinction).

**Layout.** Use `--card` sparingly: `1px solid --border`, `border-radius: var(--radius)`, no shadow. Prefer a single hairline (`--hairline`) between list rows over boxing each one. Hover = `--accent` fill, not a colour change. Keep the palette monochrome; `--primary` for links/focus/one emphasis, `--destructive`/`--success` only for genuine error/success.

**Density.** Traces is comfortable with dense tables and tight rows — prefer a compact table or list over big cards with lots of padding. Right-align and monospace numbers so columns line up. Format timestamps as short absolute dates (or relative for recent), and format bytes/tokens with thousands separators rather than raw integers.

**Sizing.** The frame is the full width of the trace page and **grows to fit your content** — the SDK measures your layout and the host resizes the iframe, so the trace page does the scrolling. Lay out naturally (`body { margin: 0 }`, no fixed-height wrapper) and do not build your own inner scroll container: a `height: 100vh` root or an internally scrolling shell gets you a nested scrollbar, and the host deliberately refuses to grow past the viewport for surfaces whose height tracks the viewport. Minimum height is 32rem, maximum 1000000px. Don't add your own outer border or page background that would double up the frame's. **Don't cap the surface with a `max-width` and center it — use the full width you're given.** A centered column reads as a floating card rather than the tab's own content.

**Motion.** Almost none. If you animate, keep it under ~150ms and wrap it in `@media (prefers-reduced-motion: reduce)` so it can be turned off.

Do not publish, install, set a current version, commit, or change the API/frontend runtime unless the user explicitly asks.

## Trying it on a real trace

A surface can only be seen on a trace, so the loop is: upload a version, try it, then release it. Uploading does not change what anyone else sees — a version is invisible until it is made current.

1. Prepare an upload for the surface key and a new semver (`traces_surfaces_prepare_upload` via the Traces MCP, or `POST /v1/surfaces/:key/versions/uploads` with an API key). You get back a short-lived, single-use `upload` destination.
2. Send the raw HTML file to it directly, e.g. `curl -fsS -X POST -H 'Content-Type: text/html' --data-binary @surface.html <url>`. The response contains `artifactId`. Never paste the HTML into a tool call.
3. Complete the upload (`traces_surfaces_complete_upload`, or `POST …/uploads/complete`) with that `artifactId`. Leave `release` off.
4. Open the version on a trace: `https://traces.com/s/<trace-id>?surface=<key>&version=<version>`. The MCP tool returns this link for the namespace's latest trace as `previewUrl`; share it with the user. Only namespace members can open a version that is not current.
5. Fix, bump the version, repeat. When the user is happy, release it (`traces_surfaces_release_version`, or `PATCH /v1/surfaces/:key`) to make it current.

## Now build

Use the user's requested surface description to decide what the surface should show and how. If that description is missing or leaves a consequential semantic choice unresolved, ask a focused question before building.

# Started building math proof visualizations

- ID: 189903e3-2f4c-44dd-8020-b4552bf40364
- URL: https://traces.com/s/189903e3-2f4c-44dd-8020-b4552bf40364
- Agent: claude-code

## User

<local-command-caveat>The command below was run directly in Claude Code, not sent to you as a request, and its output goes straight to the user. It's recorded here as context for later messages.</local-command-caveat>

## User

<command-name>/mcp</command-name>
            <command-message>mcp</command-message>
            <command-args></command-args>

## User

<local-command-stdout>Authentication successful. Connected to traces.</local-command-stdout>

## User

Look at this repo: https://github.com/openai/math I'd like to build some surfaces that help people visualize and explore the concepts set out in these proofs. They should make these complex ideas accessible and interactive to people who maybe have never heard of these concepts before. Use the traces mcp to build a surface. Start by selecting one of the proofs and we can explort from there.

## Agent

I'll start by grabbing the surface build instructions and looking at the repo in parallel.

## User

Also, if useful... incorporate elements from blender since we have the MCP

## Agent

pypdf is available. I'll extract the reasoning summary and the paper intros.

## Agent

Next I'm checking that Blender is connected, then I'll build the 3D shape pairs.

## Agent

Your saved preference is to confirm where new files go, scratchpad included. One heads-up: I already cloned the repo into the session scratchpad to read the proofs, before checking that note.

## Agent

I'll put everything in `~/Desktop/mahler-surface/`. Writing the surface template now.

## User

Remind me how to open this in blender

## Agent

To connect Blender:

1. **Open Blender** and stay in the default scene (Layout workspace).
2. **Start the addon's server.** Hover over the 3D viewport and press **N** to open the sidebar. Find the MCP for Blender tab and click its connect/start button. It should then show the server as running.
3. **If you can't find the tab,** the addon may not be installed or enabled. Install it with:
   ```
   ! uvx mcp-for-blender install-addon
   ```
   Then enable it under **Edit → Preferences → Add-ons** (search "MCP") and repeat step 2.

I'm going from the addon's install command and typical setup here, so the tab and button names may differ a little in your version.

Tell me once it's running. I'll check the connection, build the cube/octahedron and second shape pair, and render the turntable frames to drop into the surface. Meanwhile I'll keep building the 2D playground and the proof animation in `~/Desktop/mahler-surface/`.

## User

MCP tool for blender is connected

## Agent

The cube/octahedron compound renders correctly. I'm zooming in and thickening the edges, then I'll render the turntable frames.

## Agent

Fast, about 6 seconds for 36 frames. Now the icosahedron/dodecahedron pair.

## Agent

Rendering is done: 72 frames, about 770 KB. Now the main piece, the surface HTML.

## Agent

Bug: the build's placeholder string also appears inside the JS, so the frame data would be injected there too. Fixing that.

## Agent

All values match the theory, including an off-center triangle, which shows the Santaló-point search works. Now uploading to Traces.

## User

[Request interrupted by user]

## User

wait... what are you uploaded to traces? As I think this through a surface can only be opened if there's a trace to apply it to so it wouldn't be possible to use this surface... tho useful to have the structure of the file as you've built it

## Agent

You're right, and I should have raised it before uploading. A surface is a lens on one trace's data, and this page never reads the trace. On any trace it shows the same standalone explainer, so it's in the wrong container.

**What I uploaded:** one new surface in your goldfishandswedishfish namespace:
- Key `mahler-shadow-twin`, version 0.1.0, about 1.1 MB.
- **Private and not released**, so no one sees it on any trace. The only way to view it is the preview link, which renders it on your latest trace.
- Nothing else was changed or published. I can archive it whenever you like (archived surfaces stay in version history but drop out of discovery).

**What's reusable, all in `~/Desktop/mahler-surface/`:**
- `template.html`: the page, with the 2D playground, the lens animation and the dimension table.
- `frames/`: the 72 Blender turntable renders.
- `build.py`: inlines the renders into the page as one self-contained `surface.html`.

The geometry is checked: square = 8, hexagon = 9, triangle = 6.75, and the Blender cube/octahedron pair comes out at exactly 4³/3!.

**Two directions that fit what you're after:**

1. **Publish it as a standalone page.** Make it a shareable claude.ai Artifact, or host it elsewhere. The file already runs offline with no dependencies, so this is mostly a re-wrap, and each proof becomes its own explorable page. This matches "help people who've never heard of these concepts".
2. **Make it a real trace surface.** The repo ships `reasoning_traces/`, which …[truncated]

## User

Give me the preview link

## Agent

Here's the preview link: https://traces.com/s/52797cc3-0c4f-4505-b3fa-3bb171d53396?surface=mahler-shadow-twin&version=0.1.0

It opens version 0.1.0 on your latest trace. Only members of your namespace can open it, and it stays unreleased until you say otherwise.

The trace underneath has nothing to do with the page, since the page doesn't read it. You'll see:
- **Symmetric / Any shape modes:** drag corners, double-click to add one, and hover edges to see the matching parts of the twin.
- **3D turntables:** drag to rotate them.
- **Lens animation:** "Plain circle" switches to the comparison.

## User

Oh this is interesting... the surface  still works even tho it's sitting on a trace that doesn't have relevant info, right?

## Agent

Yes. A surface is just an HTML page in a sandboxed iframe. The host offers trace data through `window.traces.getTrace()`, but nothing forces the page to use it. This page makes that call once so the handshake completes, then ignores the result, so it renders the same on any trace in your namespace.

That suggests three things:

- **Traces works as a host for standalone pages.** Any self-contained HTML (explainers, demos, tools) can live as a surface and open as a tab next to Highlights and Full Trace. You get the sandboxing, versioning, previews and namespace access control for free.
- **The limit is discoverability.** It only appears where someone opens a trace and picks the tab. A trace about something else is an odd front door, and the page will look out of place to anyone who finds it there.
- **The interesting middle ground is a page that works either way.** It can stand alone, then get richer when the trace is relevant. For example, if the trace mentions "Mahler", "polar" or "Hanner", it could show those moments next to the playground: "here's where the agent computed this." On an unrelated trace it falls back to the plain explainer, the way it does now.

That last pattern could work for every proof in the repo: one explainer per proof that gets smarter on traces of someone working with it. Want me to add that trace-aware layer to this one as a test?

## User

What I think would be useful in this trace is to add a explainer at the top re: what the major arguments of the proof. For someone who hasn't deeply explored this help them understand the signifigance of the argument

## Agent

I'll add an overview at the top in two parts. The first is a short orientation (the question, why it matters, why it was hard). The second walks through the proof's five major moves, each in plain language paired with the paper's technical term. It ends with the other routes in the family, how the model found the proof, and its verification status.

## Agent

I added the overview at the top and uploaded it as version 0.2.0: https://traces.com/s/52797cc3-0c4f-4505-b3fa-3bb171d53396?surface=mahler-shadow-twin&version=0.2.0

It's still private and unreleased. I checked that the file builds and its scripts parse, but I haven't viewed the new section in a browser, so look for layout problems when you open it.

**What's new at the top:**
- **A one-paragraph opener:** the question is which shapes are the "least round". It was open since 1939, and these papers settle it in every dimension, including exactly which shapes tie.
- **Three short cards:**
  - **The question:** a shape, its twin, and multiplying their volumes as a fair score of form.
  - **Why it matters:** where the score shows up (lattice points, normed spaces, symplectic geometry), and that before this only an approximate bound was known (Bourgain–Milman 1987).
  - **Why it was hard:** many shapes tie for last, so the usual "smooth toward one winner" proofs don't work.
- **The proof in five moves.** Each has a plain-language explanation, with the paper's own term beside it in small monospace:
  1. Build the shape out of slabs.
  2. Build a fair measuring device (the lens).
  3. Turn geometry into chance: the probabilities sum to at least 1.
  4. Turn chance back into volume, which gives the bound.
  5. Find every tie, which leads to the Hanner polytopes.

  Moves 1, 2 and 5 point to the interactive section below that shows them.
- **Three closing notes:**
  - The other two ro…[truncated]

## User

Let's make the 'how the proof works section' roll up able so that the user doesn't have to scroll so far to try the shape, let them fold that up

## Agent

"How the proof works" now folds up, and **it starts closed**, so the playground is right under the opener and the three cards. Version 0.3.0, still private and unreleased: https://traces.com/s/52797cc3-0c4f-4505-b3fa-3bb171d53396?surface=mahler-shadow-twin&version=0.3.0

- **When closed** it shows as one row: a chevron, the title, and "5 moves · show" on the right. Clicking anywhere on the row opens it, and the label changes to "hide".
- **The fold also holds** the three closing notes (other routes in the family, how the model got there, how sure this is), so the top stays compact. One trade-off: the "How sure is this?" verification caveat is now hidden until someone opens the fold.

The file builds and its scripts parse, but I haven't opened this version in a browser, so check the fold there.

Two changes you might want:
- **Start it open**, so first-time readers see the argument by default and the fold is just for collapsing it.
- **Move "How sure is this?" out of the fold**, so the caveat stays visible.

# Started building math proof visualizations

- ID: 189903e3-2f4c-44dd-8020-b4552bf40364
- URL: https://traces.com/s/189903e3-2f4c-44dd-8020-b4552bf40364
- Agent: claude-code

## Agent

"How the proof works" now folds up, and **it starts closed**, so the playground is right under the opener and the three cards. Version 0.3.0, still private and unreleased: https://traces.com/s/52797cc3-0c4f-4505-b3fa-3bb171d53396?surface=mahler-shadow-twin&version=0.3.0

- **When closed** it shows as one row: a chevron, the title, and "5 moves · show" on the right. Clicking anywhere on the row opens it, and the label changes to "hide".
- **The fold also holds** the three closing notes (other routes in the family, how the model got there, how sure this is), so the top stays compact. One trade-off: the "How sure is this?" verification caveat is now hidden until someone opens the fold.

The file builds and its scripts parse, but I haven't opened this version in a browser, so check the fold there.

Two changes you might want:
- **Start it open**, so first-time readers see the argument by default and the fold is just for collapsing it.
- **Move "How sure is this?" out of the fold**, so the caveat stays visible.

>ls -la ~/Desktop/mahler-surface/ && ls ~/Desktop/mahler-surface/frames | head -3 && wc -c ~/Desktop/mahler-surface/*.html ~/Desktop/mahler-surface/build.py

total 2272
drwxr-xr-x@  6 karamelton  staff      192 Oct 11 21:41 .
drwx------@ 25 karamelton  staff      800 Oct 11 21:38 ..
-rw-r--r--@  1 karamelton  staff      959 Oct 11 21:41 build.py
drwxr-xr-x@  4 karamelton  staff      128 Oct 11 21:38 frames
-rw-r--r--@  1 karamelton  staff  1107488 Oct 11 21:55 surface.html
-rw-r--r--@  1 karamelton  staff    49011 Oct 11 21:55 template.html
cube
ico
 1107488 /Users/karamelton/Desktop/mahler-surface/surface.html
   49011 /Users/karamelton/Desktop/mahler-surface/template.html
     959 /Users/karamelton/Desktop/mahler-surface/build.py
 1157458 total

>S=/private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad; mkdir -p $S && git clone -q --depth 1 https://github.com/openai/math $S/repo 2>&1 | tail -2; find $S/repo -maxdepth 3 -not -path '*/.git*' | sed "s|$S/||" | head -80

Command did not complete within its 120s timeout and was moved to the background (ID: b9pjt7j1t). Output is being written to: /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/tasks/b9pjt7j1t.output. You will be notified when it completes. To check interim output, use Read on that file path.

>S=/private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad; ls $S/repo 2>&1; du -sh $S/repo 2>/dev/null; cat $S/../tasks/b9pjt7j1t.output

CONTENTS.md
history.md
lean
LICENSE
overview.pdf
overview.tex
preprints
README.md
reasoning_traces
3.1G	/private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad/repo
repo
repo/reasoning_traces
repo/lean
repo/LICENSE
repo/overview.tex
repo/overview.pdf
repo/history.md
repo/README.md
repo/CONTENTS.md
repo/preprints
repo/reasoning_traces/relativistic-vlasov-maxwell.pdf
repo/reasoning_traces/kaplansky-direct-finiteness-characteristic-two.pdf
repo/reasoning_traces/free-group-factor-isomorphism.pdf
repo/reasoning_traces/spontaneous-magnetization-quantum-heisenberg-ferromagnet.pdf
repo/reasoning_traces/quasipolynomial-arithmetic-progressions.pdf
repo/reasoning_traces/mezard-parisi-formula.tex
repo/reasoning_traces/irrationality-exponent-of-pi.pdf
repo/reasoning_traces/symmetric-and-general-mahler-conjectures.pdf
repo/reasoning_traces/mezard-parisi-formula.pdf
repo/reasoning_traces/ordinary-two-point-correlations.pdf
repo/reasoning_traces/basic-semidefinite-threshold-np-hardness.pdf
repo/lean/LICENSE
repo/lean/ComparatorChallenges
repo/lean/patches
repo/lean/OAI.lean
repo/lean/lean-toolchain
repo/lean/formalization.yaml
repo/lean/docs
repo/lean/README.md
repo/lean/OAI
repo/lean/lake-manifest.json
repo/lean/lakefile.lean
repo/preprints/Quantum-Depletion-for-Fixed-Bounded-Repulsive-Potentials-October-5-2026
repo/preprints/An-FPRAS-for-Common-Integer-Polymatroid-Bases-with-Binary-Capacities-October-5-2026
repo/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026
repo/preprints/Zero-Plane-Rigidity-for-Einstein-Four-Manifolds-October-4-2026
repo/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026
repo/preprints/The-canonical-massive-continuum-limit-of-the-two-dimensional-O3-model-October-4-2026
repo/preprints/Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026
repo/preprints/Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four-September-24-2026
repo/preprints/The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026
repo/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026
repo/preprints/Irrational-cubic-fourfolds-with-geometric-K3-categories-September-24-2026
repo/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026
repo/preprints/A-sharp-entropy-bound-and-the-simplex-inequality-for-isotropic-constants-October-5-2026
repo/preprints/Algebraicity-of-Weil-classes-on-split-abelian-eightfolds-September-18-2026
repo/preprints/Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026
repo/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026
repo/preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026
repo/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026
repo/preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026
repo/preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026
repo/preprints/The-Critical-Spin-Field-of-the-Planar-XY-Model-October-5-2026
repo/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026
repo/preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026
repo/preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026
repo/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026
repo/preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026
repo/preprints/An-Exact-Semidefinite-Lift-of-a-Nonspectrahedral-Hyperbolicity-Cone-October-5-2026
repo/preprints/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026
repo/preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026
repo/preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026
repo/preprints/Deforming-hypersymplectic-four-manifolds-to-hyperkahler-triples-September-23-2026
repo/preprints/Maximal-Seshadri-Constants-on-Arbitrary-Polarized-Surfaces-September-23-2026
repo/preprints/Deterministic-quasipolynomial-time-mean-payoff-games-September-25-2026
repo/preprints/Bounded-klt-complements-for-Fano-contractions-September-25-2026
repo/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026
repo/preprints/Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three-October-5-2026
repo/preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026
repo/preprints/Thomason-Model-Structures-in-Every-Strict-Higher-Dimension-September-25-2026
repo/preprints/Conditional-coordinate-sweeps-and-analytic-transfer-September-26-2026
repo/preprints/The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026
repo/preprints/The-Quasi-Riemann-Hypothesis-October-5-2026
repo/preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026
repo/preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026
repo/preprints/A-power-saving-for-planar-unit-distances-September-23-2026
repo/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026
repo/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026
repo/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026
repo/preprints/Log-abundance-in-numerical-dimension-one-for-threefolds-in-positive-characteristic-October-5-2026

[exited with code 0]

>S=/private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad; grep -E '^\*\*[0-9]{3}\.' $S/repo/CONTENTS.md | grep -iE 'color|kakeya|unit distance|heilbronn|unbiased|littlewood|percolation|sphere|packing|tiling|knot|polytope|graph|ramsey|sofa|triangle|isoperimetric' | cut -c1-400

**028. Uniformly bounded components of Gaussian-prime graphs.** Proves the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have uniformly bounded steps. More strongly, for every distance bound <i>D</i>, the graph joining Gaussian primes at distance at most <i>D</i> has uniformly bounded finite component sizes, depending only on <i>D</i>, including primes on the coor
**060. The Global Spherical Shell conjecture.** Every connected minimal compact complex surface of class VII with $`b_2\gt 0`$ contains a global spherical shell, proving the positive-<i>b</i><sub>2</sub> Global Spherical Shell conjecture. Such a shell is a holomorphically embedded neighborhood of the standard three-sphere in $`\mathbb C^2\setminus\{0\}`$ whose complement is connected.
**071. Koebe’s circle-domain conjecture.** Resolves the existence part of Koebe's circle-domain conjecture: every domain in the Riemann sphere is conformally equivalent to a domain whose complementary components are round disks or points. It also proves that circle domains with conformally removable boundary are rigid, meaning every conformal equivalence to another circle domain is Möbius, establi
**074. Kakeya in three and four dimensions.** Resolves the Kakeya maximal conjecture in three dimensions and the Hausdorff-dimension conjecture in four. In three dimensions, the radius-<i>δ</i> tube maximal operator maps $`L^3(\mathbb R^3)`$ to $`L^3(S^2)`$ with norm $`O_\varepsilon(\delta^{-\varepsilon})`$ for every <i>ε</i> &gt; 0. In four dimensions, every set containing a unit segment in every
**076. Real ultraflat Littlewood polynomials and unbounded binary merit factors.** Constructs polynomials with <i>N</i> consecutive coefficients in $`\{-1,1\}`$ whose modulus is $`(1+o(1))\sqrt N`$ uniformly on the entire unit circle, for every sufficiently large integer length <i>N</i>. Thus real Littlewood polynomials are ultraflat, including at the real endpoints. Their binary merit factors ten
**087. The Mahler conjectures, functional inequalities and polar-product symplectic width.** Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The corresponding sharp functional Mahler inequalities also hold. For <i>n</i> ≥ 2, every symmetric pol
**089. Bounded-distortion <i>L</i><sub>1</sub> embeddings of planar and bounded-treewidth graphs.** Resolves the planar and bounded-treewidth cases of the Gupta–Newman–Rabinovich–Sinclair conjecture. Shortest-path metrics of finite connected graphs with arbitrary positive edge lengths embed into real <i>L</i><sub>1</sub> with universal distortion for planar graphs, and distortion depending only on
**102. The Unique Games Conjecture and optimal approximation thresholds.** Proves Khot's Unique Games Conjecture. Independent direct reductions also establish NP-hardness, on unweighted graphs, of approximation beyond the Goemans–Williamson ratio for Max-Cut, below factor two for Vertex Cover, and within any fixed constant factor for Min-UnCut and directed feedback vertex set. These direct proofs 
**106. Hardness of coloring three-colorable graphs.** It is NP-hard to color a three-colorable graph using any fixed number <i>c</i> ≥ 3 of colors. More strongly, for every fixed $`0\lt \delta\lt 1/3`$, a deterministic polynomial-time reduction from 3SAT produces simple unweighted graphs that are three-colorable in the satisfiable case and have no independent set of size $`\delta n`$ otherwise, wh
**113. Approximate counting and entropy of perfect matchings.** Gives a fully polynomial randomized approximation scheme for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts. Also proves the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant, bounding the maximum entropy of a matching law at every feasible edge-marginal vector i
**118. Bin packing and unbounded configuration-LP gaps.** Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant. Approximating the optimum within any fixed additive constant is also NP-hard, even when every item exceeds 1/6 and each bin holds at mo
**120. Almost-linear-time exact matching and prescribed-degree factors in general graphs.** Gives a randomized algorithm finding an exact maximum-cardinality matching in any simple undirected graph in $`(n+m)^{1+o(1)}`$ word time, with success probability at least 2/3. The time bound holds on every computation path. The same guarantees apply to finding a spanning subgraph with prescribed admissibl
**124. Polynomial-time scheduling on three identical machines.** Resolves the three-processor unit-job scheduling problem of Garey and Johnson: a deterministic polynomial-time algorithm minimizes makespan for nonpreemptive unit-length jobs with arbitrary precedence constraints on three identical parallel machines. For an explicitly given precedence graph, it decides deadline feasibility exactly an
**126. Exponential semidefinite complexity of perfect matching.** Proves that every exact semidefinite lift of the perfect matching polytope has exponential size, answering Rothvoss's polynomial-size lift question negatively. The bound holds even for the positive semidefinite rank of its odd-cut slack matrix after any fixed shift $`0\lt \rho\lt 1`$, allowing arbitrary real positive semidefinite fa
**131. Rapid mixing of graph switches for every degree sequence.** Resolves the simple-undirected Kannan–Tetali–Vempala conjecture: the lazy edge-switch chain mixes in $`O(n^8)`$ time for every graphical labeled degree sequence. The same degree-constrained graphs can also be sampled exactly uniformly by an almost-surely terminating algorithm with expected polynomial bit running time. ([Lean](lean/
**133. The computational complexity of Weisfeiler–Leman refinement.** Proves unconditional $`n^{\Omega(k)}`$ deterministic time lower bounds for joint and separate <i>k</i>-dimensional Weisfeiler–Leman equivalence, for sufficiently large fixed <i>k</i> in the specified sequential adjacency-matrix models. With dimension as input, joint equivalence is EXPTIME-complete even on subcubic graphs; decidi
**150. Weak mixing of triangular billiards with an irrational angle.** Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to <i>π</i> is weakly mixing for normalized area times uniform direction. This strengthens ergodicity on the entire irrational-angle class, with no genericity or Diophantine restrictions. ([Lean](lean/docs/150.md)
**153. Arithmetic classification and non-Pisot singularity for Bernoulli convolutions.** Classifies singular and absolutely continuous unbiased Bernoulli convolutions for every $`\lambda\in(0,1)`$ by an infinite, one-sided approximation condition using explicit finite sets of algebraic units. It also proves singularity at reciprocals of every quartic Salem number in $`(1,2)`$, giving examples beyo
**155. A counterexample to periodic tiling in dimension three.** Constructs a finite translational tile in ℤ<sup>3</sup> that tiles space but admits no fully periodic tiling, disproving the periodic tiling conjecture in the smallest possible lattice dimension. Its unit-cube thickening gives the same counterexample in ℝ<sup>3</sup>, even with arbitrary real translation vectors. ([Lean](lean/docs/15
**157. Graph coloring, clique minors, and Colin de Verdière invariants.** Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy $`\chi_f(G)\gt h(G)`$, where $`h(G)`$ is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound $`\chi_f(G)\le\mu(G)+1`$. In the positive d
**158. The Euclidean plane cannot be colored with five colors.** Proves that every five-coloring of the Euclidean plane has a monochromatic pair at distance one, with no restriction on the color classes. This advances the Hadwiger–Nelson problem: together with the classical seven-coloring, only six and seven remain possible chromatic numbers of the plane. ([Lean](lean/docs/158.md))
**160. Superexponential van der Waerden numbers.** Resolves Erdős's superexponential-growth question for van der Waerden numbers. If $`W_r(k)`$ is the least interval length forcing a monochromatic <i>k</i>-term progression in every <i>r</i>-coloring, then $`W_r(k)\gt k^{ck\lfloor\log_2 r\rfloor}`$ for an absolute <i>c</i> &gt; 0, all <i>r</i> ≥ 2 and sufficiently large <i>k</i>, uniformly in <i>r<
**161. Counterexamples to Sidorenko’s conjecture and the forcing conjecture.** Disproves Sidorenko's conjecture with a connected bipartite pattern on 35 vertices and 66 edges that occurs less frequently than in a random graph of the same edge density. The same pattern disproves the forcing conjecture of Skokan and Thoma: matching its density and the edge density of a constant graphon need not forc
**162. Counterexamples to Ryser’s covering conjecture.** Disproves Ryser's covering conjecture by constructing intersecting $`(q+1)`$-partite, $`(q+1)`$-uniform hypergraphs with covering number $`q+1`$, rather than the predicted bound <i>q</i>, for every sufficiently large prime <i>q</i>. A separate construction over extension fields also disproves Gyárfás's monochromatic tree-cover conjecture. ([
**164. Hindman’s finite sums and products conjecture.** Proves Hindman's finite sums and products conjecture: every finite coloring of the positive integers contains sets of any prescribed finite size whose nonempty subset sums and nonempty subset products all have one common color.
**165. The Harary–Hill and Zarankiewicz crossing-number formulas.** Resolves the Harary–Hill conjecture and Turán's brickyard problem in the Zarankiewicz formulation, determining the crossing numbers of every complete and complete bipartite graph. The result proves the optimality of the classical drawings among all plane drawings with continuous edge arcs. ([Lean](lean/docs/165.md))
**169. Shareshian–Wachs elementary positivity.** Resolves the elementary-positivity part of the Shareshian–Wachs conjecture: the chromatic quasisymmetric function of every natural unit interval graph has elementary-basis coefficients in $`\mathbb N[q]`$. The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity. ([Lean](lean/docs/169.md))
**170. Sharp logarithmic exponents for off-diagonal Ramsey numbers.** For every fixed integer <i>s</i> ≥ 5, proves $`r(s,t)=t^{s-1}/(\log t)^{s-2+o(1)}`$ as $`t\to\infty`$, determining the logarithmic exponent and matching the classical upper bound at that scale. Here $`r(s,t)`$ is the least number of vertices forcing an <i>s</i>-clique or a <i>t</i>-vertex independent set. ([Lean](lean/docs/170.m
**171. The hypercube Ramsey conjecture.** Resolves the Burr–Erdős hypercube Ramsey conjecture: the two-color Ramsey number of the <i>n</i>-dimensional cube is $`\Theta(2^n)`$. Thus every red-blue coloring of a complete graph on a universal constant times the cube's number of vertices contains a monochromatic copy of the cube.
**172. Classification of finite Euclidean Ramsey configurations.** Classifies finite point configurations that occur monochromatically, at their original scale, in every finite coloring of sufficiently high-dimensional Euclidean space. The characterization is an algebraic condition over the coordinate field. It also disproves the Leader–Russell–Walters conjecture that every such configuration is a
**173. Seymour’s second-neighborhood conjecture.** Proves Seymour's second-neighborhood conjecture: every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one. Oriented graphs may be arbitrary apart from the exclusion of loops and oppositely directed edge pairs. ([Lean](lean/docs/173.md))
**174. Deterministic construction of strong thin spanning trees.** Resolves the strong thin-tree conjecture constructively. Every finite loopless <i>k</i>-edge-connected multigraph on at least two vertices has a spanning tree containing at most a universal $`C/k`$ fraction of the edges of every cut. Such a tree can be found deterministically in polynomial time, even with binary-encoded parallel-ed
**175. Talagrand’s expectation thresholds, discrete convexity, and graph decompositions.** Proves that integral and fractional expectation thresholds differ by at most a universal factor, and resolves Talagrand's discrete-convexity conjecture. An application proves the Ascoli–He–Park–Talagrand graph-decomposition conjecture: every graph's edges split into a universally bounded number of fixed piec
**176. The second Kahn–Kalai conjecture with an edge-count bound.** Proves the second Kahn–Kalai conjecture: for every finite simple graph <i>H</i> with <i>h</i> ≥ 1 edges and at most <i>n</i> vertices, its appearance threshold in $`G(n,p)`$ is at most $`C p_{\mathrm E}(n,H)(1+\log_2 h)`$, with universal <i>C</i>. Here $`p_{\mathrm E}`$ is the least density at which every subgraph of <i>H</i> has 
**177. Bounded-degree coboundary expanders.** Constructs arbitrarily large finite <i>d</i>-dimensional simplicial complexes, for every <i>d</i> ≥ 3, with uniformly bounded vertex degrees and uniform 𝔽<sub>2</sub> coboundary expansion in every degree below <i>d</i>. Together with the known graph and two-dimensional cases, this establishes the existence of such expanders in every positive dimension.
**178. Deterministic nonbipartite Ramanujan graphs in every fixed degree.** For every fixed <i>d</i> ≥ 3, constructs a simple <i>d</i>-regular nonbipartite Ramanujan graph on every sufficiently large even number <i>n</i> of vertices, with every nonconstant adjacency eigenvalue strictly between $`-2\sqrt{d-1}`$ and $`2\sqrt{d-1}`$. A deterministic algorithm outputs the full adjacency list in polyno
**180. Barnette’s Hamiltonian-cycle conjecture.** Proves that every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle, resolving Barnette's conjecture. Equivalently, every three-edge path in a finite simple cubic 3-vertex-connected bipartite Pfaffian graph lies in a Hamiltonian cycle. ([Lean](lean/docs/180.md))
**181. The Erdős–Gallai cycle-decomposition conjecture.** Proves that the edges of every finite simple undirected graph on <i>n</i> vertices can be partitioned into at most $`Cn`$ simple cycles and single edges, for an absolute constant <i>C</i>. This resolves the Erdős–Gallai cycle-decomposition conjecture, bounding the number of pieces linearly even for dense graphs. ([Lean](lean/docs/181.md))
**184. Correspondence coloring with a fixed forbidden subgraph.** Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph <i>F</i> need $`O_F(\Delta/\log\Delta)`$ colors when their maximum degree Δ is sufficiently large. Also proves the Ajtai–Erdős–Komlós–Szemerédi independence conjecture: for fixed <i>r</i> ≥ 4, every <i>n</i>-ve
**185. Counterexamples to infinite matroid intersection and packing/covering.** Disproves the unrestricted infinite matroid intersection and packing/covering conjectures in ZFC, using two self-dual partitional matroids on a countably infinite ground set. The same examples answer Joó’s partitional-matroid question negatively. They are neither finitary nor cofinitary, so Nash-Williams’ original fini
**186. Uniform influence and sharp thresholds for graph and hypergraph properties.** Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed $`0\lt \varepsilon\lt 1/2`$, every nontrivial increasing relabeling-invariant property crosses from probability <i>ε</i> to $`1-\varepsilon`$ within width $`O((\log n)^{-2})`$ for graphs and $`O_r((\log n)^
**188. The sharp terminal leave in random triangle removal.** Starting from the complete graph on <i>n</i> vertices, repeatedly delete a uniformly chosen remaining triangle. The terminal edge count is asymptotic to $`n^{3/2}/(2\sqrt2)`$, with mean-square convergence after normalization by <i>n</i><sup>3/2</sup>. This proves the triangle case of the Joos–Kühn sharp-constant conjecture. ([Lean](lean
**189. Cycle–clique Ramsey numbers.** Proves the Erdős–Faudree–Rousseau–Schelp conjecture: $`R(C_m,K_n)=(m-1)(n-1)+1`$ for every $`m\ge n\ge3`$, except $`R(C_3,K_3)=6`$. This is the exact threshold forcing a red <i>m</i>-cycle or a blue <i>n</i>-clique in every red–blue coloring of a complete graph. ([Lean](lean/docs/189.md))
**191. A power improvement in the Heilbronn triangle lower bound.** For every sufficiently large <i>n</i>, constructs <i>n</i> points in the unit square such that every triangle has area at least $`n^{-2+c}`$ for one absolute <i>c</i> &gt; 0. This disproves the conjectured almost-<i>n</i><sup>−2</sup> upper bound in Heilbronn's triangle problem, which asks how large the smallest determined triangl
**211. The geometric phase diagram, diffusion, and spectra of random planar maps.** Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for $`0\lt q\le4`$ and to the Brownian continuum random tree for <i>q</i> &gt; 4, establishing the surface-to-tree geometric transition. For FK–Ising and spanning-tree-weighted maps, stationary random walks converge to Liouville Br
**212. Planar first-passage geometry and the absence of bigeodesics.** Proves that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment. For exponential weights, the limit shape is strictly convex with <i>C</i><sup>1</sup> boundary. Differentiability also holds for every Gamma law with 
**213. Critical percolation on every quasi-transitive graph.** Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with $`p_c\lt 1`$: at the critical probability, there is almost surely no infinite cluster. The family also establishes this conclusion for both nearest-neighbor bond and site percolation on ℤ<sup
**214. The Benjamini–Schramm nonuniqueness conjecture.** Proves $`p_c\lt p_u`$ for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range of probabilities with infinitely many infinite clusters. A stronger operator bound also establishes the critical triangle 
**219. GOE bulk universality for regular graphs with weak Anderson disorder.** For every fixed degree <i>d</i> ≥ 3, the bulk adjacency-eigenvalue point process of a uniform simple random <i>d</i>-regular graph converges to the Gaussian orthogonal ensemble law, including for cubic graphs. The same fixed-energy universality persists under sufficiently weak fixed iid uniform diagonal disorder, throug
**224. Critical and quenched near-critical universality for Poisson–Voronoi percolation.** Proves Cardy's formula for annealed critical Poisson–Voronoi crossing probabilities in every bounded Jordan quadrilateral. With each model normalized by its own expected unit-square pivotal count, the conditional joint near-critical crossing-threshold laws for rational polygonal quads converge in environment
**231. The free uniform spanning forest is a factor of IID.** On every infinite connected locally finite simple unweighted graph, the free uniform spanning forest is a factor of independent vertex labels, by one isomorphism-equivariant rule using no root. Translation-invariant strongly Rayleigh binary processes on every countable group, including invariant determinantal processes with Hermitian po
**266. Exactly three mutually unbiased bases in dimension six.** Proves $`N(6)=3`$, resolving Zauner's dimension-six mutually unbiased bases conjecture: three such bases exist in ℂ<sup>6</sup>, but four cannot. The exclusion is a complete certified computation under the stated binary64 arithmetic and compiler conditions. An independent companion proves the Matolcsi–Ruzsa–Weiner Fourier-vanishing c
**267. Positive-temperature Bose–Einstein condensation and exact quantum depletion.** Proves Bose–Einstein condensation for the exact canonical Gibbs state of the three-dimensional hard-sphere gas: each fixed exclusion distance and sufficiently small fixed density admit a strictly positive temperature, independent of volume, with positive condensate fraction in the thermodynamic limit. At zero tem
**269. Uniform Laughlin gap and stability under bounded scalar disorder.** Proves the fermionic Laughlin spectral-gap conjecture for the full <i>V</i><sub>1</sub> interaction at filling 1/3 on the round sphere. The unique ground state remains uniformly gapped under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. Both the gap and disorder threshold ar
**281. QAOA attains the SK optimum in the thermodynamic-first limit.** Proves that QAOA approaches the ground-state energy of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity before circuit depth. For every accuracy, finite depth and deterministic angles independent of size and disorder achieve the required limiting expected energy per spin. This also yields
**305. Four-dimensional disk embedding and Wall's conjecture.** The unrestricted four-dimensional disk-embedding conjecture fails: framed algebraic dual spheres do not suffice to obtain disjoint locally flat spanning disks. In particular, the free group <i>F</i><sub>2</sub> is not good in the sense of Freedman–Quinn. Also constructs a finitely presented integral Poincaré duality group of dimension
**306. The purely cosmetic surgery conjecture.** Distinct Dehn surgery slopes on a nontrivial smooth knot in <i>S</i><sup>3</sup> never produce orientation-preservingly homeomorphic manifolds, proving the purely cosmetic surgery conjecture. The statement includes the meridional slope.
**307. Failure of rational injectivity for maximal coarse assembly.** Constructs a uniformly discrete bounded-geometry space whose maximal coarse assembly map is not rationally injective. The example is a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order kernel class. A companion gives the analogous failure for reduced coarse assembly, disproving 
**313. Finite generation for the $`K(n)`$-local sphere.** Answers the degreewise finiteness question of Hovey and Hovey–Strickland: every homotopy group of the $`K(n)`$-local sphere is a finitely generated ℤ<sub><i>p</i></sub>-module, for every prime, positive height and integer degree. The same conclusion holds after $`K(n)`$-localizing any finite <i>p</i>-local spectrum.
**316. Curtis’s conjecture.** Proves Curtis’s conjecture: the positive-degree mod-two stable Hurewicz image of the sphere is spanned by the images of the Hopf-invariant-one classes <i>η</i>, <i>ν</i>, <i>σ</i> and the Kervaire-invariant-one classes that exist.
**318. Chromatic splitting: filtrations and counterexamples.** Disproves strong chromatic splitting at height three for primes <i>p</i> ≥ 5, and weak splitting for the derived <i>p</i>-completed sphere at heights <i>p</i> (<i>p</i> ≥ 5) and $`p+1`$ (<i>p</i> ≥ 7). Nevertheless, for <i>n</i> ≥ 1 and $`p\gt n+1`$, the overlap $`L_{n-1}L_{K(n)}S_p^\wedge`$ admits a $`2^n`$-stage filtration by the pre
**322. Tingley’s sphere-isometry problem.** Resolves Tingley's problem: every surjective isometry between the unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry of the whole spaces. No dimension restriction is imposed, so the metric geometry of the unit sphere determines the Banach space up to linear isometry. ([Lean](lean/docs/322.md))
**343. Symplectic ball packing in higher dimensions.** Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension $`2n\ge6`$. Finitely many closed symplectic balls of capacities $`R_1,\ldots,R_k`$ embed disjointly into an open ball of capacity <i>R</i> exactly when $`\sum_iR_i^n\lt R^n`$ and $`R_i+R_j\lt R`$ for every distinct pair <i>i</i>, <i>j</i>. ([Lean](lean/docs/343.md))
**345. Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds.** Proves that every smooth Riemannian metric on <i>S</i><sup><i>n</i></sup>, <i>n</i> ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The same conclusion holds on every closed manifold admitting a finite smooth spherical cover and on every closed three-manifold, without orientab
**346. Sharp singular-set bounds for stationary integral varifolds.** Proves that every stationary integral <i>m</i>-varifold in a Euclidean open set has singular set of Hausdorff dimension at most $`m-1`$, a sharp bound in every positive dimension and codimension. On round spheres, the family also proves almost-everywhere regularity: the singular set has zero <i>m</i>-dimensional Hausdorff measur
**348. Nonnegative-curvature Einstein classification and an <i>L</i><sup>2</sup> topological gap.** Classifies closed connected Einstein four-manifolds with positive Einstein constant and nonnegative sectional curvature: up to scaling, their universal Riemannian covers are the round <i>S</i><sup>4</sup>, Fubini–Study $`\mathbb{CP}^2`$, or a product of equal round two-spheres. A closed simply conne
**349. The Solomon–Yau least-volume conjecture.** Proves the Solomon–Yau least-volume conjecture for minimal hypersurfaces of round spheres. For every <i>m</i> ≥ 2, a closed connected minimal immersion into the unit sphere $`S^{m+1}`$ with non-totally-geodesic image has volume at least that of the smallest minimal Clifford product, counting covering multiplicity.
**353. Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample.** Proves that every smooth locally uniformly convex affine-maximal graph of dimension three through nine, complete for its induced Euclidean metric, is an elliptic paraboloid. A smooth entire nonquadratic example in dimension ten makes this range sharp. In dimensions three through nine, the paraboloi
**354. The isoperimetric profile of the cubic three-torus.** Determines the isoperimetric profile of the unit cubic flat three-torus and classifies every finite-perimeter minimizer: balls, circular tubes around shortest closed geodesics, coordinate slabs, and their complements. The transition volumes are $`4\pi/81`$ and $`1/\pi`$, with exactly the adjacent two types minimizing at each transition. 
**363. Nonuniqueness with local conservation for the hard-sphere Boltzmann equation.** Constructs two distinct global entropy solutions of the three-dimensional periodic hard-sphere Boltzmann equation from the same nonnegative initial density, with bounded velocity support and finite mass, energy and absolute entropy. Both are strongly continuous in <i>L</i><sup>1</sup> and satisfy exact local con
**364. Kinetic limits and fluctuations over the Boltzmann lifespan.** Derives the nonlinear Boltzmann equation from three-dimensional grand-canonical Newtonian gases throughout every regular kinetic interval with uniform Gaussian decay. Stable finite-range radial potentials may have attractive wells and a singular repulsive core; initial pair exclusion and spatially summable Gaussian density and g

The user answered: "Which proof should we build next? These four lend themselves best to hands-on visuals for newcomers."="Let's do them all!", "Where should the new surface's files go? (Yesterday's Mahler files are in ~/Desktop/mahler-surface/.)"="Let's make a repo here: https://github.com/goldfishandswedishfish". Read the answers carefully — they may request clarification, changes, or that you not proceed — and follow what they actually say.

>gh auth status 2>&1 | head -8; gh repo list goldfishandswedishfish --limit 50 2>&1 | head -30

github.com
  ✓ Logged in to github.com account goldfishandswedishfish (keyring)
  - Active account: true
  - Git operations protocol: https
  - Token: gho_************************************
  - Token scopes: 'gist', 'read:org', 'repo'
goldfishandswedishfish/standing-order-hub		private	2026-08-10T20:52:23Z
goldfishandswedishfish/context-highlighter	A personal knowledge layer that captures highlights from anywhere, organizes them, and helps feed context to AI agents. [learning by building]	public	2026-05-30T14:38:13Z
goldfishandswedishfish/reachy-coat-check	Reachy Mini app — watches your outfit, logs the weather, and learns whether you need a coat. [learning by building]	public	2026-05-25T14:44:19Z
goldfishandswedishfish/router	A lightweight CLI that automatically routes messages to the right Claude model (Haiku / Sonnet / Opus) based on task complexity, with real-time cost tracking. [learning by building]	private	2026-05-24T20:40:11Z
goldfishandswedishfish/posy	iOS app to encourage us all to stop scrolling and touch a bit of metaphorical grass. Exploring: iOS development. [learning by building]	private	2026-05-23T18:48:39Z
goldfishandswedishfish/threadwatch	threadwatch is a CLI that runs the same agentic task against multiple LLM providers at the same time and shows you exactly how each one handles it. [learning by building]	public	2026-05-23T18:48:33Z
goldfishandswedishfish/crossroads	An interactive web app that connects places to music. Exploring: open source, secrets injection, and working with maps. [learning by building]	public	2026-05-23T18:48:21Z

>cd ~/Desktop/math-surfaces/mahler && wc -l template.html && grep -n -E '<(style|script|section|h2|details)|getTrace|^\s*function ' template.html | head -60

     714 template.html
7:<style>
171:<section id="overview">
189:  <details class="fold" id="proof-fold">
219:<section id="play">
220:  <h2>Try it: a shape and its twin</h2>
257:<section id="dims">
258:  <h2>Up a dimension</h2>
272:<section id="trick">
273:  <h2>The trick at the heart of the proof</h2>
292:  <details>
298:<section>
302:<script id="tt-data" type="application/json">/*TURNTABLES*/</script>
303:<script>
310:function readColors() {
316:function fitCanvas(cv) {
323:function alpha(color, a) { // works for hex / rgb / oklch via color-mix
329:function hull(pts) { // monotone chain, CCW, returns indices
337:function area(poly) { let s = 0; for (let i = 0; i < poly.length; i++) { const a = poly[i], b = poly[(i + 1) % poly.length]; s += a[0] * b[1] - a[1] * b[0]; } return Math.abs(s) / 2; }
339:function polar(poly, c) {
350:function santalo(poly) { // minimise |(K - z)°| over interior z, Nelder–Mead
388:function fullPoints() { return S.mode === "sym" ? S.free.concat(S.free.map(([x, y]) => [-x, -y])) : S.free.slice(); }
389:function compute() {
402:function view(w) { const s = w / (2 * R); return { s, X: (x) => w / 2 + x * s, Y: (y) => w / 2 - y * s, ix: (px) => (px - w / 2) / s, iy: (py) => (w / 2 - py) / s }; }
404:function drawAxes(ctx, w, v) {
409:function pathPoly(ctx, poly, v) { ctx.beginPath(); poly.forEach(([x, y], i) => (i ? ctx.lineTo(v.X(x), v.Y(y)) : ctx.moveTo(v.X(x), v.Y(y)))); ctx.closePath(); }
411:function drawPlay() {
461:function fmt(x, d = 4) { return Number.isFinite(x) ? x.toFixed(d) : "–"; }
464:function drawGauge() {
472:function setNote() {
491:function updatePlay() {
498:function renderPresets() {
506:function setMode(m) {
518:function localPt(cv, ev) { const r = cv.getBoundingClientRect(); return [ev.clientX - r.left, ev.clientY - r.top, r.width]; }
519:function segDist(p, a, b) { const dx = b[0] - a[0], dy = b[1] - a[1], t = Math.max(0, Math.min(1, ((p[0] - a[0]) * dx + (p[1] - a[1]) * dy) / (dx * dx + dy * dy || 1))); return Math.hypot(p[0] - a[0] - t * dx, p[1] - a[1] - t * dy); }
520:function hitPoint(px, py, w) {
599:function gammaHalf(k) { // Γ(k/2) for positive integer k
623:function Fr(r, th) { let re = 0, im = 0, rp = r; for (let j = 0; j < 60; j++) { const m = 2 * j + 1, s = (j % 2 ? -1 : 1) / (m * m); re += s * rp * Math.cos(m * th); im += s * rp * Math.sin(m * th); rp *= r * r; } return [(8 / (Math.PI * Math.PI)) * re, (8 / (Math.PI * Math.PI)) * im]; }
631:function drawLens() {
676:function sample(k) {
679:function resetHist() { L.bins.fill(0); L.n = 0; }
688:function loop() {
704:function redrawAll() { readColors(); drawPlay(); drawLens(); }
710:if (window.traces && typeof window.traces.getTrace === "function") { window.traces.getTrace().catch(() => {}); }

>S=/private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad; mkdir -p $S/txt; cd $S && python3 -I -c "
import pypdf,sys
P={'028':'Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/paper.pdf','155':'A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf','158':'The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf','191':'A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf'}
for k,p in P.items():
  r=pypdf.PdfReader('repo/preprints/'+p)
  t='\n'.join((pg.extract_text() or '') for pg in r.pages)
  open('txt/'+k+'.txt','w').write(t); print(k,len(r.pages),'pages',len(t),'chars')
"; ls repo/preprints | grep -iE 'five-color|plane-is-not|heilbronn|gaussian|periodic-til|translational'

Traceback (most recent call last):
  File "<string>", line 2, in <module>
ModuleNotFoundError: No module named 'pypdf'
A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026
A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026
A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026
Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026
Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-October-6-2026
Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-September-25-2026
Localization-costs-and-information-growth-for-exact-Gaussian-observations-September-27-2026
Memory-and-precision-in-noiseless-Gaussian-regression-September-27-2026
Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026
Replacing-Gaussian-observations-in-memory-constrained-inference-September-27-2026
Subsphere-methods-for-memory-sample-lower-bounds-in-noiseless-Gaussian-regression-September-27-2026
The-Euclidean-plane-is-not-five-colorable-September-23-2026
The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-October-6-2026
The-Gaussian-free-field-limit-of-integer-Lipschitz-heights-with-two-arc-boundary-data-September-25-2026
The-Gaussian-free-field-limit-of-the-balanced-six-vertex-model-with-variance-multiplier-1-over-arcsin-c-over-2-September-23-2026
The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026
Shell cwd was reset to /Users/karamelton

>S=/private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad/repo/preprints; for d in Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026 A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026 The-Euclidean-plane-is-not-five-colorable-September-23-2026 A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026; do echo "== $d"; ls -la $S/$d | awk '{print $5, $9}'; done

== Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026
 
160 .
24032 ..
320 build
489172 paper.pdf
546 README.md
== A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026
 
160 .
24032 ..
192 build
465691 paper.pdf
705 README.md
== The-Euclidean-plane-is-not-five-colorable-September-23-2026
 
160 .
24032 ..
224 build
450880 paper.pdf
564 README.md
== A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026
 
160 .
24032 ..
160 build
443755 main.pdf
622 README.md

>S=/private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad/repo/preprints; for d in Bounded-Step* A-translational-tile* The-Euclidean-plane-is-not* A-power-improvement-in-the-Heil*; do echo "== $d"; find $S/$d/build -type f | sed "s|$S/||" | head; done; which pdftotext; python3 -c "import pypdf" 2>&1 | tail -1

Exit code 1
(eval):1: no matches found: Bounded-Step*

>cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad/repo/preprints && for d in Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026 A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026 The-Euclidean-plane-is-not-five-colorable-September-23-2026 A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026; do echo "== $d"; find $d/build -type f | head; done; which pdftotext; python3 -c "import pypdf" 2>&1 | tail -1

== Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026
Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/build/coverage-transfer.tex
Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/build/information.tex
Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/build/main.tex
Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/build/geometry.tex
Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/build/schedule.tex
Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/build/references.tex
Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/build/periodicity.tex
Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/build/preliminaries.tex
== A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026
A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/build/references.bib
A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/build/paper.tex
A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/build/sections/01-introduction.tex
A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/build/sections/06-stacking.tex
A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/build/sections/05-common-solution.tex
A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/build/sections/02-word-rule.tex
A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/build/sections/03-cyclic-encoding.tex
A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/build/sections/07-three-dimensions.tex
A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/build/sections/04-activation.tex
A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/build/figures/proof-route.tex
== The-Euclidean-plane-is-not-five-colorable-September-23-2026
The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/preamble.tex
The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/references.bib
The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/main.tex
The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/transfer.tex
The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/angular.tex
The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/transitions.tex
The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/introduction.tex
The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/rigidity.tex
The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/interfaces.tex
The-Euclidean-plane-is-not-five-colorable-September-23-2026/build/sections/spectral.tex
== A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026
A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/references.bib
A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/main.tex
A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/01-introduction.tex
A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/07-zero-determinants.tex
A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/05-auxiliary-cap.tex
A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/09-prime-interval.tex
A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/03-norm-digits.tex
A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/08-alteration.tex
A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/06-sampling.tex
A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections/04-orbits.tex
pdftotext not found
Shell cwd was reset to /Users/karamelton

>cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad/repo && for n in 028 155 158 191; do echo "=== $n"; head -c 1500 lean/docs/$n.md; echo; done; sed -n '1,40p' README.md

=== 028
# Uniformly bounded components of Gaussian-prime graphs

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

- [Bounded-Step Walks on Gaussian Primes](../../preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/paper.pdf)

## Scope

The Gaussian moat problem asks whether an infinite walk through distinct Gaussian primes can have bounded step lengths. The formalized result gives a negative answer for every real step bound $D$. More strongly, one finite bound depending only on $D$ limits the size of every connected component of the bounded-step graph and the length of every injective bounded-step walk. Axis primes and all associates are included. No explicit function of $D$ is supplied.

## Comparator links

| Result | Comparator statement |
| --- | --- |
| Uniform bounds for bounded-step Gaussian-prime walks | [GaussianMoat.lean](../ComparatorChallenges/GaussianMoat.lean) |

=== 155
# A counterexample to periodic tiling in dimension three

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

- [A translational tile with no fully periodic tiling in dimension three](../../preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/paper.pdf)

## Scope

The periodic-tiling question asks whether a finite tile that tiles a lattice must admit a periodic tiling. The formalized counterexample is a finite tile in $\mathbb Z^3$ that tiles but has no complement invariant under a finite-index subgroup. Its unit-cube thickening also tiles $\mathbb R^3$ almost everywhere but admits no fully periodic tiling, even with arbitrary real translation vectors. The result also establishes that dimension three is the least lattice dimension where this failure occurs.

## Comparator links

| Result | Comparator statement |
| --- | --- |
| Aperiodic translational tile in dimension three | [PeriodicTilingThree.lean](../ComparatorChallenges/PeriodicTilingThree.lean) |

=== 158
# The Euclidean plane cannot be colored with five colors

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

- [The Euclidean plane is not five-colorable](../../preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf)

## Scope

The Hadwiger–Nelson problem asks for the fewest colors needed to color the plane so that points at distance one have different colors. The formalized results prove that five colors do not suffice and that seven colors do suffice. The lower bound applies to arbitrary colorings, with no measurability or continuity assumption; the upper bound includes every boundary point of the coloring regions.

## Comparator links

| Result | Comparator statement |
| --- | --- |
| No proper five-coloring | [EuclideanFiveColor.lean](../ComparatorChallenges/EuclideanFiveColor.lean) |
| Proper seven-coloring | [PlaneColoring.lean](../ComparatorChallenges/PlaneColoring.lean) |

=== 191
# A power improvement in the Heilbronn triangle lower bound

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

- [A power improvement in the Heilbronn triangle lower bound](../../preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/main.pdf)

## Scope

Heilbronn's triangle problem asks how large the smallest triangle determined by $n$ points in the unit square can be. The formalization constructs an unbounded sequence of sizes $n$ and point sets for which every triangle has area at least $n^{-2+\eta}$, for one fixed $\eta>0$. It consequently refutes the proposed upper bound of order $n^{-2+\varepsilon}$ for every $\varepsilon>0$. The linked construction is for an unbounded sequence of sizes; the paper's statement for every sufficiently large $n$ is broader.

## Comparator links

| Result | Comparator statement |
| --- | --- |
| Power improvement for the smallest Heilbronn triangle | [HeilbronnTriangle.lean](../ComparatorChallenges/HeilbronnTriangle.lean) |

# Readme

This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. 

As part of model development, we evaluate our models on open research problems. We expanded these evaluations after performance on our existing mathematical evaluations saturated. Some outputs build upon earlier results produced by the models.

This collection includes results at different stages of verification. Not all have accompanying Lean formalizations. We will continue to update this repository with Lean formalizations as we obtain them. 

Some of the unformalized results could have issues. We will endeavor to fix any such issues quickly.
We are also exploring community-hosted repositories for these materials. 

## Navigating the collection

The current catalogue contains 719 manuscripts organized into 372 families. A family groups related papers, which may include a principal result, companion arguments, consequences, or alternative proofs. Each family is classified by mathematical discipline.

- Start with the [overview](overview.pdf) for descriptions of the families.
- Use the [manuscript map](CONTENTS.md) to find individual papers and their supporting materials.
- The [`preprints/`](preprints/) directory contains PDFs, source files, and manuscript-specific citation and build instructions.
- The [Lean library](lean/README.md) and [formalization catalogue](lean/formalization.yaml) describe the available formal proofs, their associated papers, and verification configurations. See the [Comparator instructions](lean/ComparatorChallenges/README.md) for additional checking instructions. The repository has ~42% top-line results formalized.
- Updates to the repo are described in the [history](history.md).

### Reasoning summaries

We are also releasing abridged summaries of the model's reasoning, covering the following results:

| Family | Subject |
|---|---|
| 007 | [Ordinary two-point correlations of multiplicative functions](reasoning_traces/ordinary-two-point-correlations.pdf) |
| 017 | [The irrationality exponent of π](reasoning_traces/irrationality-exponent-of-pi.pdf) |
| 087 | [Symmetric and general Mahler conjectures](reasoning_traces/symmetric-and-general-mahler-conjectures.pdf) |
| 102 | [Ordinary NP-hardness at the basic semidefinite threshold](reasoning_traces/basic-semidefinite-threshold-np-hardness.pdf) |
| 159 | [Quasipolynomial bounds for arithmetic progressions](reasoning_traces/quasipolynomial-arithmetic-progressions.pdf) |
| 197 | [Kaplansky's direct-finiteness conjecture in characteristic two](reasoning_traces/kaplansky-direct-finiteness-characteristic-two.pdf) |
| 221 | [The Mézard–Parisi formula for diluted spin glasses](reasoning_traces/mezard-parisi-formula.pdf) |
| 271 | [Spontaneous magnetization in the quantum Heisenberg ferromagnet](reasoning_traces/spontaneous-magnetization-quantum-heisenberg-ferromagnet.pdf) |
| 287 | [Isomorphism of free group factors](reasoning_traces/free-group-factor-isomorphism.pdf) |
| 362 | [The three-dimensional relativistic Vlasov–Maxwell system](reasoning_traces/relativistic-vlasov-maxwell.pdf) |


## How the results were produced
Shell cwd was reset to /Users/karamelton

>ls /Applications | grep -iE 'chrome|chromium|edge|brave'; which npx playwright 2>/dev/null; ls ~/Library/Caches/ms-playwright 2>/dev/null

Google Chrome.app
/Users/karamelton/.local/bin/npx
playwright not found
chromium_headless_shell-1208
chromium_headless_shell-1243
chromium-1208
ffmpeg-1011

>cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad && for scheme in light dark; do "/Applications/Google Chrome.app/Contents/MacOS/Google Chrome" --headless=new --disable-gpu --hide-scrollbars --window-size=1200,3400 $( [ $scheme = dark ] && echo --force-dark-mode --blink-settings=preferredColorScheme=0 ) --virtual-time-budget=4000 --screenshot=$PWD/pc-$scheme.png "file://$HOME/Desktop/math-surfaces/plane-coloring/surface.html" 2>/dev/null; done; ls -la pc-*.png

-rw-r--r--@ 1 karamelton  wheel  409265 Oct 12 11:34 pc-dark.png
-rw-r--r--@ 1 karamelton  wheel  409265 Oct 12 11:34 pc-light.png
Shell cwd was reset to /Users/karamelton

{"query":"surface upload","results":[{"name":"traces_surfaces_complete_upload","description":"Step 2 of 2 to publish surface HTML: finalize an uploaded artifact as an immutable surface version. The result includes a previewUrl that renders this exact version on the user's latest trace (namespace members only) so they can try it before it goes live; always share it with the user. Make it current with release: true here or, after the user has tried it, via traces_surfaces_release_version, which also controls visibility.","inputSchema":{"$schema":"https://json-schema.org/draft/2020-12/schema","type":"object","properties":{"surface":{"type":"object","properties":{"key":{"type":"string","minLength":1}},"required":["key"],"additionalProperties":false},"version":{"type":"string","minLength":1},"artifactId":{"type":"string","minLength":1,"description":"data.artifactId from the upload response."},"release":{"default":false,"description":"Make this version current immediately. Default false: the version is uploaded but users keep seeing the current one until traces_surfaces_release_version.","type":"boolean"}},"required":["surface","version","artifactId","release"],"additionalProperties":false},"annotations":{"readOnlyHint":false,"destructiveHint":true,"idempotentHint":false,"openWorldHint":true}},{"name":"traces_surfaces_prepare_upload","description":"Step 1 of 2 to publish surface HTML: reserve a version and get a single-use upload destination, creating the surface first if the key is new (pass name). Send the raw HTML file body to the returned url with the returned method and headers (for example: curl -fsS -X POST -H 'Content-Type: text/html' --data-binary @surface.html <url>); the response contains data.artifactId. Then call traces_surfaces_complete_upload. The destination expires quickly and needs no bearer token. Never pass HTML through this tool. Uploading does not change what users see: complete_upload leaves the current version untouched and returns a previewUrl for trying the new one.","inputSchema":{"$schema":"https://json-schema.org/draft/2020-12/schema","type":"object","properties":{"surface":{"type":"object","properties":{"key":{"type":"string","minLength":1}},"required":["key"],"additionalProperties":false},"name":{"description":"Display name. Required only when the surface key does not exist yet; creates it.","type":"string","minLength":1},"description":{"type":"string"},"icon":{"type":"string"},"version":{"type":"string","minLength":1,"description":"New version label, e.g. 1.0.0. Must not already exist."},"byteSize":{"type":"integer","exclusiveMinimum":0,"maximum":9007199254740991,"description":"Exact byte size of the HTML file (wc -c). The upload is rejected if it differs."}},"required":["surface","version","byteSize"],"additionalProperties":false},"annotations":{"readOnlyHint":false,"destructiveHint":false,"idempotentHint":false,"openWorldHint":true}},{"name":"traces_surfaces_create","description":"Create a new private surface in this connection's namespace. The new surface has no uploaded versions or current version.","inputSchema":{"$schema":"https://json-schema.org/draft/2020-12/schema","type":"object","properties":{"key":{"type":"string","minLength":1},"name":{"type":"string","minLength":1},"description":{"type":"string"},"icon":{"type":"string"}},"required":["key","name"],"additionalProperties":false},"annotations":{"readOnlyHint":false,"destructiveHint":false,"idempotentHint":false,"openWorldHint":true}},{"name":"traces_surfaces_release_version","description":"Release an already uploaded surface version as current, or change visibility. The version must be completed through traces_surfaces_complete_upload first. This operation does not accept HTML or upload files. Omit visibility to preserve the current visibility; omission never implies public visibility.","inputSchema":{"$schema":"https://json-schema.org/draft/2020-12/schema","type":"object","properties":{"surface":{"type":"object","properties":{"key":{"type":"string","minLength":1}},"required":["key"],"additionalProperties":false},"version":{"type":"string","minLength":1},"visibility":{"type":"string","enum":["private","public"]}},"required":["surface","version"],"additionalProperties":false},"annotations":{"readOnlyHint":false,"destructiveHint":true,"idempotentHint":true,"openWorldHint":true}},{"name":"traces_surfaces_archive","description":"Archive an existing surface. Archived surfaces remain in version history but are excluded from normal discovery and public availability.","inputSchema":{"$schema":"https://json-schema.org/draft/2020-12/schema","type":"object","properties":{"surface":{"type":"object","properties":{"key":{"type":"string","minLength":1}},"required":["key"],"additionalProperties":false}},"required":["surface"],"additionalProperties":false},"annotations":{"readOnlyHint":false,"destructiveHint":true,"idempotentHint":true,"openWorldHint":true}},{"name":"traces_surfaces_get","description":"Get the complete management record for an existing surface, including metadata, current version, and immutable version history.","inputSchema":{"$schema":"https://json-schema.org/draft/2020-12/schema","type":"object","properties":{"surface":{"type":"object","properties":{"key":{"type":"string","minLength":1}},"required":["key"],"additionalProperties":false}},"required":["surface"],"additionalProperties":false},"annotations":{"readOnlyHint":true,"destructiveHint":false,"idempotentHint":true,"openWorldHint":true}},{"name":"traces_surfaces_restore","description":"Restore an archived surface without changing its current version or visibility.","inputSchema":{"$schema":"https://json-schema.org/draft/2020-12/schema","type":"object","properties":{"surface":{"type":"object","properties":{"key":{"type":"string","minLength":1}},"required":["key"],"additionalProperties":false}},"required":["surface"],"additionalProperties":false},"annotations":{"readOnlyHint":false,"destructiveHint":false,"idempotentHint":true,"openWorldHint":true}},{"name":"traces_surfaces_search","description":"List surfaces in this connection's namespace, including private, archived, and unapproved surfaces. Use this to discover a surface key before performing mutations.","inputSchema":{"$schema":"https://json-schema.org/draft/2020-12/schema","type":"object","properties":{"query":{"type":"string"},"includeArchived":{"default":false,"type":"boolean"},"limit":{"default":50,"type":"integer","minimum":1,"maximum":200}},"required":["includeArchived","limit"],"additionalProperties":false},"annotations":{"readOnlyHint":true,"destructiveHint":false,"idempotentHint":true,"openWorldHint":true}}]}

>wc -c < ~/Desktop/math-surfaces/plane-coloring/surface.html

   41622

{"surface":{"key":"plane-coloring"},"created":true,"version":"0.1.0","upload":{"url":"https://actions.traces.com/v1/internal/surface-artifact-uploads/surface_upload_f2d5e8d467b0f497a31a623cf8c27ff6cf5e8abd9759802c8d58c8eaea66d253","method":"POST","headers":{"Content-Type":"text/html"}},"nextStep":"Upload the file, then call traces_surfaces_complete_upload with version \"0.1.0\" and the returned artifactId."}

>curl -fsS -X POST -H 'Content-Type: text/html' --data-binary @$HOME/Desktop/math-surfaces/plane-coloring/surface.html https://actions.traces.com/v1/internal/surface-artifact-uploads/surface_upload_f2d5e8d467b0f497a31a623cf8c27ff6cf5e8abd9759802c8d58c8eaea66d253

{"ok":true,"data":{"artifactId":"surface_artifact_53eb079c71849060c02bf0a58285a90bd612d603f182f10c0857457294a30057"}}

{"surface":{"createdAt":1791819291899,"createdBy":"kd7ekkrfd435k459fw0hnpgq3187e6sg","currentVersion":null,"description":"Interactive explainer: the Euclidean plane is not five-colorable (openai/math family 158).","id":"ph78kjv4nd2d72ayb9967w2w218g48df","key":"plane-coloring","name":"Coloring the plane","namespaceId":"k977fb4w5km4539pw9bsq3dm4d87ffwc","publishStatus":"private","updatedAt":1791819291899,"versions":[{"approvalStatus":"not_requested","createdAt":1791819297788,"htmlByteSize":41622,"htmlSha256":"b01918fc617054edaec12d82fe129db3c7b4e9616e7168e7e5666f88d6c8cc53","sdkVersion":"surface-sdk.v1","version":"0.1.0","sourceUrl":null}],"canManage":true,"icon":null,"archivedAt":null},"version":"0.1.0","released":false,"previewUrl":"https://traces.com/s/52797cc3-0c4f-4505-b3fa-3bb171d53396?surface=plane-coloring&version=0.1.0","nextStep":"Ask the user to try previewUrl; when happy, call traces_surfaces_release_version with version \"0.1.0\" to make it current."}

# Started building math proof visualizations

- ID: jn79aj7farn8rk8apg0h5j0g7d8g4fw8
- URL: https://traces.com/s/189903e3-2f4c-44dd-8020-b4552bf40364
- Agent: claude-code

## User

<local-command-caveat>The command below was run directly in Claude Code, not sent to you as a request, and its output goes straight to the user. It's recorded here as context for later messages.</local-command-caveat>

## User

<command-name>/mcp</command-name>
            <command-message>mcp</command-message>
            <command-args></command-args>

## User

<local-command-stdout>Authentication successful. Connected to traces.</local-command-stdout>

More matching events are available. Read another window.

>cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad/repo/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections && grep -n -i -E 'eta *=|explicit|admissible exponent|10\^\{' 08-alteration.tex | head -15

91: \beta=1515k-\frac{k-1}{2}=\frac{3029k+1}{2},
101: \qquad \alpha=1+30\beta=45435k+16,
105: \eta=\frac2\alpha=\frac{2}{45435k+16}.
135:$n$. The formula \eqref{alt:exponent} is explicit; no attempt has been
Shell cwd was reset to /Users/karamelton

>cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad/repo/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections && sed -n 75,140p 08-alteration.tex

$n_r\ge a_r/2$.  Combining this with \eqref{alt:area-bound} yields
\begin{equation}\label{alt:scaled-area}
 n_r^2\Delta(P_r)\ge\frac{r^2}{64}.
\end{equation}

We now express both the cardinality and the area bound in terms of $r$.
The parameter choices give
\[
 \begin{gathered}
  100k^2r^{30}<B\le200k^2r^{30},\qquad h=B^k,\\
  h^{100}<q\le2h^{100},\qquad N^{3/2}=(hq)^{15},\\
  B^{k-1}/3\le\tau\le B^{k-1}/2.
 \end{gathered}
\]
Consequently, if
\[
 \beta=1515k-\frac{k-1}{2}=\frac{3029k+1}{2},
\]
then
\[
 \sqrt{2}\,rB^\beta<a_r\le2^{15}\sqrt{3}\,rB^\beta.
\]
Since $a_r/2\le n_r\le a_r$ for sufficiently large $r$, there are fixed
constants $A_k,D_k>0$ such that
\begin{equation}\label{alt:cardinality-growth}
 A_kr^\alpha\le n_r\le D_kr^\alpha,
 \qquad \alpha=1+30\beta=45435k+16,
\end{equation}
for every sufficiently large prime $r$. In particular, put
\begin{equation}\label{alt:exponent}
 \eta=\frac2\alpha=\frac{2}{45435k+16}.
\end{equation}
The upper bound in \eqref{alt:cardinality-growth} gives
$r^2\ge D_k^{-\eta}n_r^\eta$. Combining this with
\eqref{alt:scaled-area}, we obtain
\begin{equation}\label{alt:prime-area}
 \Delta(P_r)\ge c_*n_r^{-2+\eta},
 \qquad c_*=\frac1{64D_k^\eta}>0.
\end{equation}
Thus the same cardinality estimate that will permit interpolation also
converts the factor $r^2$ into a fixed power of the number of points.

For a sufficiently large integer $n$, put
$m=\lceil(n/A_k)^{1/\alpha}\rceil$.
Lemma~\ref{norm:prime-interval} supplies a prime $r$ with $m<r\le2m$.
Once $(n/A_k)^{1/\alpha}\ge1$, we have
$m\le2(n/A_k)^{1/\alpha}$, so \eqref{alt:cardinality-growth} gives
\[
 n\le n_r\le C_kn,
 \qquad C_k=\max\left\{1,\frac{4^\alpha D_k}{A_k}\right\}.
\]
This uses only the two-sided cardinality bounds, not monotonicity of $n_r$.
Retain any $n$ points of $P_r$. Deleting points cannot decrease the minimum
triangle area. Since $0<\eta<2$, \eqref{alt:prime-area} therefore yields
\[
 \Delta(n)\ge c_* n_r^{-2+\eta}
 \ge c_*C_k^{-2+\eta}n^{-2+\eta}.
\]
Taking $c_1=c_*C_k^{-2+\eta}$ proves Theorem~\ref{intro:main}.
All constants are absolute because $k$ is fixed independently of $r$ and
$n$. The formula \eqref{alt:exponent} is explicit; no attempt has been
made to optimize the construction parameters.
\end{proof}
Shell cwd was reset to /Users/karamelton

>cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad/repo/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections && grep -n -E 'k *= *[0-9]|fix(ed)? .*\$k|\$k\\ge|d *= *[0-9]' 0*.tex | head -15

03-norm-digits.tex:14: d=41,\qquad M=\binom{4d-1}{d},\qquad
04-orbits.tex:10:Throughout this section, let $B$ be a prime, let $k\geq 1$ be an
04-orbits.tex:213:Thus, for fixed $k$ and all sufficiently large $r$,
08-alteration.tex:7:All constants below may depend on the fixed integer $k$; none depends on
08-alteration.tex:16:Keep the parameters constructed above, with $d=41$, and set
08-alteration.tex:20:Here $k\ge2$ and $B$ is an odd prime, so $\tau\ge1$.  Take $2n_r$ samples
Shell cwd was reset to /Users/karamelton

>cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad/repo/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026/build/sections && sed -n 10,24p 03-norm-digits.tex; sed -n 1,30p 08-alteration.tex

\subsection{The norm polynomial}

Fix
\begin{equation}\label{norm:fixed-parameters}
 d=41,\qquad M=\binom{4d-1}{d},\qquad
 T=\binom{M}{3},\qquad k=T^2+1.
\end{equation}
Let $r$ be an odd prime.  We recall the elementary finite-field facts
needed here; see \cite[Chapter~2]{LidlNiederreiter1994} for background.
In a splitting field over $\mathbb F_r$, the roots of
$X^{r^d}-X$ form a field: the Frobenius identity shows closure under
addition, subtraction and multiplication, and nonzero roots are closed
under inversion.  The derivative is $-1$, so there are exactly $r^d$
distinct roots.  This field, denoted $K=\mathbb F_{r^d}$, consequently
has dimension $d$ over $\mathbb F_r$.  Its automorphism $t\mapsto t^r$
\section{Deletion and projective normalization}\label{alt:section}

The preceding estimates allow us to remove every repeated projected point
and every small triangle while retaining many columns. We first construct
a set for each sufficiently large prime $r$, compare its area bound with
its cardinality, and then pass to every sufficiently large cardinality.
All constants below may depend on the fixed integer $k$; none depends on
the growing prime $r$.

This final step is an elementary alteration argument. The earlier
probabilistic lower bound of Koml\'os, Pintz and
Szemer\'edi~\cite{KPS1982} uses a stronger hypergraph independence lemma;
no such lemma is needed below.

\begin{proof}[Proof of Theorem~\ref{intro:main}]
Keep the parameters constructed above, with $d=41$, and set
\begin{equation}\label{alt:sample-size}
 a_r=r\sqrt{N^3/\tau},\qquad n_r=\lfloor a_r\rfloor.
\end{equation}
Here $k\ge2$ and $B$ is an odd prime, so $\tau\ge1$.  Take $2n_r$ samples
with the shared random choices and independent conditional draws specified
in Section~\ref{sam:section}.  Let $Z$ count the unordered pairs of sample
indices with equal projected points, plus the unordered triples whose
projected points are pairwise distinct and whose determinant has absolute
value at most $\tau$.  Linearity of expectation, Lemma~\ref{sam:proportional},
and Corollary~\ref{sam:bad-triple} give
\begin{equation}\label{alt:expected-events}
 \frac{\mathbb EZ}{n_r}
 \ll n_r(hq)^6N^{-3}
    +n_r^2(\log(2N))^2\frac{h}{N^3r^d}.
Shell cwd was reset to /Users/karamelton

>cd ~/Desktop/math-surfaces && python3 - <<'EOF'
p='heilbronn/template.html'; s=open(p).read()
s=s.replace('[1951, "Roth", "impossibility", "First proof that you must do worse than 1/n: x > 1.", 1.0],','[1951, "Roth", "impossibility", "First proof that the smallest triangle must shrink faster than 1/n.", null],')
open(p,'w').write(s)
EOF
python3 build.py heilbronn && node -e "const s=require('fs').readFileSync('heilbronn/surface.html','utf8');new Function(s.match(/<script>([\s\S]*)<\/script>/)[1]);console.log('parses')" && cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad && "/Applications/Google Chrome.app/Contents/MacOS/Google Chrome" --headless=new --disable-gpu --hide-scrollbars --window-size=1200,2600 --virtual-time-budget=4000 --screenshot=$PWD/hb.png "file://$HOME/Desktop/math-surfaces/heilbronn/surface.html" 2>/dev/null; ls -la hb.png

heilbronn/surface.html 34 KB
parses
-rw-r--r--@ 1 karamelton  wheel  303876 Oct 12 11:39 hb.png
Shell cwd was reset to /Users/karamelton

>cd ~/Desktop/math-surfaces && python3 - <<'EOF'
p='heilbronn/template.html'; s=open(p).read()
s=s.replace('ctx.textAlign = x === 2 ? "right" : "left"; ctx.fillText(label, X(x) + (x === 2 ? -6 : 6), yAx + 22 + row * 14);','const right = x === 2 || row === 1; ctx.textAlign = right ? "right" : "left"; ctx.fillText(label, X(x) + (right ? -6 : 6), yAx + 22 + row * 14);')
s=s.replace('[8 / 7, "8/7 (1981)", 0, 1], [7 / 6, "7/6 (2025)", 0, 2]','[8 / 7, "8/7 (1981)", 0, 1], [7 / 6, "7/6 (2025)", 0, 1]')
s=s.replace('.line canvas { aspect-ratio: auto; height: 190px; }','.line canvas { aspect-ratio: auto; height: 160px; }')
open(p,'w').write(s)
EOF
python3 build.py heilbronn && wc -c < heilbronn/surface.html

heilbronn/surface.html 34 KB
   34358

>cd ~/Desktop/math-surfaces && sed -i '' 's|const right = x === 2 \|\| row === 1;|const right = x === 2 \|\| x === 8 / 7;|' heilbronn/template.html && grep -c 'x === 8 / 7' heilbronn/template.html && python3 build.py heilbronn && cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad && "/Applications/Google Chrome.app/Contents/MacOS/Google Chrome" --headless=new --disable-gpu --hide-scrollbars --window-size=1200,2600 --virtual-time-budget=4000 --screenshot=$PWD/hb.png "file://$HOME/Desktop/math-surfaces/heilbronn/surface.html" 2>/dev/null; sips -c 220 1200 --cropOffset 1980 0 hb.png --out hb-line.png >/dev/null

1
heilbronn/surface.html 34 KB
Shell cwd was reset to /Users/karamelton

>curl -fsS -X POST -H 'Content-Type: text/html' --data-binary @$HOME/Desktop/math-surfaces/heilbronn/surface.html https://actions.traces.com/v1/internal/surface-artifact-uploads/surface_upload_495d912d6782751281bff73880b83fa993a6b1189322f50e027e2cfe54612795

Exit code 22
curl: (22) The requested URL returned error: 400

{"surface":{"key":"heilbronn-triangles"},"created":false,"version":"0.1.0","upload":{"url":"https://actions.traces.com/v1/internal/surface-artifact-uploads/surface_upload_5983e8df2271143085a47e3671694b70e2afc4e7e9419385efbc717597c0271d","method":"POST","headers":{"Content-Type":"text/html"}},"nextStep":"Upload the file, then call traces_surfaces_complete_upload with version \"0.1.0\" and the returned artifactId."}

>curl -fsS -X POST -H 'Content-Type: text/html' --data-binary @$HOME/Desktop/math-surfaces/heilbronn/surface.html https://actions.traces.com/v1/internal/surface-artifact-uploads/surface_upload_5983e8df2271143085a47e3671694b70e2afc4e7e9419385efbc717597c0271d

{"ok":true,"data":{"artifactId":"surface_artifact_9c2e51b711194d4637c8fd61373e00572248e321903179d8694009ac824f8db5"}}

{"surface":{"createdAt":1791819598175,"createdBy":"kd7ekkrfd435k459fw0hnpgq3187e6sg","currentVersion":null,"description":"Interactive explainer: a power improvement in the Heilbronn triangle lower bound (openai/math family 191).","id":"ph7ak12m5xybxw3qnctahtnanx8g4w25","key":"heilbronn-triangles","name":"Heilbronn's triangles","namespaceId":"k977fb4w5km4539pw9bsq3dm4d87ffwc","publishStatus":"private","updatedAt":1791819598175,"versions":[{"approvalStatus":"not_requested","createdAt":1791819612120,"htmlByteSize":34360,"htmlSha256":"a0f0872ff7d7a300bf91b8e59ef615ac89d4bb191ceb39a9ad99b729b12fd8a4","sdkVersion":"surface-sdk.v1","version":"0.1.0","sourceUrl":null}],"canManage":true,"icon":null,"archivedAt":null},"version":"0.1.0","released":false,"previewUrl":"https://traces.com/s/52797cc3-0c4f-4505-b3fa-3bb171d53396?surface=heilbronn-triangles&version=0.1.0","nextStep":"Ask the user to try previewUrl; when happy, call traces_surfaces_release_version with version \"0.1.0\" to make it current."}

>cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad/repo/preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/build && wc -c *.tex && grep -n 'input\|section' main.tex | head -20 && sed -n '/begin{abstract}/,/end{abstract}/p' main.tex

   10968 coverage-transfer.tex
   18990 geometry.tex
   16641 information.tex
    9033 main.tex
    2735 periodicity.tex
    7239 preliminaries.tex
    3131 references.tex
   22586 schedule.tex
   91323 total
15:\newtheorem{theorem}{Theorem}[section]
35:\numberwithin{equation}{section}
53:\section{The Gaussian moat problem}\label{sec:introduction}\label{sec:intro}
108:\subsection{A finite obstruction and the route to its proof}
128:implies Theorem~\ref{thm:main}. The remaining sections construct it.
180:\input{periodicity}
181:\input{preliminaries}
182:\input{geometry}
183:\input{coverage-transfer}
184:\input{schedule}
185:\input{information}
186:\input{references}
\begin{abstract}
We prove the Gaussian moat conjecture: no infinite walk through distinct
Gaussian primes can have bounded steps. More strongly, for each fixed finite
step bound, the connected components of the Gaussian-prime graph have
uniformly bounded size. This bound applies to every starting prime, including
primes on the coordinate axes, and is nonexplicit. The proof constructs a
finite periodic sieve obstruction using geometric sampling and
information-theoretic estimates.
\end{abstract}
Shell cwd was reset to /Users/karamelton

>cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad/repo/preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026/build && sed -n 53,178p main.tex; cat periodicity.tex | head -60

\section{The Gaussian moat problem}\label{sec:introduction}\label{sec:intro}
A Gaussian prime is an irreducible element of $\Z[i]$.
We identify $a+bi$ with $(a,b)\in\Z^2$, write $N(a+bi)=a^2+b^2$,
and use the Euclidean norm $|z|=\sqrt{N(z)}$.
For a finite real $D$, let $G_D$ have all Gaussian primes as vertices,
with an edge between distinct $z,w$ if $|z-w|\le D$.
The Gaussian moat problem asks whether some $G_D$ contains an infinite
path with no repeated vertex. Large gaps along a line do not settle the
question in the plane, since a path may go around them.

\begin{theorem}[Uniform component bound]\label{thm:main}\label{cor:uniform}
For every finite real $D$ there is a finite $B_D$ such that every
connected component of $G_D$ has at most $B_D$ vertices.
The bound is independent of the starting prime and includes primes on
the coordinate axes. Consequently every sequence of distinct Gaussian
primes whose successive distances are at most $D$ has at most $B_D$ terms.
\end{theorem}

This proves the uniform form of the Gaussian moat conjecture and answers
the infinite-walk question negatively. The bound $B_D$ is nonexplicit:
the proof chooses
sufficiently large scales after fixing $D$. For $D<1$, distinct lattice
points cannot be adjacent, so $B_D=1$ suffices.

Gethner and Stark trace the question to Basil Gordon at the 1962
International Congress of Mathematicians in Stockholm
\cite[p.~289]{GethnerStark}. Erd\H{o}s later credited Gordon and Motzkin,
recalling that Motzkin had told him the problem at the Pasadena number
theory meeting in November 1963 \cite[p.~69]{Erdos1977}.

The early work separates two obstacles. Large prime-free disks obstruct
motion locally, but a planar walk may go around them. Gethner, Wagon,
and Wick constructed such disks with centers on any line containing two
Gaussian integers, and constructed arbitrarily isolated real Gaussian
primes; they credit Vardi with an independent proof of the latter result
\cite[Theorems 4.1 and 4.4]{GethnerWagonWick}.
Computations instead surround a particular starting region:
Tsuchimura obtained a finite bound on the distance reachable with steps
of length at most $6$ from the origin, adjoined as an initial vertex
\cite{Tsuchimura}. This is a result about the component of that vertex.

The closer predecessor of the present argument is finite periodic
sieving. Gethner and Stark formulated a bound on the length of a
distinct-prime walk that is uniform over its starting point, established
periodic obstructions for step bounds $\sqrt2$ and $2$, and proposed
sieving by small Gaussian primes for larger bounds
\cite[pp.~290--292]{GethnerStark}. Vardi studied the associated periodic
coprimality graphs and the passage from the absence of an infinite walk
to a uniform bound on component sizes
\cite[Section 6, Proposition 6.2]{Vardi}.
Our finite-sieve theorem supplies congruence restrictions for each fixed
step bound; the periodic reduction below then converts those restrictions
into the bound on all Gaussian-prime components.


\subsection{A finite obstruction and the route to its proof}
For a rational prime $p\equiv1\pmod4$, choose conjugate Gaussian prime
factors $\pi_p,\overline{\pi_p}$, each of norm $p$.
For a finite set $\mathcal P$ of such rational primes, put
\[
\mathcal A(\mathcal P)=\{z\in\Z[i]:
 z\not\equiv0\pmod{\pi_p},\quad
 z\not\equiv0\pmod{\overline{\pi_p}}\quad(p\in\mathcal P)\}.
\]
Changing the chosen associates does not change this set. It is periodic
under translations by $Q\Z^2$, where $Q=\prod_{p\in\mathcal P}p$.

\begin{theorem}[Finite sieve obstruction]\label{thm:sieve}
For every $D\ge1$ there is a finite set $\mathcal P_D$ of rational
primes congruent to $1$ modulo $4$ such that $\mathcal A(\mathcal P_D)$
contains no infinite sequence of distinct points with successive distances
at most $D$. The set $\mathcal P_D$ depends only on $D$.
\end{theorem}

Section~\ref{sec:consequences} first proves that this finite obstruction
implies Theorem~\ref{thm:main}. The remaining sections construct it.
We fix an arbitrary deterministic infinite self-avoiding lattice walk
with steps at most $D$. Randomness enters only through sampled times
and auxiliary choices of one Gaussian factor over each split prime.
The aim is to show that such a walk cannot avoid all the selected zero
classes.

The geometric sampling operation in Section~\ref{sec:geometry} chooses
a difference uniformly from the distinct differences of a long walk
segment and then chooses one of its endpoints. There are many distinct
differences, and a random collection of signed prime factors usually
separates them. These facts force the sampled endpoint to carry more
joint entropy in its residues than the earlier position did. The proof
uses a torus-degree argument for the differences, followed by determinant
divisibility and concentration for the factor choices. Disjoint
neighborhoods in the Boolean cube avoid a separate count of all lattice
directions.

The next task is coverage: for most factors of norm $p$, the terminal
residue must have probability at least $\tau/p$ outside fewer than
$p^{1-\beta}$ exceptional residues. Joint entropy alone, at the accuracy
available early in the construction, does not give this conclusion.
Section~\ref{sec:transfer} supplies the additional argument before the
large schedule of parameters is introduced. Repeated continuations from
an exact later checkpoint give one displacement vector shared by every
residue coordinate. If coverage failed, this vector would put the true
residue on a short list with appreciable probability. Its entropy cost
is too small to create that deficit in many coordinates at once.

Section~\ref{sec:schedule} arranges the geometric operations in a common
schedule with decreasing scales. It first establishes the entropy and
displacement estimates, then applies the coverage transfer backward
through the checkpoints. Every prime batch uses the same terminal time
law, denoted by $t_*$. An independent uniform offset makes this law
nearly invariant under the time shifts needed later.

Finally, assume that the walk avoids the chosen zero classes.
Section~\ref{sec:information} uses short increment words---finite lists
of consecutive increments---to test candidate starting residues.
Coverage and self-avoidance make these tests informative. Each prime
batch near $T$ incurs conditional information at least $c/\log T$ per
increment, where $c>0$ depends only on $D$. Section~\ref{sec:budget}
adds these costs using nested residue vectors and dyadic word lengths
at the common time law. Block subadditivity bounds their sum by the
entropy available in one bounded increment, whereas the selected batches
make the lower bounds diverge. This proves the finite sieve theorem.
The use of approximate translation invariance and block subadditivity
to bound accumulated information is methodologically related to Tao's
entropy-decrement argument~\cite[Section 3]{TaoEntropyDecrement}.
Here the signed residue batches, coverage transfer, and costs forced by
zero avoidance are established in full within this paper.
\section{Periodicity and the uniform bound}\label{sec:consequences}
The finite sieve theorem will be proved independently of the component
bound. The following conditional reduction explains why that theorem
suffices, and records the contribution of the finitely many Gaussian
primes removed by the sieve. The periodicity principle also appears in
Vardi~\cite[Section 6, Proposition 6.2]{Vardi}.

\begin{proposition}[From a periodic obstruction to a uniform bound]
\label{prop:periodic}
Fix $D\ge1$ and a finite set $\mathcal P$ of rational primes congruent
to $1$ modulo $4$. Suppose that $\mathcal A(\mathcal P)$ contains no
infinite self-avoiding walk with steps at most $D$. Put
\[
 Q=\prod_{p\in\mathcal P}p,\qquad
 K_D=\#\{v\in\Z[i]:|v|\le D\},
\]
and let $E$ be the set of associates of the selected Gaussian factors.
For every component $C$ of $G_D$,
\[
 \#C\le\max\{Q^2,\ |E|+(K_D-1)|E|Q^2\}.
\]
\end{proposition}
\begin{proof}
The graph on $\mathcal A(\mathcal P)$ with this step bound is locally
finite and invariant under translations by $Q\Z^2$. An infinite
component would contain an infinite simple path: the tree of finite
simple paths from one vertex is finitely branching, has vertices at
arbitrarily large depths, and one can successively choose a child with
arbitrarily deep descendants. Hence every component is finite.

Suppose that two distinct points $x,y$ of a component differ by
$v\in Q\Z^2$. Translation by $v$ sends that component to a component
containing $y$, so sends it to itself. A finite nonempty subset of
$\Z^2$ cannot be invariant under a nonzero translation. Each component
therefore injects into $(\Z/Q\Z)^2$ and has size at most $Q^2$.

A Gaussian prime divisible by a selected factor is an associate of that
factor. Thus, after deleting $E$, the Gaussian-prime graph is a subgraph
of the avoiding graph. A component meeting $E$ can meet at most
$(K_D-1)|E|$ components of the graph with $E$ removed: each such
component has an edge to $E$, whose total degree is at most that number.
Adding back $E$ gives the claimed bound.
\end{proof}


Applying Proposition~\ref{prop:periodic} to the set supplied by
Theorem~\ref{thm:sieve} proves Theorem~\ref{thm:main} for $D\ge1$;
the case $D<1$ was settled in the introduction. For the weaker
infinite-walk conclusion, one can simply delete an initial segment
containing the finitely many exceptional Gaussian primes: the remaining
tail would lie in $\mathcal A(\mathcal P_D)$.

Periodicity is what makes the bound uniform in the starting point.
Absence of an infinite path in a general locally finite graph does not
bound the sizes of its finite components. We now turn to the arithmetic,
entropy, and geometry needed to produce the finite obstruction.
Shell cwd was reset to /Users/karamelton

>cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad && cat >> sieve.js <<'EOF'
console.log('--- with 1+i (keep a+b odd)');
function analyze2(Ps,D2){const [Q0,A0]=allowed(Ps);const Q=Q0*2,A=new Uint8Array(Q*Q);for(let x=0;x<Q;x++)for(let y=0;y<Q;y++)A[x*Q+y]=A0[(x%Q0)*Q0+(y%Q0)]&&((x+y)%2===1)?1:0;
 const O=offsets(D2),seen=new Int32Array(Q*Q).fill(-1),LX=new Int32Array(Q*Q),LY=new Int32Array(Q*Q);let inf=false,maxSize=0,comp=0;
 for(let s=0;s<Q*Q;s++){if(!A[s]||seen[s]>=0)continue;let st=[s];seen[s]=comp;LX[s]=Math.floor(s/Q);LY[s]=s%Q;let size=0;
  while(st.length){const u=st.pop();size++;for(const [dx,dy] of O){const nx=LX[u]+dx,ny=LY[u]+dy,t=((nx%Q+Q)%Q)*Q+((ny%Q+Q)%Q);if(!A[t])continue;if(seen[t]<0){seen[t]=comp;LX[t]=nx;LY[t]=ny;st.push(t);}else if(LX[t]!==nx||LY[t]!==ny)inf=true;}}
  maxSize=Math.max(maxSize,size);comp++;}
 return {Q,inf,maxSize};}
for(const Ps of [[5],[13],[5,13],[5,17],[13,17],[5,13,17]])for(const D2 of [2,4,5])console.log('2·'+Ps.join('·'),'D²='+D2,JSON.stringify(analyze2(Ps,D2)));
EOF
node sieve.js 2>&1 | sed -n '/---/,$p'

--- with 1+i (keep a+b odd)
2·5 D²=2 {"Q":10,"inf":true,"maxSize":24}
2·5 D²=4 {"Q":10,"inf":true,"maxSize":32}
2·5 D²=5 {"Q":10,"inf":true,"maxSize":32}
2·13 D²=2 {"Q":26,"inf":true,"maxSize":288}
2·13 D²=4 {"Q":26,"inf":true,"maxSize":288}
2·13 D²=5 {"Q":26,"inf":true,"maxSize":288}
2·5·13 D²=2 {"Q":130,"inf":false,"maxSize":580}
2·5·13 D²=4 {"Q":130,"inf":true,"maxSize":4608}
2·5·13 D²=5 {"Q":130,"inf":true,"maxSize":4608}
2·5·17 D²=2 {"Q":170,"inf":true,"maxSize":3816}
2·5·17 D²=4 {"Q":170,"inf":true,"maxSize":8192}
2·5·17 D²=5 {"Q":170,"inf":true,"maxSize":8192}
2·13·17 D²=2 {"Q":442,"inf":true,"maxSize":73648}
2·13·17 D²=4 {"Q":442,"inf":true,"maxSize":73728}
2·13·17 D²=5 {"Q":442,"inf":true,"maxSize":73728}
2·5·13·17 D²=2 {"Q":2210,"inf":false,"maxSize":80}
2·5·13·17 D²=4 {"Q":2210,"inf":true,"maxSize":1176592}
2·5·13·17 D²=5 {"Q":2210,"inf":true,"maxSize":1176592}
Shell cwd was reset to /Users/karamelton

>cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad && cat > table.js <<'EOF'
const PR={5:[2,1],13:[3,2],17:[4,1]};
function build(par,Ps){const Q=(par?2:1)*Ps.reduce((a,b)=>a*b,1),A=new Uint8Array(Q*Q).fill(1);
 for(let x=0;x<Q;x++)for(let y=0;y<Q;y++){let ok=!par||((x+y)&1);for(const p of Ps){const[a,b]=PR[p];const m=(v)=>((v%p)+p)%p;
  if((m(x*a+y*b)===0&&m(y*a-x*b)===0)||(m(x*a-y*b)===0&&m(y*a+x*b)===0)){ok=0;break;}}A[x*Q+y]=ok?1:0;}return[Q,A];}
function offsets(D2){const o=[];for(let dx=-2;dx<=2;dx++)for(let dy=-2;dy<=2;dy++)if((dx||dy)&&dx*dx+dy*dy<=D2)o.push([dx,dy]);return o;}
function run(par,Ps,D2){const[Q,A]=build(par,Ps),O=offsets(D2),N=Q*Q,seen=new Int32Array(N).fill(-1),LX=new Int32Array(N),LY=new Int32Array(N);let inf=false,max=0,c=0,kept=0;
 for(let s=0;s<N;s++){if(!A[s])continue;kept++;if(seen[s]>=0)continue;const st=[s];seen[s]=c;LX[s]=(s/Q)|0;LY[s]=s%Q;let sz=0;
  while(st.length){const u=st.pop();sz++;for(const[dx,dy]of O){const nx=LX[u]+dx,ny=LY[u]+dy,t=((nx%Q+Q)%Q)*Q+((ny%Q+Q)%Q);if(!A[t])continue;if(seen[t]<0){seen[t]=c;LX[t]=nx;LY[t]=ny;st.push(t);}else if(LX[t]!==nx||LY[t]!==ny)inf=true;}}
  if(sz>max)max=sz;c++;}
 return{Q,kept:+(kept/N).toFixed(3),inf,max:inf?null:max};}
const out={};
for(const par of [0,1])for(let m=0;m<8;m++){const Ps=[5,13,17].filter((_,i)=>m>>i&1);for(const D2 of[1,2,4]){const k=(par?'2':'')+(Ps.length?'|'+Ps.join(','):'|')+'|'+D2;out[k]=run(par,Ps,D2);}}
console.log(JSON.stringify(out));
EOF
node --max-old-space-size=4096 table.js > table.json && node -e "const t=require('./table.json');for(const[k,v]of Object.entries(t))if(!v.inf)console.log(k,JSON.stringify(v))"

|5,13|1 {"Q":65,"kept":0.545,"inf":false,"max":580}
|5,13,17|1 {"Q":1105,"kept":0.483,"inf":false,"max":80}
2||1 {"Q":2,"kept":0.5,"inf":false,"max":1}
2|5|1 {"Q":10,"kept":0.32,"inf":false,"max":1}
2|13|1 {"Q":26,"kept":0.426,"inf":false,"max":1}
2|5,13|1 {"Q":130,"kept":0.273,"inf":false,"max":1}
2|5,13|2 {"Q":130,"kept":0.273,"inf":false,"max":580}
2|17|1 {"Q":34,"kept":0.443,"inf":false,"max":1}
2|5,17|1 {"Q":170,"kept":0.283,"inf":false,"max":1}
2|13,17|1 {"Q":442,"kept":0.377,"inf":false,"max":1}
2|5,13,17|1 {"Q":2210,"kept":0.242,"inf":false,"max":1}
2|5,13,17|2 {"Q":2210,"kept":0.242,"inf":false,"max":80}
Shell cwd was reset to /Users/karamelton

>cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad && cat > origin.js <<'EOF'
const LIM=1500,NM=2*LIM*LIM+10,comp=new Uint8Array(NM+1);comp[0]=comp[1]=1;for(let i=2;i*i<=NM;i++)if(!comp[i])for(let j=i*i;j<=NM;j+=i)comp[j]=1;
const isP=(n)=>n>1&&!comp[n];
function gp(a,b){a=Math.abs(a);b=Math.abs(b);if(a===0)return isP(b)&&b%4===3;if(b===0)return isP(a)&&a%4===3;return isP(a*a+b*b);}
function bfs(D2){const r=Math.floor(Math.sqrt(D2)),O=[];for(let dx=-r;dx<=r;dx++)for(let dy=-r;dy<=r;dy++)if((dx||dy)&&dx*dx+dy*dy<=D2)O.push([dx,dy]);
 const W=2*LIM+1,seen=new Uint8Array(W*W),st=[[0,0]];seen[LIM*W+LIM]=1;let n=0,far=0,esc=false;
 while(st.length){const[x,y]=st.pop();for(const[dx,dy]of O){const nx=x+dx,ny=y+dy;if(Math.abs(nx)>LIM||Math.abs(ny)>LIM){esc=true;continue;}const k=(nx+LIM)*W+ny+LIM;if(seen[k]||!gp(nx,ny))continue;seen[k]=1;n++;far=Math.max(far,Math.hypot(nx,ny));st.push([nx,ny]);}}
 return{D:Math.sqrt(D2).toFixed(3),primes:n,far:far.toFixed(1),esc};}
for(const D2 of[1,2,4,5,8,9,10,13,16,17,18,20,25,26])console.log(D2,JSON.stringify(bfs(D2)));
EOF
node origin.js

1 {"D":"1.000","primes":0,"far":"0.0","esc":false}
2 {"D":"1.414","primes":100,"far":"11.7","esc":false}
4 {"D":"2.000","primes":720,"far":"45.3","esc":false}
5 {"D":"2.236","primes":720,"far":"45.3","esc":false}
8 {"D":"2.828","primes":2996,"far":"93.5","esc":false}
9 {"D":"3.000","primes":2996,"far":"93.5","esc":false}
10 {"D":"3.162","primes":249508,"far":"1024.4","esc":false}
13 {"D":"3.606","primes":249508,"far":"1024.4","esc":false}
16 {"D":"4.000","primes":786588,"far":"2080.0","esc":true}
17 {"D":"4.123","primes":786588,"far":"2080.0","esc":true}
18 {"D":"4.243","primes":821408,"far":"2107.9","esc":true}
20 {"D":"4.472","primes":829372,"far":"2116.4","esc":true}
25 {"D":"5.000","primes":829372,"far":"2116.4","esc":true}
26 {"D":"5.099","primes":830056,"far":"2116.4","esc":true}
Shell cwd was reset to /Users/karamelton

>mkdir -p ~/Desktop/math-surfaces/gaussian-moat && S=/private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad && cp $S/table.js ~/Desktop/math-surfaces/gaussian-moat/gen_sieve_table.js && cp $S/table.json ~/Desktop/math-surfaces/gaussian-moat/sieve_table.json && cat > ~/Desktop/math-surfaces/gaussian-moat/build_extra.py <<'EOF'
# Inline the precomputed sieve verdicts (node gen_sieve_table.js > sieve_table.json).
import os
html = html.replace("/*SIEVE_TABLE*/null", open(os.path.join(here, "sieve_table.json")).read().strip(), 1)
EOF
ls ~/Desktop/math-surfaces/gaussian-moat

build_extra.py
gen_sieve_table.js
sieve_table.json

>cd ~/Desktop/math-surfaces && python3 - <<'EOF'
p='gaussian-moat/template.html'; s=open(p).read()
a=s.index('const HIST = ['); b=s.index('];',a)+2
s=s[:a]+'''const HIST = [
  ["1962", "Basil Gordon", "Asks the question at the International Congress of Mathematicians in Stockholm. Motzkin and Erdős spread it."],
  ["1997", "Gethner and Stark", "Periodic sieves trap every walker for steps √2 and 2, from any start. Proposes sieving with more primes for longer steps."],
  ["1998", "Gethner, Wagon, Wick", "Computer search: from the origin with steps up to √10, the farthest reachable prime is 1,024.4 away (reproduced in the explorer above). Also builds arbitrarily isolated primes."],
  ["1998", "Vardi", "Studies the repeating sieve patterns, and how \\"no endless walk\\" turns into a cap on island size."],
  ["2005", "Tsuchimura", "From the origin, steps up to 6 are trapped too (a large computation)."],
  ["2026", "OpenAI model", "Every step limit, every starting prime: islands are always finite, with a size cap depending only on the step limit."],
];'''+s[b:]
open(p,'w').write(s)
EOF
python3 build.py gaussian-moat && node -e "const s=require('fs').readFileSync('gaussian-moat/surface.html','utf8');new Function(s.match(/<script>([\s\S]*)<\/script>/)[1]);console.log('parses')" && cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad && "/Applications/Google Chrome.app/Contents/MacOS/Google Chrome" --headless=new --disable-gpu --hide-scrollbars --window-size=1200,2300 --virtual-time-budget=6000 --screenshot=$PWD/gm.png "file://$HOME/Desktop/math-surfaces/gaussian-moat/surface.html" 2>/dev/null; ls -la gm.png

gaussian-moat/surface.html 36 KB
parses
-rw-r--r--@ 1 karamelton  wheel  534023 Oct 12 12:01 gm.png
Shell cwd was reset to /Users/karamelton

>cd ~/Desktop/math-surfaces && sed -i '' 's|const SV = { D2: 1, par: false, ps: new Set(), showP: true };|const SV = { D2: 1, par: false, ps: new Set([5]), showP: true };|' gaussian-moat/template.html && grep -c 'new Set(\[5\])' gaussian-moat/template.html && python3 build.py gaussian-moat && wc -c < gaussian-moat/surface.html

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>cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad/repo/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/build && sed -n 1,140p sections/02-word-rule.tex; grep -n 'begin{theorem}' -A8 paper.tex | head -30

\section{A word rule with no vertical period}
\label{sec:word}

We begin with a finite word constraint whose solutions on an infinite strip
cannot have a vertical period if every column is nonconstant.  The constraint
is the two-digit $p$-adic Sudoku rule of
\citet[Definition~4.5]{GTUndecidable}.  The affine-square and rescaling
arguments below follow the framework of
\citet[Theorem~4.7, Lemma~4.8, and Section~4.2]{GTUndecidable}; we give the
complete argument for the particular obstruction needed here.

Fix a prime $p>200$, and set
\[
    N=p^2,\qquad \Sigma=\Fp^\times,
    \qquad I=\{0,1,\ldots,N-1\}.
\]
When an integer occurs in an expression over $\Fp$, its residue modulo $p$
is understood.

\begin{definition}[Allowed words and the line rule]
\label{word:definition}
A word $w\colon I\to\Sigma$ is \emph{allowed} if there are integers $a,b$,
not both divisible by $p$, such that, for every $n\in I$,
\[
 w(n)=
 \begin{cases}
   an+b\pmod p, & p\nmid an+b,\\[2pt]
   (an+b)/p\pmod p, & p\mid an+b\text{ and }p^2\nmid an+b.
 \end{cases}
\]
At indices with $p^2\mid an+b$, the value may be any member of $\Sigma$.
An array $W\colon I\times\Z\to\Sigma$ satisfies the \emph{line rule} if
\[
    n\longmapsto W(n,dn+e)
\]
is an allowed word for every $d,e\in\Z$.
A \emph{positive vertical period} of $W$ is an integer $M>0$ such that
$W(n,m+M)=W(n,m)$ for all $(n,m)\in I\times\Z$.
\end{definition}

The allowed-word constraint depends only on the residues of $a,b$ modulo
$p^2$.  Indeed, these residues determine the divisibility alternatives
and each of the prescribed values.  In particular, it is a finite
constraint on words over the finite alphabet $\Sigma$.

For every allowed word, reduction of a witnessing pair gives a nonzero
affine form $\ell(n)=\bar a n+\bar b$ over $\Fp$.  The word agrees with
$\ell$ wherever $\ell\ne0$.  A nonzero affine form has at most one zero
residue class, so there are at most $N/p=p$ exceptional indices on $I$.
Here and below, ``nonzero affine form'' means that its coefficients are
not all zero; a nonzero constant form is permitted.

\begin{lemma}[Global affine approximation]
\label{word:affine}
If $W$ satisfies the line rule, there are $A,B,C\in\Fp$, not all zero,
such that
\[
    W(n,m)=An+Bm+C
    \quad\text{whenever }An+Bm+C\ne0.
\]
\end{lemma}

\begin{proof}
For each line $m=dn+e$ with $d\in\{-1,0,1\}$, choose the nonzero affine
form $\ell_{d,e}(n)$ supplied by an allowed-word witness.  Call an index on
that line exceptional when $\ell_{d,e}(n)=0$.

First we find a consecutive $4\times4$ square containing no exceptional
cell for any of these three line directions.  In the rectangle
$I\times\{0,\ldots,K-1\}$ there are $(N-3)(K-3)$ such squares.  The
numbers of lines of slopes $0,1,-1$ meeting the rectangle are respectively
$K,K+N-1,K+N-1$.  Each line has at most $p$ exceptional cells on the
whole strip, and each cell belongs to at most $16$ candidate squares.
Consequently at most
\[
    16p(3K+2N-2)\leq16p(3K+2N)
\]
squares are excluded.  Since
\[
    N-3-48p=p^2-48p-3>0,
\]
we can choose an integer $K\geq4$ with
\[
    K>\frac{3(N-3)+32pN}{N-3-48p}.
\]
For this choice,
\[
    (N-3)(K-3)-16p(3K+2N)
      =K(N-3-48p)-3(N-3)-32pN>0.
\]
A square of the required kind therefore exists.

Write its column indices as $r,r+1,r+2,r+3$ and its row indices as
$q,q+1,q+2,q+3$.  Each row agrees on the square with an affine function
$U_m n+V_m$.  For $m=q+1,q+2$ and $n=r+1,r+2$, the second difference
along either diagonal vanishes.  Thus, for each sign,
\[
 \begin{split}
  0={}&n(U_{m+1}-2U_m+U_{m-1})
       +(V_{m+1}-2V_m+V_{m-1})\\
     &\qquad\qquad{}\pm(U_{m+1}-U_{m-1}).
 \end{split}
\]
Subtracting the identities at the two consecutive values of $n$ gives
$U_{m+1}-2U_m+U_{m-1}=0$.  Subtracting the two signs gives
$U_{m+1}=U_{m-1}$, because $2\ne0$ in $\Fp$.  Together these imply
$U_{m-1}=U_m=U_{m+1}$.  The two overlapping triples show that all four
row slopes have a common value $A$.  The remaining identities say that
the four intercepts have zero second differences, so they have the form
$V_m=Bm+C$.  Therefore the form
\[
    H(n,m)=An+Bm+C
\]
agrees with $W$ on the square.  It is not the zero form, because every
entry of $W$ lies in $\Sigma$.

Call a cell \emph{good} if either $H(n,m)=0$ or $W(n,m)=H(n,m)$.  We claim
that four consecutive good cells on a line of slope $0,1$, or $-1$ make
the entire line good.  The restriction of $H$ to that line is an affine
form in $n$.  If it is identically zero, the claim follows from the
definition of goodness.  Otherwise, it and the chosen form
$\ell_{d,e}$ each have at most one zero among the four indices, which
are distinct modulo $p$.  At least two of the four indices remain where
both forms are nonzero.  At these indices, goodness and the word rule
identify both forms with $W$.  Two distinct residues determine an
affine form, so the two forms are equal.  Wherever this common form is
nonzero the word rule gives $W=H$, and its zeros are good by definition.
This proves the claim.

Applying the claim to the four square rows makes those entire rows good.
To adjoin a row immediately above a block of four good rows, consider
its cell in column $n$.  At least one of
\[
    n+4\leq N-1\qquad\text{or}\qquad n-4\geq0
\]
holds, since $N\geq8$.  Choose $\varepsilon\in\{-1,1\}$ so that the four
columns $n+\varepsilon,n+2\varepsilon,n+3\varepsilon,n+4\varepsilon$
are in $I$.  If the new row has index $m$, the cells
\[
    (n+j\varepsilon,m-j),\qquad 1\leq j\leq4,
79:\begin{theorem}\label{thm:main}
80-There is a finite nonempty set $T\subset\Z^3$ such that:
81-\begin{enumerate}[label=\textup{(\roman*)}]
82-\item $T$ tiles $\Z^3$ by translations, but no translation set $A$ with $A\oplus T=\Z^3$ is invariant under a subgroup of finite index in $\Z^3$;
83-\item the set $\Omega=T+[0,1]^3$ tiles $\R^3$ by translations up to null sets, but no such tiling has a translation set invariant under a lattice of full rank in $\R^3$.
84-\end{enumerate}
85-\end{theorem}
86-
87-\input{sections/01-introduction}
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>cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad/repo/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/build && grep -n -i 'last nonzero\|last-nonzero' -A12 sections/02-word-rule.tex | head -60; grep -n -i 'size\|cardinal\|how many\|number of cells\||T|' sections/07-three-dimensions.tex | head

295:residue $t$, so its last nonzero digit is $t$.
116:\cref{geom:representatives-U}.  In particular, $|T|=|U|m^3$.
399:on the size of the finite construction is required.
Shell cwd was reset to /Users/karamelton

>cd /private/tmp/claude-501/-Users-karamelton/6f7a7327-e7e7-4b6a-a638-c6310b54914e/scratchpad/repo/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026/build && sed -n 225,300p sections/02-word-rule.tex; sed -n 1,60p sections/07-three-dimensions.tex

\]
where the last equality follows from the nonzero value
$H_1(n_0+pk,k)=c$.  On the other hand,
\[
    a(n_0+pk)+b
       =p\left(\frac{an_0+b}{p}+ak\right).
\]
Whenever the parenthesized expression is nonzero modulo $p$, the second
case of the allowed-word rule forces this entry to equal
\[
    \frac{an_0+b}{p}+ak\pmod p.
\]
As $k$ runs from $0$ to $p-1$, these residues run through all of $\Fp$,
because $p\nmid a$.  Thus $p-1$ of the entries are forced to take the
$p-1$ distinct nonzero values, contradicting their common value $c$.
The remaining index, where the rule prescribes no value, is not needed
for this contradiction.

We have proved $B_1\ne0$, so every column of $W_1$ is nonconstant by
\cref{word:vertical}.  But $M/p$ is a positive vertical period of $W_1$,
since
\[
    W_1(n,m+M/p)=W(n,pm+M)=W(n,pm)=W_1(n,m).
\]
This contradicts the choice of $M$.
\end{proof}

\begin{lemma}[A model for the line rule]
\label{word:model}
For $t\in\Z\setminus\{0\}$, let $\operatorname{ord}_p(t)$ be the
largest nonnegative integer $r$ for which $p^r\mid t$, and define
\[
    f(t)=\frac{t}{p^{\operatorname{ord}_p(t)}}\pmod p,
    \qquad f(0)=1.
\]
Then $f\colon\Z\to\Sigma$, and the array $W(n,m)=f(m)$ satisfies the
line rule and has every column nonconstant.
\end{lemma}

\begin{proof}
The definition gives a nonzero residue for every nonzero integer,
including negative integers.  For every $r\geq0$ and $t\in\Z$ we have
$f(p^rt)=f(t)$; this also holds at $t=0$ by definition.

Fix integers $d,e$.  If $(d,e)\ne(0,0)$, let $r$ be the largest integer
such that $p^r$ divides both coefficients, and write $d=p^ra$, $e=p^rb$.
The pair $a,b$ is not both divisible by $p$, and
\[
    f(dn+e)=f(an+b)\qquad(n\in I).
\]
When $an+b$ has $p$-adic order zero or one, this is exactly the
respective value prescribed by the allowed-word rule.  When
$p^2\mid an+b$, including $an+b=0$, the rule leaves the value
unrestricted.  Thus $a,b$ witness that the word is allowed.  If
$(d,e)=(0,0)$, the word is constantly $1$ and has witness $a=0,b=1$.
Finally $f(1)=1$ and $f(2)=2$, so every column is nonconstant.
\end{proof}

For the encoding that follows, parametrize the lines by
$x=(x_1,x_2)\in\Z^2$ and write
\begin{equation}
\label{word:line-maps}
    L_n(x)=x_2+nx_1,\qquad u_n=(1,-n)\qquad(n\in I).
\end{equation}
The map $L_n\colon\Z^2\to\Z$ is surjective and its kernel is generated
by $u_n$: the equation $x_2+nx_1=0$ is equivalent to
$x=x_1(1,-n)$.  The model supplies the allowed word
$\bigl(f(L_n(x))\bigr)_{n\in I}$ at every $x$.  Moreover, if
$x_1\equiv0$ and $x_2\equiv t\pmod p$ for $t\in\{1,2\}$, every entry
of this word equals $t$.  Indeed, every $L_n(x)$ then has the nonzero
residue $t$, so its last nonzero digit is $t$.
\section{Passing to three-dimensional space}
\label{sec:geometry}

It remains to remove the finite coordinate in \cref{stack:cyclic}.
We construct a finite tile in $\Z^3$ whose unit-cube thickening also
excludes periodic tilings with arbitrary real translation vectors.
Quotient-to-lattice reductions are studied by \citet[Theorem~1.1]{MSSReduction},
building on earlier rigid tile constructions such as
\citet[Lemma~9.3]{GTTwoTiles}. The need to constrain real translation
vectors also appears in the Euclidean reduction of
\citet[Theorem~2.1 and Lemma~2.2]{GTPeriodic}. The argument below
proves both required statements directly for the same finite tile
and its unit-cube thickening.

Let
\[
 \pi:\Z^3\longrightarrow\Gamma,\qquad
 \pi(a_1,a_2,a_3)=(a_1,a_2,a_3\bmod Q),
 \qquad w=(0,0,Q).
\]
Thus $\ker\pi=\Z w$.  Translate the tile $F$ of \cref{stack:cyclic}
so that $0\in F$, and choose a finite set $U\subset\Z^3$ containing
$0$ on which $\pi$ is a bijection onto $F$.  In particular,
\begin{equation}
 U\cap\ker\pi=\{0\},\qquad w\notin U.
 \label{geom:representatives-U}
\end{equation}
Translating $F$ changes neither tileability nor the existence of a
fully periodic tiling.

\subsection{A residue transversal with many differences}

\begin{lemma}[Rigid representatives]
\label{geom:random-representatives}
For a sufficiently large integer $m\geq2$, one can choose
\[
 T_0=\{t(v):v\in\{0,\ldots,m-1\}^3\},\qquad
 t(v)\equiv v\pmod m,
\]
such that, with $T_*=T_0\setminus\{t(0)\}$,
\begin{equation}
 T_*-T_*\supset
 \{d\in\Z^3:\norm{d}_\infty\leq m,
                         \ d\notin m\Z^3\}.
 \label{geom:difference-property}
\end{equation}
\end{lemma}

\begin{proof}
Choose independently, for each residue $v$,
\[
 t(v)=v+mh_v,\qquad
 h_v\text{ uniform in }\{-2,-1,0,1,2\}^3.
\]
Fix a requested vector $d$ on the right-hand side of
\cref{geom:difference-property}.  Translation by $d\bmod m$ on
$(\Zmod{m})^3$ is a permutation without fixed points.  A cycle of
length $L\geq2$ contains $\lfloor L/2\rfloor\geq L/3$ disjoint
successive pairs.  Selecting these pairs in every cycle gives at
least $m^3/3$ disjoint pairs of residues $(u,v)$ with
Shell cwd was reset to /Users/karamelton

{
  "type": "tool_reference",
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}
{
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  "tool_name": "mcp__blender__get_scene_info"
}
{
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{"result":"Error checking addon status: Could not connect to Blender. Make sure the Blender addon is running."}

>node -e "
// sanity: tripod tile T={0,e1,e2,e3} tiles Z^3 with lattice x1+2x2+3x3≡0 mod 4
const T=[[0,0,0],[1,0,0],[0,1,0],[0,0,1]];const cov=new Map();const R=6;
for(let a=-R;a<=R;a++)for(let b=-R;b<=R;b++)for(let c=-R;c<=R;c++){if(((a+2*b+3*c)%4+4)%4)continue;for(const t of T){const k=[a+t[0],b+t[1],c+t[2]].join();cov.set(k,(cov.get(k)||0)+1);}}
let bad=0;for(let a=-3;a<=3;a++)for(let b=-3;b<=3;b++)for(let c=-3;c<=3;c++)if(cov.get([a,b,c].join())!==1)bad++;console.log('cells not covered exactly once:',bad);
// 1D: Newman search check for tile {0,2} and {0,1,3}
function tiles1D(T){const D=Math.max(...T);for(let L=T.length;L<=2**(D+1);L+=T.length){const used=new Uint8Array(L),A=[];
 const rec=()=>{let i=used.indexOf(0);if(i<0)return true;for(const s of T){const st=i-s;if(T.every(t=>!used[((st+t)%L+L)%L])){T.forEach(t=>used[((st+t)%L+L)%L]=1);A.push(st);if(rec())return true;A.pop();T.forEach(t=>used[((st+t)%L+L)%L]=0);}}return false};
 if(rec())return{L,A:A.map(a=>((a%L)+L)%L).sort((x,y)=>x-y)};}return null}
for(const T of [[0,2],[0,1,3],[0,1,4,5],[0,3,4,7],[0,2,3],[0,1,2,4]])console.log(JSON.stringify(T),JSON.stringify(tiles1D(T)));
"