files / notes / chat / transcript.json
{
"title": "Math",
"subtitle": "16 August 2026",
"through": {
"sha": "96faf40",
"url": "https://github.com/wustep/maths/tree/96faf401a92891926d9183202935e2ec9aebdcdf",
"note": "16 Aug 2026. Later commits only restyled this page."
},
"extraction": {
"method": "Visible chat, first occurrence only. User bubbles by [tNu]. Assistant SendMessages by text/type/attachments. Summary-replay duplicates dropped.",
"source_lines": 1507,
"unique_user_bubbles": 31,
"send_messages": 149,
"total_bubbles": 180,
"image_attachments": 14,
"images_resolved": 14,
"images_missing": 0,
"media_note": "All referenced figures live in notes/chat/media/. Five lost originals were regenerated from committed compute data (search_results.json, green_m_p.csv, rct4 reduction counts, q2 20-min UNKNOWN, m=144 / 2-violation residue). The Schur point-cloud landscape bytes were not recovered."
},
"messages": [
{
"id": "t0u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:20 AM (UTC-7)",
"text": "you're math bot!"
},
{
"role": "assistant",
"type": "text",
"id": "s0",
"text": "Math bot it is. What are we solving?"
},
{
"id": "t1u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:20 AM (UTC-7)",
"text": "please create yourself an icon similar to other bot icons"
},
{
"role": "assistant",
"type": "text",
"id": "s1",
"text": "On it. Checking the other bots' icons so I can match that look."
},
{
"id": "t2u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:21 AM (UTC-7)",
"text": "call yourself Math btw"
},
{
"role": "assistant",
"type": "text",
"id": "s2",
"text": "Math it is. Still working on the icon."
},
{
"role": "assistant",
"type": "text",
"id": "s3",
"text": "Got the style. Making a sticker icon to match."
},
{
"role": "assistant",
"type": "text",
"id": "s4",
"text": "Fox with a π. Same sticker style as the others, with room around it.",
"images": [
{
"original": "file:///workspace/math-avatar-padded.png",
"src": "media/math-avatar-padded.png",
"missing": false,
"alt": "Math bot icon: orange fox holding a pi badge"
}
]
},
{
"id": "t3u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:22 AM (UTC-7)",
"text": "Create an environment with everything you might need (any libraries?), make a git repo called /maths/ and push this to private github\n\nwe're going to solve hard math problems\n\nread https://openai.com/index/ten-advances-in-mathematics/ to learn more"
},
{
"role": "assistant",
"type": "text",
"id": "s5",
"text": "On it. Reading that OpenAI post first, then I'll set up a private maths repo with a real environment."
},
{
"id": "t5u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:23 AM (UTC-7)",
"text": "see this repo: we're going to use Lean (https://github.com/openai/ten-proofs)"
},
{
"id": "t9u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:26 AM (UTC-7)",
"text": "Then using codex, identify 50 open math problems to solve, then use judges from grok 4.6 high, claude code fable, and gpt 5.6 to narrow down that list of which 5 are most solvable, then fire subagents to solve them using GPT 5.6 sol Max"
},
{
"role": "assistant",
"type": "text",
"id": "s11",
"text": "That's a full pipeline. Reading the Codex skill first, then I'll start pulling 50 open problems."
},
{
"role": "assistant",
"type": "text",
"id": "s12",
"text": "Codex and Claude Code are both installed. Starting the 50-problem run."
},
{
"role": "assistant",
"type": "text",
"id": "s13",
"text": "Repo is landing at `/maths`. Codex and Claude both need a sign-in before I can run the 50-problem search."
},
{
"role": "assistant",
"type": "text",
"id": "s14",
"text": "Codex needs a ChatGPT sign-in. Open [auth.openai.com/codex/device](https://auth.openai.com/codex/device) and enter `UVLC-4ZFF9` (expires in 15 min). Claude login is next."
},
{
"id": "t10u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:28 AM (UTC-7)",
"text": "btw, please update your intrsuctions.md in this folder, and set up anything that may be useful to solve this (any libraries? any tools? etc)\n\nhave a grok subagent pay attention to the transcripts of gpt 5.6 when it's solving to improve this iteration loop and learn what can be improved in general. our goal is to make a self-improving math AI environment where multiple models can launch quests to explore different ideas. each math problem should have its own folder btw, unlike the ten-proofs, and have a md file that explains the problem in general"
},
{
"role": "assistant",
"type": "text",
"id": "s15",
"text": "Updating `instructions.md` and reshaping the repo so each problem gets its own folder, plus a Grok watcher on the GPT-5.6 transcripts."
},
{
"role": "assistant",
"type": "text",
"id": "s16",
"text": "`instructions.md` is in. Adding a couple more solvers and a standing Grok watcher on new transcripts."
},
{
"role": "assistant",
"type": "text",
"id": "s17",
"text": "`/maths/instructions.md` is the constitution now. One folder per problem, with `PROBLEM.md` as the brief. `python /maths/scripts/new_problem.py <slug> \"Title\"` scaffolds it."
},
{
"role": "assistant",
"type": "text",
"id": "s18",
"text": "Every GPT-5.6 solve gets a Grok watcher on its transcript. Learnings go in `loop/learnings.md`. There's also a daily check at 9am / 3pm / 9pm PT. Extra solvers in the venv: `python-sat` and PARI (`cypari2`), on top of Lean 4.32 + sympy/z3/flint."
},
{
"role": "assistant",
"type": "text",
"id": "s19",
"text": "Still blocked on Codex/Claude login for the 50-problem run, and on GitHub auth for the private push. The earlier device code may still work: [auth.openai.com/codex/device](https://auth.openai.com/codex/device) · `UVLC-4ZFF9`"
},
{
"id": "t11u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:30 AM (UTC-7)",
"text": "hmm this subagent being a routine is a bit weird, i'm just running this overnight"
},
{
"id": "t12u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:30 AM (UTC-7)",
"text": "codex should be signed in now"
},
{
"role": "assistant",
"type": "text",
"id": "s20",
"text": "Yeah, a standing schedule is the wrong shape for an overnight run. Killing that routine. Checking Codex, then starting the 50."
},
{
"role": "assistant",
"type": "text",
"id": "s21",
"text": "Codex is in. Starting the 50-problem search."
},
{
"role": "assistant",
"type": "text",
"id": "s22",
"text": "Routine's gone. Codex is writing the 50 now. I'll start the three judges as soon as that list lands."
},
{
"id": "t13u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:31 AM (UTC-7)",
"text": "btw are there other online articles that might be helpful for you to learn to create the best maths environment?\n\nother than https://openai.com/index/ten-advances-in-mathematics/\n\nmaybe we can learn from terrance tao's posts? look around and lmk what you find"
},
{
"role": "assistant",
"type": "text",
"id": "s23",
"text": "Looking around Tao's blog and a few other formal-math writeups. I'll send what actually looks useful."
},
{
"role": "assistant",
"type": "text",
"id": "s24",
"text": "Tao is the right place to look. The one that matters most is from four days ago: [A digestion of the proof of Sendov’s conjecture](https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the-proof-of-sendovs-conjecture/). An AI tool found a Lean-checked proof (~90k lines). He then spent days turning it into a human argument and a 15k-line formalization. Digestion is a required step, not polish. He also lists still-open cousins (Borcea, Schmeisser, Smale) that look more like our kind of problem than another sphere-packing."
},
{
"role": "assistant",
"type": "text",
"id": "s25",
"text": "The other Tao cluster is how to run many models at once. [Equational Theories](https://terrytao.wordpress.com/2024/10/12/the-equational-theories-project-a-brief-tour/) split a space into 22 million tiny true/false edges, kept claims “conjectured” until Lean, and used old ATPs (Prover9/Mace4, Z3) more than LLMs for the actual settling. [PFR + Blueprint](https://terrytao.wordpress.com/2023/11/18/formalizing-the-proof-of-pfr-in-lean4-using-blueprint-a-short-tour/) is the “write a human skeleton, then fill sorries” version of our PROBLEM.md. His [2026 distillation challenge](https://terrytao.wordpress.com/2026/03/13/mathematics-distillation-challenge-equational-theories/) is the next problem after the machine wins: a cheat sheet a person can read."
},
{
"role": "assistant",
"type": "text",
"id": "s26",
"text": "What I would steal for us: blueprint before Lean, conjectured-until-checked, pair Z3/Mace4 with GPT-5.6 instead of making the LLM the only searcher, and a digestion quest after any machine proof. Also [this LLM-in-the-loop note](https://quemy.info/2026-07-06-machine-checked-maths-llm-in-the-loop.html): never let a model invent a Mathlib lemma name, search first (`loogle` / LeanSearch).\n\nI put the full list in `/maths/loop/sources.md`. Codex is still on the 50."
},
{
"id": "t14u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:34 AM (UTC-7)",
"text": "oo I quite like Tao. I like his use of visualizations as well. When we write our learning reports, we should use them.\n\nWe should use md files and commit to them often. And have images in the repo as well with inline images in the md files."
},
{
"role": "assistant",
"type": "text",
"id": "s27",
"text": "Progress stays in the repo: markdown, figures inline, commit often. Updating the instructions to match."
},
{
"role": "assistant",
"type": "text",
"id": "s28",
"text": "Progress is git now. `PROBLEM.md`, `ATTACK.md`, `loop/learnings.md`, plus `figures/` next to them. Embed like ``."
},
{
"role": "assistant",
"type": "text",
"id": "s29",
"text": "Committed on `main` as `8ef8828`. Watcher reports that could have used a picture and didn't are unfinished. Shared plot helper is `src/maths/figures.py`."
},
{
"id": "t15u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:38 AM (UTC-7)",
"text": "ok i'm going to bed, but i want you to work over night with codex. if claude can't judge then just have codex and grok pick the 5 problems to work on. you can use 50% of my codex usage maximum, so be efficient. you are a god tier mathemetician and subagent orchestrator. figure out the best open problems to solve and then solve them."
},
{
"role": "assistant",
"type": "text",
"id": "s30",
"text": "Night. I'll finish the 50, Grok+Codex pick five (skipping Claude if it's still dark), then run tight solve quests and stay under half your Codex."
},
{
"role": "assistant",
"type": "text",
"id": "s31",
"text": "Claude isn't signed in, so it's Grok + Codex on the five. The 50 is still cooking. Writing the overnight playbook so we can move the second it lands."
},
{
"role": "assistant",
"type": "text",
"id": "s32",
"text": "Codex is still on the 50 (about 9 minutes in). I'll take the list the moment it lands, pick five with Grok, and start the Max quests. Sleep well."
},
{
"id": "t16u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:40 AM (UTC-7)",
"text": "oh also see this: https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf which was openai's 10 solutions\n\nwe'l want to produce similar reasoning chains and solutions, so understand it carefully"
},
{
"role": "assistant",
"type": "text",
"id": "s33",
"text": "Got it. I'll pull the OpenAI walkthroughs and fold that writeup style into how we attack the five."
},
{
"id": "t17u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:40 AM (UTC-7)",
"text": "ok i'm going to bed, i believe in you!"
},
{
"role": "assistant",
"type": "text",
"id": "s34",
"text": "Sleep well. I'll have the walkthrough style in the loop and the five under attack by morning."
},
{
"role": "assistant",
"type": "text",
"id": "s35",
"text": "Fifty is in. Writing the Grok ballot now, then one Codex judge pass."
},
{
"role": "assistant",
"type": "text",
"id": "s36",
"text": "Walkthroughs are 12 discovery notes, not cleaned proofs: false starts, the click, then the argument. Every solve tonight writes that same WALKTHROUGH.md. Judges are in flight."
},
{
"role": "assistant",
"type": "text",
"id": "s37",
"text": "Grok ballot is in. Checking it against the second writer so we don't overwrite a better slate."
},
{
"role": "assistant",
"type": "text",
"id": "s38",
"text": "Grok's five: unique-sum mod p, no-three-in-line at 71, Chowla cosine, Sidon second term, Brocard–Ramanujan. Scaffolding those folders while Codex judges."
},
{
"role": "assistant",
"type": "text",
"id": "s39",
"text": "Merged five: unique-sum mod p, no-three-in-line at 71, Brocard–Ramanujan, Schur S(7), and covering-code radius 2. Hofstadter is out (looks settled). Scaffolding now, then Max."
},
{
"role": "assistant",
"type": "text",
"id": "s40",
"text": "Walkthrough style is in. Every Max quest writes a discovery note like OpenAI's: false starts, the click, then the argument."
},
{
"role": "assistant",
"type": "text",
"id": "s41",
"text": "The seven beats every solve has to hit. Lean only certifies the argument; the walkthrough is the whole chain.",
"images": [
{
"original": "file:///maths/loop/figures/walkthrough-shape.png",
"src": "media/walkthrough-shape.png",
"missing": false,
"alt": "Seven-beat reasoning walkthrough: missing degree of freedom, named false starts, useful failure, the click, the argument, computer residue, proven vs open"
}
]
},
{
"role": "assistant",
"type": "text",
"id": "s42",
"text": "Five folders are filled. Starting all five Max quests now."
},
{
"role": "assistant",
"type": "text",
"id": "s43",
"text": "All five Max quests are running, each with a watcher. Sleep. I'll have residue in the morning."
},
{
"role": "assistant",
"type": "text",
"id": "s44",
"text": "No-three-in-line at 71: no 142-set in the time box. Codex built the rct4 SAT (1.9M clauses) and every bounded run came back UNKNOWN. Encoding is saved; the hole is still open."
},
{
"role": "assistant",
"type": "text",
"id": "s45",
"text": "The residue: rct4 reduction, then every solver timed out. Not an obstruction.",
"images": [
{
"original": "file:///maths/problems/no-three-in-line-grid/figures/n71-rct4-residue.png",
"src": "media/n71-rct4-residue.png",
"missing": false,
"alt": "n=71 rct4 SAT residue: CP-SAT and Kissat/CaDiCaL all UNKNOWN"
}
]
},
{
"role": "assistant",
"type": "text",
"id": "s46",
"text": "Brocard actually moved: replayed n≤63 (only 4,5,7), and a Lean lemma that 151 kills n=16..31. Schur S(7) timed out — no coloring of [1697], near-miss had two triples."
},
{
"role": "assistant",
"type": "text",
"id": "s47",
"text": "Unique-sum: exact \\(m(p)\\) for every prime \\(p\\le 200\\). It tracks \\(\\log p\\) better than \\((\\log p)^2\\) on this range (\\(R^2=0.974\\) vs \\(0.926\\)). Not an asymptotic claim. Covering codes still running.",
"images": [
{
"original": "file:///maths/problems/no-unique-sum-mod-p/figures/m_p.png",
"src": "media/m_p.png",
"missing": false,
"alt": "Exact m(p) for primes p≤200 vs log p and (log p)^2"
}
]
},
{
"role": "assistant",
"type": "text",
"id": "s48",
"text": "Correction on unique-sum: the table is not Green’s \\(m(p)\\). Codex only forbade unique sums (\\(r=1\\)), not \\(r\\in\\{1,2\\}\\). It matches OEIS A398173 only at \\(p=3,5,7\\). Next quest has to recode the SAT. No retry tonight."
},
{
"role": "assistant",
"type": "text",
"id": "s49",
"text": "Covering codes moved: \\(\\ell_2(10,2)\\) from 51 to 50. Watcher rebuilt \\(H\\) and re-checked all 1024 syndromes. Beats the Davydov–Marcugini–Pambianco table. Not a claim on \\(f(2)\\)."
},
{
"role": "assistant",
"type": "text",
"id": "s50",
"text": "The density plot. Blue star is the new \\(n=50\\) code. Holes at \\((8,25)\\), \\((9,38)\\), \\((10,49)\\) stayed open.",
"images": [
{
"original": "file:///maths/problems/linear-covering-code-radius-two/figures/q1_density_vs_length.png",
"src": "media/q1_density_vs_length.png",
"missing": false,
"alt": "Radius-2 covering density vs length; certified n=50 at r=10"
}
]
},
{
"role": "assistant",
"type": "text",
"id": "s51",
"text": "Overnight five, closed:\n\n1. Covering codes — real dent. \\(\\ell_2(10,2)\\le 50\\).\n2. Brocard — real dent. \\(p=151\\) kills \\(n=16..31\\), Lean builds.\n3. No-three-in-line 71 — encoding saved, SAT UNKNOWN.\n4. Schur \\(S(7)\\) — timeout, no coloring of [1697].\n5. Unique-sum — table exists, wrong predicate. Recode later.\n\nProgress is in `/maths` git. Sleep."
},
{
"id": "t18u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 3:50 AM (UTC-7)",
"text": "what’s your /usage now"
},
{
"role": "assistant",
"type": "text",
"id": "s52",
"text": "Pulling Cursor, Claude, and Codex usage now."
},
{
"id": "t19u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 3:51 AM (UTC-7)",
"text": "use 14% more to take another pass at each of these problems"
},
{
"role": "assistant",
"type": "text",
"id": "s53",
"text": "Another pass on all five, capped at about 14% more Codex. Starting with the unique-sum recode and the covering n=49 hole."
},
{
"id": "t20u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 3:51 AM (UTC-7)",
"text": "btw the git repo should record all progress, notes on each problem, attempts, research if any online research was done, etc"
},
{
"role": "assistant",
"type": "text",
"id": "s54",
"text": "Git will hold the full attack log, not just the dents. I'll backfill notes and research on each folder, then take the second pass."
},
{
"role": "assistant",
"type": "text",
"id": "s55",
"text": "Writing the five q2 quests from the q1 residues, then launching Max on the four that aren't still running."
},
{
"role": "assistant",
"type": "text",
"id": "s56",
"text": "Three q2 Max quests are up (n=71, Brocard, Schur). Retrying covering, unique-sum still finishing q1."
},
{
"role": "assistant",
"type": "text",
"id": "s57",
"text": "q2 quests are in git. Four Max runs are up; unique-sum waits on its leftover q1. Notes and research logs are being backfilled into each folder."
},
{
"role": "assistant",
"type": "text",
"id": "s58",
"text": "**Usage** — Sunday Aug 16, 3:52 AM PT\n\n**Cursor** · Ultra · [Usage](https://cursor.com/dashboard/spending)\n- Cursor Models 4% used · resets Sep 13\n- Other Models 83% used · resets Sep 13\n- Grok Bot: 35% used · resets Aug 20\n- Invoice credits: $280 remaining\n\n**Claude** · Max (20x) · [Usage](https://claude.ai/settings/usage)\n- Session: 9% used · resets Aug 16, 5:28 AM PT\n- Weekly: 12% used · resets Aug 22, 8:00 AM PT\n- Fable: 0% used\n\n**Codex** · [Usage](https://chatgpt.com/codex/settings/usage)\n- Weekly: 38% remaining · resets Aug 23, 2:30 AM PT\n- Workspace monthly: 9,487 of 6,000 credits · resets Aug 31, 5:00 PM PT"
},
{
"role": "assistant",
"type": "text",
"id": "s59",
"text": "Weekly can absorb the 14% (38% left). Workspace monthly is already overdrawn. q2 is running on the weekly pool."
},
{
"role": "assistant",
"type": "text",
"id": "s60",
"text": "Lab notebook is in git. Each of the five now has ATTACK, RESEARCH, an honest walkthrough, and a q2 quest."
},
{
"role": "assistant",
"type": "text",
"id": "s61",
"text": "Brocard q2: Lean lemma that if \\(n!+1=m^2\\) and an odd prime power \\(p^k\\) divides \\(n!\\), then \\(p^k\\) divides \\(m-1\\) or \\(m+1\\). Builds, no sorry. Not the conjecture. The remaining split is a unitary-divisor partition."
},
{
"role": "assistant",
"type": "text",
"id": "s62",
"text": "Closest split of \\(n!/4\\) vs the \\(2^{24}\\) sign assignments at \\(n=100\\). The three known solutions are the only gap-1 points.",
"images": [
{
"original": "file:///maths/problems/brocard-ramanujan-factorial-square/figures/q2-unitary-factorization.png",
"src": "media/q2-unitary-factorization.png",
"missing": false,
"alt": "Brocard q2: closest prime-power split of n!/4 and exponential sign assignments"
}
]
},
{
"id": "t21u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 4:15 AM (UTC-7)",
"text": "keep going! i believe in you"
},
{
"role": "assistant",
"type": "text",
"id": "s63",
"text": "On it. Checking the remaining four and kicking unique-sum the moment leftover q1 is dead."
},
{
"role": "assistant",
"type": "text",
"id": "s64",
"text": "Unique-sum q1 is a leftover from 2:46 AM. Killing it so the recode can start. The other three q2s are writing."
},
{
"role": "assistant",
"type": "text",
"id": "s65",
"text": "All five q2s are live now. Covering is still at 7 uncovered on n=49. Schur still at 2 violations. n=71 is on the saved CNF. Unique-sum is recoding the predicate."
},
{
"role": "assistant",
"type": "text",
"id": "s66",
"text": "n=71 q2: 20 minutes on the saved CNF, Kissat and CaDiCaL both UNKNOWN. No 142-set. Hole still open."
},
{
"role": "assistant",
"type": "text",
"id": "s67",
"text": "Same CNF, 10× the clock. Still UNKNOWN, not UNSAT.",
"images": [
{
"original": "file:///maths/problems/no-three-in-line-grid/figures/n71-rct4-q2-residue.png",
"src": "media/n71-rct4-q2-residue.png",
"missing": false,
"alt": "n=71 q2: 20 min Kissat and CaDiCaL both UNKNOWN on saved rct4 CNF"
}
]
},
{
"role": "assistant",
"type": "text",
"id": "s68",
"text": "Covering q2: no n=49. Same 7 uncovered syndromes. Bound stays \\(\\ell_2(10,2)\\le 50\\). Hill-climbing died on the unique-representation wall."
},
{
"role": "assistant",
"type": "text",
"id": "s69",
"text": "Every neighborhood stuck at 7 holes. One-swap from there jumps to 20. Not a nonexistence proof.",
"images": [
{
"original": "file:///maths/problems/linear-covering-code-radius-two/figures/q2_n49_search_residue.png",
"src": "media/q2_n49_search_residue.png",
"missing": false,
"alt": "Covering q2: n=49 search stuck at 7 uncovered syndromes"
}
]
},
{
"role": "assistant",
"type": "text",
"id": "s70",
"text": "Schur q2: still no coloring of [1697]. Better seed landscape (cyclic m=144) and 50k almost-symmetric 536 seeds all UNSAT. Record stays \\(S(7)\\ge 1696\\)."
},
{
"role": "assistant",
"type": "text",
"id": "s71",
"text": "The seed landscape: m=144 starts at 189 defects vs q1's 3841. Still not a coloring.",
"images": [
{
"original": "file:///maths/problems/schur-number-seven-bound/figures/q2-alternate-seed-landscape.png",
"src": "media/q2-alternate-seed-landscape.png",
"missing": false,
"alt": "Schur q2: cyclic-1680 multiplier landscape, best m=144"
}
]
},
{
"role": "assistant",
"type": "text",
"id": "s72",
"text": "Unique-sum q2: the SAT is the right predicate now. Official table matches all 14 A398173 terms through \\(p=47\\). q1 CSV left alone. No \\(p\\le 200\\) extension."
},
{
"role": "assistant",
"type": "text",
"id": "s73",
"text": "Green's \\(m(p)\\) through \\(p=47\\), rematched.",
"images": [
{
"original": "file:///maths/problems/no-unique-sum-mod-p/figures/green_m_p.png",
"src": "media/green_m_p.png",
"missing": false,
"alt": "Green m(p) table rematched to OEIS A398173 through p=47"
}
]
},
{
"role": "assistant",
"type": "text",
"id": "s74",
"text": "Second pass, closed:\n\n1. Brocard — new Lean lemma (odd prime power splits \\(m\\pm1\\)).\n2. Unique-sum — recoded; matches A398173 through \\(p=47\\).\n3. Covering — no n=49; still \\(\\ell_2(10,2)\\le50\\).\n4. n=71 — 20 min SAT, still UNKNOWN.\n5. Schur — better seed, still no [1697] coloring.\n\nAll of it is in `/maths` git."
},
{
"id": "t23u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 9:24 AM (UTC-7)",
"text": "also can you push this to GH for me now"
},
{
"role": "assistant",
"type": "text",
"id": "s75",
"text": "On both. Checking the n=50 covering certificate first (that's the only thing that could actually be new), then pushing the repo."
},
{
"role": "assistant",
"type": "text",
"id": "s76",
"text": "One thing, if the certificate holds: the covering code. The Nov 2025 Davydov–Marcugini–Pambianco paper still lists \\(\\ell_2(10,2)\\le 51\\) from Kaikkonen–Rosendahl 2003, and uses that \\([51,41]\\) code as the seed for their new families. They improved \\(r=18,20,28+\\), not \\(r=10\\). A 2026 crypto paper still cites the same \\([51,41]\\) matrix. So \\(51\\to 50\\) would be a real finite-record dent on a 23-year-old parameter."
},
{
"role": "assistant",
"type": "text",
"id": "s77",
"text": "Everything else is not novel. The Brocard Lean lemmas are the standard split \\(n!=(m-1)(m+1)\\) with odd prime powers going to one side. Unique-sum rematched OEIS A398173. \\(n=71\\) and Schur are timeouts, not bounds."
},
{
"role": "assistant",
"type": "text",
"id": "s78",
"text": "Can't push the overnight tree yet. `gh` isn't logged in on my computer, and `/maths` (where the q1/q2 work actually lived) isn't on this box. What's here is only the initial `/workspace/maths` scaffold. Starting GitHub login, and I'll try to recover the overnight files from transcripts."
},
{
"role": "assistant",
"type": "text",
"id": "s79",
"text": "GitHub login hit a rate limit on the device flow. Retrying."
},
{
"role": "assistant",
"type": "text",
"id": "s80",
"text": "GitHub connector is already signed in, so I can create the private repo that way instead of fighting the CLI login."
},
{
"role": "assistant",
"type": "widget",
"id": "s81",
"widget": {
"prompt": "GitHub token still can't create repos (same 403 as last night). How should I get the private repo up?",
"helpText": "Overnight files are also missing from /maths on this box. I'm reconstructing what I can from transcripts, then I'll push whatever I recover.",
"options": [
{
"label": "Re-auth GitHub connector",
"value": "Re-auth the GitHub connector with repo scope",
"style": "primary"
},
{
"label": "I'll create wustep/maths",
"value": "I'll create empty private wustep/maths, you push into it"
},
{
"label": "I'll paste a PAT",
"value": "I'll give you a GitHub PAT with repo scope"
}
]
}
},
{
"role": "assistant",
"type": "text",
"id": "s82",
"text": "Recovered the overnight tree. The n=50 covering matrix independently verifies: 50 distinct columns, rank 10, every syndrome in \\(\\mathbb{F}_2^{10}\\) is a sum of at most 2 columns (1024/1024). That's a real \\([50,40]_2\\) radius-2 code. Committing it locally, then creating the private repo."
},
{
"role": "assistant",
"type": "text",
"id": "s83",
"text": "Committed locally as `88bedc7` (98 files). Creating the private repo in the browser now. The GitHub API token still can't create repos, so this is the workaround."
},
{
"id": "u107",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 10:11 AM (UTC-7)",
"text": "Re-auth the GitHub connector with repo scope"
},
{
"role": "assistant",
"type": "text",
"id": "s84",
"text": "Re-authing the GitHub connector now. Approve the connect card with repo scope and I'll create `wustep/maths` and push."
},
{
"role": "assistant",
"type": "text",
"id": "s85",
"text": "This GitHub connector is PAT-based, so there's no OAuth card. Need a classic token with `repo` scope (create + push). Don't paste it in chat."
},
{
"role": "assistant",
"type": "secret-request",
"id": "s86",
"secret": {
"label": "GitHub personal access token",
"connector": "user-Github",
"field": "GITHUB_PERSONAL_ACCESS_TOKEN",
"description": "Classic PAT with repo scope. github.com/settings/tokens"
}
},
{
"role": "assistant",
"type": "text",
"id": "s87",
"text": "Token is in. Checking the GitHub connection, then creating `wustep/maths` and pushing."
},
{
"role": "assistant",
"type": "text",
"id": "s88",
"text": "Connector still shows needsAuth. Using the token with `gh` directly so we can create the repo."
},
{
"role": "assistant",
"type": "text",
"id": "s89",
"text": "Token is a fine-grained PAT, so the API still 403s on create. Opening github.com/new in the browser instead."
},
{
"role": "assistant",
"type": "text",
"id": "s90",
"text": "`gh` is logged in as you, but this fine-grained token still can't create repos. Waiting on the browser to finish github.com/new. Once the empty repo exists I'll push."
},
{
"id": "t24u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 12:25 PM (UTC-7)",
"text": "ok I made /maths, you can use it now"
},
{
"role": "assistant",
"type": "text",
"id": "s91",
"text": "Pushing the recovered tree to `wustep/maths` now."
},
{
"role": "assistant",
"type": "text",
"id": "s92",
"text": "Repo is up. It's public and already has a LICENSE, so I'm rebasing onto that, then pushing."
},
{
"role": "assistant",
"type": "text",
"id": "s93",
"text": "Pushed. [github.com/wustep/maths](https://github.com/wustep/maths) (`ebd5498`). The n=50 covering certificate is in `problems/linear-covering-code-radius-two/compute/`."
},
{
"role": "assistant",
"type": "text",
"id": "s94",
"text": "It's public. This token can't flip visibility, so switch it to private in the repo settings if you want that. The covering matrix is now on the open internet."
},
{
"id": "t25u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 12:28 PM (UTC-7)",
"text": "Hmm there seems to be a ton of random folders and maybe excessive devEx work. Can we clean up what's not necessary and improve the folder structure"
},
{
"role": "assistant",
"type": "text",
"id": "s95",
"text": "Looking at the tree first so I can drop the scaffold and keep the actual math."
},
{
"role": "assistant",
"type": "text",
"id": "s96",
"text": "The root is leftover Lean/Python scaffolding. Stripping that, dropping the unused stubs, and flattening to five problem folders plus a short overnight note."
},
{
"role": "assistant",
"type": "text",
"id": "s97",
"text": "Pushed [2bbfb2c](https://github.com/wustep/maths). Root is now just README, license, a Lean pin, `notes/`, and `problems/`."
},
{
"role": "assistant",
"type": "text",
"id": "s98",
"text": "Five folders: `covering`, `brocard`, `unique-sum`, `three-in-line`, `schur`. Dropped the Lean/Python package, Docker, unused stubs (Chowla, Sidon), Codex quest prompts, and plot scripts with no figures. Each problem is statement / attack / walkthrough / research / compute."
},
{
"id": "t26u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 12:31 PM (UTC-7)",
"text": "was there anything you deleted that was useful? for example the openAI reference pdf? maybe there's a good folder for these sort of things (inspo / reference?)"
},
{
"role": "assistant",
"type": "text",
"id": "s99",
"text": "The OpenAI PDF was never in the recovered tree. Checking whether it's still on disk, and what else from the old `loop/refs` we can put back."
},
{
"role": "assistant",
"type": "text",
"id": "s100",
"text": "The PDF was lost with `/maths`, not in last night's cleanup. Downloading it into `refs/` along with the other OpenAI sources."
},
{
"role": "assistant",
"type": "text",
"id": "s101",
"text": "Nothing useful went out in the cleanup. Docker, the Lean/Python package, Chowla/Sidon stubs, quest prompts, and plot scripts with no figures. The OpenAI PDF was already gone with the old `/maths` tree."
},
{
"role": "assistant",
"type": "text",
"id": "s102",
"text": "Put it back under [`refs/`](https://github.com/wustep/maths/tree/main/refs): both PDFs, the extracted walkthrough text, and a short house-style note. That's the folder for this kind of thing. Lost for good: the original git history, the n=71 CNF, and the PNG figures."
},
{
"id": "t28u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 12:45 PM (UTC-7)",
"text": "btw some of the walkthroughs have broken images, please fix"
},
{
"role": "assistant",
"type": "text",
"id": "s103",
"text": "On it, checking which walkthrough images are broken."
},
{
"role": "assistant",
"type": "text",
"id": "s104",
"text": "Four walkthroughs point at missing PNGs (covering, brocard, unique-sum, three-in-line). Regenerating them from the compute data."
},
{
"role": "assistant",
"type": "text",
"id": "s105",
"text": "Brocard n≤100 reran clean. Writing the four figures from the actual tables now."
},
{
"role": "assistant",
"type": "text",
"id": "s106",
"text": "All four walkthrough embeds were pointing at PNGs lost with the old tree. Regenerated from the compute data and pushed in `23c8479`."
},
{
"role": "assistant",
"type": "text",
"id": "s107",
"text": "Covering: certified \\(n=50\\) against the table points and the three hole residues.",
"images": [
{
"original": "file:///workspace/maths/problems/covering/figures/q1_density_vs_length.png",
"src": "media/q1_density_vs_length.png",
"missing": false,
"alt": "Covering density vs length: certified n=50 and unresolved targets"
}
]
},
{
"role": "assistant",
"type": "text",
"id": "s108",
"text": "Brocard: exact unitary gaps on \\(4\\le n\\le100\\), red stars at the three gap-one cases.",
"images": [
{
"original": "file:///workspace/maths/problems/brocard/figures/q2-unitary-factorization.png",
"src": "media/q2-unitary-factorization.png",
"missing": false,
"alt": "Brocard unitary-divisor gaps and sign-vector growth"
}
]
},
{
"role": "assistant",
"type": "text",
"id": "s109",
"text": "Unique-sum: recovered \\(m(p)\\) through \\(p=47\\) only (A398173). The old caption claimed \\(p\\le200\\); that plot is gone.",
"images": [
{
"original": "file:///workspace/maths/problems/unique-sum/figures/m_p.png",
"src": "media/m_p.png",
"missing": false,
"alt": "Recovered m(p) through p=47 against log p and log-squared p"
}
]
},
{
"role": "assistant",
"type": "text",
"id": "s110",
"text": "Three-in-line: the rct4 reduction and the five UNKNOWN runs. No \\(n=71\\) configuration exists to draw.",
"images": [
{
"original": "file:///workspace/maths/problems/three-in-line/figures/n71-rct4-residue.png",
"src": "media/n71-rct4-residue.png",
"missing": false,
"alt": "n=71 rct4 reduction schematic and UNKNOWN timeout bars"
}
]
},
{
"role": "assistant",
"type": "text",
"id": "s111",
"text": "W(2,7) just landed a PR. Checking whether it actually found a coloring."
},
{
"role": "assistant",
"type": "cursor-agent",
"id": "s112",
"bcId": "bc-9b575b07-97e3-4da8-ac81-065cc8f05755",
"label": "Cursor agent"
},
{
"role": "assistant",
"type": "text",
"id": "s113",
"text": "No dent. I verified the reconstructed Paley coloring of \\([3703]\\) (quadratic residues mod \\(617\\), six periods plus one bit). Zero mono 7-APs. Length \\(3703\\) is the published bound, not an improvement."
},
{
"role": "assistant",
"type": "text",
"id": "s114",
"text": "What blocked \\(3704\\): appending \\(0\\) completes the class-\\(2\\) difference-\\(617\\) AP; appending \\(1\\) completes six others. Cadical refutes \\(\\le 6\\) flips of that seed and also refutes rewriting the last period. No QR prime in \\(619..50000\\) has cyclic run \\(\\le 6\\)."
},
{
"role": "assistant",
"type": "text",
"id": "s115",
"text": "PR is [vdw-w27: verified 3703 residue coloring, no dent at 3704](https://github.com/wustep/maths/pull/2). I passed the stealable bits (unsound homemade counters, one-leftover local cages, few-flip SAT around a published seed) to the still-running C7 agent."
},
{
"role": "assistant",
"type": "text",
"id": "s116",
"text": "Overnight five, all Codex `gpt-5.6-sol` Max: covering, brocard, unique-sum, three-in-line, schur. That's in [notes/models.md](https://github.com/wustep/maths/blob/main/notes/models.md)."
},
{
"role": "assistant",
"type": "text",
"id": "s117",
"text": "New pair is Grok 4.6, effort high. \\(W(2,7)\\): coloring of \\([3704]+\\) with no mono 7-AP. \\(C_7^{\\boxtimes 5}\\): beat the 367-set (July 2026 already moved the capacity number via higher powers). They write stealable methods into `notes/learnings.md` so I can relay between them. Two watchers are on that."
},
{
"role": "assistant",
"type": "cursor-agent",
"id": "s118",
"bcId": "bc-3330e5e4-491e-4b2b-90c7-fbe47fecd449",
"label": "Cursor agent"
},
{
"id": "t29u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 1:20 PM (UTC-7)",
"text": "Here's a prompt from Claude, Opus 5 High on the `covering` work. Please read it and fire a subagent with Opus 5 with claude code to do this:\n\n\"Claude Code prompt — package the radius-2 covering code result\n\nPaste everything below the line into Claude Code.\n\nYou are working in my maths repo (github.com/wustep/maths). The folder problems/covering/ contains an overnight agent run that produced a binary [50,40]₂ code of covering radius 2, certifying ℓ₂(10,2) ≤ 50. That result is real — I have independently re-verified the certificate — but the folder badly undersells it and disclaims the consequence that actually matters.\n\nYour job: build problems/covering/result/ as a self-contained, independently reproducible, submission-grade artifact for the full result, including the propagation the original run missed.\n\nBackground you must internalize before writing any code\n\nNotation: ℓ₂(r,R) is the smallest length of a binary linear code with codimension r and covering radius R. For an r × n parity-check matrix H with column set S ⊂ F₂^r, covering radius ≤ 2 is equivalent to {0} ∪ S ∪ (S+S) = F₂^r.\n\nThe reference is Davydov, Marcugini, Pambianco, \"New upper bounds for binary linear covering codes\", arXiv:2511.02542 (Nov 2025). Fetch it. Read at minimum: Definition 3.2 (the (R,ℓ)-partition), Theorem 4.1 (Constructions QM₁²/QM₂²), Theorem 4.3 (the Kaikkonen–Rosendahl [51,41]₂2 code), Theorem 5.2 (their computer-searched 2-partition with p(H_KR) = 11), and Theorem 5.7 with its proof (the iterative family).\n\nThe load-bearing fact: their entire R=2 infinite family is seeded by the r=10 entry. Table 5.1 row r=10 reads n = 51, reference [50] = Kaikkonen–Rosendahl 2003, density 1.29590 — and it is not bolded, meaning even this Nov 2025 paper did not improve it. It has stood since 2003. Theorem 5.7(i) says outright that the 2-partitions of H_KR are \"very important for the next iterative process.\"\n\nSo a 50-column matrix at r=10 does not merely shave one column off one table cell. If it admits a small enough 2-partition, it re-seeds the whole family.\n\nNote: problems/covering/PROBLEM.md currently quotes Green's Problem 40 range as 1 ≤ f(2) ≤ 1.4238. That is stale — arXiv:2511.02542 already improved the upper bound to ≈1.3203. Fix this in the new note; do not propagate the error.\n\nVerified inputs (use these; do not re-derive by search)\n\nThe 50 columns of H, as integers over F₂^10 (bit i = row i+1, LSB first). These match problems/covering/compute/witness_r10_n50.json:\n\n[1, 2, 4, 15, 16, 32, 65, 86, 128, 173, 183, 202, 212, 247, 256, 297, 320, 329, 341, 366, 373, 381, 391, 403, 438, 460, 479, 491, 502, 559, 576, 608, 653, 734, 742, 754, 771, 777, 789, 821, 846, 855, 869, 881, 893, 897, 927, 981, 1003, 1004] \n\nA valid (2,0)-partition into 10 blocks — this is the new ingredient the original run never computed, and it is what unlocks everything downstream:\n\nblock 0: [2, 128, 202, 212, 771, 855, 897, 981] block 1: [86] block 2: [381, 893, 1003] block 3: [183, 297] block 4: [1, 65, 247, 256, 320, 438, 502, 734] block 5: [15, 173, 329, 366, 460, 559, 653, 846] block 6: [4, 16, 391, 491, 742, 754, 869, 881] block 7: [479, 1004] block 8: [403] block 9: [32, 341, 373, 576, 608, 777, 789, 821, 927] \n\nThis gives p(H) = 10, versus the paper's p(H_KR) = 11. Treat both the columns and the partition as claims to be verified by your own code, not as trusted input. If any check fails, stop and report — do not patch around it.\n\nWhat to build\n\nCreate problems/covering/result/ with this layout:\n\nresult/ NOTE.md the writeup (see below) RESULT.md one-page summary: claim, table, how to verify PRIORITY.md literature / priority checklist data/ H_r10_n50.txt copied from ../compute/, unchanged partition_p10.json columns + block assignment, with a schema comment H_r18_n815.txt generated H_r20_n1631.txt generated kr_r10_n51.txt Kaikkonen–Rosendahl baseline, reconstructed from hex verify/ verify.py verifier #1 — written from scratch verify.rs (or .c) verifier #2 — INDEPENDENT reimplementation build_propagation.py constructs the r=18 and r=20 matrices run_all.sh one command, exits nonzero on any failed assertion Makefile optional convenience wrapper \n\nVerifier requirements\n\nWrite two verifiers in two different languages, and write the second one without looking at the first. Two implementations that share an encoding convention are one implementation. Each must independently:\n\nParse the matrix from its text file (not from JSON).\n\nConfirm all columns are distinct, nonzero, and of the right width.\n\nCompute F₂-rank and confirm it equals r.\n\nExhaustively enumerate {0} ∪ S ∪ (S+S) and confirm it equals all of F₂^r. Enumerate honestly — no early exit, no trusting a stored certificate.\n\nConfirm covering radius is exactly 2, i.e. some syndrome is neither 0 nor a single column.\n\nReport the multiplicity histogram and the covering density as an exact rational.\n\nAdd a separate partition checker: for every syndrome s that is neither 0 nor a column of H, confirm there exists a pair hᵢ + hⱼ = s with i, j in distinct blocks. Syndromes equal to 0 or to a single column are satisfied by definition (sums of 0 or 1 column) — get this right, it's the most common place to write a wrong checker.\n\nPropagation\n\nImplement Construction QM₂² (Theorem 4.1, eq. 4.2 and 4.4) directly from the paper. Do not hand-wave it.\n\nStarting code: n₀ = 50, r₀ = 10, p(H₀) = 10.\n\nCondition to check explicitly in code: n₀ ≥ 2^m ≥ p(H₀). This permits m = 4, 5 and not m ≥ 6 (since 2^6 = 64 > 50). Assert this rather than assuming it.\n\nIndicators: assign βⱼ ∈ F_{2^m} such that columns in distinct blocks get distinct indicators, and the union of indicators used covers all of F_{2^m}. Since indicator sets must be disjoint across blocks, this needs Σ_b |I_b| = 2^m with |I_b| ≤ |block b| — write a greedy allocator and assert feasibility. Block sizes here are [1,1,2,2,3,8,8,8,8,9], summing to 50.\n\nColumns of A(hⱼ, βⱼ): (hⱼ, ξ, βⱼ·ξ) for all ξ ∈ F_{2^m} including ξ=0.\n\nD = D₁(2): the 2^m − 1 columns (0_{r₀}, 0_m, w) for w ranging over the nonzero m-bit vectors (parity-check matrix of the Hamming code).\n\nField arithmetic: GF(16) with x⁴+x+1 (0x13), GF(32) with x⁵+x²+1 (0x25). Include a small self-test that multiplication is associative and that every nonzero element has an inverse.\n\nThen run the full exhaustive covering check on each result. Do not sample.\n\nHard assertions\n\nrun_all.sh must assert every one of these and fail loudly otherwise:\n\nquantity expected r=10 matrix shape 10 × 50, rank 10, 50 distinct nonzero columns r=10 coverage 1024 / 1024 r=10 density exactly 319/256 = 1.24609375 r=10 representation multiplicity histogram {1:859, 2:129, 4:24, 5:9, 6:3} syndromes needing a pair 973 pair-only multiplicity histogram over those 973 {1:821, 2:123, 4:19, 5:8, 6:2} forced-split pairs (unique representation) 821 partition validity valid (2,0)-partition, 10 blocks KR baseline 51 distinct columns, 1024/1024, density 1327/1024 r=18 propagation (m=4) n = 815, 262144 / 262144, density 332521/262144 ≈ 1.26847 r=20 propagation (m=5) n = 1631, 1048576 / 1048576, density 1330897/1048576 ≈ 1.26925\n\nReconstruct the KR baseline from the hex in Theorem 4.3 (H_KR = [I₁₀ M_KR], M_KR = [1B6, 193, 1CC, 187, 1F6, F7, 16E, 140, 3C, 296, 22F, 303, 381, 365, 11D, 1A3, 274, 2F2, 254, 56, F, 41, 357, 208, 34, 329, 28D, 31D, 3D5, 129, 3D7, B7, 3EC, 2E2, 23C, AD, 34E, 155, 2E6, 371, D4]). Verifying the baseline is what makes the improvement meaningful — include it.\n\nThe family\n\nFrom the recurrence n_{new} + 1 = 2^m(n₀ + 1), r_new = r₀ + 2m, derive and state the closed form n = 51·2^{r/2−5} − 1 for the reachable even r, with asymptotic covering density → 51²/2^{11} = 2601/2048 ≈ 1.26999.\n\nCompare against: the paper's new μ̄(2) ≈ 1.32031 and the prior μ̄(2) ≈ 1.4238 that had stood since 1992. Since Green's f(2) is a liminf over r, and the even-r subsequence attains this density, note that this also bounds f(2) ≤ 2601/2048.\n\nBe careful and explicit about which even r are reachable by iterating QM₂² from this seed (each step requires n₀ ≥ 2^m ≥ p(H₀), and QM₂² gives p(H_C) ≤ 2^{m+1} + 1). Produce a generated table of reachable r up to 64 with n, density, and the corresponding published value, and mark which rows are improvements. Generate this table with code, not by hand.\n\nNOTE.md — the writeup\n\nAim for something a coding-theory referee could act on in ten minutes. Structure:\n\nStatement. ℓ₂(10,2) ≤ 50, with the matrix inline.\n\nWhy it matters. The r=10 seed argument. This is the section the original walkthrough is missing entirely.\n\nThe 2-partition. p(H) = 10 ≤ p(H_KR) = 11.\n\nPropagation. QM₂² applied at m=4, 5, with the exhaustively verified results, then the family and asymptotic density.\n\nVerification. How to re-run; state plainly that both verifiers ignore any stored certificate and enumerate from the matrix text.\n\nWhat is not claimed. Be aggressive here:\n\n50 is not shown optimal. The sphere-covering bound only gives ℓ₂(10,2) ≥ 45, so there is real room below.\n\nThe n=49 run leaving 7 uncovered syndromes is search residue, not a lower bound. Say this in those words.\n\nThe (8,25) and (9,38) gaps remain open.\n\nThe f(2) bound is an upper bound only; whether f(2) = 1 is untouched.\n\nProvenance. State plainly that the matrix was found by targeted simulated annealing driven by an LLM agent (Codex gpt-5.6-sol, overnight 2026-08-16), seeded from the KR matrix; that the partition and propagation were derived subsequently; and that every claim is exhaustively machine- checkable from the committed matrices. Do not bury this and do not apologize for it — checkable certificates are exactly what makes this domain safe for machine discovery, and that framing is part of what makes the note interesting.\n\nAlso draft note.tex (article class, ~3 pages, no fancy packages) suitable for arXiv math.CO / cs.IT, with the matrix as an appendix listing.\n\nPRIORITY.md\n\nA checklist, not prose:\n\n[ ] Check Lobstein's online covering-radius bibliography for any ℓ₂(10,2) ≤ 50.\n\n[ ] Search ACCT proceedings (Davydov–Drozhzhina-Labinskaya lineage) — these are poorly indexed and are the most likely place a 50 could already be hiding.\n\n[ ] Check Cohen–Honkala–Litsyn–Lobstein Covering Codes (1997) length tables.\n\n[ ] Check for any known lower bound on ℓ₂(10,2) better than the sphere-covering bound of 45.\n\n[ ] Email Davydov ([redacted]) and Marcugini/Pambianco (unipg.it) with the matrix and verifier before posting anything. Draft that email into PRIORITY.md as a fenced block.\n\nTable 5.1 is explicitly \"best as far as the authors know\" — treat priority as unresolved until a human confirms it.\n\nRules\n\nVerify, don't trust. Every number in NOTE.md must be emitted by code in verify/, not typed by you. If a number in this prompt disagrees with what your code computes, the code wins — stop and tell me.\n\nNever fabricate. If a paper fetch fails, or a value can't be confirmed, write UNVERIFIED and move on. Do not invent citations, table entries, or page numbers.\n\nNo overclaiming. Upper bounds only. No optimality, no f(2) resolution.\n\nDeterministic. No RNG in the committed pipeline. run_all.sh must produce byte-identical output across runs.\n\nDon't modify problems/covering/compute/. Copy what you need.\n\nUpdate problems/covering/ATTACK.md with a dated entry pointing at result/, and update the repo README.md covering row to reflect the propagation. Keep the existing house style.\n\nDo not delete the honest negative results (the n=49 residue, the (8,25) and (9,38) failures). They are part of the record and belong in the note.\n\nStart by fetching arXiv:2511.02542 and reading the sections listed above, then show me your plan for verify/ before writing it.\"\n\nAlso here's a package of stuff from Claude Opus too:"
},
{
"role": "assistant",
"type": "text",
"id": "s119",
"text": "On it. Reading the Claude Code skill, then starting the covering-result package with Opus."
},
{
"id": "t30u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 1:20 PM (UTC-7)",
"text": "Also see my Downloads/files (2) folder with this info from calude:\n\n\"Yes — I already have working code, and there are a couple of cheap checks worth running first that would strengthen the note.Two useful findings before I package anything.\n\nThe code is minimal (locally optimal). Deleting any one of the 50 columns breaks covering radius 2 — the best single deletion leaves 9 uncovered syndromes. In Davydov's terminology this makes it an LO code, equivalently a minimal 1-saturating 50-set in PG(9,2). That's a named property with its own literature and it's worth a sentence in the note: it means the result isn't a padded construction that someone can trivially shorten.\n\nIt satisfies the condition the r=28 branch needs. The matrix has minimum distance 3 (10 linearly dependent triples), and the triple (491, 734, 821) sits in three distinct blocks of the partition. That's the exact analogue of the paper's Theorem 5.2(ii) — they needed h₅+h₂₇+h₂₉ = 0 across distinct subsets to run the QM₅² step at r=28. So the new seed inherits that too, which matters for how far the family extends.\n\nLet me package everything.Everything passes end-to-end. Packaging it up.The tarball contains the generated matrices plus two scripts. Drop it at problems/covering/_reference/.\n\nThe one thing to get right: oracle_check.py is a working verifier, so if Claude Code reads it before writing its own, you've got one implementation wearing two hats. Tell it explicitly not to read _reference/scripts/ until both of its verifiers are written and passing, then diff. data/ is safe to read anytime — those are inputs, not verification logic.\n\nThree additions to the prompt now that these are computed:\n\nAdd minimality to the assertion table (min uncovered after deleting any column = 9) and to the note. \"Minimal 1-saturating set in PG(9,2)\" / \"LO code\" is the vocabulary Davydov et al. use, and it kills the obvious referee question about padding.\n\nAdd the dependent-triple check (10 triples, one spanning blocks 6/4/9). This is the analogue of their Theorem 5.2(ii), which the r=28 branch needs.\n\nAdd the encoding warning. The KR hex in the paper is MSB-first with row 1 as MSB; everything else here is LSB-first. gen_data.py handles the reversal, but a from-scratch verifier that misses it will produce a spurious mismatch on the baseline and send someone down a rabbit hole.\n\nThe H_r18_n815.txt and H_r20_n1631.txt files are also worth handing over rather than having Claude Code regenerate them — if its independent construction produces different matrices that both verify, that's a stronger result (two distinct witnesses), but if they differ and only one verifies, you want to know immediately rather than after the note is drafted.\n\nThe soft spot is unchanged and flagged in the README: m=4 and m=5 are exhaustive, everything past that inherits the paper's p(H_C) ≤ 2^{m+1}+1 bound rather than computing partitions directly. That's the claim a referee will push on first.\""
},
{
"role": "assistant",
"type": "text",
"id": "s120",
"text": "Got it. Pulling the Downloads package, then starting Claude Code on the covering result."
},
{
"role": "assistant",
"type": "text",
"id": "s121",
"text": "Kit is at `problems/covering/_reference/`. Scripts stay quarantined until both verifiers pass. Starting Claude Code on Opus 5, high effort."
},
{
"role": "assistant",
"type": "text",
"id": "s122",
"text": "Claude Code is on it: Opus 5, high effort, branch `covering-result`. Extra checks from the kit are in the prompt (minimality / LO code, the (491,734,821) triple, LSB vs KR hex). Scripts stay unread until both verifiers pass."
},
{
"role": "assistant",
"type": "text",
"id": "s123",
"text": "Claude Code finished the covering package. Checking PR #3 and the assertion list."
},
{
"role": "assistant",
"type": "text",
"id": "s124",
"text": "Claude Code packaged it. [PR #3](https://github.com/wustep/maths/pull/3) is open on `covering-result`. I re-ran `run_all.sh`: 74 checks, exit 0."
},
{
"role": "assistant",
"type": "text",
"id": "s125",
"text": "The overnight run only certified \\(\\ell_2(10,2)\\le 50\\). The package is the seed argument: \\(p(H)=10\\) (paper had 11), then QM₂² at \\(m=4,5\\) gives verified \\(n=815\\) at \\(r=18\\) and \\(n=1631\\) at \\(r=20\\), against published 831 and 1663. Family density \\(2601/2048 \\approx 1.27002\\), so \\(f(2)\\le 2601/2048\\) if the inherited partitions hold."
},
{
"role": "assistant",
"type": "text",
"id": "s126",
"text": "Two extra facts landed. Deleting any column leaves at least 9 holes (LO / minimal 1-saturating set). The triple \\((491,734,821)\\) sits in three distinct blocks, the Thm 5.2(ii) analogue for the \\(r=28\\) branch. The \\(r=18\\) and \\(r=20\\) matrices are distinct from the kit and both verify."
},
{
"role": "assistant",
"type": "text",
"id": "s127",
"text": "Do not merge yet. Priority is unresolved. `PRIORITY.md` has the literature checklist and a drafted email to Davydov / Marcugini / Pambianco. One prompt slip: \\(2601/2048\\) is \\(1.27002\\), not \\(1.26999\\). Fraction is right."
},
{
"id": "t31u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 1:56 PM (UTC-7)",
"text": "Clean up anything worth cleaning up, then merge"
},
{
"role": "assistant",
"type": "text",
"id": "s128",
"text": "Looking at PR #3 for leftover scaffold, then I'll merge."
},
{
"role": "assistant",
"type": "text",
"id": "s129",
"text": "Dropping the Claude prompt and the leftover `_reference` folder. Alternate \\(r=18/20\\) matrices stay as second witnesses."
},
{
"role": "assistant",
"type": "text",
"id": "s130",
"text": "Merged. [PR #3](https://github.com/wustep/maths/pull/3) is on `main` as `0f79fcc`. Dropped the Claude prompt and `_reference/` kit. Alternate \\(r=18/20\\) matrices are in `problems/covering/result/data/alt/`. Pipeline still 74/74."
},
{
"role": "assistant",
"type": "text",
"id": "s131",
"text": "Left the two drafts alone: [W(2,7) #2](https://github.com/wustep/maths/pull/2) (no dent) and [C7 #1](https://github.com/wustep/maths/pull/1) (still the Grok run). Priority checklist is still unchecked."
},
{
"id": "t32u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 1:59 PM (UTC-7)",
"text": "Add this explainer as well to covering: file:///Users/wustep/Downloads/covering-explainer-v2_1.html (written by Claude Opus & Fable 5)"
},
{
"role": "assistant",
"type": "text",
"id": "s132",
"text": "On it. Copying the explainer into covering."
},
{
"role": "assistant",
"type": "text",
"id": "s133",
"text": "It's on `main` as [`problems/covering/explainer.html`](https://github.com/wustep/maths/blob/main/problems/covering/explainer.html). Linked from the covering PROBLEM and the repo README. Credit is Claude Opus and Fable 5."
},
{
"id": "t33u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:01 PM (UTC-7)",
"text": "Can you generate a pdf explaining the problem, process, result, and verification via latex and throw it in there too?"
},
{
"role": "assistant",
"type": "text",
"id": "s134",
"text": "On it. Writing a LaTeX explainer from the verified numbers, then compiling the PDF into covering."
},
{
"role": "assistant",
"type": "text",
"id": "s135",
"text": "TeX install failed on the first try. Retrying that while I write the explainer."
},
{
"role": "assistant",
"type": "text",
"id": "s136",
"text": "Explainer source is written. Compiling it now."
},
{
"role": "assistant",
"type": "text",
"id": "s137",
"text": "The first compile died silently. Running Tectonic again with the log on disk."
},
{
"role": "assistant",
"type": "text",
"id": "s138",
"text": "It's on `main`: [`problems/covering/explainer.pdf`](https://github.com/wustep/maths/blob/main/problems/covering/explainer.pdf). Source is `explainer.tex`. Four sections: problem, process, result, verification. Every number is from `run_all.sh`."
},
{
"role": "assistant",
"type": "attachment",
"id": "s140",
"attachment": {
"original": "file:///workspace/maths/problems/covering/explainer.pdf",
"src": "media/explainer.pdf",
"missing": false,
"filename": "explainer.pdf"
}
},
{
"id": "t34u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:06 PM (UTC-7)",
"text": "update the authorship to be \"Grok Bot (4.6), Claude Opus 5, Claude Fable 5, GPT Sol 5.6\", with \"with instructions from Stephen Wu\" even smaller than the models"
},
{
"role": "assistant",
"type": "text",
"id": "s141",
"text": "Updating the byline on the explainer and the note, then rebuilding the PDFs."
},
{
"id": "s143b",
"role": "assistant",
"type": "text",
"text": "Byline is now **Grok Bot (4.6), Claude Opus 5, Claude Fable 5, GPT Sol 5.6**, with “with instructions from Stephen Wu” on its own smaller line (`\\tiny` in the PDFs, 9px in the HTML)."
},
{
"role": "assistant",
"type": "text",
"id": "s144",
"text": "C7 just landed. Checking whether it beat the 367-set."
},
{
"role": "assistant",
"type": "text",
"id": "s145",
"text": "No dent. It reconstructed the Polak–Schrijver 367-set and verified it. Nothing of size 368. \\(367^{1/5}\\approx 3.25787\\), still below the July 2026 capacity \\(3.258805\\)."
},
{
"role": "assistant",
"type": "text",
"id": "s146",
"text": "Useful residue: Cadical plus a real at-most encoder refuted every odd Hamming distance \\(\\le 9\\) from that seed (including 4-out/5-in). Distance 11 timed out. That is a local cage, not \\(\\alpha\\le 367\\). Min-conflicts stalled at ~800 leftover pairs."
},
{
"role": "assistant",
"type": "text",
"id": "s147",
"text": "PR is [C7 fifth power: verified 367-set, search for 368](https://github.com/wustep/maths/pull/1). Both Grok 4.6 runs are done. Neither moved a published bound."
},
{
"role": "assistant",
"type": "text",
"id": "s148",
"text": "W(2,7) merge is in. Only `notes/learnings.md` conflicted; keeping both the C7 and W(2,7) sections, then I'll add the two README rows."
},
{
"role": "assistant",
"type": "text",
"id": "s149",
"text": "Also finished the leftover merge. PR #2 is on main. README now has rows for `vdw-w27` (verified 3703, no dent) and `c7-shannon` (verified 367-set, no dent). Covering is still the only finite record."
},
{
"role": "assistant",
"type": "text",
"id": "s150",
"text": "Starting the leftover Fable 5 pass so the covering explainer stands alone."
},
{
"role": "assistant",
"type": "text",
"id": "s151",
"text": "Fable 5 standalone pass is on main (`3cd0b3d`). The PDF is still 7 pages. A reader who only has the PDF now gets `[n,k]_2`, the `(2,0)`-partition, \\(\\bar\\mu(2)\\) vs Green’s \\(f(2)\\), and a one-line QM₂² gloss."
},
{
"role": "assistant",
"type": "text",
"id": "s152",
"text": "HTML also lost the bad `1.26999` (now `1.27002`). Byline unchanged. No invented numbers."
},
{
"id": "t43u",
"role": "user",
"type": "text",
"timestamp": "Sunday, Aug 16, 2026, 2:30 PM (UTC-7)",
"text": "btw the PDF should assume that the user is not reading it from the repo, so there should be references to the repo, including in the How to re-run section"
},
{
"role": "assistant",
"type": "text",
"id": "s162",
"text": "Adding the GitHub repo to the PDF so a reader who only has the file can still find the code and re-run it."
},
{
"role": "assistant",
"type": "text",
"id": "s163",
"text": "Repo links are in. Tightening one overflowing URL, then I’ll push."
},
{
"role": "assistant",
"type": "text",
"id": "s164",
"text": "The PDF now assumes you only have the file. How to re-run starts with `git clone https://github.com/wustep/maths.git`, and the abstract plus the file list point at the repo. On main as `96faf40`."
}
]
}