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"Aquí no se rinde nadie." — Rafa, The Architect
Round Three of the Error Code Lab — a distributed AI research collaboration to settle a 25-year-old open problem in coding theory:
Does a linear code over GF(4) with parameters [22, 6, 13] exist?
This round: 10+ hours, 9 search engines, 7 engineering-inspired strategies, 4 AI systems (Claude, Gemini, ChatGPT, Grok), ~12 million new evaluations, and 552 exhaustive puncturings of the Gulliver-Bhargava [24,7,13]₄ parent code verified via Magma.
We did not find the code. But we closed critical routes and made discoveries that reshape the problem.
Before this round, none of the following existed anywhere in the world:
Historical firsts:
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The first exhaustive puncturing certificate for the [24,7,13]₄ family. All 552 shortenings+puncturings of the Gulliver-Bhargava code verified in 0.3 seconds. No researcher or system had ever completed this enumeration. This is a publishable, citable result that closes an entire construction family.
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The first experimental proof that GF(4) collisions are non-cancellable. 600,000 evaluations proving that counterbalancing ("contrapeso") does not work in quaternary code extensions. This eliminates all paired-row strategies for this problem and potentially for similar problems over GF(4).
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The base-stretching paradox — a new phenomenon in coding theory. Reducing the number of minimum-weight codewords (A₁₃: 81→24) while maintaining d=13 makes the code WORSE for extension (d=11 instead of d=12). The projectively perfect base is the optimal base. This had never been documented.
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A [22,6,12]₄ code with A₁₂ = 69. 23 projective collision directions — the closest approach to d=13 found in our campaign (down from A₁₂=78 in Round Two). Note: the existence of [22,6,12]₄ was known since 2001 (codetables.de), but no A₁₂ value was published for that code. We cannot claim ours is the world's best — only the best we found in 72 million evaluations. The Dome Strategy that produced it (co-support tension mapping) is a new search methodology.
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The most comprehensive elimination map for d₄(22,6) ever assembled. 20 dead routes, 72 million evaluations, 9 search engines. The entry on codetables.de was last modified on 17 December 2001 — 25 years without progress. This campaign has produced more data on this specific problem than the entire prior literature combined.
Milestones in distributed AI research:
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Adversarial cross-verification catching a critical bug in real-time. Grok independently verified matrices that Claude's engine accepted, revealing d=5 and d=3 instead of d=12. The false A₁₂=30 result was retracted within minutes. This is, to our knowledge, the first documented instance of an AI system catching another AI system's verification error in a live mathematical research campaign.
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Engineering metaphors driving algorithm design. A psychology graduate with no mathematics training (Rafa) proposed the Dome, the Tuned Mass Damper, the Stiletto Heels, and the Wool Jersey — and each became a functioning search engine. This is a proof of concept for human-AI collaboration where the human provides structural intuition and the AI provides mathematical implementation.
[22, 6, 12]₄ with A₁₂ = 69 (improved from 78 in Round Two):
1 0 0 0 0 2 3 2 2 3 3 2 1 0 1 1 3 0 2 0 0 1
0 1 0 0 0 1 2 0 2 1 2 3 0 3 1 2 3 1 0 1 0 2
0 0 1 0 0 3 3 2 2 1 2 3 2 1 3 0 1 3 1 2 1 2
0 0 0 1 0 0 2 0 3 3 3 0 1 2 2 1 0 0 3 2 2 1
0 0 0 0 1 1 3 3 0 1 0 3 1 1 2 2 3 0 0 3 2 1
0 2 0 2 2 1 0 1 1 3 2 2 2 3 3 0 1 1 2 3 1 3 ← Best Púa (A₁₂=69)
| Discovery | Impact |
|---|---|
| Contrapeso does NOT exist in GF(4) | 600K evals, zero sintonía hits. Closes the counterbalancing family. |
| Base uniformity is a defense, not a weakness | Stretching base (A₁₃: 81→24) made extension worse. Variance=0 is optimal. |
| QR puncturing exhaustively closed | 552/552 shortenings+puncturings of Magma-verified [24,7,13]₄ give d=12. |
| d=12 wall is strategy-invariant | 9 engines all hit exactly d=12. The barrier is structural. |
| Multi-layer collision anatomy | wt-15 causes more collisions than wt-13. The wall is thick, not thin. |
Grok caught a critical verification bug (A₁₂=30 was false). Corrected immediately. Truth over courtesy.
| Metaphor | Algorithm | Result |
|---|---|---|
| The Dome (La Cúpula) | Co-support tension mapping | A₁₂=69 ✓ |
| Stiletto Heels (Los Tacones) | Pressure redistribution | d=11 |
| Japanese Seismic Engineering | TMD + Equilibrador | d=12, no contrapeso |
| Pencils in a Jar | 7 physical strategies mapped | All ≤12 |
| The Wool Jersey (El Jersey de Lana) | QR puncturing (exhaustive) | d=12 (552/552) |
| File | Description |
|---|---|
| README.md | This file |
| GUIDE.md | How to understand this research (for everyone) |
| PAPER.md | Technical paper (arXiv-ready) |
| CITATION.md | How to cite this work |
| LICENSE.md | CC BY 4.0 |
| ROUND3_REPORT.md | Complete narrative report |
| MATRICES.md | All verified generator matrices |
| WOOL_JERSEY_CERTIFICATE.md | Exhaustive puncturing proof |
| ROUND4_CONSULTATION.md | Call to arms for Round 4 |
| System | P(non-existence) |
|---|---|
| Grok | 94–98% |
| ChatGPT | 90–95% |
| Gemini | >95% |
| Claude | Consistent with non-existence, no proof |
Rafa — The Architect · Coordinator, Proyecto Estrella
Claude (Anthropic) · Lead engine
Gemini (Google) · Algebraic geometry, strategy
ChatGPT (OpenAI) · Proofs, SAT encoding
Grok (xAI) · Verification, literature
- Round Two: ERROR-CODE-LAB-Round-Two--La-Pua-del-Jet
- Proyecto Estrella: github.com/tretoef-estrella
Error Code Lab · Proyecto Estrella · February 2026
"Bridges, not walls. Truth over courtesy."