Optimality of the added-vector code in the 19-dimensional kissing construction
Let \(D\subseteq\mathbb F_2^{19}\) be the fixed 4096-word ambient binary linear code used in Ho’s 19-dimensional improvement of the Cohn–Li kissing construction. Is every subset \(A\subseteq D\) with minimum Hamming distance at least \(5\) of size at most \(1280\)? No linearity assumption is imposed on \(A\).
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Computation
- Field
- Coding theory; algebraic graph theory; discrete geometry
- Posed by
- Gonzalez, Conjecture 16 (preprint, version 3, 2026), after Ho's 1280-word construction
- Year posed
- 2026
- Years open
- 0y
- Solved
- 2026-09-06
- Model
- ChatGPT (OpenAI, model version unstated)
- Vendor
- OpenAI
- Collaborators
- Alexey M. Kolosov
- Verification
- Site-confirmed
- Publication
- Preprint
- Significance
- 8 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
The manuscript proves the upper bound \(|A|\le1280\) for every admissible subset of the fixed ambient code \(D\). Together with Ho’s existing construction, this determines the exact maximum. The proof partitions \(D\) into 256 cosets of a 16-word subspace whose induced forbidden-distance graph is the Clebsch graph; each coset contributes at most five words. This does not determine the unrestricted kissing number \(k(19)\) or improve the known 11948-point configuration. Independent expert review is pending.
What the AI did
The manuscript's Section 5: "ChatGPT (OpenAI) was used for literature search, the optimization-based search for the five-word certificate, development and checking of the proof, preparation of the verification program, and drafting and translation of the manuscript. The mathematical argument and the exact verification data are given explicitly; a language-model output or a numerical solver status is not used as a substitute for proof." Certificate found and proof developed with the model under the author's direction: co-developed.
Verification
Site-confirmed: the author's exact verifier (anc/verify_golay_1280_optimality.py, Python standard library, exact arithmetic) was replayed here on 6 September 2026 from a clone of cheptil/kissing-number-19-dimensions at 0a90b8e and passed every check: the five generator identities and rank certificate, the weight distribution of the 16-word subspace, the exact Fourier dual certificate, the independence number 5 of the 16-vertex certificate graph by exact subset DP, the 256 cosets giving 1280, and the explicit matching constructions with minimum distances 6 and 5. The manuscript's Section 5 discloses ChatGPT's role. No independent expert statement; the tier records a replayed certificate, not a referee.
Sources
Submitted by cheptil on