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Parity obstruction in the minimum-determinant problem for Latin squares

A Mathematics Stack Exchange question posted on 3 August 2014 asks when the standard divisibility lower bound for determinants of Latin square matrices is attained. For an n×nn\times n Latin square LL with entries 1,…,n1,\ldots,n, let
bn={n2(n+1)/2,n odd,n2(n+1)/4,n even. b_n=\begin{cases} n^2(n+1)/2,&n\text{ odd},\\ n^2(n+1)/4,&n\text{ even} \end{cases}.
For which positive integers nn does there exist such an LL with ∣det⁡L∣=bn|\det L|=b_n? The question conjectures that n=4,6n=4,6 are the only orders for which this minimum cannot be attained.

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Latin squares; determinant divisibility
Posed by
Mathematics Stack Exchange user "Peter"
Year posed
2014
Years open
12y
Solved
2026-04-22
Model
GPT-5.4
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Significance
10 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For even nn, let q(L)=det⁡(L)/bnq(L)=\det(L)/b_n. The work proves that q(L)q(L) is even exactly when the stronger centered divisibility n2∣det⁡(Estd)n^2\mid\det(E_{\mathrm{std}}) holds. For n≡2(mod4)n\equiv2\pmod4, this is equivalent to rank⁡F2(A mod 2)<n−1\operatorname{rank}_{\mathbb F_2}(A\bmod2)<n-1; for n≡0(mod4)n\equiv0\pmod4, it is equivalent to adj⁡(A mod 2)1=0\operatorname{adj}(A\bmod2)\mathbf1=0. An explicit family gives odd q(L)q(L) for every n≡2(mod4)n\equiv2\pmod4, n≥6n\ge6. This removes a universal extra-factor-two obstruction, but it does not prove ∣q(L)∣=1|q(L)|=1. Exact minimum attainment and the separate singularity question remain open.

What the AI did

Under the author's direction, OpenAI's ChatGPT, using the GPT-5.4 model, generated the central mathematical development of this work, including the ordinary-to-centered determinant reduction, the exact binary rank and adjugate criteria governing the additional factor of two, and the all-order construction producing an odd determinant quotient for every n≡2(mod4)n\equiv2\pmod4, n≥6n\ge6. It also assisted with the development of the exact verification code and the manuscript. The author selected the research direction, checked the mathematical derivations and certified outputs, established the public claim boundaries, commissioned adversarial reviews, and takes responsibility for the final content.

Verification

The paper and public repository contain complete proofs, exact certified datasets, and a deterministic verifier that currently passes all 12 public artifacts. The release also received an artifact-oriented adversarial audit. These checks establish internal consistency and reproducibility, not independent expert endorsement of the headline theorem; no domain expert has yet endorsed it. The appropriate VibeMathed verification label is therefore Unreviewed.

Sources

Submitted by VividMarten473 on

Changelog2 changes

Discussion