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Exact rank and Smith profile of affine incidence over Z/p3Z\mathbb Z/p^3\mathbb Z

Let pp be prime, k,n≥1k,n\geq 1, and R=Z/pkZR=\mathbb Z/p^k\mathbb Z. For each primitive direction b∈P(R)n−1b\in\mathbb P(R)^{n-1} modulo multiplication by units and each λ∈R\lambda\in R, let
Hb,λ={x∈Rn:⟨b,x⟩=λ}, H_{b,\lambda}=\{x\in R^n:\langle b,x\rangle=\lambda\},
and let A(pk,n)A(p^k,n) be the 00-11 matrix whose rows are the indicators of these distinct affine hyperplanes and whose columns are the points of RnR^n. What is rank⁡FpA(pk,n)\operatorname{rank}_{\mathbb F_p}A(p^k,n)? Łaba and Trainor explicitly recorded the residue-ring point-hyperplane rank question as open and proved upper bounds. Dvir later used the normalized distinct-row matrix above and obtained further bounds. The field case k=1k=1 is known, but the exact rank remains open in general for k≥2k\geq2. This entry concerns the depth-three plane specialization, (k,n)=(3,2)(k,n)=(3,2).

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Finite geometry; Smith normal forms
Posed by
Izabella Łaba and Charlotte Trainor (arXiv:2403.05719, 2024)
Year posed
2024
Years open
2y
Solved
2026-09-01
Model
GPT-5.6 Sol
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Significance
7 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For (k,n)=(3,2)(k,n)=(3,2), the normalized distinct-row matrix B3B_3 equals A(p3,2)A(p^3,2) up to row and column ordering. For every prime pp,
rank⁡FpB3={240,p=3,p(p+1)(3p4+4p3+3p−1)18,p≡1(mod3),p2(p+1)2(3p2+p+1)18,p≡2(mod3). \operatorname{rank}_{\mathbb F_p}B_3= \begin{cases} 240,&p=3,\\ \dfrac{p(p+1)(3p^4+4p^3+3p-1)}{18},&p\equiv1\pmod3,\\ \dfrac{p^2(p+1)^2(3p^2+p+1)}{18},&p\equiv2\pmod3. \end{cases}
The paper also determines the complete pp-primary Smith profile of coker⁡(B3T)(p)\operatorname{coker}(B_3^\mathsf T)_{(p)}: its exponent is p5p^5, and all six multiplicities are explicit for every prime. The repeated-row Łaba–Trainor matrix has the same Fp\mathbb F_p-rank, but the integral Smith claim applies only to B3B_3. Arbitrary depth, higher dimension, projective Hjelmslev incidence and the generalized-polynomial characterization remain open. Finite computations certify exceptional cases; they do not prove the uniform formulas.

What the AI did

Under the author's direction, OpenAI Codex using GPT-5.6 Sol generated the central mathematical development: the depth-three reduction, the tripotent-sector decomposition, the truncated q-Pascal and q-Lucas block analysis, the q-Newton determinantal-minor argument, the cross-chart divided-carry closure, and the resulting all-prime Smith formulas. It also assisted with the exact companion software, exceptional certificates and manuscript drafting. The author selected the research direction, iteratively challenged and checked the derivations and certified outputs, established the public claim and source boundaries, and takes responsibility for the final content. Adversarial machine reviews were produced within the same OpenAI Codex workflow and are not human peer review or independent expert verification.

Verification

The public source contains a complete all-prime argument, an exact-arithmetic companion, deterministic release checks and an explicit AI-use disclosure. On 2026-09-08 the isolated normal and optimized release verifiers both passed, as did the targeted mathematical and release-assurance suites (66/66 tests). These checks establish reproducibility, artifact integrity and internal consistency; they are not independent validation of the mathematics because the checking was performed within the same model-assisted workflow. No named independent domain expert has yet checked or endorsed the theorem, so Unreviewed is the correct tier.

Sources

Submitted by Oleksiy Babanskyy on

Changelog2 changes

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