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Exact rank and Smith profile of affine incidence over Z/p2Z\mathbb Z/p^2\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.

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
2026-08-25
Model
GPT-5.6 Sol
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Significance
8 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For the plane at depth two, (k,n)=(2,2)(k,n)=(2,2), the paper's B2B_2 is A(p2,2)A(p^2,2) up to row and column ordering. For every prime pp it proves
rank⁡FpB2=p2(p+1)24. \operatorname{rank}_{\mathbb F_p}B_2=\frac{p^2(p+1)^2}{4}.
More strongly, it determines the complete nonunit pp-primary Smith profile:
coker⁡(B2T)(p)≅(Z/pZ)p3(p−1)/2⊕(Z/p2Z)p(p−1)2(p+2)/4⊕(Z/p3Z)p(p−1)/2. \operatorname{coker}(B_2^\mathsf T)_{(p)}\cong (\mathbb Z/p\mathbb Z)^{p^3(p-1)/2}\oplus (\mathbb Z/p^2\mathbb Z)^{p(p-1)^2(p+2)/4}\oplus (\mathbb Z/p^3\mathbb Z)^{p(p-1)/2}.
It also proves that the canonical depth-two transfer extension is nonsplit. The theorem includes p=5p=5; only the memory-intensive full-matrix companion computation for (p,depth)=(5,2)(p,\mathrm{depth})=(5,2) is not run, and it is not used in the proof. Arbitrary depth, higher ambient dimension, the adjacent projective Hjelmslev problem and the separate generalized-polynomial characterization question remain open.

What the AI did

Under the author's direction, OpenAI Codex using GPT-5.6 Sol generated the central mathematical development: the maximal-order and depth-transition framework, the reduction of the depth-two Smith profile to three invariants, the modular-rank and point-fibre-transfer arguments, the relative-shell and two-chart carry analysis, and the resulting all-prime formulas. It also assisted with the exact companion software 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 proof, an exact-arithmetic companion, deterministic release checks and an explicit AI-use disclosure. Clean normal and optimized replays passed, as did the hostile verification suite (59/59 tests); the final source and PDF were also subjected to adversarial same-workflow checks. These checks establish reproducibility and internal consistency, not independent mathematical endorsement. No named independent domain expert has yet endorsed the theorem, so Unreviewed is the correct tier.

Sources

Submitted by Oleksiy Babanskyy on

Changelog2 changes

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