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

Precise statement

Let $p$ be prime, $k,n\geq 1$, and $R=\mathbb Z/p^k\mathbb Z$. For each primitive direction $b\in\mathbb P(R)^{n-1}$ modulo multiplication by units and each $\lambda\in R$, let $$ H_{b,\lambda}=\{x\in R^n:\langle b,x\rangle=\lambda\}, $$ and let $A(p^k,n)$ be the $0$-$1$ matrix whose rows are the indicators of these distinct affine hyperplanes and whose columns are the points of $R^n$. What is $\operatorname{rank}_{\mathbb F_p}A(p^k,n)$? 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 $k=1$ is known, but the exact rank remains open in general for $k\geq2$. This entry concerns the depth-three plane specialization, $(k,n)=(3,2)$.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

GPT-5.6 Sol

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.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

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.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-09-01

   GitHub manuscript: Depth-Three Smith Profiles and Newton-Minor Geometry for Affine Hjelmslev–Radon Incidence

   For $(k,n)=(3,2)$, the normalized distinct-row matrix $B_3$ equals $A(p^3,2)$ up to row and column ordering. For every prime $p$, $$ \operatorname{rank}_{\mathbb F_p}B_3= \begin{cases} 240,&p=3,\\ \dfrac{p(p+1)(3p^4+4p^3+3p-1)}{18},&p\equiv1\pmod3,\\ \dfrac{p^2(p+1)^2(3p^2+p+1)}{18},&p\equiv2\pmod3. \end{cases} $$ The paper also determines the complete $p$-primary Smith profile of $\operatorname{coker}(B_3^\mathsf T)_{(p)}$: its exponent is $p^5$, and all six multiplicities are explicit for every prime. The repeated-row Łaba–Trainor matrix has the same $\mathbb F_p$-rank, but the integral Smith claim applies only to $B_3$. 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.

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.