Exact rank and Smith profile of affine incidence over $\mathbb Z/p^2\mathbb Z$
partialconfidence 70%
VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
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 $k\geq2$.
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 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.
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.
GitHub manuscript: Depth-Two Smith Profiles and Carry Geometry for Affine Hjelmslev–Radon Incidence
For the plane at depth two, $(k,n)=(2,2)$, the paper's $B_2$ is $A(p^2,2)$ up to row and column ordering. For every prime $p$ it proves
$$
\operatorname{rank}_{\mathbb F_p}B_2=\frac{p^2(p+1)^2}{4}.
$$
More strongly, it determines the complete nonunit $p$-primary Smith profile:
$$
\operatorname{coker}(B_2^\mathsf T)_{(p)}\cong
(\mathbb Z/p\mathbb Z)^{p^3(p-1)/2}\oplus
(\mathbb Z/p^2\mathbb Z)^{p(p-1)^2(p+2)/4}\oplus
(\mathbb Z/p^3\mathbb Z)^{p(p-1)/2}.
$$
It also proves that the canonical depth-two transfer extension is nonsplit. The theorem includes $p=5$; only the memory-intensive full-matrix companion computation for $(p,\mathrm{depth})=(5,2)$ 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.
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.