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  "problem_id": "vibemathed:exact-rank-and-smith-profile-of-affine-incidence-over-mathbb-z-p-3-mathbb-z",
  "title": "Exact rank and Smith profile of affine incidence over $\\mathbb Z/p^3\\mathbb Z$",
  "canonical_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\n\n$$\nH_{b,\\lambda}=\\{x\\in R^n:\\langle b,x\\rangle=\\lambda\\},\n$$\n\nand 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, $(k,n)=(3,2)$.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
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      "id": "vibemathed:exact-rank-and-smith-profile-of-affine-incidence-over-mathbb-z-p-3-mathbb-z:event",
      "problem_id": "vibemathed:exact-rank-and-smith-profile-of-affine-incidence-over-mathbb-z-p-3-mathbb-z",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-09-01T00:00:00.000Z",
      "title": "GitHub manuscript: Depth-Three Smith Profiles and Newton-Minor Geometry for Affine Hjelmslev–Radon Incidence",
      "summary": "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 $p$,\n\n$$\n\\operatorname{rank}_{\\mathbb F_p}B_3=\n\\begin{cases}\n240,&p=3,\\\\\n\\dfrac{p(p+1)(3p^4+4p^3+3p-1)}{18},&p\\equiv1\\pmod3,\\\\\n\\dfrac{p^2(p+1)^2(3p^2+p+1)}{18},&p\\equiv2\\pmod3.\n\\end{cases}\n$$\n\nThe 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.",
      "ai_contribution": "ai_discovered",
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      "model": "GPT-5.6 Sol",
      "provider": "OpenAI",
      "attempted_at": "2026-09-01T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "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.",
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          "url": "https://github.com/aconsciousfractal/FCIG-Depth-Three-Smith-Profiles-and-Newton-Minor-Geometry-for-Affine-Hjelmslev-Radon-Incidence/tree/0807e47a7ad8791a861dcc76e47c70e764f6f3b3",
          "kind": "code",
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          "label": "Łaba–Trainor: the open residue-ring incidence-rank question",
          "url": "https://arxiv.org/abs/2403.05719",
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          "label": "Dvir: normalized incidence matrix and later rank bounds",
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        {
          "label": "Depth-two affine incidence rank — previous parameter case",
          "url": "https://vibemathed.com/problem/exact-rank-and-smith-profile-of-affine-incidence-over-mathbb-z-p-2-mathbb-z",
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      "note": "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.",
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          "label": "VibeMathed: Exact rank and Smith profile of affine incidence over $\\mathbb Z/p^3\\mathbb Z$",
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    "VibeMath has not independently verified the mathematical claim.",
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