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

Lineage and corrections

Łaba–Trainor: the open residue-ring incidence-rank question evidence for this event

Exact-arithmetic companion and reproducibility repository evidence for this event

Dvir: normalized incidence matrix and later rank bounds evidence for this event

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

Exact rank and Smith profile of affine incidence over $\mathbb Z/p^2\mathbb Z$ evidence for this event

Exact rank and Smith profile of affine incidence over $\mathbb Z/p^2\mathbb Z$ parent of this event

This event attributed to GPT-5.6 Sol

VibeMathed record: Exact rank and Smith profile of affine incidence over $\mathbb Z/p^2\mathbb Z$ evidence for this event

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. parent of this event