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

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.

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.