Columns of a Square-Block Partition of the Integers: Parity Obstruction, a Modular Characterisation of Column Divisibility, and Column Prime Density

A single formula, N(m, x) = m² + m + 1 + x for odd m and m² + m + 1 − x for even m, maps the triangular array |x| ≤ m onto the integers 2, 3, 4, … without repetition. We prove the rows are the blocks (m², (m+1)²], so every integer has a unique address (m, c). A fixed offset c gives a column, interlacing two quadratic subsequences, one per row parity. The following results follow. First, N(m, c) ≡ 1 + c (mod 2) for every m, so the odd columns hold no odd number and at most one prime each; this fixed divisor makes them inadmissible. Second, confirmed in 57,910 trials without exception: for an odd prime p, the number of multiples of p in each parity class over any 2p consecutive rows is 1 + χ(−(3 + 4εmc)); for composite odd n, it is the product ∏ ρpa(D) over the prime powers pa ∥ n. The local profiles describe a nearly fourfold spread (3.9×), from 5.10% to 20.01%. Finite-range Bateman–Horn predictions, with truncated singular series and no fitted parameters, agree with observed proportions. Full proofs and the admissibility classification are in Appendices C and D. Version 1.6 adds Appendix E: Exact Displacement Fibers and a Reducibility Obstruction in the Shadow Map. This appendix corrects an earlier equal-branch interpretation by introducing the exact fiber decomposition, proves that when the displacement I is a perfect square the infinite tail is completely composite, and provides extended numerical verification. It also incorporates the cyclotomic generalization for Repunit centers (d+1 prime).