Empirical Diagnostics for the Shadow-Sieve Model in Square-Centered Intervals

We give a computational audit of a “shadow-sieve” model for prime detection in the square-centered intervals J_n = [4n²-n, 4n²+n]. For a roughness threshold B = floor(n^α), the model writes π(J_n) = |A_B(n)| - |C_B(n)| exactly, where A_B is the set of B-rough survivors and C_B the subset represented by the bilinear shadow map (r, s) ↦ 2n(s-r) - rs. The identity is exact, but the question is whether the geometry supplies usable analytic leverage.

Five diagnostics answer that question. First, the ratio |C_B|/|A_B| is empirically insensitive to whether the interval is centered at an even square, an odd square, or shifted away from a square. Second, the naive Mertens prediction for the shadow density misses two opposing effects: a Buchstab-range excess in the factor-pair count and a collision discount from multiple admissible factor pairs. Third, P_3 composites account for 78–95% of that collision discount across the tested range. Fourth, the Selberg/linear-sieve main-term band remains a factor 4–9 wider than the empirical ratio, while a rigorous Brun–Titchmarsh bound for P_3 is within 4% of the trivial lattice-point count. Finally, at the natural threshold B = 2n the shadow disappears pointwise (up to one explicit square in the adjacent epoch block), but the lower-bound linear-sieve coefficient has already degenerated.

The model therefore isolates the standard parity obstruction rather than evading it. Its value is as a geometric reparameterization and a quantitative diagnostic target for stronger averaged prime-distribution methods, not as an independent proof mechanism.