Exact Congruence Counts on L1-Shells and the Limits of Fixed-Period Prime Prefilters

We prove an exact crystallisation formula for a ternary quadratic congruence on three-dimensional L1-shells. For every prime p > 3, the number of points x on the shell ||x||₁ = p satisfying x₁² + x₂² + x₃² ≡ 0 (mod p) is exactly 6 when p ≡ 2 (mod 3), and exactly 8p − 2 when p ≡ 1 (mod 3). We further derive the correct critical Abel-weighted density for any fixed finite set of odd primes and prove that every sound fixed-period prime prefilter must retain all reduced residue classes. Consequently, an ordinary wheel has the minimum possible candidate density among sound lookup filters of the same fixed period. This work is a corrective and substantially strengthened successor to “Geometric Forbidden Sieve (GFS): Theory and SIMD-Parallel Extreme Optimization for Prime Generation” (DOI: 10.5281/zenodo.20480555). It preserves and extends the valid shell-congruence theory while correcting the earlier valley, density, fixed-period filtering, and implementation claims. The deposit includes the manuscript, LaTeX source, reproducibility code, citation metadata, licenses, and cryptographic checksums.