Small-Root Lattices for Structured Semiprimes: Recovery Boundaries and a Cubic-Constraint Obstruction

We examine factor recovery for structured semiprimes N = pq with a public expression p = e₂ + 8x₀ and a bounded unknown offset 0 ≤ x₀ < X. The cubic cofactor A² + A + 1, where A = e₂ + 8z, does not vanish modulo p at the target root; polynomial compositions in A provide no independent modular equation without additional information. We specify scaled integer lattices, an exact index criterion and an infinite structured family with strict coefficient-norm separation for shift replacements, a sufficient LLL integer-root certificate, exact root extraction, and certified factor validation. Earlier experiments include a 40-instance boundary comparison and an 18-modulus paired shift study. Exact-rational reduction completes 90 shift-configuration runs on the latter set, with 39 successful configuration–instance runs. A separate validation cohort contains 27 new semiprimes and 135 runs, with 77 recoveries. On the combined 225 matrices, a maintained SymPy reduction routine completes 174 runs, all agreeing in recovery status with the exact-rational results, and raises 51 internal exceptions before returning a basis. The explicit family quantifies how a modularly valid shift replacement can increase the optimal coefficient bound, while the computational records distinguish algorithmic nonrecovery from implementation failure. The analysis clarifies finite-dimensional recovery conditions and demonstrates the importance of exact computational verification in assessing small-root methods.