Generators of stability-preserving semigroups, spectral gaps, and classical ground-state computation

We classify generators of complex stability-preserving semigroups on finite polynomial boxes and prove polynomial-time recognition from Gaussian-rational binary one-site and two-site inputs. Damping bounded below by μ > 0 gives strict zero-freeness and a sharp, capacity-independent real-part spectral gap, extending the Hermitian 2μ gap of Bravyi, Gosset, Liu, and Wong. At bounded degree and local strength, deterministic classical algorithms approximate ground energies and relative amplitudes against stable product states for Suzuki–Fisher Hamiltonians with fixed positive longitudinal fields; field perturbation gives zero-field energy-value polynomial-time approximation schemes for Einstein–Podolsky–Rosen (EPR) and bipartite Quantum MaxCut on bounded-degree unweighted graphs. An all-temperature converse for full Gibbs tensors characterizes Hermitian qubit Hamiltonians, in a fixed basis, as precisely the edgewise phase-rotated Suzuki–Fisher family up to scalars. Together with strict positive-field Gibbs radii, this answers two questions of Wong, Bravyi, Gosset, and Liu. Weighted token-graph spectral concavity resolves the adjacency and signless-Laplacian conjectures of Apte, Parekh, and Sud. An exact certificate disproves the stronger proposed EPR ground-state radius. This record contains the 69-page manuscript and the supplementary archive certificates-8552777.zip. The archive provides two exact rational verification procedures for the six-vertex EPR counterexample in Appendix C, together with execution instructions and SHA256 checksums. The verification code corresponds to GitHub commit 85527775f5de8af8911941fc422dbe140a63d11a. The manuscript includes a disclosure of AI assistance.