The Third Verdict at Strong Coupling: Is the Interacting Entanglement Hamiltonian Irreducible to the Bisognano–Wichmann Local Energy Density? (Phase VII, and its certification Phase VII-b)

Scope label (load-bearing). Genuinely interacting quantum field theory in flat 1+1 Minkowski space. No gravity: not dynamical, not backreacting, not quantum. This is Exit (a) of Proposition 2 — the coupling axis, isolated from the gravity axis. Archived as Phase VII (with its certification Phase VII-b) ; not merged into the free-field Minkowski corpus of Phases IV–V, nor into the fixed-curved-background Phase VI. Phase VI closed the fixed-background branch with a structural obstruction (Proposition 2): for a free field on a fixed classical background the state is a functional of the background data, hence so is every modular observable, and grade (ii) is the ceiling — a genuine positive is structurally unreachable there. Exactly two exits survive: (a) genuine interaction, and (b) dynamical gravity. This paper attacks Exit (a), the minimal scope extension: still flat space, no gravity, only the coupling switched on. The object is the interval entanglement Hamiltonian KA = −log ρA in the DMRG/MPS ground state of the mixed-field Ising chain — genuinely interacting, non-integrable, gapped, non-conformal — and the reference is the model's own interacting Bisognano–Wichmann family (identity plus a freely weighted local energy density plus first descendants), never the free stress tensor. The same structural logic caps the attainable outcome at a distinction inside grade (ii): REDUCE (the third verdict extends to coupling) versus IRREDUCIBLE (the first crack of the enclosure). Both were preregistered as publishable and hash-frozen before any production run. Phase VII returned a substantive but uncertifiable result: all four target residuals sit 5–30× below the controls, the residual anti-tracks the coupling, and the co-primary spectral reconstruction echoes a correlation-length ordering — yet the frozen multiplicative threshold (θ = 10 × floor = 1. 62) exceeded the maximum response of any bounded foil (≈ 0. 45), because the free-control floor was dominated by Jordan–Wigner representation mismatch, an instrument bias ~30× the target signals. The teeth precondition failed by arithmetic, not by physics, and the verdict was left with an asterisk. Phase VII-b removes it: an independently frozen rerun in the model's natural variables (the fermionic Jordan–Wigner-bilinear BW family, whose free-control residual is exactly zero at full range — the calibration the earlier instrument lacked), an additive threshold, and satisfiable contrast teeth. Teeth pass; all four targets lie 8–200× below the certified threshold and below every control, in both representations. Verdict: COLLAPSE (Exit (a) ), certified. The interacting entanglement Hamiltonian reduces to the local Bisognano–Wichmann energy density; the reduction is representation-independent, and increasing the coupling makes KA more BW-local, not less. The strong-coupling axis carries no grade- (ii) -irreducible modular orientation. No grade- (i) claim is made or implied (Proposition 2 forbids it for a fixed-Hamiltonian ground state). The genuine positive, if it exists, is pushed onto Exit (b) — dynamical geometry — named and out of scope here. Preregistration and reproducibility. Both investigations were governed by preregistrations frozen and hashed before any production run — Phase VII: SHA-256 (§1–§6) = 5c53a0c0…fc09f (frozen 2026-06-30 22: 45: 34 UTC) ; Phase VII-b: SHA-256 (§1–§6) = c63f8a10…39d46 — with append-only deviation logs (DEV-1…DEV-6 and DEV-1…DEV-4 (VII-b) ) below the hash boundary, and cautionary records of the two failed instruments preserved rather than discarded. Code, preregistrations, per-claim ledgers, and all emitted CSV/JSON data: https: //github. com/PauDD/Simulador-Fase-VII Continues Phase IV (10. 5281/zenodo. 21261576), Phase V (10. 5281/zenodo. 21302338) and Phase VI (10. 5281/zenodo. 21321084) of the programme Rigidez de la Incidencia Geométrica (Geometric Incidence Rigidity).