The Finite-Binary Möbius Conjugate-Response Law

Exact Gibbs certification, covariance closure, and all-order Hamiltonian reconstruction in correlated binary equilibrium. This work establishes the complete finite-binary Möbius conjugate-response closure C_FB. It joins six measurable sectors—thermodynamic pressure, contextual insertion energy, conditional odds, covariance, total correlation, and Möbius interaction coefficients—through three compact equilibrium identities, a finite-residual robustness certificate, and a correlation-completed information balance. For finite classical binary systems in full-support isothermal constrained equilibrium, and for the jointly diagonal quantum sector in which the Hamiltonian and all binary record projectors share one commuting algebra, the local law is Pi_i + D_iH(b) + k_BT ell_i(b) = 0 and the matrix law is -(partial Pi/partial p) Cov(X) = k_BT I. The complete edge family is necessary and sufficient for the finite Gibbs law. A nonzero residual localizes the violated equation to a bit and conditioning context without uniquely identifying which physical input caused the violation; uniformly bounded residuals yield explicit Kullback–Leibler and total-variation guarantees. Boolean/Möbius differentiation reconstructs every nonconstant Hamiltonian interaction order through redundant conditional-logit routes, with the one-body sector completed by an independently calibrated source. An exact total-correlation term closes the information balance for arbitrary differentiable correlated paths and makes the independent-product reduction a sharp boundary. The contribution is the complete overdetermined closure: a new proof and Hamiltonian-reconstruction method, an exact domain-specific physical law, and a reusable cross-field measurement architecture. Coherent noncommuting systems, hidden reservoirs, driven steady states, and general nonequilibrium dynamics remain outside the theorem. Shared executable final seal: this package carries the complete public Cross-Field Spectral Capacity Probe, not merely a companion citation. SCP instantiates the normalized pressure/logit and product-root composition route developed here alongside the Silver spectral-capacity and Boolean/Möbius carrier routes. Its hard integrated benchmark reports PASS at 100.0 readiness. In the declared twelve-case deterministic ablation, capacity alone resolves 9/12 cases and the complete diagnostic resolves 12/12; the pressure/logit route is separately tested through exact product composition and raw-probability false-map rejection. The engine, frozen outputs, hashes, and replay instructions are included in SCP_TRIAD_FINAL_SEAL/. The co-released triad is The Möbius–Obstruction Instrument: From Response Surfaces to Irreducible Supports and Minimal Interventions (doi:10.5281/zenodo.21745631); The Finite-Binary Möbius Conjugate-Response Law (doi:10.5281/zenodo.21745774); and The Silver Web: A Cross-Field Carrier–Capacity Correspondence (doi:10.5281/zenodo.21746428). Each is a stand-alone result. Neither companion is a premise of this theorem, and no evidence battery is pooled across the papers. Their complementary roles are response-to-support-to-intervention, finite-binary physical closure, and cross-field carrier-to-capacity correspondence. The deposit includes the paper in PDF, HTML, and Markdown, an embedded standard-library Python verifier with byte-matched recorded output, release metadata, licenses, manifests, and a deterministic reproducibility archive. First-party publication materials are CC BY 4.0; first-party executable code is Apache-2.0. Mixed files are licensed by portion, and third-party components retain their upstream terms.