Conditional Strong-CP Phase Closure from QCD Response Certification: Gauge–Determinant Phase Descent, CP Fixed-Point Identification, and π-Branch Thermodynamic Ordering

The strong-CP problem concerns the empirical smallness of the circle-valued QCD phase θ̄ despite nontrivial topological sectors and anomalous chiral rephasings. This paper establishes a conditional Strong-CP phase-closure theorem separating five mathematically distinct questions: descent of the total gauge–determinant phase, identification of that phase from a retained response record, reduction of a CP-invariant selected response to the CP fixed-point set, ordering of the fixed points [0] and [π], and completion-level selection on the fixed candidate set. On a regular positive-mass source stratum, anomaly-compatible representative changes act anti-diagonally on the topological and determinant phases, so their sum descends through the declared source identity relation to a circle-valued quotient invariant. For a finite-volume fixed-topology response, the exact phase stabilizer is the annihilator of the subgroup generated by the supported charges. Primitive charge support therefore identifies the descended phase uniquely; a finite Bézout witness reconstructs its circle coordinate explicitly; and a positive cofinal separation modulus prevents loss of identifiability in the continuum limit. An exact phase-independent extraction transfers this record-level identification theorem to observable data. The CP theorem distinguishes three structures that are often conflated: oddness of the descended phase, covariance of the response law, and invariance of the selected response. Together with phase identifiability, these hypotheses imply that the selected phase belongs to the CP fixed-point set {[0], [π]}. A quantitative version bounds the distance to this fixed set by the inverse response-identifiability modulus. The theorem also distinguishes invariance of a selected response from invariance of a state, and a CP-invariant mixture from an invariant extremal vacuum. The fixed points are compared through the normalization-invariant parity ratio ZΛ(π)/ZΛ(0) and its exponential rate δπ. A relative second-Chern gluing theorem gives integral additive block charges and exact parity factorization. Source-fixed minorization, quotient-spectral, and projective-cone criteria each yield the same operator-order parity certificate. The certification route may vary from block to block when all routes act on the same completed parity-resolved response. A positive averaged contraction rate per physical volume implies δπ > 0, so the resulting global suppression theorem depends only on the certified blockwise parity floors and not on the analytic route used to obtain them. A positive δπ gives strict Euclidean 0-versus-π rate ordering. Under the additional hypotheses of reflection positivity, exact slab composition, Osterwalder–Schrader range completeness, positive transfer, and the stated limits, a finite retained lower bound gives strict ground-state energy-density ordering. Completion-level zero selection is treated separately: fixed positive priors and one common rate functional on the candidate set {c₀, cπ} convert the independently proved rate gap into exponential concentration on [0]. The selector is a static law on the candidate set and does not define real-time evolution between QCD theories with different fixed values of θ̄. The paper also proves common-source invariance. The response, determinant or spectral reading, descended phase, rate, and selector factor through one fixed parent datum, and admissible fixed-datum automorphisms preserve the linked readings. This work is the quantitative response-certification theorem in a three-paper sequence. “The QCD θ-Phase as a Gauge–Determinant Circle Datum: A Technical Note on Topological Characters, Determinant Phases, and Admissible CP Comparison” defines the QCD phase source object and its anomaly-compatible descent data. “Strong CP as Completion-Locked Phase Closure: A Conditional Structural Theorem for Admissible QCD θ-Invariants” proves the abstract CP fixed-point, zero-branch propagation, common-source invariance, and Ward-deformation results. The present paper adds quantitative phase identification, relative topological-charge gluing, heterogeneous mixed-route parity suppression, thermodynamic ordering, completion-level selection, and explicit regulator-admissibility conditions. License note: Distributed under CC BY-NC-ND 4.0.