Strong CP as Completion-Locked Phase Closure: A Conditional Structural Theorem for Admissible QCD θ-Invariants

In quantum chromodynamics (QCD), the strong charge-parity (CP) problem is usually framed as the question of why the circle-valued phase θ̄ is zero or extraordinarily small despite nontrivial topological sectors and anomalous chiral rephasings. This paper proves a conditional structural closure theorem for admissible QCD θ-invariants. It asks when a representative-level phase descends through a specified source identity relation, becomes identifiable through a declared response, lies in the CP fixed-point set, and remains invariant under transformations preserving the same completed source datum. The fixed-point theorem separates four hypotheses: CP oddness of the descended phase, covariance of the response law, invariance of the selected response, and phase identifiability. Together they imply Θ̄(z) ∈ {[0], [π]} ⊂ 𝕋 = ℝ/2πℤ. These conditions are mathematically distinct. Covariance of a response family is not invariance of a selected response or state, and a CP-invariant mixture need not be an invariant extremal vacuum. Zero-branch closure is treated separately. A zero conclusion follows under either a declared zero-branch datum or a connected zero-anchor criterion on one regular no-crossing branch. Neither condition is derived from CP invariance or ordinary QCD dynamics. The theorem does not itself establish thermodynamic ordering between the 0 and π theories, define a completion-level probability law on those candidates, or supply real-time phase relaxation. The paper also proves common-source invariance. When the descended phase and its linked response and determinant or spectral readings factor through one declared joint image and one fixed parent datum, every admissible fixed-parent automorphism preserves those readings. The substantive hypotheses are construction of the common parent, exact factorization of the linked readings, and identification of the fixed-parent automorphism class with the source-preserving transformations of the realized theory. When the joint response–spectral map is a smooth embedding, Ward admissibility supplies the corresponding descended infinitesimal deformation law. The conditional QCD corollary applies the theorem to a gauge–determinant phase datum. The topological phase is treated as a character of the integer charge group, while exp(iθQ) is its evaluation on a specified topological sector. The total phase combines the topological and determinant or chiral phases on one source space together with representative changes, boundary and regularization data, branch data, response laws, and the nonphase quantities held fixed in the comparison. Under the stated descent, CP, response-identification, and branch hypotheses, the descended phase lies in {[0], [π]}; a separate zero-branch condition is required to conclude [0]. This work is the abstract closure 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. The present paper proves the abstract CP fixed-point, zero-branch propagation, common-source invariance, and Ward-deformation results. “Conditional Strong-CP Phase Closure from QCD Response Certification: Gauge–Determinant Phase Descent, CP Fixed-Point Identification, and π-Branch Thermodynamic Ordering” adds quantitative phase identification, relative topological-charge gluing, mixed-route parity suppression, thermodynamic ordering, and completion-level selection. License note: Distributed under CC BY-NC-ND 4.0.