The QCD θ-Phase as a Gauge–Determinant Circle Datum: A Technical Note on Topological Characters, Determinant Phases, and Admissible CP

This paper develops a source-datum formulation of the quantum chromodynamics (QCD) θ-phase. It addresses a structural question that precedes claims about observability, compensation, relaxation, or closure: what is the circle-valued phase object being compared, which source data define it, and through which identity relation does it become invariant? The QCD phase is formulated as a gauge–determinant circle datum combining a theory-level topological character, its evaluation on integer topological sectors, the determinant and chiral phase, boundary and regularization data, representative-change rules, and the nonphase quantities held fixed in a phase comparison. The paper distinguishes the character [θ] ∈ Hom(ℤ, U(1)) ≅ U(1) ≅ 𝕋 from the sector weight exp(iθQ). It places the topological and determinant phases on one source space and defines their sum as the total gauge–determinant phase. Anomaly-compatible chiral changes act anti-diagonally on the two phase coordinates, and a circle-valued cocycle condition gives the exact criterion for descent of the total phase through the declared source identity relation. The paper also separates three CP structures: source-level oddness of the total phase, covariance of the response law, and invariance of a selected response. Together with a phase-identifying response, these are the interfaces required for a CP fixed-point conclusion. Common-source invariance is formulated through exact factorization of linked response and determinant or spectral readings through one fixed parent datum. Admissible automorphisms preserving that parent preserve the linked phase readings. Massless loci, determinant zeros, index jumps, boundary changes, and other singular domains require explicit strata or transition data. Peccei–Quinn, axion, and other source-space extensions define enlarged phase data rather than internal retunings of the original QCD source. This work supplies the source object for a three-paper Strong-CP theorem sequence. “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 for such a datum. “Conditional Strong-CP Phase Closure from QCD Response Certification: Gauge–Determinant Phase Descent, CP Fixed-Point Identification, and π-Branch Thermodynamic Ordering” adds quantitative fixed-topology phase identification, relative topological-charge gluing, heterogeneous mixed-route parity suppression, thermodynamic ordering, completion-level selection, and explicit regulator-admissibility conditions. The three papers form a source-datum, abstract-closure, and quantitative response-certification sequence for conditional Strong-CP phase closure. License note: Distributed under CC BY-NC-ND 4.0.