In quantum chromodynamics, the strong charge-parity problem is usually framed as the problem of explaining why the physical theta phase is zero or extremely small despite the presence of topological sectors and anomalous chiral phase shifts. This paper reformulates that problem as a structural closure problem for a circle-valued phase. The relevant question is not only whether a QCD theta-like phase can be shifted among representatives, but whether the descended total phase is fixed by the specified source datum, forced to a CP fixed branch, and protected against post-hoc one-branch retuning. The paper proves a conditional structural closure theorem for admissible QCD theta-invariants. The theorem has two logically distinct components. The first is a CP fixed-point theorem: when a descended phase is CP-odd, the observable law is CP-invariant, and the observable law is phase-complete on the selected branch, the phase is forced into the CP fixed set consisting of the zero and pi branches. This gives fixed-branch closure, distinct from zero-branch closure; selecting the zero branch requires an independent zero-branch certificate or an explicitly specified zero-branch rule. The second component is a completion-locked non-retuning theorem. In that theorem, response and determinant or spectral phase readings are not compared as independent downstream quantities. They must factor through a common downstream dual image and a common specified source datum, with quotient, comparison data, response law, branch data, and source relation fixed before the readings are compared. Under these hypotheses, no operation preserving the same locked datum can tune only one phase reading. The conditional QCD corollary applies the closure theorem to the gauge-determinant circle datum developed in the companion source-datum paper. The topological sector is represented by a circle character of the quantized charge group, while the sector weight is the evaluation of that character on a field configuration. The theorem acts on the descended total phase coordinate formed from the topological phase and the determinant/chiral phase, with determinant representatives, chiral representatives, boundary terms, regularization choices, comparison data, branch data, and response laws included in the source space. When the specified QCD verification hypotheses are satisfied, the descended phase lies in the CP fixed set. When a zero-branch certificate or specified zero-branch rule is also supplied, the selected descended phase is zero. Residual CP-odd phase is therefore not an unexplained leftover parameter. The closure theorem identifies the precise source of any residual phase: a failed hypothesis, a specified nonzero CP fixed branch, a singular or stratified phase domain, or an enlargement of the source space, such as a Peccei-Quinn or axion datum. The result supplies a theorem-level closure framework for strong-CP phase claims while identifying source-datum certification, phase-completeness, and zero-branch verification as explicit requirements for QCD closure. As a companion to The QCD θ-Phase as a Gauge–Determinant Circle Datum: A Technical Note on Topological Characters, Determinant Phases, and Admissible CP, this work establishes the closure consequences for the QCD phase datum identified there. The two papers form a source-datum/theorem pair: the companion paper identifies and certifies the QCD phase object together with its verification conditions, while the present paper proves the closure consequences for that datum under the stated hypotheses. Together they place strong-CP closure on explicit source data, phase descent, observable identification, common-source non-retuning, and branch certification. License note: Distributed under CC BY-NC-ND 4.0.
Salimah H. Meghani (Sun,) studied this question.