Structural Vanishing of the Strong CP Phase in the Coupled Dirac–Λ Framework: A Record-Admissibility and Positivity-Based Resolution

The strong CP problem asks why the QCD vacuum angle theta satisfies |theta| < 10^-10 despite being a free parameter in the Standard Model. Conventional resolutions invoke additional symmetries (such as Peccei–Quinn symmetry), axions, or model-building mechanisms. This paper presents a structural resolution within the Euclidean spectral-action realization of the coupled Dirac–Lambda framework. The argument proceeds in three logically independent layers: No dynamical generation. The bosonic spectral action, defined as the trace of a function of the squared Euclidean Dirac operator divided by a cutoff scale, is CP-even and cannot generate the CP-odd topological term theta times Tr (F wedge F) at any order in the heat-kernel expansion. Within-sector inertness. On any fixed topological sector, an externally added theta term contributes only a constant phase and does not affect variational dynamics or KKT stationarity conditions. Within a single sector, theta is dynamically invisible. Record-admissibility obstruction (decisive layer). The coupled Dirac–Lambda programme is record-bearing. Record-bearing admissibility requires the existence of a positive coarse-graining map (a conditional expectation) on the observable star-algebra, from which positivity of the induced state is derived. For theta not equal to zero (mod 2 pi), the Euclidean weight exp (-SYM - i theta Qₜop) is not a positive measure in the cross-sector theta-vacuum regime. Consequently, no positive state exists, no positive conditional expectation can be defined, and Osterwalder–Schrader reconstruction fails. Thus, within the record-bearing Euclidean Dirac–Lambda framework, theta not equal to zero is structurally inadmissible. The result is unconditional inside the framework and does not require axions, Peccei–Quinn symmetry, or additional dynamical assumptions. The CKM CP-violating phase remains fully compatible, as it enters through a real fermionic action and preserves Euclidean measure positivity. This work situates the strong CP problem within a broader structural framework in which Euclidean positivity, spectral geometry, operator-algebraic conditional expectations, and Osterwalder–Schrader reconstruction jointly constrain admissible parameters.