A Coupled Dirac–Λ Dynamical System: Constrained Spectral Unification with Structural Consequences

This paper presents a constrained operator-dynamical system unifying gravity, gauge fields, and fermions within a spectral framework coupled to an independent irreversible entropy channel. The carrier of the theory is a Dirac-type operator on a compact Euclidean spin 4-manifold, generating the standard spectral action. Its low-energy expansion reproduces the Einstein–Hilbert term, Yang–Mills kinetic terms, and fermionic dynamics via the heat-kernel expansion. The novel ingredient is a scale-by-scale capacity inequality coupling the Dirac spectral sector to a positive selfadjoint dissipative generator. The coupling is enforced through a semi-infinite Karush–Kuhn–Tucker (KKT) variational framework with a nonnegative measure-valued multiplier. A single ultraviolet anchor fixes the unique normalization parameter, and all determinant prescriptions, filters, and regularization conventions are globally fixed and non-adjustable. Within this constrained system: • Fermion masses arise as structural outputs from finite active-scale saturation under a Jacobian nondegeneracy condition. • The Born rule exponent p = 2 is shown to be the unique value compatible with contractive saturation geometry and modular flow identification. • A quantitative measurement dissipation bound is derived, exhibiting 1/T decoherence suppression in the infrared regime. • The strong CP phase θ is structurally excluded within the Osterwalder–Schrader reconstructible Euclidean formulation; θ = 0 is the unique value consistent with CP-even spectral action and reflection positivity. The framework is falsified if:(i) the ultraviolet anchor is unsolvable,(ii) the capacity inequality fails on the observationally realized configuration class,(iii) the infrared admissibility bound is violated,(iv) the predicted fermion masses disagree with experiment,(v) decoherence fails to exhibit the derived large-scale suppression,or (vi) a nonzero neutron electric dipole moment implies θ ≠ 0. All spectral schemes are fixed globally and are not retuned per background. This work is formulated independently of broader law-selection considerations and stands as a concrete constrained dynamical proposal within spectral geometric physics.