CPT--Coherence Theory: A $\tau$-Generated Spectral--Geometric Framework for Operator Selection, Phase Structure, and Emergent Dynamics

We formulate CPT-Coherence Theory as a spectral-geometric framework in which physical structure arises from a $\tau$-generated operator flow on a covariant Hilbert–geometric arena[cite: 8]. The theory is defined by three fundamental elements: A representation-independent operator substrate: Utilizing a separable Hilbert space where all physical statements are covariant under unitary transformations[cite: 10, 35, 38]. A strongly continuous $\tau$-semigroup: Generated by a dissipative-unitary operator that indexes the structural evolution of operator-geometry configurations[cite: 10, 42, 44]. A spectral stationarity condition: A selection principle that identifies dynamically viable and stable configurations[cite: 10, 47, 49]. Stable configurations correspond to fixed points of a Lyapunov-monotone coherence functional[cite: 11, 52]. Near these attractors, the operator algebra exhibits finite-rank spectral dominance, where high-frequency modes decouple and low-lying eigenbranches control effective dynamics[cite: 12, 56]. This mechanism induces phase boundaries manifesting as mass hierarchies, symmetry restoration, geometric curvature dynamics, and dissipative collapse regimes[cite: 12, 57]. By restricting the geometric sector to Lorentzian metrics and Laplace-type operators, we demonstrate that low-energy stationarity of the spectral functional reproduces Einstein dynamics in the semiclassical limit[cite: 13, 60]. Independently, spectral filtering and leakage mechanisms generate hierarchical flavor structure—such as the CKM and PMNS matrices—without introducing ad hoc parameters at the structural level[cite: 14, 61, 700]. The theory does not quantize gravity nor modify quantum mechanics[cite: 15]. Instead, it replaces external sourcing and collapse postulates with a unified spectral–geometric selection principle, where known physical regimes arise as controlled limits of $\tau$-generated structural flow[cite: 16, 17, 64].