Principal Dynamics: Admissible Gated Gradient Flows with Interaction-Induced Geometry and Projection

This paper develops a class of admissible gated gradient flows on interaction-induced configuration spaces within a domain-neutral axiomatic framework called Principal Dynamics. Five axioms fix the structural commitments; explicit realization-layer hypotheses supply the regularity required for theorem-bearing analysis. An interaction kernel induces a metric-measure structure and, under smooth-upgrade hypotheses, a coherence manifold equipped with a Fisher information metric, so that geometry is derived from interaction data rather than postulated. A five-term free energy generates the Coherence Transformation Equation (CTE) as a gated gradient flow. The gate — a state-dependent mobility modulator tied to a disorder threshold — enforces admissibility directly within the variational structure. The paper proves a six-theorem analytic chain: variational derivation of the CTE, hard-gate forward invariance, soft-gate overshoot suppression, local and global well-posedness via semilinear parabolic theory, and existence of a compact global attractor under asymptotic compactness. The gated gradient-flow class admits exact correspondences with Allen–Cahn, time-dependent Ginzburg–Landau, gated reaction–diffusion, and Cahn–Hilliard-type equations. Observables are defined by projection from configuration-level dynamics. Additional foundation layers covering symbolic structure, homological closure, regime classification, and falsification are recorded in compact theorem form. The paper is self-contained and does not claim domain-specific closure or empirical adequacy. 48 pages, 52 theorem-like items (all proved or labeled as standard imported results), 6 figures, 28 references. MSC 2020 codes (put in additional notes or description): 35K58 (primary); 35B41, 37L30, 58J35, 53C21 (secondary) Self-contained flagship foundation paper for Principal Dynamics. Develops the analytic/geometric realization layer in full proof-bearing detail; records the broader foundation shell in compact theorem form. MSC 2020: 35K58 (primary); 35B41, 37L30, 58J35, 53C21 (secondary). LaTeX source included.