Paper T5: Audit-Grade Noether Validation for Triadic Phase Dynamics

Paper T5 delivers an audit-grade Noether validation lane for a triadic, real-valued phase-field model on a graph. The core objective is governance: demonstrate that conserved quantities are conserved when symmetry-breaking terms are disabled, and that any departures are quantitatively attributable when damping and/or external torque are explicitly introduced. We operationalize the Noether charge J(t)J(t)J(t), its quadratic Casimir C2(t)=∥J(t)∥2C_2(t)=\|J(t)\|^2C2(t)=∥J(t)∥2, and a program-reported energy proxy E(t)E(t)E(t) as decision-grade KPIs, with pass/fail gates defined a priori. A three-run control ladder is presented: (1) a conservative control showing invariant preservation within numerical tolerance; (2) a damped run with a rectangular torque pulse showing controlled, model-attributed drift with low residual; and (3) a damped run with rotating-axis sinusoidal forcing demonstrating stable accounting with near machine-precision residual closure. The paper also defines a loop non-closure diagnostic for defect/closure studies and scopes all gauge-language to code-checkable invariances rather than a continuum gauge-theory derivation. Collectively, T5 establishes a reproducibility contract (manifests, deterministic artifacts, fixed filenames) and a validated baseline for scaling, heterogeneity, and future upgrades.