Formal Mathematical Structure of the Causal Chain Axioms

I present the complete formal mathematical structure underlying the Causal Rate Invariance (CRI) framework. Starting from one equation---$T = \sum_i (d_i/C + \tau_i)$---and five structural axioms (C0--C5), we derive: complex-valued amplitudes, exponential propagation phases, the processing amplitude with dissipation, the full path integral measure, gauge invariance, anomaly cancellation, the Born rule via Gleason's theorem, the mass gap for non-Abelian Yang--Mills theories, the Einstein field equations with derived cosmological constant $\Lambda = 2H^2/C^2$, the equivalence principle, the geodesic equation, the spin-statistics theorem, the Pauli exclusion principle, and Bose-Einstein enhancement. Each derivation is traced to specific axioms.