The Fractal Heterogeneity Coefficient α as a Universal Phase-Dynamic Invariant: First-Principles Derivation, Cross-Domain Universality Theorem, and Application to Climate-Geodynamic Coupling

The CL5D Hybrid Model (Papers 1–8, Chakraborty 2026) has been applied across six physical domains. In each, the Fractal Heterogeneity Coefficient α has appeared as the central parameter governing phase classification and Benchmark Score topology, defined operationally as a normalised standard deviation. This paper closes that gap by proving two theorems and one proposition. Theorem 1 establishes that α = H[φ]/H_max = −Σ p_i ln p_i / ln N — the normalised Shannon entropy of the system's micro-event phase-space distribution — and that the empirical α of Papers 1–8 is its large-N Gaussian limit, with residuals below 0.3% across all 57 SEISGLOB1 depth levels. Theorem 2 (CL5D Universality) states that any bounded dynamical system with well-defined first four moments admits a unique CL5D Benchmark Score BS ∈ [0, 1], and that BS is invariant under bijective transformation of the observable. The cross-domain numerical consistency of Papers 7 and 8 (isolated stellar collapse black hole at BS = 0.889 vs. Earth's lower mantle at BS = 0.868 ± 0.023) is a corollary. Proposition 3 (D″–Climate Coupling) demonstrates that D″-layer Phase III excursions (α > 0.30) predict anomalous volcanic CO2 outgassing with lag Δt ≈ 1–5 Myr, consistent with the Siberian Traps (252 Ma), CAMP (201 Ma), Karoo (183 Ma), Deccan Traps (66 Ma), and NAIP (56 Ma) large igneous province events. Current α = 0.247 predicts no anomalous outgassing for the present epoch. The CMB anisotropy field (α ≈ 0.08, BS ≈ 0.921) is identified as the Phase I cosmological origin point of the universe, providing CL5D with a timeline spanning from t → 0⁺ to the present.