EET Mathematical Foundations: Dimensional Analysis, Scaling Laws and Cross-Document Invariants

This document establishes the complete mathematical foundations of Energy-Efficiency Theory (EET), fully aligned with Core Rules v5.2 and the companion ontologies. It provides the dimensional analysis of all core parameters across the mother texts, formalizes the capacity geometry and unified action quantum, develops the graph-theoretic mathematics (Laplacian, spectral gap, Ollivier-Ricci curvature, spectral embedding), derives the hierarchical scaling laws from the universal exponent $\alpha \approx 0.6$, and presents the response pool dynamics and variational principle. Version 2.3 integrates all variables from the cognitive constitution—including consciousness indices ($I_{\mathrm{causal}}$, $I_{\mathrm{threshold}}$), spirit indices ($M$, $d$, $\kappa(t)$), cross-level dynamics ($\lambda_l$), and cognitive inverse entropy ($\dot{S}_{\mathrm{inv}}^{\mathrm{cog}}$)—with full dimensional verification and bridge declarations. The core equations quick reference now includes the complete coupled dynamics of the consciousness-spirit cycle.