Mathematical Extension of the Entropy-Decay Framework: Memory Kernel Derivation, Multi-System Generalisation, Structural Identity Imprinting, Sign-Regime Analysis, and Universal Equilibrium Point

This paper presents a self-contained mathematical extension of the entropy-decay curvature framework, building from the foundational Tsang equation through memory kernel derivation, multi-system effective kernel generalisation, and structural identity theory. The central contributions are: the identification of K_eff as the decay fingerprint of a physical subsystem — the curvature-memory record encoding its propagation character, transition susceptibility, and structural history; a complete combinatorial regime map of K_eff, dS/dτ, and d²S/dτ² covering all observable physical states from light and radiation through stellar burning, planetary formation, and black hole lifecycle; the formal definition of the Universal Equilibrium Point (dS/dτ = 0) as the sole universal trigger for structural transition, with the local K_eff value determining whether the outcome is planet formation, stellar ignition, black hole collapse, or a Big Bang class cascade; and a proposed two-dimensional structural correspondence pathway {τ_m/τ_P, Ψ_m/Ψ_ref} connecting framework coordinates to observable SI projections. The framework is applied to the steady-state cosmological picture of correlated local τ = 1 events, replacing dark matter, dark energy, and ghost particles with structural memory conditions already present in the regime map. The graphene transport application of prior work provides the primary empirical anchor. Two UK patent applications are identified as the first concrete experimental predictions of the framework. This document serves as a year-one mathematical milestone and public theoretical record of the entropy-decay framework.