Book V — Pending Derivations (Radial Coherential Dynamics)

Book V — Pending Derivations continues the Radial Coherential Dynamics (RCD) research program by addressing the explicit technical gaps identified in Book IV. The volume focuses on three unresolved elements:(i) the formal construction of an effective entropy functional S_eff[C] for the coherence field,(ii) its energetic interpretation via Landauer’s principle, and(iii) the coupling of coherence entropy to spacetime geometry through Jacobson’s thermodynamic derivation of gravity. Chapter I derives a minimal, thermodynamically compatible ansatz for S_eff[C] based on coarse-graining and non-negotiable physical constraints (monotonicity, extensivity, and Landauer compatibility).Chapter II connects this functional to the minimum energetic cost of persistence, showing how state geometry determines the thermodynamic “price” of maintaining coherence.Chapter III extends Jacobson’s framework by introducing coherence-dependent corrections Δ_{μν}[C] to Einstein’s equations, explicitly recovering standard gravity in the classical limit.Chapter IV formalizes the transition to the N3 regime using non-Markovian dynamics and a memory kernel K(t,s), providing an operational criterion for structural persistence: τ_memory ≫ τ_noise. No claim of completeness is made. Several parameters remain free, and no new experimental predictions are asserted. Instead, the book establishes explicit mathematical structure where only qualitative compatibility existed before, clearly separating what is derived, what is proposed, and what remains open. Book V is intended as a technical bridge between empirical coherence persistence (Book III–IV) and future work on geometry, gravity, and living systems. It emphasizes methodological honesty: technical debts are acknowledged and paid incrementally, without overclaim.