The Tri-Scale Synthesis: Deriving Macroscopic Cosmological Handover Protocols from Local Quantum Relative Entropy via Kodama–Hayward Horizon Integration

This paper establishes the first-principles mathematical derivation for the Tri-Scale Transition cosmology by bridging local algebraic quantum field theory (AQFT) with global horizon thermodynamics. By mapping local quantum relative entropy across expanding spacetimes using Hideo Kodama’s canonical conserved vector field and integrating via Sean Hayward’s Unified First Law, we provide the underlying microscopic engine for macroscopic cosmological handover protocols. Key Breakthroughs: Microscopic Semiclassical Foundation: Successfully bridges Dorau & Much's (2026) local Rindler horizon relative entropy to global Friedmann-Lemaître-Robertson-Walker (FLRW) dynamics within the adiabatic limit, proving that macroscopic boundary impedance laws emerge directly from quantum informational distinguishability. Vacuum Catastrophe Resolution: Executing a definite integral between the Planck scale and the apparent causal horizon analytically proves an exact volumetric (a^-3) dilution of primordial vacuum energy by a factor of ~10^-183, organically crushing the 120-order-of-magnitude discrepancy without arbitrary UV cutoffs or fine-tuning. Emergent Friedmann Holography: Rigorously derives Friedmann’s holographic area scaling (H² ∝ ρₕolo) as an emergent boundary property. The resulting 2ρcrit boundary density is shown to be the direct mathematical manifestation of Padmanabhan’s holographic equipartition theorem for dynamic spacetimes. Thermodynamic Impedance Matching: Proves that the transitional Hubble rate (Hₜrans = √ (Hₑarly · Hₗate) ≈ 70. 2 km/s/Mpc) is the mandatory algebraic equilibrium demanded by the Maximum Entropy Production Principle (MEPP) when the open-system channel impedance ratio reaches unity (Z = 1) at redshift z ≈ 1. 5. Global Energy Conservation: Demonstrates that step-function Hubble transitions rigorously obey contracted Bianchi identities and local continuity equations, as macroscopic energy differentials are identically balanced by boundary heat flux and gravitational work across the Kodama horizon. Strict Empirical Falsifiability: Reaffirms the framework's testability against unfalsifiable dark-sector parameterizations, offering concrete laboratory targets—most notably a spontaneous gravitational decoherence threshold for isolated 50 µg silica or diamond spheres in optomechanical levitation traps. By synthesizing local quantum operator algebras with dynamic spacetime geometry, this work proves that the Hubble tension and vacuum catastrophe are not observational errors or dark sector particles, but the verifiable thermodynamic signatures of an expanding information-theoretic universe.