Gravitational Phase Transitions: The Crystallization of Time at the Event Horizon

Part II of the Tri-Parametric Information Synthesis (TPIS) Framework. Sequel to: Tri-Parametric Information Synthesis (TPIS): A Unification of MOND, the Holographic Principle, and Entropic Gravity through Information Resolution. Abstract: Current cosmological models struggle to provide a first-principles derivation for the observed tight correlation between the mass of supermassive black holes and the velocity dispersion of their host galaxies (the M-sigma relation). Furthermore, the classical description of black holes necessitates a singularity, which violates the principles of discrete information storage. Building upon the Tri-Parametric Information Synthesis (TPIS), which successfully derived the galactic acceleration scale a_th ≈ cH0/2π from entropic principles, this paper extends the framework to the regime of strong gravity. We propose that the event horizon represents a saturation of information density, governed by the same critical acceleration threshold a_th that dictates galactic dynamics. By equating the entropic acceleration at the black hole's sphere of influence to the cosmic information limit, we analytically derive the M ∝ σ^4 scaling relation without invoking feedback mechanisms. Additionally, we argue that the quantization of spacetime at the horizon prevents the formation of a singularity, suggesting black holes are stable, maximum-entropy states of spacetime storage. This "Crystallization" mechanism provides a physical basis for the "Firewall" phenomenon and resolves the Information Paradox. The model is empirically validated against the Magorrian relation (deriving the mass ratio ξ^2 ≈ 0.16%) and offers a precise, falsifiable prediction for Gravitational Wave Echoes (e.g., Δt ≈ 0.273s for GW150914), which aligns with tentative detections in LIGO data.