A Unified Equipartition Framework for Holographic Information Bounds under Planck-Scale Modified Dispersion

Heuristic attempts to incorporate Planck-scale Modified Dispersion Relations (MDR) into holo-graphic information quotients frequently introduce severe derivational pathologies, includingtautological variable cancellation, contradictory mass assumptions, and catastrophic temper-ature divergences during late-stage black hole evaporation. In this paper, we formulate anobserver-independent, dimensionally homogeneous framework for boundary information ca-pacity governed by a differential thermodynamic equipartition relation: dI = dE_total/ε_mode. Byestablishing a robust effective field theory model for quantum-gravitational mode energy suppression near the causal horizon, we replace algebraic piecewise quotients with a continuousthermodynamic area integration. For macroscopic black holes, framing the fundamental bit costas a generalized Landauer erasure cost at the invariant causal boundary harmonizes directlywith the Bekenstein-Hawking area law, naturally preserving the exact 1/4 coefficient upon un-perturbed calibration. Integrating the regularized differential framework analytically generatesa rigorous, positive logarithmic area correction (+∆I ∝ ln A) resulting from kinematic phase-space deformation, distinguishing it from standard negative statistical corrections. Furthermore,evaluating the heat capacity proves that the black hole undergoes a continuous Davies-typephase transition, switching from a thermodynamically unstable negative heat capacity (CM < 0)at macroscopic scales to a strictly stable positive heat capacity (CM > 0) below a critical massthreshold. Crucially, the entropy derivative possesses an asymptotic pole ( dS/dx → ∞) at the criticalarea limit, driving the underlying semiclassical Hawking temperature vector field toward anabsolute zero attractor (T → 0) and stabilizing the physical system at a finite discrete remnantcutoff. This ensures complete thermodynamic stability, predicts a cold Planck-mass remnantholding a non-vanishing residual microstate capacity, and provides a geometrically regularizedframework for tracking information recovery without infinite state degeneracy.