Quantum Lattice Model: Planck Energy Density Cap and Minimal Saturated-Core Completion

This project presents a self-contained derivation and application, within the Quantum Lattice Model (QLM), of a fundamental constraint implied by action-limited lattice dynamics: a universal, local Planck energy density cap. In QLM, physics is formulated as deterministic phase–action evolution on a discrete Planck-scale spacetime lattice, where each lattice four-volume transports exactly one quantum of reduced action per Planck tick. From this structure, a maximal local energy density follows directly, without invoking phenomenological cutoffs or additional physical assumptions. Treating this Planck energy density as a strict local constraint, the work demonstrates that unlimited gravitational compression is forbidden. Instead, gravitational collapse must terminate in a finite, fully saturated interior configuration. The resulting interior completion is a compact saturated core whose radius scales as the cube root of the total mass, while the exterior gravitational field remains exactly Schwarzschild. All results are expressed purely in terms of QLM primitives and standard Planck quantities, with no modification of exterior general relativity. The paper further explores the dynamical implications of this constraint near the Schwarzschild radius. Gravitational time dilation is interpreted as an impedance that throttles lattice action transport relative to an asymptotic observer. As the impedance diverges near the horizon, inward-propagating gravitational perturbations experience partial reflection, producing delayed ringdown echoes, while outward action leakage becomes intrinsically bandwidth-limited. This leads naturally to a late-time breakdown of semiclassical evaporation and a finite evaporation endpoint. Black-hole temperature and entropy are recovered with their standard scaling laws, but are reinterpreted in QLM as emergent measures of throttled and frozen lattice action capacity rather than thermodynamic properties of interior microstates. Numerical examples spanning terrestrial, stellar, and supermassive mass scales confirm all analytic scaling relations. A detailed appendix provides a lattice-level derivation of gravitational radiation as propagating impedance perturbations, formulated in a Maxwell-like first-order system. Together, these results show that singular collapse, horizon-scale echoes, and evaporation endpoints arise as direct consequences of action-limited lattice dynamics, without introducing exotic matter, new degrees of freedom, or modifications to the exterior Schwarzschild solution.