The Constraint-Relaxation Energy Model: Recovering Mass-Energy Equivalence, Thermodynamic Laws, and the Bekenstein Bound as Limiting Cases of Constraint Geometry

Abstract The Constraint-Relaxation Energy Model (CREM) proposes that energy is not a fundamental quantity but an emergent consequence of imposing and relaxing constraints on an underlying phase space of possible states. Prior work established the core formalism — stored energy E_s proportional to degrees-of-freedom reduction, released energy E_r proportional to the time-derivative of induced relational geometry, and the Constraint Density Limit as an upper bound on energy storage density. The present paper completes the foundational CREM program by demonstrating formal consistency with three cornerstone results of established physics: (1) mass-energy equivalence (E=mc²), recovered by identifying mass with integrated constraint density and c² with maximum constraint propagation rate at the Reflective Interface; (2) the laws of thermodynamics, recovered by identifying entropy with constraint-space volume and thermodynamic processes with constraint-geometry evolution; and (3) the Bekenstein-Hawking entropy bound, recovered by identifying holographic information limits with maximum constraint density at spatial boundaries. These recoveries do not constitute derivations of physical constants from CREM alone — c is imported from special relativity, and G from general relativity, as boundary conditions. What is demonstrated is formal consistency and subsumption: that CREM provides the deeper structural account from which these results emerge as special cases. Implications for the Bidirectional Constraint Closure (BCC) framework, the Constraint Density Limit, and future empirical directions are discussed. Keywords: constraint-relaxation energy model, mass-energy equivalence, thermodynamics, Bekenstein bound, degrees of freedom, phase space, constraint density, Bidirectional Constraint Closure, UCST, emergent energy