Entropy Has Phases: A Constraint-Geometry View of Black-Hole Evaporation

This paper presents a geometric constraint framework for analyzing entropy partitioning during black hole evaporation. Rather than proposing a microscopic quantum gravity mechanism, the approach derives kinematic constraints that any unitary evaporation process must satisfy. By partitioning total entropy into scalar (thermal, structureless) and vector (geometric, holographic) components under the normalization condition a+b=1, all allowed configurations are shown to lie on a concave constraint manifold in mediator–asymmetry space. Within this framework, the mediator h=\sqrt{ab} provides a symmetric, bounded measure of cross-channel coherence. A distinguished efficiency extremum at asymmetry \chi=1/e corresponds to a 90% Shannon-efficient partition (a,b)=(0.316,0.684). The semiclassical Hawking estimate (0.27,0.73) is shown to lie on the same manifold, deviating strongly in asymmetry and response ratio but by only 4.5% in mediator value, indicating substantial preservation of mediation capacity despite increasing asymmetry. The paper proposes an interpretive picture in which entropy exhibits dimensional phases—tensor (correlated), vector (holographic), and scalar (thermal)—with black hole evaporation corresponding to a progressive dimensional reduction. Within this view, thermal Hawking radiation represents scalar-phase entropy, while correlations are not destroyed but integrated into a structureless constant. The framework yields falsifiable predictions, including a possible correction to the terminal entropy partition, phase-dependent deviations from strict thermality, and residual correlations at the final evaporation stage. The results are explicitly kinematic and model-independent, offering a geometric scaffold that complements, but does not replace, dynamical quantum gravity approaches to the black hole information problem. Any system subject to constraint necessarily generates a directional response; when the constraint is relational coexistence itself, that response is gravitational.