The Heat Equation as a Resolution Geometry Theorem: Deriving Diffusion as the Minimal Exchange-Consistent Smoothing Operator Under Finite Capacity

This companion paper demonstrates that the heat equation is a geometric inevitability: the unique second-order smoothing dynamics permitted by identity preservation under finite distinguishability capacity.

In Resolution Geometry terms, a scalar field u(x,t)—whether temperature, concentration, probability density, or feature activation—is a ledger of distinguishable states distributed across a scaffold. The system's fundamental pressure is to eliminate gradient tension (costly local distinguishability) while conserving identity content. The minimal energetic principle that achieves this is membrane tension minimization (Dirichlet energy). The resulting evolution is the constrained gradient flow of that energy, and the variational derivative produces the Laplacian.

Thus the heat equation, ∂u/∂t = D∇²u, emerges as the lowest-order exchange-consistent dynamics that (i) reduces gradient cost, (ii) preserves total identity content, and (iii) respects finite capacity. Higher-order smoothing would require additional structure and violates minimal overflow; first-order evolution cannot dissipate tension. The heat equation is therefore the canonical 'resolution relaxation' law—appearing across physics, finance, biology, and machine learning because it is geometry, not domain.