We develop a radical inversion of the traditional relationship between logic and physics: rather than assuming logic as a pre-given foundation upon which physics is built, we derive a canonical logical semantics as emergent from a physical cost functional. The canonical reciprocal cost J(x) = 1/2 (x + x^(-1)) - 1 is uniquely determined by normalization, a composition law, and calibration—it is not chosen but forced. We prove three fundamental identifications: Logical consistency is a low-energy (zero-defect) state of a physical system. Proof is a path of zero-cost transitions (geodesic) from premises to conclusion. Mathematical existence is physical stability: defect(x) = 0 if and only if x = 1. Contradictions are expensive (no stable witness); consistency is cheap (a stable witness exists). The cost landscape is the logical landscape. This framework resolves the question "Why is reality logical?" by showing that classical logical structure arises as the induced Boolean semantics on the unique stable minimizer. The results are machine-verified in Lean 4.