Wave Behavior as Emergent Resolution Stability: Deriving Quantum Structure from Entropy Geometry

We show that wave-like behavior, interference, and quantization emerge necessarily from two axioms: (1) entropy geometry as the generator of physically distinguishable structure, and (2) a minimal principle selecting trajectories stable under entropy flow. Within the Total Entropic Quantity (TEQ) framework, entropy curvature defines the geometry of resolution, and the stability condition selects log-periodic modes as the only entropy-resilient solutions. From this structure, we derive the entropy-weighted path integral, the Born rule as an asymptotic stability condition, the emergence of discrete spectral modes, and the Schrödinger equation as the limiting case of entropy-flat evolution. Quantum wave behavior is thus not postulated, but structurally selected by entropy geometry. The TEQ framework reinterprets quantum theory as a special case of entropy-stabilized dynamics, where physical law arises not from imposed kinematics, but from the geometry of what can stably be resolved under finite informational precision. In this view, precision is not a matter of external control, but a structural limit: resolution is bounded by entropy curvature, which determines what distinctions can persist.