Emergent Mathematics from a Scalar Coherence Field

This paper introduces and proves a unifying hypothesis: that core structures of mathematics — including constants like π and e, arithmetic, logic, irrationality, and prime numbers — emerge as invariant patterns from a physically grounded scalar coherence field. The framework is based on a nonlinear partial differential equation derived from a variational principle, modeling curvature minimization and nonlinear resonance. We show that classical mathematical results arise deterministically as solutions to this equation under symmetry and boundary constraints. Each result is supported by analytic proof, falsifiable observables, and numerical certification. Moreover, we demonstrate that the same field equation emerges as a macroscopic limit of real physical systems including nonlinear optics, Bose–Einstein condensates, reaction–diffusion systems, and elastic media. This work replaces axiomatic assumptions with structural invariants, offering a physically derivable and testable foundation for mathematics.