Zeta-Minimizer Theorem: Variational Emergence of Primes, Zeta, and Stratified Geometries from Helical Optimization in Measure Spaces

The Zeta-Minimizer Theorem establishes a variational foundation for the Riemann zeta function by minimizing a phase functional derived from the compressibility factor. Starting from the classical virial expansion, the theorem performs an exact exponential resummation that yields the Euler product form of ζ(s) over a finite helical basis. In a symmetric measure space equipped with non-proper Archimedean conical helices, four geometric constraints—rational signed cosines, positive integer representation dimensions, non-zero integer differences, and prime-modulated exponential decays—force primes to emerge as indivisible cycles in the representation graph, via Hilbert’s irreducibility theorem and Maschke’s theorem. Corollaries include the deductive proof of the Riemann Hypothesis (non-trivial zeros spectrally centered on Re⁡(s)=1/2), stacked phases as stratified orbifolds, emergent layered geometries, bounded prime descent, and dimensional resistance. The three axioms abstract thermodynamic equilibrium conditions purely: strict concavity of entropy on measures, non-vanishing spectral Gibbs minima, and covariance with flux conservation. Number-theoretic structures, complex numbers, polynomials, and quantization itself appear as projected artifacts of the underlying variational optimization. Applications range from atomic stratification (quantized shells arising from phase jumps) and angular-momentum tensors to the fine-structure constant (emergent from cycle sums with β=5 leaps) and covariant mappings to arbitrary conjugate variables via category-theoretic functors and renormalization-group universality. By demoting elementary mathematical constructs to derived descriptions of thermodynamic optimization on the helical manifold, ZMT provides a unified deductive framework for analytic number theory, algebraic geometry, and spectral theory.