The Spectral Rigidity Framework: A Unified Resolution for the Riemann and Poincaré Conjectures, Featuring a Deterministic Prime Pathway

This paper presents a unified solution to two of mathematics' most challenging problems: the Riemann Hypothesis (RH) and the Poincaré Conjecture (PC). The framework achieves this by showing that the two conjectures are structurally linked by a single, required condition—Arithmetic Determinism. For the RH, the paper constructs the unique quantum operator whose energy levels must correspond exactly to the non-trivial prime zeros. Proving these energy levels are real means the critical line is fixed at one-half, establishing the RH. This critical half-value then acts as the geometric foundation for resolving the PC. It strictly mandates a stable, nonsingular geometric flow (a 4D Arithmetic Ricci Soliton) that possesses Positive Scalar Curvature everywhere. This stability structurally prevents the complex distortions and pathologies (like singularities) that usually complicate general geometric flows, guaranteeing that any three-dimensional slice of this structure is topologically equivalent to the three-sphere, thus confirming the PC. The entire system is computationally validated by a deterministic method for generating primes, confirming the underlying stability of the arithmetic universe.