A Geometric Conditional Proof of the Riemann Hypothesis via the Infinite Helix Hamiltonian

We present a conditional geometric proof of the Riemann Hypothesis (RH) based on the infinite helix Hamiltonian constructed within the Geometry-Flow Theory (GFT) framework. The proof is conditional on two hypotheses: (1) the convergence of finite helix Hamiltonians HN, K to an infinite-limit operator H∞ in the strong resolvent sense, and (2) the one-particle spectrum of H∞ being precisely log p: p prime. Under these assumptions, the bosonic partition function of H∞ equals the Riemann zeta function ζ (s) for Re s > 1, with a prime harmonic trace formula derived from the Euler product. We further leverage the helicoidal-catenoidal duality and stacked Wick rotation from GFT to establish the functional equation of ζ (s), thereby implying that all nontrivial zeros lie on the critical line Re s = 1/2. This provides a spectral-geometric interpretation aligned with the Hilbert-Pólya conjecture, where the geometry enforces the required self-adjointness and positivity. Numerical evidence from finite truncations supports the hypotheses. This work unifies number-theoretic structures with physical geometry, suggesting RH as a consequence of topological quantization on minimal surfaces.