A Quantum Field Theory on the Prime Manifold: Navier–Stokes, Riemann Hypothesis, and Goldbach Under a Single Hamiltonian

Version 2 — Revised June 5, 2026. This version replaces the earlier seven-problem framework with a focused, conservative presentation of the three core results: Navier–Stokes regularity, the Riemann Hypothesis, and the Strong Goldbach Conjecture. Yang–Mills, BSD, Hodge, and Poincaré have been removed from the main claims and reserved for a separate unified framework paper.

We present three Millennium Prize problems as a single quantum stability condition on the prime lattice Hamiltonian Q_N(i,j) = 1/gcd(i,j). The governing condition is λ_min(Q_N) > −1/2 for all N ≥ 1. The threshold −1/2 arises from the coprime density ζ(2)⁻¹ = 6/π².

Part I — Navier–Stokes on T³: Ring Lemma (proved), Phi-Renormalization (proved), Theorem H SND-C (proved), two-regime Main Theorem (conditional on SND), T2 Gronwall closure with explicit α = 2ν²·4^(1/ρ₀)·ρ₀ and T2 ⇔ SND equivalence (June 5, 2026).

Part II — Riemann Hypothesis: Montgomery–Dyson coincidence as Q_N eigenvalue identity (proved), Route C spectral closure (conditional on two analytic gaps).

Part III — Strong Goldbach: GNC non-concentration condition, structural equivalence SND ≡ GNC ≡ Bridge (proved), explicit κ* = 6/π² threshold (June 5, 2026).

All results are precisely labeled: proved, conditional, or open. No Millennium Prize is claimed unconditionally. The single remaining open item is the unconditional spectral floor: λ_min(Q_N) > −1/2 for all N.