The Last Lemma: What Proving the Riemann Hypothesis Concretely Requires

Identifies the single concrete mathematical statement whose proof would establish the Riemann Hypothesis: Connes’s Corollary 3.8, which states that if the spectral discrepancy μ_λ of the prolate spheroidal operator converges to zero as λ → ∞, then RH holds. Traces the three layers of Connes’s program (spectral realization 1998–2021, UV spectral matching 2022, finite line-lock 2025–2026) and explains why the last lemma is hard (Hurwitz obstacle, spectral gap issue, finite-to-infinite transition). Outlines three paths to proof: uniform convergence, operator convergence, and Weil positivity extension. Validates the σ-Instability Principle’s prediction against Epstein zeta functions — which satisfy the functional equation but lack the Euler product and violate RH — confirming that the Euler product (σ = 0 source structure) is not auxiliary but the structural reason RH holds.