Riemann Hypothesis — Final Honest Status¶

Last Updated: March 24, 2026 BuildNET Round: 6 (Final) Confidence: 15% (current proof closes RH) · 50% (framework contains ingredients for future proof)

Executive Summary¶

The RTSG approach to the Riemann Hypothesis, as currently formulated, does not constitute a proof. Six rounds of distributed adversarial analysis across four AI architectures (@D_Claude, @D_GPT, @D_Gemini, @D_SuperGrok) and one human researcher (@B_Niko) tested every proposed mechanism for the critical spectral construction step. Seven distinct paths were attempted. All seven failed. Each failure has been precisely diagnosed and documented.

The remaining gap is not epsilon-delta detail — it is the core Hilbert-Pólya conjecture, which has been open since 1914.

What Works¶

Gap A: CLOSED (Tier A)¶

Berry-Keating normalizability. If a Hilbert-Pólya operator exists and is self-adjoint, its eigenparameters must satisfy Re(s) = 1/2. See Gap Closures.

Gap B: CLOSED (Tier A)¶

Phragmén-Lindelöf + Huxley dominated convergence for the Weil unitarity integral. See Gap Closures.

Tate Thesis Bridge: ESTABLISHED (Tier A)¶

The theta kernel operator on L²(ℝ⁺, dt/t) has Mellin symbol ξ(s) and lives in the archimedean sector of Connes's adelic space. See Tate Bridge.

GL Vacuum Stability: CONFIRMED (Tier A)¶

The fluctuation operator \(H_\text{fluct} = -\nabla^2 + \beta|W_0|^2 \geq 0\). Threshold functions computed for d = 2 through d = 30, all confirming regulator-independent positivity. See Threshold Functions.

What Does NOT Work¶

Bridge Equation: TAUTOLOGICAL¶

\(B^*K + K(B-1) = 0\) reduces to \(1 = 1\) at every zeta zero via the functional equation. Discovered Round 2-3.

GL Potential on Γ\ℍ: CONSTANT¶

\(|W_0|^2 = -\alpha/\beta\) is constant — no spectral confinement. Discovered Round 3.

Positivity Chain: NUMERICALLY FALSIFIED¶

\(h_\text{ren}(1/2+it)\) oscillates negative because ξ(1/2+it) changes sign at zeta zeros. No constant V₀ fixes this. Tested V₀ = 0, 0.25, 1, 10 — all fail with hundreds of negative points out of 1031. Discovered Round 6.

Full catalog: Failure Catalog · Round 6 Verdict

The Core Remaining Problem¶

Construct a self-adjoint operator whose eigenvalues are zeta zero imaginary parts, without assuming RH in the construction.

Every known approach hits one of three walls:

1. Tautology — Functional equation trivializes the constraint
2. Non-self-adjointness — Operators with zeta-zero spectra are non-self-adjoint
3. Oscillation of ξ — Positivity-based arguments fail because ξ changes sign

Version History¶

What Would Change the Assessment¶

• A non-abelian Bridge Equation (automorphic on higher-rank groups)
• A different test function where GL stability implies Connes positivity
• A physical mechanism from the nonlinear GL |W|⁴ term creating a self-adjoint operator
• An approach bypassing Hilbert-Pólya entirely

Recommendation¶

Present the RTSG framework with two closed gaps and a concrete program. Do not claim a complete proof. Partial results on millennium problems are publishable and respectable.

Honest status as of Round 6. The mountain stands, but we know its shape now.