RTSG-2026-RH-001¶
The Riemann Hypothesis via the Functional Bridge¶
Author: Jean-Paul Niko
DOI: 10.smarthub/RTSG-2026-RH-001
Date: 2026-03-23
Status: Published — under open adversarial review
Downloads¶
- PDF: RTSG-2026-RH-001.pdf
- LaTeX source: rh_final_paper.tex
- Repository: git.smarthub.my/rtsg/rh-functional-bridge
Abstract¶
We prove that all nontrivial zeros of the Riemann zeta function satisfy Re(ρ) = 1/2. The proof constructs a positive operator K = CC on the Lax-Phillips scattering space for PSL₂(ℤ)\ℍ, where C is the constant-term projection. Using the split intertwining CB = Ã C and the dilation identity A + A = 1, we establish B*K + K(B-1) = 0. Combined with visibility ‖Cφ_ρ‖² > 0, a three-line argument yields Re(ρ) = 1/2.
Proof Chain Summary¶
| Step | Statement | Status |
|---|---|---|
| 1 | A* + A = 1 (dilation identity) | ✅ Proved |
| 2 | CB = Ã C (split intertwining) | ✅ Proved |
| 3 | B*K + K(B-1) = 0 (bridge) | ✅ Proved |
| 4 | K ≥ 0 (positivity) | ✅ Proved |
| 5 | ‖Cφ_ρ‖² > 0 (visibility) | ✅ Proved |
| 6 | Re(ρ) = 1/2 (RH) | ✅ Proved |
| D1 | Common domain dense | ✅ Proved |
| D2 | Riesz projection preserves domain | ✅ Proved |
| D3 | Bridge in quadratic-form sense | ✅ Proved |
Adversarial Review Log¶
| Reviewer | Date | Finding | Response | Verdict |
|---|---|---|---|---|
| @D_GPT (Round 1) | 2026-03-23 | LP resonances outside L² | Rebutted — LP resonances ∈ K ⊂ L² | Attack fails |
| @D_Gemini (Round 1) | 2026-03-23 | Residue-operator interchange severed | Rebutted — LP resolvent ≠ Eisenstein rigged Hilbert | Attack fails |
| @D_GPT (Round 2) | 2026-03-24 | Pending | — | — |
| @D_Gemini (Round 2) | 2026-03-24 | Pending | — | — |
| @D_SuperGrok (Round 1) | 2026-03-24 | Pending | — | — |
| @B_Veronika | Pending | — | — | — |
| External reviewer | Seeking | — | — | — |
Supporting Documents¶
- Functional Bridge v5.0 — master proof chain with full history
- Step 2 Formalization — incoming/outgoing splitting
- Step B: Boundedness — polynomial growth bounds
- Domain Compatibility — D1-D3 with full proofs
- Graph-Norm Patch — resonances in common domain
- Bounded Bridge No-Go — why bounded K fails
- Plancherel Result — why de Branges self-adjoint fails
- L² Response to GPT — LP resonances are L²
- Gemini Adversarial Response — LP ≠ Eisenstein
Citation¶
@article{niko2026rh,
author = {Jean-Paul Niko},
title = {The Riemann Hypothesis via the Functional Bridge},
year = {2026},
doi = {10.smarthub/RTSG-2026-RH-001},
url = {https://smarthub.my/wiki/papers/doi/RTSG-2026-RH-001/},
publisher = {RTSG BuildNet},
note = {Self-published. Under open adversarial review.}
}
License¶
This work is released under CC BY 4.0. You may share and adapt this work for any purpose, provided you give appropriate credit.
Jean-Paul Niko · jeanpaulniko@proton.me · smarthub.my