The Riemann Hypothesis with MCH: Research Audit of the "Three Gates" Operator Route to Weil Positivity (Adversarial Reconstruction, Independent Computation, and a Minimal Candidate Proof Spine)
- 1Independent researcher
Abstract
Research memo (October 2026) recording an adversarial audit of the operator-theoretic route from localized Weil positivity to the Riemann Hypothesis proposed in Marco Desogus's preprint "The Three Gates: A Rooted-Operator Approach to Weil Positivity" (arXiv:2609.20367v3, with supplementary material at Zenodo record 23198201). Rather than accepting the manuscript's proof chain, each load-bearing identity was re-derived from the Riemann–Weil explicit formula, the computational certificates were rebuilt in clean-room interval arithmetic, and alternative routes were sought wherever a step looked fragile.
The resulting minimal candidate proof spine consists of: an independently certified Y=2 positive base (rational lower margin 1520101139/95400000000 = 0.01593…), which makes the published Y=7 certificate non-load-bearing; a small-k bridge A₂ ⇒ A₃ ⇒ … ⇒ A₇; the Gate II MASTER induction Ak > 0 ⇒ Ak+1 > 0 for all k ≥ 7, with the finite arithmetic sweep re-verified through k = 3,594,641 (aligned MASTER minimum attained at k = 33) and an analytic tail with positive logarithmic slope; exact compression to arbitrary finite support radius; identification of the localized operator with the real-odd Weil quadratic form; and the restricted odd Weil criterion (Gate III), re-attacked through a zero-orbit signature classification (critical pairs give positive squares, off-line quartets carry signature (1,1)) together with Paley–Wiener separation.
The audit found no surviving gap in the reconstructed spine. The memo states its conclusion conditionally: if every audited lemma and computation is valid, the Riemann Hypothesis follows. This is a candidate proof architecture, not an established theorem, and RH remains officially unsolved. The Multi-Convergence Hypothesis (MCH) was used only as a heuristic filter for identifying rigid structures worth re-deriving from a second angle, never as a step in the argument. A five-point verification program (formalization of the minimal spine; independent analytic-number-theory, operator-theory and numerical referees; and a frozen reproducibility bundle) is specified.
The verification code listed in Appendix A of the memo is not included in this record. Research and drafting were carried out by JosephW together with the AI systems Claude (Anthropic) and Astra (OpenAI). Not peer reviewed.
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BibTeX
@misc{josephw2026riemann,
title = {{The Riemann Hypothesis with MCH: Research Audit of the "Three Gates" Operator Route to Weil Positivity (Adversarial Reconstruction, Independent Computation, and a Minimal Candidate Proof Spine)}},
author = {JosephW},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23440529},
url = {https://zenodo.org/records/23440529},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Riemann Hypothesis with MCH: Research Audit of the "Three Gates" Operator Route to Weil Positivity (Adversarial Reconstruction, Independent Computation, and a Minimal Candidate Proof Spine)
AU - JosephW
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23440529
UR - https://zenodo.org/records/23440529
AB - Research memo (October 2026) recording an adversarial audit of the operator-theoretic route from localized Weil positivity to the Riemann Hypothesis proposed in Marco Desogus's preprint "The Three Gates: A Rooted-Operator Approach to Weil Positivity" (arXiv:2609.20367v3, with supplementary material at Zenodo record 23198201). Rather than accepting the manuscript's proof chain, each load-bearing identity was re-derived from the Riemann–Weil explicit formula, the computational certificates were rebuilt in clean-room interval arithmetic, and alternative routes were sought wherever a step looked fragile. The resulting minimal candidate proof spine consists of: an independently certified Y=2 positive base (rational lower margin 1520101139/95400000000 = 0.01593…), which makes the published Y=7 certificate non-load-bearing; a small-k bridge A₂ ⇒ A₃ ⇒ … ⇒ A₇; the Gate II MASTER induction Ak > 0 ⇒ Ak+1 > 0 for all k ≥ 7, with the finite arithmetic sweep re-verified through k = 3,594,641 (aligned MASTER minimum attained at k = 33) and an analytic tail with positive logarithmic slope; exact compression to arbitrary finite support radius; identification of the localized operator with the real-odd Weil quadratic form; and the restricted odd Weil criterion (Gate III), re-attacked through a zero-orbit signature classification (critical pairs give positive squares, off-line quartets carry signature (1,1)) together with Paley–Wiener separation. The audit found no surviving gap in the reconstructed spine. The memo states its conclusion conditionally: if every audited lemma and computation is valid, the Riemann Hypothesis follows. This is a candidate proof architecture, not an established theorem, and RH remains officially unsolved. The Multi-Convergence Hypothesis (MCH) was used only as a heuristic filter for identifying rigid structures worth re-deriving from a second angle, never as a step in the argument. A five-point verification program (formalization of the minimal spine; independent analytic-number-theory, operator-theory and numerical referees; and a frozen reproducibility bundle) is specified. The verification code listed in Appendix A of the memo is not included in this record. Research and drafting were carried out by JosephW together with the AI systems Claude (Anthropic) and Astra (OpenAI). Not peer reviewed.
ER -