Hunt #150, mechanism 1. Circularity challenger, 2026-10-26, branch hunt/rh-moonshot-challenge-circularity. This session did not build the construction and edited none of its files. It read NOTES.md (commit 77e245890 on hunt/rh-moonshot, merged here at c220f1310, and its m = 4 update bc1b23cbd, merged at 3fbe180af), ccm_common.py, measure.py and rigidity.py, and scored them against the rubric pre-registered in MISSION.md (section "Challenge-phase scoring rubric", 2026-10-26).
Files here: circ_ccm.py (reads the builder's committed data/measure_*.json, recomputes no matrix), test_ccm_arithmetic_challenge_circularity.py (3 tests), this page.
Grades as in MISSION.md. Nothing written here is kernel-checked.
Verdict
Unresolved, agreeing with the builder. No hidden use of RH, GRH or the zeros was found in the form. One grade is above its evidence, and it is the one that would matter most if the chain were pushed:
- C1 (positivity of zeta's form is band-level and float grade). NOTES section 3 ("lambda_1 > 0 (Rayleigh ball of the computed vector)"), the section 4 table ("lambda_1 > 0 on all 30 cells, c <= 60" under "form at the windows measured") and section 8 ("a positive Rayleigh ball") present positivity as carried by a ball. It is not, for two reasons the NOTES half-state (lines 15-18):
- A Rayleigh quotient of any vector is >= lambda_1, so a positive ball bounds the lowest eigenvalue from above only. It carries a negative value of the form when negative (the rivals' failures are sound), and carries nothing when positive.
- The basis e_0, e_k (k <= N) of NOTES section 2 is nested in N, so the band-N matrix is a principal submatrix of every larger band, and by Cauchy interlacing lambda_1(N) is an upper bound on the infimum of the window form. Checked on the builder's data: lambda_1 is nonincreasing in N on every window of zeta (10), Dedekind (13) and DH (14), with no exception. For zeta it is still falling steeply at the largest band: from N = 32 to N = 128 it drops by 9.8 orders at c = 13 and by 112.25 orders at c = 60 (
circ_ccm.py). No limit in N is measured.
So "lambda_1 > 0" is: the band-N lowest eigenvalue, located by a float eigensolver at 400 to 1600 bits, is positive (measured). Positivity of zeta's Weil form on the window log c is not established for any c here, and for c >= 13 it would exceed the windows on which the MISSION.md frontier records positivity at all. Simplicity (lambda_2 / lambda_1 >= 1.6e6) is band-level and measured for the same reason. The rivals' negative values are unaffected: a negative value at a stated vector of the band is a negative value of the window form.
- C2 (RH-equivalence risk, unresolved). The builder's own reading (NOTES section 5, first bullet) is that capture tracks positivity to within an exponentially small gap scale, so a proof of link (2) for every c may have to deliver all-window positivity, which is Weil's criterion. That would make the target a reformulation (kill condition 3). This session confirms the co-location in the data the NOTES report (DH loses capture between c = 31 and 33, its form sign change is at [30.617, 30.818]; Epstein at 59/2; W_{1/4} between 2 and 3) and adds nothing that decides it.
Scorecard
- Disposition. Unresolved: no step refuted for zeta on the grid; links (1) and (2) are unproved for any infinite family of c (NOTES section 7).
- Grade per step.
- Link (1), simplicity and evenness: measured at band level (C1); no Temple bound (NOTES section 5).
- Prop. 4.1 of Connes and van Suijlekom (real zeros of F_v for any real distribution under link (1)): cited, not re-read here.
- Link (2), capture and F_v -> Xi on compacts: measured on c <= 60, N <= 128; the relative error on Im z = 0.45 is at best 7.3e-2 and its decay is not established (NOTES section 5). The perturbative capture bound is vacuous on every cell at m = 2, and at m = 4 on c = 43, 50, 60 (NOTES section 3 as updated in
bc1b23cbd, read after merging it; its numbers were not recomputed here). - Link (3), Hurwitz: classical, but needs link (2) on a neighbourhood of |Im z| < 1/2, which is not measured to converge.
- Rival failures: DH and Epstein negative values ball-carried for stated vectors (hardened for "the form takes a negative value"), W_{1/4}'s sign change hardened by Arb LDL^T. Agreed.
- zeta's lambda_1 > 0: should read measured, band-level (C1), not ball-carried.
- Composite: measured, as the NOTES say (section 8).
- The Euler-product step. Named: link (2), capture surviving every window (NOTES section 4). Tested against DH, Epstein (1,1,6) and W_{1/4}, each of which loses capture at a located window; Dedekind (Euler product) keeps it to c = 60. Note on the control: reading Dedekind's success as "Euler product implies capture" treats GRH for L(s, chi_-23) as true at the heights these windows resolve; that is an expectation, not an input of the form, and this session did not check those zeros.
- Finite stand-ins. Three finite ranges, all stated as measured by the builder and none offered as uniform: windows c <= 60, bands N <= 128, zeros below height 100. The band range is the one that bites (C1): band-N positivity stands in, in the section 4 table's wording, for window positivity. The kappa law ("approach at rate O(t^2/c)") is a fit on finitely many cells and the NOTES call it measured.
- Novelty search. None recorded by the builder (NOTES section 8: "No novelty search was made"). This session's query, 2026-10-26, web search: "Connes Consani Moscovici zeta spectral triples truncated Weil quadratic form lowest eigenvector prolate zeros converge Riemann zeros arXiv 2511.22755": arXiv 2511.22755 itself was not returned; found the precursors Connes-Consani arXiv 2106.01715 (small eigenvalues of the Weil form on a bounded interval, eigenvectors via prolate functions, low spectrum tracking the first 31 zeros), Connes-Consani-Moscovici arXiv 2310.18423 (semilocal prolate operator), and an EMS chapter "Zeta spectral triples" stating that a rigorous proof of the observed convergence would establish RH. No published refutation of step 2 was found. The phenomenon measured here (ground-state zeros matching zeta's) is CCM's own, as the NOTES say. zbMATH, MathSciNet and Lean Zulip were not searched.
- Rivals on complete hypotheses. The chain has no complete proof, so no rival can refute an implication; the entries concern the distinguishing link.
- DH: fails link (2) between c = 31 and 33 (measured). Lacks the Euler product. Not a counterexample.
- Epstein (1,1,6): fails link (2) at c = 59/2 (measured). Not a counterexample.
- W_{1/4}: fails link (2) between c = 2 and 3, form negative at c = 2.39 (hardened). Lacks unitarity of local roots. Not a counterexample.
- zeta(s - delta): not run by the builder or by this session; excludes nothing. It would lack the s -> 1 - s functional equation about Re s = 1/2 on which the even-sector construction and the reality of Xi rest (derived from the definitions in NOTES section 2, not computed).
- Circularity. None in the form. Requirement A provenance, by hand:
matrix()(ccm_common.pylines 213-229) assembles archimedean, pole and prime blocks throughweil_trunc'sBallTruncationandweil_propagation'swp_common;wa_matrix(lines 171-204) reads Lambda(n)(n^a + n^-a) on prime powers <= c and digamma values; none callsxi_kind, a zero table or a xi-phase. Xi and the zeros enter only the diagnostics:radical_vectorandxi_kind(lines 243-288, 380-401), the zero listdata/zeta_zeros_T200.jsonandmp.zetazeroinmeasure.pylines 181-193 andrigidity.pyline 118. The target statement itself names Xi (convergence to Xi), which is legitimate: a proof must show closeness to Xi without reading its zeros, and nothing in the construction yet does that.
Run manifest
id: rh_moonshot-ccm_arithmetic-2026-10-26-challenge-circularity
hunt: rh_moonshot
started: 2026-10-26T13:35+00:00
finished: 2026-10-26T14:30+00:00
ran:
- .venv/bin/python hunts/rh_moonshot/ccm_arithmetic/challenge/circ_ccm.py
- .venv/bin/python -m pytest -q -n0 hunts/rh_moonshot
outcome: no hidden RH or zero import in the form; zeta's lambda_1 > 0 is band-level and float grade, since a positive Rayleigh ball and a band-N eigenvalue both bound the window infimum from above and lambda_1 falls by up to 112 orders from N = 32 to 128; unresolved
artifacts:
- hunts/rh_moonshot/ccm_arithmetic/challenge/circ_ccm.py
- hunts/rh_moonshot/ccm_arithmetic/challenge/test_ccm_arithmetic_challenge_circularity.py
- hunts/rh_moonshot/ccm_arithmetic/challenge/CHALLENGE_circularity.mdCompute: under a second; no estimate block needed. The builder's tests pass in this checkout: .venv/bin/python -m pytest -q -n0 hunts/rh_moonshot gave 142 passed in 97 s, the whole hunt with the four challenge directories in one process. The full grid at c = 60, N = 128 was not rebuilt here.
Amendment 2026-10-26 (follow-up): field 5, prior art and published refutations
Added after the scorecard above was accepted; the original text is kept unchanged. All queries were web searches on 2026-10-26; hits are as the result pages and arXiv abstract pages returned them, and no paper was read beyond its abstract or excerpts.
| # | query (abridged) | hits that bear on the mechanism |
|---|---|---|
| 1 | Connes Consani Moscovici "zeta spectral triples" arXiv 2511.22755, truncated Weil form, ground state | A. Connes, C. Consani, H. Moscovici, Zeta Spectral Triples, arXiv 2511.22755 (27 Nov 2025): rank-one perturbations of a spectral triple built from Euler products over p <= x, numerical agreement with the low zeros, and "A rigorous proof of this convergence would establish the Riemann Hypothesis" (abstract); no mention of DH or of functions without an Euler product on the abstract page. A. Groskin, High-Precision Approximation of Riemann Zeros via the Truncated Weil Form, arXiv 2605.20224 (2026, v4): an implementation of the Connes-van Suijlekom Galerkin matrix at sixteen cutoffs, no proof claimed, convergence called open; no rivals tested on the abstract page. A Zenodo record (https://zenodo.org/records/23441946) claiming evenness from simplicity, unreviewed and not assessed here |
| 2 | Connes van Suijlekom arXiv 2511.23257, lowest eigenvalue simple, even eigenfunction, real zeros | A. Connes and W. D. van Suijlekom, Quadratic Forms, Real Zeros and Echoes of the Spectral Action, arXiv 2511.23257, Commun. Math. Phys. 406 (2025) 312: the cited Prop. 4.1 (real zeros of the ground state's transform for any real distribution, under simplicity, isolation and evenness), proved through a Caratheodory-Fejer corollary and Hurwitz; the authors flag verifying that the minimal eigenvalue is simple as the key difficulty |
| 3 | Davenport-Heilbronn, Weil quadratic form, small eigenvalue, prolate approximation, off-line zeros | A. Connes, C. Consani, H. Moscovici, Zeta zeros and prolate wave operators, arXiv 2310.18423 (2024); A. Connes and C. Consani, Spectral triples and zeta-cycles, arXiv 2106.01715; A finite Guinand-Weil dictionary and archimedean tail order for the truncated Weil quadratic form, arXiv 2607.02828; X. Zhu, Weil positivity in compact windows: a finite reduction, [one word elided under the lexical rule] two-sided bounds, and a Landau-Widom decay law, arXiv 2608.24827 (2026): lower bounds at window L = 0.8, variational upper bounds beyond, and the empirical law -ln lambda_min(L) ~ 2 pi^2 N(T*)/ln N(T*) with T* = 2 pi e^(2L); under RH lambda_min(L) <= exp(-L e^L) for large L. No published test of the CCM steps on DH or another function without an Euler product was found |
| 4 | (from the positivity_geometry searches) Connes-Consani, Weil positivity and trace formula, the archimedean place, arXiv 2006.13771 | positivity of the Weil form at the archimedean place via Hermitian Toeplitz matrices and prolate functions |
| 5 | Lean Zulip, CCM, truncated Weil form | no thread returned by the searches above; the Zulip archive was not browsed |
What changes.
- C1 is consistent with the published picture. Zhu (hit 3) finds that the window infimum lambda_min(L) is positive but plunges doubly exponentially in the window, and establishes a lower bound only at L = 0.8. The ground-state eigenvalues reported here (1e-192 at c = 60, N = 128) are of that kind. They are band-level upper bounds on a quantity whose positivity is open at these windows. C1 stands, now with a citation.
- The cited Prop. 4.1 is published (hit 2, Commun. Math. Phys.); its proof's own flagged difficulty is link (1), simplicity, which this mechanism measures only at band level.
- CCM's own statement that convergence would establish RH (hit 1) is the mechanism's target; no published proof or refutation of step 2 was found, and no published rival test of it. The rival table of NOTES section 4 is, as far as these searches reach, not in the literature (novelty not established; this is a search result, not a claim).
- Disposition: unchanged (unresolved).
Correction to the run manifest above. Its finished time, 14:30, was written before the work finished and is later than the clock at the time; it is not a measured time, and its started time is an estimate as well. The follow-up's finished time below is read from the clock and its started time is approximate (to about 10 minutes).
id: rh_moonshot-ccm_arithmetic-2026-10-26-challenge-circularity-field5
hunt: rh_moonshot
started: 2026-10-26T13:30+00:00
finished: 2026-10-26T13:38+00:00
ran:
- web searches 1-5 of the table above and the arXiv abstract pages of 2511.22755, 2608.24827 and 2605.20224
- .venv/bin/python -m pytest -q -n0 hunts/rh_moonshot/ccm_arithmetic/challenge
outcome: no published proof, refutation or rival test of CCM step 2 found; Zhu's Landau-Widom law is consistent with C1; disposition unchanged
artifacts:
- hunts/rh_moonshot/ccm_arithmetic/challenge/CHALLENGE_circularity.md