CHALLENGE (circularity): probe_tower

Hunt #150, mechanism 6. 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 and tower_template.py as committed at 6c3b90e4b and scored them against the rubric pre-registered in MISSION.md (section "Challenge-phase scoring rubric", 2026-10-26).

Files here: circ_probe.py (independent recomputation, shares no code with tower_template.py), test_probe_tower_challenge_circularity.py (7 tests, every number below), this page. The builder's 29 tests pass in this checkout, with the pinned paper at ~/.cache/zeta-lab/openai-math-adc7f124/paper.tex matching the recorded SHA-256 42a5ee0f...6a3.

Grades as in MISSION.md, plus the builder's cited and inferred (NOTES lines 9-14). Nothing written here is kernel-checked.

Verdict

Restricted class obstructed, as the builder says, for the class as section 9 defines it (the order-k template), with composite grade inferred. No hidden use of RH, GRH or the zeros was found in the template; the formal order k = 1 is Littlewood's RH-equivalent Moebius bound, and the NOTES flag it as a restatement and use it as nothing more (lines 183-194). The LP walls are recomputed here by a separate route (exact vertex enumeration in rationals) and the dual-multiplier identity is rechecked symbolically: they hold. What limits the result is not circularity but one inferred reading, and one place where agreement is counted as more independent than it is:

• P1 (shared inferred reading). Every wall rests on reading each 6 of the paper as the character order k (NOTES 2.1, line 89). The builder's own planted fault (a pole k/36, NOTES lines 122-128) shows that the k = 6 reproduction cannot pin this. The NOTES then count agreement with hunt #136's probe lane as a cross-check (lines 175-176, 436-441). Both start from the same reading, so their agreement checks algebra, not the reading. This session's recomputation starts from the NOTES' section 2.2 formulas and shares the reading too. The reading is structurally well motivated (lines 3758 and 4045: the principal frequencies are a^k, and the sum over them is zeta_F(kz), with its pole at 1/k), but it remains inferred, and the class's relevance to any actual order-k theta system takes that grade.
• P2 (sources that were uncommitted when cited). Sections 4 to 6 grade as cited some statements read from worktrees that were uncommitted at the time (hunt #136 PROBE.md, hunt #148 S4.md, RIVAL.md, LOCAL_IDENTITY.md). They have since been checkpointed: 3f6ab82fd on hunt/ff-probe (pushed; S4.md line 79 has the quoted 234 / 52 split) and 57c7024c1 on hunt/below-seven-eighths (local only, not on origin when checked). Until the latter is pushed, an outside reader cannot check those citations; they feed only the small-k doors (sections 4 and 5), not the walls.
• P3 (no novelty search). The NOTES record none (line 536). This session's searches found no published treatment of the order-k tower (queries below), so nothing published covers the exact form as far as these searches reach.

Scorecard

1. Disposition. Restricted class obstructed. The theorem is the LP of NOTES 3.2 with its dual multipliers (derived, self-reviewed; rechecked here). Its hypotheses apply to the class by definition (section 9 defines the class as the template), so the transfer from the class to real probes is exactly the inferred reading of P1. Within the class the mission's tower premise is refuted: sigma_0(k) increases to 1.
2. Grade per step.
3. The paper's constants at k = 6: cited (49 lines checked by verify_citations against the pinned file; builder's tests pass here). Agreed.
4. sigma_0(k) = 1 - 1/k + (M - 2 ell)/(2k): derived. Rebuilt here symbolically from the signal offset c = l_x/2 - 1 + h/k and the direct bound l_x/2 + b/(2k) (NOTES 2.2): difference from the closed form 0, value 7/8 at (ell, M, b) = (1/6, 5/6, 1/8), k = 6 (test_sigma0_closed_form_and_paper_value). Agreed.
5. The k-reading: inferred (P1). Agreed with the NOTES' own grade.
6. Sufficiency of M + ell <= 1 for a general geometry: inferred, from hunt #134 (NOTES line 92). Agreed.
7. Walls (NOTES table, section 3.2): derived. Rebuilt here by exact vertex enumeration, no LP solver: for k in {2, ..., 8, 12, 100}, dual-length wall 1 - 2/k, floor wall 1 - 4/(5k) at delta_0 -> 0 with minimiser (ell, M, b) = (1/5, 4/5, 0) for every k, and 1 - 25/(32k) at delta_0 = 1/50; all equal to the NOTES table. Floor wall strictly increasing for k = 2..40, minimum 3/5 at k = 2. Dual-multiplier identity (M - 2 ell) - m* = y1 (-floor) + y2 (1 - M - ell) + y3 b: residual 0 symbolically in k and delta_0; a 1 percent perturbation of y1 breaks it (planted fault). With the second route the walls at the listed k are two-route exact; the all-k statement stays derived (symbolic identity, self-reviewed by the builder, rechecked here).
8. Rival measurements at zeros (7.3-7.5): measured. DH recomputed here by a Hurwitz route that does not use zeta.epstein: the zero refined by findroot to 0.80851718245663738555 + 85.699348485377592172i, |DH| = 5.5e-41, |L(chi)| = |L(chibar)| = 0.70219735, |R| = 0.77831832, |H - 1| = 1.0000 (to 1e-20), and the closed form c1 c2 (L1 - L2)^2/(L1 L2) agrees to 2.4e-41. Agreed.
9. Saias-Weingartner (7.3): cited from the abstract, proof not checked. I re-read the abstract (arXiv 0807.0783): for periodic a not of the form P(s) L(s, psi) there is eta(a) > 0 with c1 T <= N_a(sigma_1, sigma_2, T) <= c2 T for all 1/2 < sigma_1 < sigma_2 < 1 + eta. The NOTES quote it correctly.
10. Composite: inferred (P1), as the NOTES say (line 481).
11. The Euler-product step. Named: (E_k), the principal row's Moebius-weighted squarefree sum must equal H/L with sup |H - 1| <= 1/2 (NOTES 7.1, from lines 3967, 4043-4047, 413). It is k-independent. Tested on rivals lacking the Euler product: DH fails it at its zero (recomputed above) and, for every finite deletion set, by Saias-Weingartner; Epstein (1,1,6) fails it at 0.953260 + 16.290216i (builder's measurement, not recomputed here). The step distinguishes.
12. Finite stand-ins. None in the claims. The tower's direction, the walls and the best order are closed forms symbolic in k (rechecked); the random-geometry scan for k = 1..64 (NOTES 3.5) is a supplementary test, not the argument. The NOTES' point that deleting small primes hides DH's failure from every finite scan (lines 361-366) is itself a correct warning against a finite stand-in, and the all-height failure is carried by the cited theorem, not by the scan.
13. Novelty search. Absent in the NOTES (line 536). This session's queries, 2026-10-26, web search: (a) "Quasi-Riemann hypothesis 7/8 Patterson cubic theta higher order metaplectic theta zero-free region generalization order k": returned David, Dunn, Hamieh and Lin (arXiv 2306.11875) and Patterson-bias papers, nothing on an order-k zero-free tower; (b) a grep of the pinned paper for "higher order", "order $k$", "n-th order", "barrier", "limitation": only lines 134 and 4728, both on Blomer, Goldmakher and Louvel's higher-order large-sieve recursion. arXiv listing search, zbMATH and Lean Zulip were not searched.
14. Rivals on complete hypotheses.
15. DH: lacks (E_k) (|H - 1| = 1.0000 at its zero; measured here by a second route). Not that counterexample.
16. Epstein (1,1,6): lacks (E_k) at its zero (builder's measurement; not recomputed here, so from this session: not run, excludes nothing beyond the builder's record).
17. W_{1/4}: passes (E_k), violates RH's analogue, lacks unitarity, which the direct bound and the local identity use (NOTES 7.5). Not that counterexample (builder's measurement and derivation).
18. zeta(s - delta), delta = 1/4 and 1/10 (computed here, not in the NOTES): f(n) = n^delta is completely multiplicative, so the squarefree Moebius sum is 1/zeta(s - delta) exactly and H = 1: it passes (E_k) (measured at s = 3 to n = 20000: 0.79352844955 against 0.79352844951 for delta = 1/4, and 0.817571979528 against 0.817571979518 for delta = 1/10, inside the tail bound) and violates RH (zeros at 1/2 + delta + i gamma). It lacks unitarity, |f(n)| = n^delta, and the NOTES' change-of-variables argument for the factor zeta(s - 1/4) of W_a (lines 412-418) applies to it verbatim with 1/4 replaced by delta: every wall rises by delta, to at least 3/5 + delta > 1/2 + delta. Not that counterexample; it is the cleanest rival for the unitarity hypothesis, cleaner than W_a because it has degree 1 and no second factor.
19. Circularity. None found. Requirement A provenance, by hand: tower_template.py imports only the standard library at module level; the template routines (wall, the sigma_0 formulas, the LPs) use Fractions and sympy and read no zero. Zeros enter only the rival diagnostics: dh_at_offline_zero (from zeta.epstein's OFFLINE_ZERO_RE/IM), epstein_at_offline_zero (a hard-coded zero) and wa_at_offline_zero (mp.zetazero(1) to place W_a's zero). That is evaluation of a rival at its known zero, which is what the gate asks for, and nothing feeds back into a wall. The target implies RH and GRH for Dirichlet L-functions (NOTES section 1); no input assumes either.

What this challenge does not settle

The rival-gate challenger's computation on Epstein (1,1,6) and W_a is separate. Whether an order-k theta system exists for k in {2, 4, 5} with a scalar Euler identity, and whether the k-reading survives a reading of the paper's proof at k != 6, are open.

Run manifest

id: rh_moonshot-probe_tower-2026-10-26-challenge-circularity
hunt: rh_moonshot
started: 2026-10-26T12:20+00:00
finished: 2026-10-26T13:30+00:00
ran:
  - .venv/bin/python -m pytest -q -n0 hunts/rh_moonshot/positivity_geometry hunts/rh_moonshot/probe_tower
  - .venv/bin/python hunts/rh_moonshot/probe_tower/challenge/circ_probe.py
  - .venv/bin/python -m pytest -q -n0 hunts/rh_moonshot/probe_tower/challenge
outcome: no hidden RH or zero import; walls rebuilt by exact vertex enumeration and the dual identity rechecked; DH correction at its zero rebuilt by a Hurwitz route; zeta(s - delta) passes the Euler step and lacks unitarity; composite grade stays inferred on the k-reading
artifacts:
  - hunts/rh_moonshot/probe_tower/challenge/circ_probe.py
  - hunts/rh_moonshot/probe_tower/challenge/test_probe_tower_challenge_circularity.py
  - hunts/rh_moonshot/probe_tower/challenge/CHALLENGE_circularity.md

Compute: about 3 s of one core for the script, under 10 s for the tests; no estimate block needed.

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.

What changes.

• Liu's optimality theorem (hit 2) is a published restricted-class result at k = 6. It proves the 0.874957 endpoint optimal within its exponent comparison system, which matches hunt #134's 0.874958 that the NOTES reproduce at k = 6 (lines 177-178, 284). It covers the k = 6 member of the template, not the tower over k, and is consistent with the NOTES' floor wall 13/15 being a lower bound under weaker bookkeeping. It strengthens the k = 6 column; it does not reach the k-reading of P1.
• Liu's mu_12 paper exists (hit 3) with the arXiv number the NOTES give (line 272); that it "keeps zeta_F^S(6z)" is not on its abstract page and stays a reading quoted from hunt #136, unverified here.
• No published order-k tower, and no published barrier at 1/2 for this method, was found (hits 2, 4, 6). The nearest structural analogue is Bettin and Gonek's conditional tower (hit 4), where a mean-value conjecture at level theta is tied to the zero-free abscissa and only theta = infinity gives RH; it goes the right way in its parameter, where the order-k template goes the wrong way. It is a different mechanism (not a refutation).
• Disposition: unchanged (restricted class obstructed for the order-k template, composite grade inferred).

Correction to the run manifest above. Its finished time, 13:30, was written before the work finished and is not a measured time, nor is its started time; the follow-up below has its finished time read from the clock and an approximate started time.

id: rh_moonshot-probe_tower-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 1-6 of the table above and the arXiv abstract pages of 2610.12234, 2610.15291, 2610.26835 and 1604.02740
  - .venv/bin/python -m pytest -q -n0 hunts/rh_moonshot/probe_tower/challenge
outcome: Liu proves the k = 6 endpoint 0.874957 optimal within its comparison system; no published order-k tower or barrier at 1/2 found; disposition unchanged
artifacts:
  - hunts/rh_moonshot/probe_tower/challenge/CHALLENGE_circularity.md