CHALLENGE (circularity): li_arithmetic

Hunt #150, mechanism 3. 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 1fe59177e on hunt/rh-moonshot, merged here at c220f1310), li_arith_prime.py, li_arith_wa.py, run_li_arith_prime.py and the committed artifacts/, and scored them against the rubric pre-registered in MISSION.md (section "Challenge-phase scoring rubric", 2026-10-26).

Files here: circ_li.py (independent prime-side integral and Li coefficients of the shifted completion; shares no code with li_arith_*.py), test_li_arithmetic_challenge_circularity.py (5 tests), this page.

Grades as in MISSION.md. Nothing written here is kernel-checked.

Verdict

Unresolved, agreeing with the builder; the target is recorded as a reformulation (kill condition 3) until an estimate is offered. P_n <= M(n) for every n is Li's criterion rewritten (NOTES section 1 says so), and no inequality proved from the Euler product is proposed. No hidden RH enters any positive claim. RH-conditional inputs enter only the two bound routes, which are refuted, and one cross-check. Findings:

• L1 (RH inputs in the refuted routes, scoped). The pointwise route uses Schoenfeld's envelope above 10^8 and the mean-square route Cramer's mean square, whose constant CRAMER_MEAN_SQUARE is "sum over all nontrivial zeros of 1/|rho|^2 under RH" (li_arith_prime.py lines 55-62). Both are labelled conditional, and their failures stand a fortiori for every envelope at least as large, which includes every unconditional one in use. They do not refute an oracle route that uses the exact |E(x)| beyond 10^8. The positive half, "at most M(n) for n in [32, 1258]" (NOTES section 3, mean-square bullet), is conditional on RH and on a model and is no evidence for P_n <= M(n) on that range. NOTES section 5 says these routes are "circular as written"; the section 3 bullet should carry that hedge at the point of use.
• L2 (provenance of P_n, cleared by a second route). The artifact's P_exact is not computed from primes. It is M(n) - lambda_n with lambda_n from the xi side (run_li_arith_prime.py: Pn = M - L[:n_max]; checked here to 1e-9 at n = 1, 7, 100, 5000). So "the prime integral satisfies P_n <= M(n)" on n <= 5000 (NOTES section 8) is lambda_n >= 0 restated, through a derived identity checked by primes only at n <= 6. This session computed the prime integral itself, truncated at X, in closed form (antiderivatives of e^(-u) L^{(2)} and L^{(2)}, its own sieve). It reproduces the artifact to within 5e-3 for every n <= 60 at X = 10^6, 10^7 and 10^8, and to within 0.04 at X = 10^8 for n = 100, 150, ..., 1000, where M(n) >= 46. The identity and the xi-side values therefore cohere through n = 1000 by a second, prime-side route. That route omits x > 10^8 with no bound on the omitted part, so it is measured, not hardened. For 1000 < n <= 5000 the statement rests on the xi side alone. An own bug, the antiderivative written with L^{(3)} in place of L^{(1)}, was caught by this comparison (it agrees at n = 1 only, since L_0 = 1); it is kept as a planted fault.
• L3 (a cross-check that reads zeros). W_{1/4}'s second route sums zeta's first 1000 zeros from mpmath and replaces the rest by the smooth on-line density (li_arith_wa.py lines 33-39, 111-126). It therefore assumes on-line zeros above height about 1419, which is verified to 3e12 (Platt and Trudgian 2021) and conditional above. Under any placement of zeros above 3e12 their contribution at n = 8260 is far below the margin |lambda_8260| = 122 (heuristic, not computed). The sign's "hardened" grade stands on the unconditional Cauchy route plus this check. The Cauchy radius rule for the same function reads the zero-free region at low height (lines 16-30), which is a verified finite fact.
• L4 (DH's failure needs no extrapolation). The rival gate says a uniform bound must fail for DH "at the latest at DH's first negative index", taken as n = 328997, extrapolated (NOTES section 4). That some lambda_n(DH) < 0 is a theorem: DH's zero multiset contains off-line zeros, and Bombieri and Lagarias's multiset theorem applies, as li_arith_wa.py lines 13-15 already use it for W_a. Only the index is extrapolated.
• L5 (prior art). The measured smallness of the arithmetic part (|T(n)| <= 13.5 for n <= 5000) matches Maslanka's observation that the finite-place term stayed below 20 in absolute value over his computed range (arXiv math/0402168), which the NOTES do not cite.

Scorecard

1. Disposition. Unresolved. The target is an RH-equivalent restatement with no estimate (kill condition 3: reformulation). The two bound routes are refuted at the builder's stated parameters (NOTES section 7), scoped as in L1.
2. Grade per step.
3. The split lambda_n = A(n) + T(n): derived, self-reviewed, defect 8.1e-28 measured (builder). Agreed.
4. The Laguerre identity lambda_n = M(n) - P_n: derived. Now checked by a prime route for n <= 1000 (sampled, L2): measured, second session.
5. "P_n <= M(n) for n <= 5000": measured, xi side; equal to lambda_n >= 0 on that range (L2). Not a prime-side measurement past n = 1000.
6. W_{1/4}: lambda_8260 < 0, hardened (Cauchy at two radii plus the zero route, L3). Agreed, with the L3 note.
7. Pointwise and mean-square refutations: measured on sampled trapezoid grids, scoped as in L1.
8. Onset heuristic: inferred (builder). Agreed.
9. Composite: measured, as the NOTES say (section 8).
10. The Euler-product step. Named: P_n <= M(n) through psi (NOTES section 4). Tested: W_{1/4} (Euler product, nonnegative coefficients, functional equation) fails by n = 8260, so those hypotheses do not suffice. DH is not tested in the range where it fails (n <= 5000 only; its failure exists by L4). No candidate inequality exists to test, so the step is named but carries no inequality.
11. Finite stand-ins. n <= 5000 everywhere, x <= 10^8 for primes; the NOTES state that a finite range is a finite height (section 5) and offer no uniform claim. Agreed. The one place a finite computation is read too strongly is L2's "prime integral" wording.
12. Novelty search. None recorded by the builder (NOTES section 8). This session's query, 2026-10-26, web search: "Li coefficients arithmetic formula Bombieri Lagarias Laguerre polynomial psi(x) - x integral unconditional bound Coffey Maslanka arithmetic part": found Lagarias arXiv math/0404394 (the explicit-formula split for automorphic L-functions), Maslanka arXiv math/0402168 and math/0406312 (numerics; the finite-place term small, L5) and Coffey's lower bound on the archimedean part; nothing on the Laguerre-kernel integral form of P_n or on the failure of absolute-value routes. zbMATH, MathSciNet and Lean Zulip were not searched.
13. Rivals on complete hypotheses.
14. DH: lambda_n > 0 for n <= 5000 (measured, builder); passes every finite-range bound tested, a shared lemma, no verdict. Its lambda_n fails somewhere by a theorem (L4).
15. Epstein (1,1,6): not run by the builder or by this session; excludes nothing.
16. W_{1/4}: meets Euler product, nonnegative log-derivative coefficients and the functional equation, and violates the conclusion at n = 8260 (hardened): refutes any implication from those hypotheses alone.
17. zeta(s - delta), delta = 1/4 (computed here, not in the NOTES): its completion xi(s - delta) has every zero at real part 1/2 + delta, so |1 - 1/rho| < 1 for each and lambda_n = sum_rho [1 - (1 - 1/rho)^n] > 0 for every n (derived, self-reviewed by this session; the paired Hadamard product of xi(s - delta) has no exponential factor); measured positive for n <= 30 by a Cauchy integral (lambda_1 = 0.0115513352, against 0.0230957090 at delta = 0, which reproduces zeta's lambda_1). It has an Euler product with coefficients Lambda(n) n^delta >= 0 and violates RH. It lacks the s -> 1 - s functional equation, so it is not a counterexample; what it shows is that Li positivity implies RH only through the functional equation. Any proof of P_n <= M(n) must therefore use the functional equation, which in this split enters through the archimedean part A(n), not through psi.
18. Circularity. The target is RH-equivalent by design and stated so. No positive claim uses an RH input. RH or a zero list enters: Schoenfeld's envelope and Cramer's constant in the refuted routes (L1); zeta's zeros and an on-line density in W_{1/4}'s cross-check (L3). The xi-side lambda_n are Cauchy integrals of log xi with Arb zeta and read no zeros (li_arith_split.py). The builder's next step 1 (bound P_n zero by zero through the explicit formula) would read the zeros, and the NOTES say so.

Run manifest

id: rh_moonshot-li_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/li_arithmetic/challenge/circ_li.py
  - .venv/bin/python -m pytest -q -n0 hunts/rh_moonshot/li_arithmetic/challenge
outcome: no RH input in any positive claim; the artifact's P_n is the xi-side lambda_n restated, and an independent prime-side route to 10^8 reproduces it within 0.04 for n <= 1000; zeta(s - delta) has lambda_n > 0 for every n and violates RH, so the functional equation carries Li's criterion; unresolved, target a reformulation
artifacts:
  - hunts/rh_moonshot/li_arithmetic/challenge/circ_li.py
  - hunts/rh_moonshot/li_arithmetic/challenge/test_li_arithmetic_challenge_circularity.py
  - hunts/rh_moonshot/li_arithmetic/challenge/CHALLENGE_circularity.md

Compute: the script, with its sieve to 10^8 and the n = 100..1000 scan, took 4 min 40 s of one core (timed); the tests run the same routines at X = 10^6 in seconds. Under one core-hour, so no estimate was sent.

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 abstracts returned them, and no paper was read beyond its abstract or excerpts. Journal volumes and years for Keiper, Voros and Johansson are from memory and unchecked; the arXiv numbers were returned by the searches.

What changes.

• The onset heuristic of NOTES section 5 is a known mechanism. Voros (hit 4) proves the dichotomy: tempered growth n (A log n + B) exactly under RH, a non-tempered oscillation otherwise, driven by the off-line zeros. The builder's root of n log R = log((n/4) log n) is a quantitative reading of that dichotomy and should cite it. Not a refutation.
• The measured range is small beside the record. Johansson (hit 5) has lambda_n with interval enclosures to n = 100000; this mechanism's n <= 5000 and the laboratory's earlier n <= 58 rows are far inside it. Not a refutation; field 4 (finite stand-ins) is unaffected, since no step claims uniformity.
• The split is Bombieri-Lagarias's Theorem 3 and Coffey's S_inf / S_f split (hits 1, 2), as the NOTES say. Coffey's lower bound on the archimedean part is the published ingredient nearest to "an inequality on the arithmetic part"; no published uniform bound on S_f(n) was found, and none that would finish.
• No published refutation of the exact form "P_n <= M(n) for all n from the Euler product" was found, nor any treatment of the Laguerre-kernel prime integral (hit 6).
• Disposition: unchanged (unresolved; the target a reformulation under kill condition 3).

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). The follow-up manifest below has its finished time read from the clock and an approximate started time.

id: rh_moonshot-li_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 1-7 of the table above
  - .venv/bin/python -m pytest -q -n0 hunts/rh_moonshot/li_arithmetic/challenge
outcome: Voros's sharpening is the published form of the onset dichotomy; Johansson has lambda_n to n = 100000; no published refutation or uniform arithmetic-part bound found; disposition unchanged
artifacts:
  - hunts/rh_moonshot/li_arithmetic/challenge/CHALLENGE_circularity.md