Role: rival-gate challenger, 2026-10-26. I did not build this mechanism. rival_gate.py imports none of the mechanism modules. It independently extracts small Li coefficients by a roots-of-unity Cauchy sum, reconstructs log-derivative coefficients by a local Dirichlet recurrence, and recomputes the high-index W_1/4 value from zero quadruples and a smooth zero-density tail. Public completed functions and the public zeta zero table are used only for the stated rival diagnostics.
Scorecard
1. Disposition
Unresolved. The exact target is the all-n inequality P_n <= M(n) (NOTES.md lines 47-56). The construction gives no uniform signed estimate for P_n. Its pointwise absolute-value route fails throughout the tested range where M(n) is positive, and its mean-square route fails outside n in [32, 1258] (NOTES.md lines 92-108). Those two sub-routes are refuted at the reported parameters, but their failure does not refute P_n <= M(n). The finite zeta calculation through n = 5000 cannot establish the all-n target.
The W_1/4 result does refute a weaker proposed implication: a scalar Euler product, nonnegative prime-power log-derivative coefficients, and a central functional equation do not imply nonnegative Li coefficients. A stronger zeta-specific statement, for example one adding unitary local roots, remains unformulated and unproved.
2. Grade per step
| Step | Evidence in the construction | Rival-challenge grade |
|---|---|---|
| Li's all-n criterion is equivalent to RH | classical sources cited at NOTES.md lines 17-27 | derived in the cited literature; not independently reviewed here |
| The split lambda_n = A(n) + T(n) | formulas and tests at NOTES.md lines 29-45 | derived, self-reviewed by the builder; finite identities hardened by multiple numerical routes |
| The prime-side identity reduces positivity to P_n <= M(n) | derivation at NOTES.md lines 47-56 | derived, self-reviewed by the builder; finite prime sums measured |
| Pointwise or mean-square control proves P_n <= M(n) | measured bounds at NOTES.md lines 84-108 | refuted for the stated routes and ranges; neither supplies the target estimate |
| P_n <= M(n) holds for every n | only n <= 5000 computed at NOTES.md lines 84-91 | unresolved; no all-n grade exists |
| All Li coefficients are nonnegative, hence RH | depends on the preceding all-n inequality | unresolved |
The complete proof chain has no composite grade because its decisive all-n step is absent. The strongest positive arithmetic statement remains measured on n <= 5000. The sign of lambda_8260 for W_1/4 is hardened: the builder's two Cauchy radii and its separate zero-side calculation agree (NOTES.md lines 75-82), and test_wa_negative_li_value_is_reproduced_from_zeros recomputes the zero-side value without mechanism imports.
3. The Euler-product step
The construction names the correct step: P_n <= M(n), with the Euler product entering through psi and the prime-power coefficients Lambda(m) (NOTES.md lines 47-56 and 110-127). The independent recurrence in test_log_derivative_coefficients_separate_the_rivals finds no negative or off-prime-power coefficient through index 60 for zeta, W_1/4, or zeta(s - 0.1). It finds the first negative coefficient at 3 and the first off-prime-power coefficient at 6 for DH. For Epstein (1,1,6), those indices are 48 and 6.
This challenge also independently obtains
lambda_8260(W_1/4) = -122.315821669551207
from 1000 zeta zeros plus the smooth tail. Thus positivity and prime-power support of the log derivative, even together with the functional equation, are insufficient. The missing discriminator identified by the construction is unitarity of the local roots (NOTES.md lines 123-127). No estimate using that stronger input is constructed.
4. Finite stand-ins
The zeta and DH split is evaluated only for n <= 5000, the zeta prime-side integral only for n <= 5000 and x <= 10^8 before modeled tails, and W_1/4 only for n <= 9000 (NOTES.md lines 58-108). The rival recurrence here ends at coefficient index 60; the independent Cauchy extraction ends at n = 12; the W_1/4 zero sum uses 1000 zeros and a smooth tail. These computations test identities and reject weak hypotheses. None stands in for the required inequality at every n. In particular, zeta positivity through n = 5000 says nothing about a later off-line zero, as the onset discussion at NOTES.md lines 129-146 explains.
5. Novelty search
Absent. NOTES.md lines 166-178 identify the classical sources for the split and state that no search was made for the measured failure of the absolute-value routes. Absence changes no grade.
6. Rival results on complete hypotheses
| Rival | Complete-hypothesis result |
|---|---|
| DH | Lacks the named scalar Euler product and nonnegative prime-power coefficients. The recurrence through index 60 finds a negative coefficient at 3 and off-prime-power support at 6. Independent Cauchy values are positive for n = 1 through 12, agreeing with the builder's finite positivity through 5000. It lacks a named hypothesis and is not that counterexample. Range and grade: n <= 12 and coefficient index <= 60, measured. |
| Epstein (1,1,6) | Lacks a scalar Euler product. Its recurrence has off-prime-power support at 6 and a negative coefficient at 48. Its reported off-line zero gives a completed-function residual 4.31e-11. It lacks a named hypothesis and is not that counterexample. Range and grade: coefficient index <= 60 and one approximate zero, measured. |
| W_1/4 | Meets the weaker stated package of a scalar Euler product, nonnegative prime-power log-derivative coefficients, and central functional equation, yet lambda_8260 is negative. This refutes the implication from that package. It lacks unitary local roots, since their moduli are p^(1/4) and p^(-1/4), so it is not a counterexample to a stronger theorem that explicitly assumes unitarity. Range and grade: recurrence index <= 60 and n = 8260, with the sign hardened by independent routes. |
| zeta(s - 0.1) | Has a scalar Euler product with nonnegative prime-power coefficients, but its local modulus at p = 2 is 2^0.1 = 1.0717734625 and its symmetry center is 0.6. At 0.37 + 7.25i, the defect about 0.6 is 6.26e-53 while the defect about 0.5 is 0.0103207778. It lacks unitarity and the fixed-half normalization, so it is not that counterexample. Range and grade: coefficient index <= 60, one symmetry point and one shifted zero, measured. |
The construction has not stated a complete sufficient-hypothesis theorem beyond zeta itself. Consequently the W_1/4 row refutes the explicit coarse package, while the other rows locate hypotheses a stronger theorem would need. No rival here refutes a theorem combining fixed-half normalization, unitary local roots, and whatever signed estimate is still missing.
7. Circularity
The target is explicitly RH-equivalent (NOTES.md lines 17-27). Rewriting it as P_n <= M(n) is not progress unless a new unconditional estimate proves the inequality. The pointwise tail uses Schoenfeld's RH-conditional envelope and the mean-square tail uses Cramer's RH mean square as a model (NOTES.md lines 92-102). Either would be circular if used in an RH proof, and neither reaches the required inequality on the full tested range anyway.
The builder does not use a zero list as positive evidence for zeta. This challenge's zero-side W_1/4 calculation is only an independent falsification check for a negative rival value. It is not an input to the proposed prime-side proof. The small Cauchy calculation evaluates completed functions directly and imports no zero list.
Run record
id: rh_moonshot-li_arithmetic-rival-2026-10-26
hunt: rh_moonshot
started: 2026-10-26T13:19:00Z
finished: 2026-10-26T13:22:20Z
ran:
- nice -n 10 .venv/bin/python hunts/rh_moonshot/li_arithmetic/challenge/rival_gate.py --write hunts/rh_moonshot/li_arithmetic/challenge/results_rival.json
- nice -n 10 .venv/bin/python -m pytest -q -n0 hunts/rh_moonshot/li_arithmetic/challenge/test_rival_gate.py
outcome: the independent recurrence separates non-Euler rivals, the zero-side route reproduces the negative W_1/4 value, and shifted zeta fails unitarity and fixed-half normalization; the zeta all-n signed estimate remains unresolved
artifacts:
- hunts/rh_moonshot/li_arithmetic/challenge/rival_gate.py
- hunts/rh_moonshot/li_arithmetic/challenge/results_rival.json
- hunts/rh_moonshot/li_arithmetic/challenge/test_rival_gate.py
- hunts/rh_moonshot/li_arithmetic/challenge/CHALLENGE_RIVAL.md