Role: rival-gate challenger, 2026-10-26. I did not build this mechanism. rival_gate.py imports no mechanism or other-hunt module. It evaluates the Hermite-Biehler defect directly from the public completed functions in zeta.core and zeta.epstein, at a failure point and a control point for each required rival.
Scorecard
1. Disposition
Unresolved. The Lagarias family is valid, but its decisive condition is an exact reformulation. For the stated symmetry and Hadamard hypotheses,
E_h is Hermite-Biehler
is equivalent to having no zero with real part greater than 1/2 + h (NOTES.md lines 97-111). Therefore proving the condition for every h > 0 is quasi-RH at every abscissa, not an estimate toward it. The known arithmetic input only supplies the already known zero-free range, h >= 1/2 classically and h > 3/8 from the 7/8 strip (NOTES.md lines 84-88 and 113-120).
This is a located obstruction and triggers the mission's reformulation kill condition, but it is not a theorem that RH cannot be proved by establishing those zero-free half-planes. I therefore do not upgrade the obstruction to a restricted-class impossibility result.
2. Grade per step
| Step | Evidence in the construction | Rival-challenge grade |
|---|---|---|
| Define E_h(z) = Lambda(1/2 + h - iz) and its defect | NOTES.md lines 68-82 | derived, self-reviewed |
| E_h is Hermite-Biehler exactly when Re rho <= 1/2 + h for every zero | Lagarias's factor argument plus the zero of E_h exhibited at NOTES.md lines 97-111 | derived, self-reviewed; the rival extension assumes the stated Hadamard normalization |
| The 7/8 zero-free strip gives the property for h > 3/8 | NOTES.md lines 113-120 | derived, self-reviewed, conditional on the recorded 7/8 result and its review conditions |
| Hermite-Biehler gives real, simple, interlacing zeros of A and B | Lagarias Lemma 2.2 and Theorem 2.1, recorded at NOTES.md lines 42-52 | cited primary-source theorem; source not independently rechecked by this challenger |
| The property for every h > 0 and Hurwitz give RH | NOTES.md lines 79-82 and 103-111 | derived, self-reviewed, but its all-h premise is RH-equivalent and unproved |
The finite rival defects below are measured by one independent numerical route. The complete RH chain has no composite grade because its all-h premise is absent.
3. The Euler-product step
The construction names the arithmetic input as a zero-free half-plane derived from the Euler product (NOTES.md lines 84-88). It also shows that individual Euler factors do not have the Hermite-Biehler sign (NOTES.md lines 122-128), so the desired inequality is not assembled locally from positive factors.
The independent cells confirm the actual discriminator. DH and Epstein lack a scalar Euler product and their defects turn positive at their off-line zeros. W_1/4 has a scalar Euler product, nonnegative log-derivative coefficients and the fixed functional equation, yet its defect is positive for h = 0.2. Thus those coarse Euler-product properties do not imply the needed zero-free half-plane. Its nonunitary local roots move a zero to real part 0.75.
4. Finite stand-ins
The builder's grid has |x| <= 100, except Epstein ends at 40, with seven h values and two main y values (NOTES.md lines 130-145). Negative defects on that grid do not establish the inequality throughout the upper half-plane. The independent challenge uses one failure point and one control point per rival, all at y = 0.05. Those points test the exact zero-offset prediction but do not replace an all-z or all-h statement. The exact equivalence, not the finite grid, supplies the obstruction.
5. Novelty search
The builder records searches for Conrey and Li and two Lagarias papers, then reads the available arXiv sources (NOTES.md lines 17-56). It finds that Conrey and Li refute different Hilbert-space positivity conditions, not this E_h family (NOTES.md lines 90-95). No novelty search beyond those sources is recorded for the extension to h > 3/8 or for the exact equivalence (NOTES.md lines 113-120). No novelty claim is made here.
6. Rival results on complete hypotheses
| Rival | Complete-hypothesis result |
|---|---|
| DH | It has the fixed-half functional equation and reality, but no scalar Euler product. At x = 85.6993484853775922 and y = 0.05, the defect is 2.3441483868663243899 for h = 0.25 and -2.1334132025776454372 for h = 0.375. Its completed-function residual at the stated zero with real part 0.808517182456637386 is 5.12e-77. It fails the zero-free conclusion-producing hypothesis exactly where predicted, so it is not a counterexample to the equivalence. Range and grade: two points, measured. |
| Epstein (1,1,6) | It has the fixed-half functional equation and reality, but no scalar Euler product. At x = 16.2902157204 and y = 0.05, the defect is 1.3494121383316399218 for h = 0.375 and -3.5506433820889641886 for h = 0.5. The completed residual at real part 0.9532604748 is 2.28e-21. It lacks the zero-free hypothesis and is not that counterexample. Range and grade: two points, measured. |
| W_1/4 | It has the fixed functional equation and a scalar Euler product, but nonunitary local roots. At the first zeta ordinate and y = 0.05, the defect is 62.923988502644232857 for h = 0.2 and -63.336207588322219641 for h = 0.3. The off-line zero at real part 0.75 has residual 3.62e-35. This refutes any implication from only the scalar Euler product, nonnegative coefficients and functional equation to the Hermite-Biehler condition. It lacks the required zero-free half-plane. Range and grade: two points, measured. |
| zeta(s - 0.1) | Its symmetry center is 0.6, not 0.5. The fixed-half functional-equation defect at 0.37 + 7.25i is 0.010320777772574, while the defect about center 0.6 is 6.91e-53. At the first zeta ordinate and y = 0.05, its Hermite-Biehler defect changes from 63.150172698616547874 at h = 0.05 to -63.150172698616547874 at h = 0.15. It lacks the fixed functional equation, so the equivalence's complete hypotheses refuse it. Range and grade: three points, measured. |
No rival meets the conclusion-producing zero-free hypothesis and violates the Hermite-Biehler conclusion. W_1/4 does meet the weaker Euler-product package and refutes that weaker implication.
7. Circularity
The construction's critical input is already the desired zero-free statement. By the exact equivalence at NOTES.md lines 97-111, proving the Hermite-Biehler condition for each h is neither weaker than nor independent of proving the corresponding quasi-RH half-plane. Taking every h > 0 is RH. The family therefore provides a correct reformulation, not a new estimate.
The challenger evaluates completed functions directly. Known rival zeros are used only to choose falsifying points and to check residuals. The zeta control at h = 0.1 has defect -1.0992890467837299957 at one point; it is not evidence for the all-z inequality.
Run record
id: rh_moonshot-de-branges-rival-2026-10-26
hunt: rh_moonshot
started: 2026-10-26T14:37:00Z
finished: 2026-10-26T14:39:43Z
ran:
- nice -n 10 .venv/bin/python hunts/rh_moonshot/de_branges/challenge/rival_gate.py --write hunts/rh_moonshot/de_branges/challenge/results_rival.json
- nice -n 10 .venv/bin/python -m pytest -q -n0 hunts/rh_moonshot/de_branges/challenge/test_rival_gate.py
outcome: every off-line rival flips the defect at its predicted offset; W_1/4 refutes the coarse Euler-product package, shifted zeta lacks the fixed symmetry, and the Lagarias family remains an exact unresolved zero-free reformulation
artifacts:
- hunts/rh_moonshot/de_branges/challenge/rival_gate.py
- hunts/rh_moonshot/de_branges/challenge/results_rival.json
- hunts/rh_moonshot/de_branges/challenge/test_rival_gate.py
- hunts/rh_moonshot/de_branges/challenge/CHALLENGE_RIVAL.md