CHALLENGE (circularity): de_branges

Hunt #150, mechanism 2. 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 dbr_common.py as committed at c2449a4b9 (merged here at d89c63d6d) and scored them against the rubric pre-registered in MISSION.md (section "Challenge-phase scoring rubric", 2026-10-26).

Files here: circ_dbr.py (an own Chowla-Selberg route for the Epstein zeta, argument- principle counts, a census, Conrey and Li's numbers by an own xi, a zero trap around the builder's defect), test_de_branges_challenge_circularity.py (9 tests), this page.

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

Verdict

The builder's disposition stands: a reformulation under kill condition 3, with the located obstruction that the Hermite-Biehler property of E_h is exactly a zero-free half-plane; unresolved as a route; Conrey and Li do not cover this form. No hidden use of RH or of the zeros was found in the definitions; RH enters only as Lagarias's conditional statements, which the NOTES label. Findings:

• B1 (prior art for statement (a)). Statement (a), "E_h is Hermite-Biehler if and only if Lambda has no zero with Re rho > 1/2 + h", is proved in the NOTES by Lagarias's factor-by-factor argument plus a one-line converse. The same mechanism, the paired Hadamard product with the functional equation, gives Sondow and Dumitrescu's theorem that |xi| is strictly increasing along every horizontal half-line in a zero-free right half-plane, and their reformulation of RH as monotonicity of |xi(sigma + it)| on sigma > 1/2 (A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis, Period. Math. Hungar. 60 (2010) 37-40, arXiv 1005.1104; restated with sigma_0 at least every zero's real part by Matiyasevich, Saidak and Zvengrowski, as arXiv 2201.08599 reports). Lagarias's chapter Hilbert spaces of entire functions and Dirichlet L-functions (Frontiers in Number Theory, Physics, and Geometry I, Springer 2006, 365-377) gives the h -> 0 end: a de Branges space attached to L exists if and only if RH holds for L. The exact form (a) at a given h was not found in print, but it is a short step from these; novelty is not established, and nothing in the disposition depends on it.
• B2 (the rival grid's high side is settled by theorems, not by the grid). The NOTES say the grid passes DH at h = 3/8 and 1/2 only on |x| <= 100 and that DH has zeros in Re s > 1 by a quoted 1936 result (NOTES section 5). That is citable now: DH's coefficients are periodic and DH is not a Dirichlet polynomial times one L-function, so by Saias and Weingartner (Acta Arith. 140 (2009) 335-344, arXiv 0807.0783; abstract re-read by this session on 2026-10-26: c_1 T <= N(sigma_1, sigma_2, T) for every 1/2 < sigma_1 < sigma_2 < 1 + eta) DH has zeros with real part in (1, 1 + eta), and by (a) E_h^DH fails for every h <= 1/2. The grid's "passes at 3/8 and 1/2" is a finite stand-in that the theorem overrides. For Epstein (1,1,6) the analogous statement (zeros in Re s > 1, Davenport and Heilbronn 1936 for class number > 1) is quoted, not checked here.
• B3 (the "second Epstein off-line zero" is new against the record and is not enclosed). See the section below; it is the second of at least five (B5).
• B5 (the Epstein rival has more off-line zeros than the NOTES list, and the grid missed one inside its range). This session's census finds five zeros of zeta_Q in [0.55, 1.6] x [0.5, 60.5], not two: besides 0.953260 + 16.290216i and the builder's 0.797997 + 29.551761i, three more, at 0.663652 + 33.650101i, 0.809992 + 43.858664i and 0.782379 + 46.960994i (offsets 0.1637, 0.3100, 0.2824; details below), none of them in the record. The one at height 33.65 lies inside the builder's Epstein grid (x <= 40), yet the grid lists no positive defect there: at h = 1/20, y = 1/20 the defect is +0.457 at x = 33.6501 but negative at the grid's own points x = 33.5 (-0.013) and x = 34.0 (-0.056) (this session's route). So a step of 1/2 in x is too coarse to see a failure region of this size, the NOTES' Epstein row ("h <= 3/8 at x = 16.290, and h <= 1/4 also at x = 29.5") is a statement about the sample, not about the function, and section 5's "fails for h below its offsets 0.4533 and 0.2980" omits three offsets. None of this contradicts (a), which predicts the failure at every one of these zeros; it is a finite stand-in read as complete. The same caution applies to zeta's "0 positive in 2828 points": that is a sampled statement, as the NOTES grade it.
• B4 (zero configuration only). By (a) the defect reads the real parts of the zeros and nothing about primes; the NOTES say so (section 6) and place the family in the MISSION.md dead-route class "mechanisms that read only the zero configuration". The Euler product enters only through the zero-free half-plane it is known to give (Re s >= 1, and Re s > 7/8 from the OpenAI strip, kernel-checked here, pending human review). Agreed.

The second Epstein off-line zero

New against the record. Searched on 2026-10-26: git grep for 29.551, 29.552 and 0.79799 under hunts/, zeta/ and docs/ on every local and remote branch (the only hits are unrelated numbers in hunts/prime_pair_error/results_residue.json); zeta.epstein pins only DH's zero (OFFLINE_ZERO_RE, OFFLINE_ZERO_IM); hunts/weil_propagation (theory RESULTS.md, numerics epstein_offline.py) pins only 0.953260 + 16.290216i; hunts/plocal_rivals (hunt #149, branch hunt/plocal-rivals, ZEROS.md and zeros.json) is a census of the F_q rivals, not of Epstein (1,1,6); and hunt #120 (hunts/qrh_rival_step/RESULTS.md section 3.2) counted the D = -23 function only in [0.8751, 1.5] x [0.5, 60.5], finding the zero at 16.29, so a zero with real part 0.798 was outside every searched box. The zero is therefore new to this tree. Whether it is in the literature was not settled: the search "Epstein zeta function x^2+xy+6y^2 discriminant -23 zeros off critical line computed Hejhal" returned Hejhal's ICM 1986 survey (Zeros of Epstein zeta functions and supercomputers, cited, not retrieved) and papers on other forms (arXiv 1907.06387, 1909.07112, 2110.09368, 1811.01613, 1102.0367), none listing this zero.

Recomputed by a third route. circ_dbr.epstein_cs is this session's Chowla-Selberg series (Poisson summation in m, re-derived here; the Bessel-sum prefactor is 2^(s + 5/2) pi^s / (sqrt(a) Gamma(s)) Delta^((1/2 - s)/2), and the 2^(s + 3/2) of a common statement is off by a factor 2, kept as a planted fault). It agrees with zeta.epstein to 1e-20 at s = 3 and 0.7 + 5i, and with a direct lattice sum at s = 4. Results:

• Newton from the builder's printed value: 0.7979971571786801862221945 + 29.55176109862911477071554i, 2.5e-14 from the builder's printed value; |zeta_Q| = 2.2e-50 there against 0.037 at distance 0.01 (dps 50).
• Argument principle (own implementation, every argument increment below pi/4): winding 1 on the square of half-side 0.02 around rho (largest increment 0.126, 68 evaluations), 1 around the mirror 1 - conj(rho) (0.138), 0 on the square displaced by 0.1 (0.029).
• Census of zeros of zeta_Q in [0.55, 1.6] x [0.5, 60.5] by windows of height 10 (the left edge as hunt #120's strip floor; argument principle on this route, measured): 0, 1, 1, 1, 2, 0 for the windows starting at 0.5, 10.5, 20.5, 30.5, 40.5, 50.5. Recursive bisection plus Newton (circ_dbr.locate, each root accepted only inside its box) puts the three unrecorded ones at

Each is measured by two routes (this session's Chowla-Selberg series and zeta.epstein, whose 1e-18 is its precision floor at dps 30 at these heights) and a winding count; none is enclosed, for the reason below. A git grep of every branch for their leading digits finds no record of them (one unrelated coincidence, 0.782375 in hunts/frontier_math/repulsion_trade.py). Hunt #120's D = -23 statement "one zero beyond 7/8 below 60.5" is consistent with this: all three have real part below 7/8.

Not enclosed. The location is measured by three routes (the builder's Arb route through hunt #120's Bessel series, the builder's zeta.epstein lattice route, and this Chowla-Selberg route) and by a winding count, but no step carries a ball enclosure of the zero: the builder's evidence is a Newton root plus |s(s-1) Lambda_Q| = 9e-47 against 1.3e-18 at distance 0.01, which is a small value, not an enclosure; and the in-tree ball route cannot evaluate this function on segment balls (hunt #120 measured an input radius of 1e-4 inflating route B's output radius to 1e67 at height 15, probe.route_c_ball docstring), so the segment-enclosed winding that hunt #120 ran for D = -15 is not available for D = -23. The NOTES grade it "measured, by two routes", which is correct; it should not be quoted as enclosed.

Scorecard

1. Disposition. Agreed with NOTES section 7: the target is recorded as a reformulation (kill condition 3); for the Lagarias family E_h the Hermite-Biehler property is exactly a zero-free half-plane, so the route is as hard as quasi-RH at every abscissa above 1/2 (unresolved); Conrey and Li's refutation concerns the positivity conditions on H(xi(1 - iz)) and does not cover this form.
2. Grade per step.
3. (a) the equivalence: derived, self-reviewed; this session rechecked the converse (a zero 1/2 + delta + i gamma with delta > h gives E_h(-gamma + i(delta - h)) = 0 in the upper half-plane). Prior art B1.
4. (b) h > 3/8: derived, conditional on the 7/8 strip (kernel-checked, pending human review). Agreed; not RH-conditional.
5. (c) the local factor is not Hermite-Biehler: derived and pinned by the builder. Agreed.
6. Zeta grid, 2828 points, defect negative: hardened at the points (two routes); a sampled statement, not a region statement. Agreed.
7. Rival grid: hardened at the points; the high-h passes for DH are overridden by B2.
8. Conrey and Li's numbers: reproduced by this session's own xi to 12 digits (-5.389100507182945e-69 at the 34th zero; -0.000131957 at 1 + 282i).
9. Second Epstein zero: measured, three routes, not enclosed (B3); the three further zeros of B5: measured, two routes, not enclosed.
10. Composite: derived, conditional on the 7/8 strip, as the NOTES say (section 8).
11. The Euler-product step. Named: the zero-free half-plane the Euler product supplies (NOTES section 2). Tested on DH, Epstein (1,1,6), W_{1/4} and zeta(s - 1/5); each fails where (a) predicts from its zeros. The step is not new arithmetic: the Euler product enters only through Re s >= 1 (and 7/8), and the local factors are not Hermite-Biehler (c).
12. Finite stand-ins. The grid (|x| <= 100, seven values of h, two of y) is finite and the NOTES say it says nothing above x = 100. One place it was read too far is B2 (DH "passes" at h = 3/8 and 1/2), now overridden by a theorem.
13. Novelty search. The builder recorded three searches and read Conrey-Li and Lagarias 2005 at section (NOTES section 1). This session's queries, web search, 2026-10-26: (a) "Lagarias Zero spacing distributions for differenced L-functions structure function xi(1/2 + h - iz) de Branges Hermite-Biehler equivalent zero-free half-plane": the paper (Acta Arith. 120 (2005) 159-184; arXiv math/0601653), with Lemma 2.1 as the NOTES read it; a ResearchGate item A family of deformations of the Riemann xi-function (not read). (b) "Hermite-Biehler property E(z) = xi(1/2 + h - iz) if and only if no zeros Re s > 1/2 + h; de Branges space exists iff RH; Lagarias Hilbert spaces of entire functions and Dirichlet L-functions": Lagarias's 2006 chapter (365-377), the title the NOTES give, so the NOTES' correction of rogue_frontier/FRONTIER_MAP.md is confirmed; a Canadian J. Math. paper On the Hilbert space derived from the Weil distribution; Lagarias's slides (https://websites.umich.edu/~lagarias//TALK-SLIDES/benasque-riemann2012jun.pdf). (c) "Sondow Dumitrescu monotonicity property of Riemann's xi function ...; Matiyasevich Saidak Zvengrowski horizontal monotonicity": B1. (d) "Conrey Li note on some positivity conditions ... de Branges conjecture refuted follow-up": Conrey and Li, arXiv math/9812166; A positivity conjecture related to the Riemann zeta function, arXiv 2011.02847 (not read); no refutation of the form (a) found. (e) The Epstein query of B3. zbMATH and the Lean Zulip archive were not browsed.
14. Rivals on complete hypotheses.
15. DH: meets every hypothesis of (a) and agrees with its conclusion (it fails exactly where its zeros say), so it does not refute (a); it lacks the Hermite-Biehler property for every h <= 1/2 by B2 (theorem) and for h <= 3/10 on the grid (hardened).
16. Epstein (1,1,6): same reading; fails at offsets 0.4533 and 0.2980 on the grid, and by (a) also at the unrecorded offsets 0.3100, 0.2824 and 0.1637 (B5).
17. W_{1/4}: has an Euler product with non-unitary local roots and fails for h < 1/4: an Euler product alone does not supply the half-plane. Not a counterexample to (a).
18. zeta(s - 1/5): lacks the s -> 1 - s functional equation, so (a) does not apply; builder's measurement.
19. Circularity. None in the definitions. dbr_common.py evaluates completed functions by two routes (mpmath through zeta.core.xi and zeta.epstein; Arb through python-flint, li_arith_split and hunt #120's probe), the defect from their values, and reads mp.zetazero(34) only in conrey_li_numbers to reproduce Conrey and Li's printed number. A run-time trap (mpmath's zero routines on mp and on the module, every public function of zeta.zeros) around the builder's defect_mp for zeta, DH and Epstein records no call (test_builder_defect_reads_no_zero; the trap catches a planted zetazero call). RH appears only in Lagarias's conditional statements, which the NOTES never use as inputs.

Run manifest

id: rh_moonshot-de_branges-2026-10-26-challenge-circularity
hunt: rh_moonshot
started: 2026-10-26T14:30+00:00
finished: 2026-10-26T16:09+00:00
ran:
  - .venv/bin/python hunts/rh_moonshot/de_branges/challenge/circ_dbr.py
  - .venv/bin/python -m pytest -q -n0 hunts/rh_moonshot/de_branges
outcome: no hidden RH or zero import; (a) is close to Sondow-Dumitrescu and Lagarias; DH fails for every h <= 1/2 by Saias-Weingartner; the second Epstein zero is new to the tree and only measured, and a census finds three more unrecorded off-line zeros at heights 33.65, 43.86 and 46.96, one of them missed inside the builder's grid
artifacts:
  - hunts/rh_moonshot/de_branges/challenge/circ_dbr.py
  - hunts/rh_moonshot/de_branges/challenge/test_de_branges_challenge_circularity.py
  - hunts/rh_moonshot/de_branges/challenge/CHALLENGE_circularity.md

Compute: the script took 11 minutes of one core (15:22 to 15:33, mostly the census), and locate on [30.5, 50.5] under 25 minutes of a second core; under one core-hour together, so no estimate was sent. A first census attempt with its left edge at Re s = 0.52 ran for 41 minutes with buffered output and was stopped, unfinished; its results were never read. Times are read from the clock, except started, which is approximate (to about 5 minutes).