de_branges: a Hermite-Biehler structure function built from xi

Hunt #150 (rh_moonshot), mechanism 2. Built 2026-10-26. Status: constructed, measured and pushed to a located obstruction; not challenged yet; unreviewed by any person. Nothing here is a result until the case log in hunts/README.md says how this hunt ended. Files: dbr_common.py (the completed functions by two routes, the defect, the local factor, Conrey and Li's numbers), dbr_grid.py (the grid), test_de_branges.py (the pins), data/ (every number below), RUNS.md, this page.

Grades are the hunt's (MISSION.md, top): measured, hardened, derived with its review status, kernel-checked; cited marks a statement read in a primary source at the stated place; quoted marks one taken from a secondary source and not checked. A sign read from an Arb ball that excludes 0 is ball-carried, and with the mpmath route agreeing it is hardened. Nothing here is kernel-checked. A composite claim takes the grade of its weakest step.

1. Sources, read at section, and the search

Searched on 2026-10-26: three web searches (Conrey-Li's title; Lagarias "Hilbert spaces of entire functions and Dirichlet L-functions"; Lagarias "Zero spacing distributions for differenced L-functions"), then the arXiv TeX sources were fetched into the session scratch directory and read as data:

• Conrey and Li, "A note on some positivity conditions related to zeta and L-functions", Internat. Math. Res. Notices 2000, no. 18, 929-940; arXiv math/9812166 (e-print SHA-256 c458dcc9b7eb8a93f78d7d76b8f5a4687e9a9aea49d464cfd7c424ca863ba29f). Cited. Section 2, Theorem 1 (de Branges's condition): for E with no real zeros, |E(conj z)| < |E(z)| on Im z > 0, conj E(conj z) = eps E(z - i) and |E(x + iy)| increasing in y > 0, if Re <F(z), F(z + i)>_{H(E)} >= 0 for every F in H(E) with F(z + i) in H(E), then the zeros of E lie on Im z = -1/2. Theorem 2 is the companion condition on a space F(W). Section 3.1 takes E(z) = xi(1 - iz) with xi(s) = s(s-1) pi^{-s/2} Gamma(s/2) zeta(s) (no factor 1/2), proves it Hermite-Biehler from 0 < Re rho < 1, and refutes condition (3.1) by -Re{xi'(rho) xi(1 + rho)} = -5.389100507182945e-69 at the 34th zero rho = 1/2 + 111.0295...i, and condition (3.3) by Re{xi(1 + 282i)/xi(2 + 282i)} = -0.000131957; section 3.2 does the same for L(s, chi_4); the closing remark records Sarnak's computation-free proof that (3.3) fails. Both printed numbers are reproduced here by two routes (test_conrey_li_numbers_two_routes).
• de Branges, "The Riemann hypothesis for Hilbert spaces of entire functions", Bull. AMS 15 (1986) 1-17; "The convergence of Euler products", J. Funct. Anal. 107 (1992) 122-210; "A conjecture which implies the Riemann hypothesis", J. Funct. Anal. 121 (1994) 117-184. Quoted (titles and places from the two bibliographies above; not read).
• Lagarias, "Zero spacing distributions for differenced L-functions", Acta Arith. 120 (2005) 159-184; arXiv math/0601653 (e-print SHA-256 7f648d2824d17e24b4c1a4fce249dde77744f0d70d9fcb7e163e6d5eb74de4d5). Cited. Lemma 2.1: (1) for h >= 1/2, |xi(h + s)| > |xi(h + 1 - conj s)| when Re s > 1/2; (2) under RH the same for each h > 0; proved factor by factor on the Hadamard product. Lemma 2.2 (de Branges 1959, Lemma 5): such an E has A = (E + E^#)/2 and B with only real, interlacing zeros. Theorem 2.1: so A_h and B_h have only simple zeros on the critical line for |h| >= 1/2, and for all h != 0 under RH. Section 6 restates Lemma 2.1 as: E_h(z) = xi(1/2 + h - iz) is a de Branges structure function for h >= 1/2, and for all h != 0 under RH; h = 1/2 is de Branges's E(z) = xi(1 - iz). The same section cites Conrey and Li as showing de Branges's hypotheses fail for H(xi_chi(1 - iz)), chi trivial or chi_{-4}.
• Lagarias, "Hilbert spaces of entire functions and Dirichlet L-functions", in Frontiers in Number Theory, Physics, and Geometry I, Springer 2006. Quoted from the 2005 paper's section 6: a de Branges space attached to L(s, chi) exists if and only if RH holds for it. Not read.

A citation in this tree is wrong. hunts/rogue_frontier/FRONTIER_MAP.md (territory F, "de Branges / canonical-system route") says "Lagarias (2006) proved RH holds iff E = A - iB built from xi is a Hermite-Biehler function", gives the title as "Hilbert spaces of entire functions and the Riemann zeta function" (2006), and dates the work "(2025)". The checked record: the 2006 chapter's title ends "and Dirichlet L-functions"; the Hermite-Biehler family is in the 2005 Acta Arith. paper, which proves the property for h >= 1/2 unconditionally and for every h != 0 under RH, and does not state the converse (section 3 below supplies it in one line); no 2025 item by Lagarias on this was found. That file is outside this hunt's write scope, so the correction is recorded here for the operator.

2. Target

The structure function, for any entire Lambda with Lambda(1 - s) = Lambda(s) and Lambda(conj s) = conj Lambda(s):

E_h(z) = Lambda(1/2 + h - i z), h > 0,

so for z = x + iy, E_h(z) = Lambda(1/2 + h + y - ix) and E_h(conj z) = Lambda(1/2 + h - y - ix). The Hermite-Biehler defect is D_h(z) = log|E_h(conj z)| - log|E_h(z)|; E_h is Hermite-Biehler when D_h < 0 on Im z > 0. For Lambda = xi:

• if E_h is Hermite-Biehler, A_h = (E_h + E_h^#)/2, which on the line is Re xi(1/2 + h + it), has only real zeros (Lagarias, Lemma 2.2 and Theorem 2.1);
• if E_h is Hermite-Biehler for every h in (0, 1/2], A_h -> xi(1/2 - iz) = Xi(z) locally uniformly as h -> 0, and Hurwitz's theorem gives RH.

The named arithmetic input is the zero-free half-plane that the Euler product supplies: zeta(s) != 0 on Re s >= 1 (Hadamard and de la Vallee Poussin, from the Euler product) gives h >= 1/2, which is Lagarias's Lemma 2.1(1); the strip Re s > 7/8 (OpenAI 2026, kernel-checked here on two kernels, pending human review of its argument, docs/08 section 1.1) gives h > 3/8 (section 3).

How this differs from what Conrey and Li refuted. They take the one structure function E_{1/2}(z) = xi(1 - iz), which is Hermite-Biehler unconditionally, and refute de Branges's positivity conditions on its space: (3.1) Re <F(z), F(z + i)> >= 0 on H(E), and (3.3) on F(1/xi(1 - iz)). The form here asks instead for the Hermite-Biehler property of E_h itself at h < 1/2, a condition on the structure function and not an inner-product positivity on a space. Conrey and Li do not cover it. Section 3 shows what does.

3. Construction: what the Hermite-Biehler property of E_h is (derived, self-reviewed)

(a) It is a zero-free half-plane, exactly. For Lambda as above with a Hadamard product pairing rho with 1 - rho and no exponential factor left over (Lagarias's B* = 0 for xi; assumed, not checked, for the three rivals with the functional equation) and h > 0:

E_h is Hermite-Biehler if and only if Lambda has no zero with Re rho > 1/2 + h.

If: Lagarias's factor-by-factor proof of Lemma 2.1(1) uses only beta - h - 1/2 < 0 for every zero, which is this hypothesis with the functional equation (a zero exactly on Re rho = 1/2 + h gives equality in its own factor and leaves the others strict). Only if: a zero rho = 1/2 + delta + i gamma with delta > h makes E_h vanish at z = -gamma + i(delta - h) in the upper half-plane, where |E_h(conj z)| < |E_h(z)| = 0 is impossible. So "E_h Hermite-Biehler for every h > 0" is RH, and "for a given h" is the quasi-Riemann hypothesis with abscissa 1/2 + h.

(b) The furthest statement the arithmetic input reaches. With Re s > 7/8 zero-free, every zero has Re rho <= 7/8 < 1/2 + h for h > 3/8, so E_h is Hermite-Biehler for every h > 3/8, and Lagarias's proof of Theorem 2.1 then gives: for h > 3/8 and every theta, A_{h,theta} and B_{h,theta} have only simple zeros, all on the critical line, interlacing (the simplicity step needs a zero with real part 1/2 + h > 7/8, which does not exist). Grade: derived, self-reviewed, conditional on the 7/8 strip (kernel-checked, pending human review), so it extends Lagarias's unconditional range h >= 1/2 to h > 3/8. Novelty: not searched beyond the two papers read.

(c) The Euler product is not Hermite-Biehler factor by factor. For one local factor E_p(z) = (1 - p^{-(1/2 + h - iz)})^{-1}, with r_{+-} = p^{-(1/2 + h +- y)}, |E_p(conj z)| / |E_p(z)| = sqrt(f(r_+)/f(r_-)), f(r) = 1 - 2r cos(x log p) + r^2, which exceeds 1 exactly when cos(x log p) > (r_+ + r_-)/2. Since r_{+-} < 1, every p violates the inequality on a set of x of positive density. So the property of E_h does not come from the Euler product locally: it comes only from where the zeros are, through the Hadamard product (pinned by test_local_euler_factor_is_not_hermite_biehler).

4. Tests run and measurements

Grid (dbr_grid.py): h in {1/2, 3/8, 3/10, 1/4, 1/5, 1/10, 1/20}, y in {1/20, 1/5}, x on [0, 100] in steps of 1/2 plus 14.1347 and the rival's off-line ordinate (Epstein: x <= 40), both routes at every point (Epstein's mpmath lattice route on 16 points per h, at 12 to 18 s each), and an Arb column at x = gamma, y = 0.01 .. 0.60. test_de_branges.py: Conrey and Li's numbers, route agreement and the functional equation for each function, the defect's sign at chosen points, the local-factor criterion, and every grid number below.

A second off-line zero of Epstein (1,1,6). The grid's failure at x = 29.5 for h <= 1/4 but not at 3/10 pointed at a zero of offset between 1/4 and 3/10, and Newton's method on the Arb route found it at 0.7979971571787 + 29.5517610986291i (offset 0.2980); the lattice route gives |s(s-1)Lambda_Q| = 9e-47 there against 1.3e-18 at distance 0.01 (test_epstein_second_offline_zero). Measured, by two routes; the record of this tree pinned only the zero at height 16.29; whether this one is in the literature was not searched.

Every positive region sits where (a) puts it: E_h^DH vanishes at y = delta - h with delta = 0.3085 (0.0085 at h = 3/10, 0.0585 at h = 1/4), Epstein's delta is 0.4533, W's 1/4 (boundary case h = 1/4 passes), and zeta(s - 1/5) is symmetric about 1/2 + 1/5, so its defect changes sign at h = 1/5 identically in x. The Arb undecided points are removable points the Arb routes evaluate literally: s = 1 (zeta, DH and Epstein at h = 3/10 or 1/5, y = 1/5, x = 0) and, for Epstein's Fourier-Bessel route, s = 1/2 (h = 1/20, y = 1/20, x = 0, which also makes that h's recorded maximum NaN). The mpmath route handles both (Epstein's lattice route by a circle mean at s = 0 and 1); its values there are negative.

5. The distinguishing step and the rival gate

The step that separates zeta from the rivals is the zero-free half-plane Re s > 1/2 + h, and nothing else: by (a) the defect reads only the real parts of the zeros. The rivals' results on complete hypotheses:

• DH meets every hypothesis of (a) and is not a counterexample to it: (a) predicts its E_h fails exactly for h < 0.3085 at its pinned zero, and the grid shows that. It does not meet the construction's conclusion-producing hypothesis (Hermite-Biehler), so the kill "the DH analogue meets the complete hypotheses" is not triggered. The grid passes DH at h = 3/8 and 1/2, but only on |x| <= 100: DH has zeros in Re s > 1 (Davenport and Heilbronn 1936, quoted, not verified here; li_dh_onset found none in [1, 4] x [0, 120]), which by (a) make E_{1/2}^DH fail somewhere higher. A finite grid stands in here for the whole half-plane, and says nothing above x = 100.
• Epstein (1,1,6), likewise, fails for h below its offsets 0.4533 and 0.2980, each at its own zero.
• W_{1/4} has a scalar Euler product, nonnegative coefficients and the functional equation, and fails for h < 1/4 at every zeta ordinate on the grid: an Euler product alone does not supply the input; its non-unitary local roots move the zeros, as in ccm_arithmetic and li_arithmetic.
• zeta(s - delta) lacks the s -> 1 - s functional equation, so (a) does not apply; its defect sees its symmetry centre directly.

6. Known obstructions

• Conrey and Li cover de Branges's positivity conditions on H(xi(1 - iz)); they do not cover the form here (section 2).
• Reformulation (kill condition 3 of MISSION.md). By (a), "E_h Hermite-Biehler for all h > 0" is RH restated as a family of zero-free half-planes, and the arithmetic input enters only as such a half-plane. The form adds no estimate: every h it reaches is a zero-free strip proved some other way (h >= 1/2 classically, h > 3/8 from the 7/8 strip). It does not reduce to Weil positivity; it reduces to quasi-RH.
• Zero-configuration only. The defect reads the zero locus and nothing about primes (c); it is the de Branges analogue of MISSION.md's dead route "mechanisms that read only the zero configuration".
• Finite grid. No measurement here bears on |x| > 100 (section 5).

7. Disposition (proposed by the builder; the challengers decide)

Recorded as a reformulation under kill condition 3, with a located obstruction: for the Lagarias family E_h, the Hermite-Biehler inequality is exactly the zero-free half-plane Re s > 1/2 + h, so it can deliver nothing beyond the zero-free strips it is given. On ALIGNMENT.md section 5's scale the route to RH through this family is unresolved (it is as hard as quasi-RH for every abscissa above 1/2) and adds no estimate. Conrey and Li's refutation does not apply to it.

8. Furthest true statement, with its grade

For every h > 3/8, E_h(z) = xi(1/2 + h - iz) is a de Branges structure function, and A_{h,theta}, B_{h,theta} have only simple, interlacing zeros on the critical line: derived, self-reviewed, conditional on the 7/8 strip (kernel-checked here, pending human review); unconditional for h >= 1/2 (Lagarias 2005, cited). On the grid, zeta's defect is negative at all 2828 points for h from 1/20 to 1/2 with |x| <= 100 (hardened: Arb ball signs with the mpmath route agreeing to 4.3e-11), and each rival's defect turns positive exactly at its off-line zero (hardened at the points listed).

9. Next step

Nothing further along this family: (a) exhausts it. A structure function whose Hermite-Biehler property is not decided by the zero locus alone would be a different construction: Suzuki's de Branges space of the Weil form (arXiv:2301.00421, which assumes RH) or his screw-function form (arXiv:2606.09096) are the candidates the record names, and either would first have to say which inequality on the screw function would be new. The h-range itself moves only with a new zero-free strip (probe_tower, hunt #136).

Correction (2026-10-26, synthesis session)

Added by the synthesis session of hunt #150, which did not build this mechanism; the text above is left as written. The circularity challenge (challenge/CHALLENGE_circularity.md, findings B2 and B5) shows that two statements above are about the sample, not about the functions:

• Epstein (1,1,6) has more off-line zeros than section 4's table row and section 5's Epstein bullet list. An argument-principle census of [0.55, 1.6] x [0.5, 60.5] on an independent Chowla-Selberg route finds five, not two: besides 0.953260 + 16.290216i and 0.797997 + 29.551761i, also 0.663652 + 33.650101i, 0.809992 + 43.858664i and 0.782379 + 46.960994i (offsets 0.1637, 0.3100, 0.2824), each measured by two routes and a winding count, none enclosed. The zero at 33.650 lies inside this page's grid (x <= 40); the grid's step of 1/2 in x missed its failure region (defect +0.457 at x = 33.6501, negative at the grid points 33.5 and 34.0, h = 1/20, y = 1/20). So the table's "h <= 3/8 at x = 16.290, and h <= 1/4 also at x = 29.5" and section 5's "fails for h below its offsets 0.4533 and 0.2980" should read: by (a), E_h fails below each of the offsets 0.4533, 0.3100, 0.2980, 0.2824 and 0.1637 (and below any offset of a zero outside the census box); the grid sampled two of them. Statement (a) is unaffected.
• DH fails for every h <= 1/2, not only h <= 3/10. By Saias and Weingartner (Acta Arith. 140 (2009), abstract), DH has zeros with real part in (1, 1 + eta), so by (a) E_h^DH is not Hermite-Biehler for any h <= 1/2. Section 5's "the grid passes DH at h = 3/8 and 1/2" is a finite stand-in that the theorem overrides.

The Epstein zeros are recorded in ../RESULTS.md section 9 with their grade.