DH_GATE: hunt #135's Davenport-Heilbronn rival gate

Nothing here bears on RH. The Davenport-Heilbronn function (DH) satisfies a functional equation of Riemann type and has zeros off the critical line. MISSION.md ("The rival gate, with its expected outcome stated in advance") predicted that the wedge construction reads only the weight's double-exponential decay, which DH shares, and so succeeds for DH as for zeta. A pass shows, measured in this tree, that the construction is shared structure with no RH content. Grade: measured throughout.

Reproduce:

.venv/bin/python hunts/jensen_wedge/dh_gate.py --time-unit   # about 25 s
.venv/bin/python hunts/jensen_wedge/dh_gate.py               # about 8 min, writes results_dh.json
.venv/bin/python -m pytest -q -n0 hunts/jensen_wedge/test_dh_gate.py

0. Verdict: the prediction held

The prediction, written in MISSION.md on 2026-10-18 before any DH number existed and restated item by item in the brief of 2026-10-19 before this probe ran: the construction succeeds for DH exactly as for zeta. That means Holland's sign facts at R_3 and R_4, the sign map of E_F(5), and the [2/2] Pade match of R_5 with all parameters positive.

Graded: held, on every item, on both routes.

Route B (Laplace series, no quadrature) sits inside its stated error at all 21 subgrid points; at those points two routes that share no quadrature agree in sign and resolve it, which is the shape of a hardened reading. This file grades it measured, as phases 0 and 1 did. The contrast holds as required: DH's Jensen non-hyperbolicity was found only at n = 0 (section 7), and at every wedge cell measured here (n = 50 and 3200) DH's Jensen polynomials are real-rooted. The two regions do not overlap.

1. The table, and a floor in jensen_clock's omega_dh (the bug check came first)

The brief named hunts/jensen_clock/probe.py's DH table (dh_nodes, gammas_from_nodes). Built literally at this hunt's precision, jensen_clock's node sets disagreed with each other by 6 to 19 percent at n = 1600 in the scoping runs (RUNS.md). The cause is omega_dh's cutoff: it drops each theta term below an absolute floor of 10^-(dps+8), so the weight becomes exactly 0 once e^(-pi x/5) is below the floor. At the working precision here (dps 75) that is u > 2.859, and 1001 of the 3072 nodes of jensen_clock's default node set carry weight 0. The peak of the n = 3200 integrand sits at u = L_DH/2 = 3.62.

dh_gate.py keeps jensen_clock's construction (its weight, make_nodes and both of its node sets) and changes only the cutoff to a relative one (omega_dh_rel). With it the two node sets, (U, segments, Gauss-Legendre level) = (4.2, 16, 7) and (3.9, 13, 8), agree on log gamma to 4.5e-72 over m = 50..3296, and gammas_from_nodes on the same nodes is reproduced to 7.1e-74.

jensen_clock's own numbers are not touched by the floor in the ranges it used: at its dps 130 and j <= 328 the literal and relative-cutoff tables agree to 1.6e-39, and every contrast cell of section 7 gives the same Delta to 10 digits on both. Its largest table (stage pair2, dps 180, j <= 400, node set (4.2, 18, 7)) agrees with the relative-cutoff table to 1.6e-63. Its stage precision also ran at dps 110 (j <= 280) and dps 160 (j <= 360); those two tables were not checked here. jensen_clock is outside this hunt's write scope, so the floor is recorded here as an observation, not fixed there.

How much accuracy the construction needs: an alternating error of 1e-12 in log gamma at n = 400 already leaves Holland's system with no solution in the positive class, and 1e-14 flips the sign of the Pade model's R_6 defect (test_dh_gate.py, planted faults). The literal table's 1.7e-4 at n = 400 would not have produced a wrong sign; it would not have solved at all.

2. Holland's sign facts on DH (phase 0, sections 4 and 5)

L_DH is the saddle of DH's leading moment, (pi/5) e^L = 3/4 + 2n/L. It plays the part of Holland's L_n because log gamma_DH(x) = -log Gamma(x + 1/2) + log I_1(x) + (linear in x) has Holland's eq. (14) shape, and phase 1's reduction (RESULTS.md 11.1) reads the weight only through h''(x) ~ (2/L)/x. At equal n, L_DH is larger than zeta's L_n (7.25 against 5.85 at n = 3200), so DH sits further along every 1/L approach.

Every fitted parameter is positive and C > D at every n; the largest matching residual of any fit is 7.8e-67. Route B against route A on log R_j: at most 8.0e-14 against a stated 1.0e-10 at n = 50, and 3.7e-26 against 1.3e-20 at n = 3200. Route B's m >= 2 theta terms are 2.6e-20 of the total at n = 50 and 9.8e-34 at n = 100, and are carried anyway.

3. The sign map of E_F(5) over (theta, n) (phase 1, section 10)

With D = theta B_J and phase 1's solver (family_map, unchanged), route A:

The shape is zeta's: negative below theta_star, a positive window of width about 1/(6 L) (zeta: 0.145/L to 0.166/L), A passing infinity at theta_inf, and theta_star rising to 1 like 1 - 2/(3L). Route B, by direct saddle evaluation, finds theta_star equal to route A's to 2.3e-13 to 4.7e-13 at n = 100, 200, 400, 800, 1600 and 3200.

4. The [2/2] Pade point (phase 1, sections 11.3 and 11.4)

At every subgrid n the crossing theta_star equals D/B_J of the direct four-parameter solve to all 12 printed digits, so on DH too phase 0's crossing is the Pade point. Route A, every n: the solve succeeds with A > B > 0 and C > D > 0, and its R_6 defect E_S(6) is negative at all 3151 n, with route B agreeing in sign and resolving it at all 21 subgrid n.

5. Large n, route B alone

Working precision 5 log10 n + 50 digits, as in phases 0 and 1. The spread between six and five Laplace corrections is at most 1.5e-22 relative on any defect quoted (n = 1e4) and falls fast with n.

Every column moves monotonically toward the limit Holland states or phase 1 derives. At n = 1e20 the DH values sit within 0.5 percent of zeta's at the same n (zeta: 0.9088, -10.46, 0.6587, -1.872 in the matching columns), and the O(1/L) coefficient of the first column, (1 - n^2 L D_n/2) L, is 3.822 for DH against 3.816 for zeta. Measured on eight points, not derived.

6. DH's Jensen polynomials at the wedge's measured cells

Phase 1's root-geometry instrument (root_geometry, unchanged) on DH, from a dps-260 block of the same table (node set (4.2, 16, 8), agreeing with the dps-60 table to 7.8e-72 at n = 3200 and 1.2e-73 at n = 50), each polynomial evaluated at dps 60 + 2d:

(d = 12 and 48 at n = 3200 are in the JSON and sit between their neighbours.) Every DH Jensen polynomial checked here is hyperbolic, and the multiplier the stability step needs small is small. These J^{d,n} are hyperbolic for zeta by GORTTW Cor. 1.3; for DH nothing guarantees them, and they are measured.

7. The contrast: DH's non-hyperbolicity sits at n = 0

From hunts/jensen_clock/results.json: the pair of DH zeros at height 85.7 lands in J^{d,0} at d* = 20785.13 (relative spread 8.9e-5 over four quadratures), the pair at height 114.2 at d* = 146365.5, and at n = 1 the first pair is already on the real axis at d = 3e4, 1e6 and 1e8 (contour count 1, stage map2). Recomputed here with jensen_clock's instrument (pair_delta on the damped series) on both tables at dps 130:

So the measured non-hyperbolic cells of DH have n = 0; the wedge cells measured here have n >= 50, and every DH Jensen polynomial checked there is real-rooted (section 6). No overlap. It is structural: Holland's wedge n^3 log^2(n+2) >= K d^5 excludes n = 0 for every K > 0 and d >= 1, and so does every wedge of this shape, which is exactly why it cannot see the region where DH fails.

8. What this shows and what it does not

• Shows (measured): every step that phases 0 and 1 built for zeta goes through for DH, over the same shift ranges, on two routes, with the same leading constants once L is read from DH's own weight. In this range the construction cannot tell an RH-true function from an RH-false one. That is a measured, in-tree form of Farmer (Adv. Math. 411 (2022)
• and GORTTW Remark 3, for this construction.
• Does not show: anything about phase 1's conditional chain (H1, H3) for DH, though its Lemmas A to C use no property of the weight; any proof; any statement about RH. A finite computation is not a uniform statement. The gate is not one of MISSION.md's kill conditions and changes none of their statuses.
• Certainty ladder. The result is the expected one, and it still got the bug checks first: the table's own failure (section 1) was found and worked around before any construction number was read, the two node sets and route B check the table, and the planted faults of test_dh_gate.py (the floor put back, an alternating error in log gamma at two sizes, sign flips in the record) each turn the gate red.

9. Files

• dh_gate.py: the DH table (route A), the Laplace route B with DH's constants, the construction and the theta map reused from probe.py and probe_phase1.py unchanged, the root geometry, the n = 0 contrast and the verdict function.
• results_dh.json: every number above, with the floor record and the verdict.
• test_dh_gate.py: instrument tests, a live recomputation at n = 400 on both routes, planted faults (the floor put back, an alternating error in log gamma, sign flips in the record), and the pins.
• RUNS.md: unit timing, estimate and one manifest per run.