Hunt #122: the tracked pair lands at t_c = 1.0876360002295869..., and the lower endpoint reaches 1.0876359

Status: exploratory probe, complete. Nothing here is a result (hunts/README.md). Narrow frame s = 1/2 + iz throughout; the parent hunt's wide-frame values are the narrow ones times four. Numbers are in results.json and pinned by test_dh_minus_landing.py. Reproduce from the repository root:

.venv/bin/python hunts/dh_minus_landing/probe.py # about 35 s, one core .venv/bin/python -m pytest -q -n0 hunts/dh_minus_landing # 15 tests, under 3 s

1. The question and the answer

hunts/dh_minus_heat left 217/200 < Lambda_minus <= 567009/320000 and named as its first door a later rational heat time for the same conjugate pair, the one its scout followed from z = 8.918 + 1.809i at t = 0 to depth 0.0726 at t = 1.085. The question here: where does that pair land, and how close to that time can the parent hunt's own disk instrument still decide a non-real zero?

Measured landing time (two independent routes):

t_c = 1.087636000229586932217964154453984835461141904 (route A, 50 digits) x_c = 7.539944242167335224755075792589912656790177113 (the double real zero)

Route A (mpmath tanh-sinh quadrature, Newton on the double-zero system with analytic Jacobian) gives the same t_c at 30 and 50 digits to 4.6e-26, with residuals 8e-53 and 2.7e-54 in H and H'. Route B (float64 Gauss-Legendre grid, contour-moment discriminant Delta(t) = (z_1 - z_2)^2, winding number required to be exactly 2) gives 1.0876360002296344 on 480 nodes with a 128-point contour and 1.0876360002296646 on 960 and 256, both within 5e-14 of route A: the float64 limit, since H'' at the double zero is only 0.0395. Grade of t_c: hardened (two independent float routes agree); it carries no enclosure and no claim about any other pair.

Enclosure-carrying lower endpoint. Five rational heat times below t_c, each with the pair's upper zero polished to 14 digits and the parent hunt's Taylor/Rouche disk (odd_ball.rouche, radius 1/10^7) decided positive on all four of its configurations (96/128/160 bits, theta cutoffs 24/26/28, integration cutoffs 4 and 7/2):

The written margin is the exact rational r abs H'(c) - abs H(c) - M2 r^2/2 from the coarse column bounds, which are in the safe direction of every configuration's dyadic endpoints (test). The Taylor/Rouche argument and its tail accounting are the parent hunt's RESULTS sections 2 and 3, unchanged; the disk at 10876359/10^7 therefore places one simple non-real zero of H_t at that time, and the parent's section 5 (the set of real-rooted times is a closed upper ray, Dobner/de Bruijn) turns that into

10876359/10000000 = 1.0876359 < Lambda_minus <= 567009/320000 (narrow), 10876359/2500000 = 4.3505436 < Lambda_minus_wide <= 567009/80000 (wide).

Grade of the lower endpoint: enclosure-carrying at the numerical step (Arb balls with every theta and integration tail charged, exact rational margin comparison); the surrounding argument is ordinary and model-reviewed only, as the parent hunt states.

Pricing the door. The pair moves the recorded lower endpoint by t_c - 217/200 = 2.636e-3, which is 0.38% of the recorded gap 567009/320000 - 217/200 = 0.6869. The ladder stops 1.0e-7 below t_c; nothing in the instrument stops it earlier (section 4), and nothing below t_c is worth more than that 1.0e-7. For this pair the door is shut.

2. What the pair does after landing, and the other pilot pair

Measured, float route only. After t_c the discriminant is positive and the two real zeros keep separating: Delta = 0.0080 at t_c + 10^-3, 0.098 at t = 1.1, 0.49 at t = 1.15 (separation 0.70), with the mean moving slowly left from 7.5399. Near t_c the measured Delta grows like 8 (t - t_c), which is the isolated-pair law. This is a measured absence of re-merging on [t_c, 1.15], not the no-creation step; the parent hunt's argument never needed that step for the lower bound and this hunt does not supply it.

The pilot's second zero, s = 1.9437 + 18.899i (depth 1.444 at t = 0), tracked by the same two routes, lands at

t*_2 = 0.63095343954 (route B, 480 nodes), 0.63095343961 (route A, 30 digits),

the routes differing by 6.7e-11 and the 960-node grid by 1.0e-10: the float conditioning is 1600 times worse here because H'' at that double zero is 2.45e-5. It lands 0.457 before the first pair, so among the two pilot zeros the first is the binding one. The pilot used four seeds below height 20 and is not a census; whether a pair at greater height lands later than 1.0876360 is the open question that bounds this hunt's reach (section 4).

3. Controls, in the four roles of the standing checklist

1. Rival. Not applicable in the battery's sense: the subject is itself the second Davenport-Heilbronn rival, and no structural statement about zeta or RH is made. The structure-matched comparison is the parent hunt's: the plus function's threshold is at most 1/2 and has no pair above it to track.
2. Decoy. No arithmetic effect is claimed, so no null is owed.
3. Lesions, each caught by a named check. A contour of radius 0.05 at t = 1.08 (pair depth 0.124) returns winding 0 and the tracker refuses. A centre displaced by 10^-4 (a thousand radii) at 217/200 is not decided. A theta cutoff of nmax = 2 is not decided. The shared-layer fault: scaling the residue-2 coefficient by 1.01 in both routes moves the landing time to 1.0206817207544 (a shift of 0.067), and the two routes still agree with each other to 1.4e-13; the zero-time Hurwitz identity H_0(z) = -i F(1/2 + iz), which shares no theta integral with either route, reports defect 1.2e-3 against 4.9e-31 for the true kernel. Agreement of the two routes is therefore evidence about the integrators, and the identity is what vouches for the kernel; both ran.
4. Precision response. Route A: 4.6e-26 between 30 and 50 digits. Route B: 3.0e-14 between the two grids. Disks: every margin is decided at 96, 128 and 160 bits and at both integration cutoffs, with identical dyadic margins across the four configurations at each rung.

Positive controls on known values, through the same libraries: zeta(2) against pi^2/6 (defect 0), gamma_1 against 14.134725141734694 (2.1e-16), Xi(0) against 0.4971207781 (8.8e-11, the reference's own precision). The recorded 217/200 disk is rerun at 128 bits and meets its coarse bounds (abs H < 10^-14, abs H' > 28/10000, M2 < 19/10). The tracker on exact polynomial heat flows returns y_0^2/2 = 0.18 for an isolated pair to 0 and the quartic closed form 0.10041797146407168 to 1.4e-17 (docs/37, four-root control). The t = 0 gate reproduces both pilot zeros to 1.8e-13 and 7.4e-11.

4. Kill conditions and what was not done

None of the five kill conditions fired: the routes agree (4.7e-14, threshold 1e-10); t_c is stable under refinement (4.6e-26, threshold 1e-20); the t = 0 gate passes; five rational times above 217/200 are decided; the second pair lands earlier.

Not done, and why:

• No census of non-real zeros of D_minus at greater height, so no statement that this pair is the last to land. Lambda_minus is at least the supremum of all landing times (docs/37, "Landing times are unconditional lower bounds"), and the pilot found two zeros from four seeds. A census is a different information class (section 5) and was outside this hunt's budget.
• No enclosure of t_c itself. A ball enclosure of the double zero would need an interval Newton step on the 2x2 system; the parent's disk instrument encloses simple zeros only. The ladder makes this unnecessary for the lower bound: any rational t < t_c with a decided disk is a bound, and the last rung is 1.0e-7 below.
• No change to the upper endpoint. The parent's second door is priced in section 5 from a number already decided in hunts/lambda_dh_bounds; it was not rerun here.
• No novelty search beyond the tree. The landing-time instruments are docs/37 entries from flow_repair and lambda_dh_exact; the disk is the parent's. Bombieri and Ghosh (2011, section 6) publish the minus-function phase abscissa to ten digits, which strip2_results.json records; no published heat constant for this function was searched for here.

Nothing here bears on RH (docs/08); the subject is a rival function.

5. The doors

Active constraints at the optimum

The quantity is the strict lower endpoint of the Lambda_minus bracket, and "optimum" means the last rung the ladder can decide. Ranked by what binds:

1. The landing time t_c = 1.0876360002295869... of this pair. Binding absolutely: no disk at t >= t_c can decide a non-real zero of this pair because it has none. Shadow price of the whole door for this pair: 2.636e-3, now spent to within 1.0e-7.
2. Centre accuracy against the pair's depth. At t_c - delta the depth is measured as y = 1.42 sqrt(delta) (0.1238 at delta = 7.6e-3, 4.5e-4 at 1.0e-7), abs H'(c) falls in proportion (0.00286 to 1.8e-5 on the ladder), and the margin r abs H'(c) - abs H(c) - M2 r^2/2 falls with it: 2.9e-10 at 217/200, 1.8e-12 at the top rung. The saturation slope is linear in sqrt(delta); at delta = 10^-9 the margin would be about 1.8e-13 against a remainder 9.5e-15, still decidable with a 14-digit centre. The instrument does not bind before t_c does.
3. Every other pair. The second pilot pair lands at 0.631 and binds nothing. A pair landing after 1.0876360 would move the endpoint, and the hunt has no instrument that says whether one exists.

Frozen-constant inventory

The information class of each door

• This door (same pair, later time) stays inside local analytic data: one value, one derivative, one disk-wide majorant of H_t near a known zero. Its ceiling is t_c, measured and hardened, not enclosed. Shut for this pair to within 1.0e-7; a kernel-checked or enclosed t_c would change the grade, not the number.
• Other pairs (a census). Requires reading more: the non-real zeros of D_minus(s) in a box of the critical strip, by an argument-principle count (lambda_dh_bounds has the chord-tube winding instrument), then each one's landing time by the two routes here. The strip half-width 1.882 says how deep a pair can sit at t = 0; de Bruijn's y_max(t) <= sqrt(Delta^2 - 2t) gives no useful cap here since Delta^2/2 = 1.77 exceeds every landing seen. Cost unknown; the minus function's zeros at height 9 and 19 were found in 0.15 s each at 40 digits, so a census to height 100 or so is cheap to scout, and its reach, not its cost, is the question.
• The upper endpoint. Its information class is the strip half-width alone, which de Bruijn's Theorem 13 turns into Delta^2/2. The parent's second door, a sharper strip, is already decided in the tree and not consumed: hunts/lambda_dh_bounds/strip2_results.json (control_tau_minus) decides the minus-function phase abscissa sigma* <= 2.38228610898712387205 at P = 10^5 on both backends (Bombieri and Ghosh 2011 give 2.3822861089), so the upper endpoint can drop to (sigma* - 1/2)^2/2 = 1.77150049804..., from 1.771903125: worth 4.03e-4, 0.06% of the gap, and STRIP2 section 3.3 proves the phase lemma is the end of that line (the modulus constraint is free at the argument-maximising point). Repriced, not rerun. Anything further for the upper endpoint needs a different class: either a theorem sharper than Delta^2/2 for a strip of known zero density, or the Polymath-15 route of tracking every zero below some height plus an unconditional argument above it. Both are larger than this bracket.