Hunt #122: the second-DH lower endpoint, pushed to the tracked pair's landing time

Base: origin/main at ef4e056 (2026-10-08), worktree branch hunt/forage-2026-10-08. Status: exploratory probe; nothing here is a result (hunts/README.md). Narrow frame s = 1/2 + iz throughout, as in the parent hunt; wide-frame values are the narrow ones times four.

The door, and why this one

hunts/dh_minus_heat/RESULTS.md left the second conductor-five heat constant in the bracket 217/200 < Lambda_minus <= 567009/320000 and named three doors. Its first door is a later rational heat time for the same conjugate pair, information class "local analytic data: Arb enclosures of one value, one derivative, and a disk-wide second-derivative majorant". The scout (collision_scout.json) saw the pair at depth 0.0726 at t = 1.085 and a real root at t = 1.09, and the hunt refused, correctly, to promote that into a collision time.

This door is chosen because it is a precise question with a numerical answer (where does the pair land, and how close to that time can the hunt's own disk instrument decide a non-real zero), it is decidable in seconds with instruments already in the tree (the parent hunt's odd_ball.rouche, docs/37's contour-moment discriminant tracker), and its answer reprices a recorded door: after it, the door is worth exactly t_c - 217/200 for this pair and is shut for this pair, so no later session needs to climb the same ladder.

Runners-up, and why not

1. The parent hunt's second door, a sharper zero strip for the upper endpoint. Already answered in the tree and not consumed: hunts/lambda_dh_bounds/strip2_results.json, field control_tau_minus, decides the minus-function phase abscissa sigma* = 2.38228610898712387... at P = 10^5 on both backends, and STRIP2 section 3.3 proves the phase lemma is the end of that line of argument. The repricing is one subtraction (recorded in RESULTS.md, "The doors"), not a run. Worth about 4.0e-4 of the bracket.
2. A zero-surplus witness at N = 576 for paid_surplus_obstruction. A linear feasibility question, decidable, but that hunt states itself that zero surplus at another cutoff "would only identify C_N = psi(N) there" and supplies no scale-dependent estimate: closing it reprices nothing.
3. Hunt #119's registered predictions for Lambda_DH (first DH function). Open and unadjudicated, but its evaluation phase needs deep zeros above height 600 and a census, which is not a one-hour four-core question, and the contract files say whoever resumes should decide the adjudication first.
4. The third dh_minus_heat door (even versus odd conductor-five kernels). A theorem is asked for; no numerical experiment decides it cheaply.

Open branches checked on 2026-10-08 (git branch -r): depth-bound-selfterm, robin-tfree, weil-c4-s2, weil-propagation, openai-math-release, erdos-counterexample-arm, the palomar series, gate-and-setup, nobody-asked-rule, zeta-lab-research-director. None touches hunts/dh_minus_heat or this question.

What was done

probe.py measures the landing time t_c of the pair by two independent routes (an mpmath double-zero Newton solve with analytic Jacobian at 30 and 50 digits; a float64 contour-moment discriminant tracker on two grids), then climbs five rational heat times below t_c with the parent hunt's Arb Taylor/Rouche disk at four precision and cutoff configurations each. Controls: known values (zeta(2), gamma_1, Xi(0)), the recorded 217/200 disk reproduced, the t = 0 admissibility gate against the pilot zeros and the Hurwitz zero-time identity, two closed-form polynomial landings for the tracker, three instrument lesions, and one shared-layer kernel fault that the two routes cannot see and the Hurwitz identity can. The pilot's second zero is tracked to its own landing as a scout.

Resource boundary

Serial numerics only, about 35 seconds per probe run on one core; no Lean build, no CI, no paid compute, no census, no edit outside this directory and the one case-log entry.

id: dh_minus_landing
question: How far can the dh_minus_heat tracked pair push the strict lower endpoint of Lambda_minus, and at what heat time does it land?
frontier: narrow bracket 217/200 < Lambda_minus <= 567009/320000 (hunts/dh_minus_heat), scout saw the pair at depth 0.0726 at t=1.085 and a real root at t=1.09
proposed_attack: measure the landing time by two independent routes, then decide Arb Taylor/Rouche disks at rational times just below it
dead_routes:
  - promoting the float scout's last off-axis time into a collision statement (refused by the parent hunt)
  - sharpening the integer tail of the prime-phase strip for the upper endpoint (already decided in lambda_dh_bounds/strip2_results.json, control_tau_minus; worth about 4e-4)
  - coefficient domination for the strip (lambda_dh_bounds/STRIP.md, a factor 2.08 weaker than the phase bound)
required_oracles:
  - Arb disk enclosures with explicit theta and integration tails, compared as exact rational endpoints (hunts/dh_minus_heat/odd_ball.py)
  - the Hurwitz-zeta zero-time identity, which shares no theta integral with either tracking route
  - closed-form polynomial landing times (y0^2/2 for an isolated pair, the quartic formula of docs/37) on the same tracker
  - the recorded 217/200 disk and its coarse bounds, rerun
kill_conditions:
  - the two landing-time routes disagree by more than 1e-10
  - the landing time moves by more than 1e-20 between 30 and 50 digits
  - the tracker fails to reproduce a pilot zero at t=0
  - no rational time above 217/200 yields a decided disk on every configuration
  - the second pilot pair lands later than the first, so the same-pair door is not the binding one
agents_may:
  - derive
  - run bounded serial numerics
  - import the parent hunt's instruments unmodified
  - plant faults and record which check catches them
agents_may_not:
  - assign evidentiary status to this hunt's output
  - promote their own claim
  - declare novelty
  - modify hunts/dh_minus_heat, zeta/, ontology/ or harness/
  - state a landing time as the de Bruijn-Newman constant, which needs every pair and the no-creation step