NOTES — what the probe measured

Run of 2026-08-07, probe.py, raw numbers in results.json. Probe language throughout: measured, observed. Everything here is the accurate regime (mpmath floats with measured cross-route defects); no enclosure claims are made anywhere in this hunt.

0. The instrument is telling the truth (stage validate, pair1 t=0)

• Route agreement. H₀^{DH}(z) by Φ_DH-quadrature vs zeta.epstein.completed_dh(1/2+iz) by Hurwitz zeta — two code paths sharing nothing — agree to 7.2e-41 worst over real and complex probes at dps 40. The constants in H₀(z) = c·Ξ_DH(a·z) fitted from the data: |c − 1| = 4.2e-42, |a − 1| = 2.0e-18 (stencil-limited). So (c, a) = (1, 1): the DH flow normalisation is cleaner than ζ's H₀(z) = (1/8)Ξ(z/2) — no pole terms, no factor 8, no z/2.
• Φ_DH evenness (the functional equation in disguise): raw unfolded series both sides, relative defect ≤ 4.2e-51 for |u| ≤ 0.5 at dps 50.
• The same evaluator pointed at ζ's Φ reproduces zeta.heatflow.H_t to 5.4e-42 worst over four (z, t) probes, with an independently built rule (own integration limit, own panel count). Rival-as-validation: the instrument reproduces the sibling before it is trusted on the rival.
• t = 0 moments vs the pinned zero. mp.findroot from the repo's 50-digit OFFLINE_ZERO_* seed moves it by 4.2e-49 (|f| residual 5.3e-77); the contour moments then recover y₀ to all 24 compared digits, N = 2 with winding defect 8.5e-29, and Δ(0) + 4y₀² = 4.7e-29.

1. The headline: the repair times, measured

The backward-heat flow H_t = exp(−t∂²)H₀ pulls each off-line quadruple of the Davenport–Heilbronn Ξ onto the real axis at a landing time t\* — read off as the root of the pair discriminant Δ(t) = 2q₂ − q₁² (contour moments), which is analytic through the collision. Every t\* is a lower bound for Λ_DH := inf{t : H_t^{DH} real-rooted}, the DH analogue of the de Bruijn–Newman constant.

(Pair 1 is the repo's pinned zero; the other seeds are Spira 1994 and Balanzario–Sánchez-Ortiz 2007, each re-polished in-tree by mp.findroot on dh_f with a unit winding check before use.)

Measured lower bound: Λ_DH ≥ 0.0576518, attained by pair 5 — not the famous pair 1, which comes third. Prediction P4 (the crown goes to the height-240 quadruple, the deepest β in the surveyed list) held; the other deep pairs, 6 and 9, land at 0.0447 and 0.0427, well behind.

Scope, stated plainly: nine quadruples out of infinitely many. Nothing here bounds Λ_DH from above; the measured maximum is a floor, not the constant. And the strip confinement −1 < Re s < 2 (pinned by the |aₙ| argument in zeta/epstein.py) means every DH pair sits at y₀ < 1.5, so no single pair can land later than y₀²/2 < 1.125 under dynamics in which every zeta-like neighbour accelerates the landing — an observation about the surveyed mechanism, not a theorem about the sup.

2. Predictions, settled

• P1 (strictly below naive): held, 9/9. Every measured t\* is below its isolated-pair y₀²/2. But P1's quantitative clause ("shave 5–15%, growing with height") was wrong on its own terms and lost to P2: the shave tracks y₀² × local zero density (0.60% for the shallow pair 3, 15.6% for the deep pair 5), exactly what the P2 slope formula implies. Two pre-registered predictions contradicted each other; the mechanism P2 encodes is the one the data kept. (The height clause survives at fixed depth: the three y₀ ≈ 0.31–0.32 pairs shave 7.28% → 12.49% → 14.50% as γ runs 86 → 321 → 412, the density term doing exactly what it says.)
• P2 (slope): held. dΔ/dt at t = 0 measured 9.248 for pair 1 against 8 for an isolated pair; the crowding surplus matches the 16y₀²Σ1/((x−w)²+y₀²) term built from the measured neighbour positions.
• P3 (the null control): held, spectacularly. See §3.
• P4 (pair 5 wins): held. t\*₅ = 0.0576518, inside the pre-registered 0.06 ± 0.005 band.
• P5 (the moral): held. Λ_DH ≥ 0.0576518 sits comfortably inside [0, 0.2] — the interval that bracketed Λ_ζ before Rodgers–Tao proved Λ_ζ ≥ 0 and after Polymath 15 pushed the upper bound to 0.22. Measured in flow time, the counterexample's RH-failure is smaller than ζ's own historical uncertainty about which side of RH it sits on. A structural story of the form "ζ satisfies RH because its Λ is small" is therefore empty: the function where RH is false has a small Λ too. Gate #3, quantified on the flow axis.

3. The null control: the repair clock reads geometry, not arithmetic

For each pair, the probe censused the t = 0 zero configuration in a ±40 window (line zeros by phase-refined sign scan; total strip count by argument principle; the accounting closed exactly in all five windows: line + 2·quadruples = strip, e.g. 49 + 4 = 53 at pair 1), then integrated the bare N-body dynamics ż_k = 2Σ 1/(z_k − z_j) — no Dirichlet series, no character, no conductor, just the measured starting positions — in the collision-safe variable Q = Δ/4 with the universal identity dQ/dt = 2 − 4Q Σ_a 1/((x−a)² − Q).

Sub-0.04% across the board, sign scattering with the truncation knobs (window ±40, density-model tail, RK4 step) — the ~1% target of P3 beaten by a factor of ~25. The flow-repair time contains no information about the Davenport–Heilbronn function beyond where its zeros start. Any entire function with the same zero layout would repair on the same clock. Λ-style quantities measure a configuration's geometry; what would be special about ζ (if RH holds) is that its configuration needs no repair — a restatement of RH, not an explanation. This is docs/18's position-sensitivity lesson meeting docs/09's gate #3 on the flow axis.

4. Lesions: the refusals fire, and the newborn pair hides

• Clipped contour (circle around one member of the pair): refused — contour holds N=1 zeros, expected 2. The moment machinery cannot be silently fed half a pair.
• Grazing contour (radius exactly y₀, passing through both zeros): winding came back ≈ 1.8e9 — grotesquely non-integer — and was refused. Both failure modes are loud, never quiet.
• Post-landing blindness, with a clock. Just after the landing the two newborn real zeros sit closer than the default sign-scan grid (mean_gap/20 ≈ 0.0744 here). Swept through five grid phases (hunt #3's lesson): at t = 1.002·t\* (gap/step = 0.36) only 2 of 5 phases see the pair; by t = 1.03·t\* (gap/step = 1.38) all five do. The repair is invisible to the standard instrument for the first ~1–3% of t\* past the landing, while the contour count N = 2 never wavers. Sign scans bound from below; the argument principle decides — the same moral as hunt #3, now on the other side of a collision.

5. Precision response: the standing rule, passed flat

t\* for pair 1 re-measured at dps 44 / 54 / 70 and contour node counts 96 / 192, bracket tolerance 1e-10: all four runs agree to the last digit (0.0441263445516239; spread 0.0). Δ(0) + 4y₀² = 4.7e-29 against the 50-digit pinned zero. The quantity is pinned; nothing wanders.

6. Literature position (searched 2026-08-07, two passes)

No tabulated de Bruijn–Newman-type constant for the Davenport–Heilbronn function was found; adjacent work exists (generalized Newman constants for other families; the de Bruijn/Newman/Ki–Kim–Lee flow theory; the Rodgers–Tao lower bound and the Polymath 15 upper bound for ζ; Spira's and Balanzario–Sánchez-Ortiz's DH zero computations, which supplied the survey seeds). The measurement itself — Λ_DH ≥ 0.0577, repair times for nine off-line quadruples, the null-control agreement — appears unpublished as of this search. One search is a check, not a survey; the numbers above are measurements made for this tree, with no novelty claim attached.

Calogero–Moser connection. The ODE ż_k = 2Σ 1/(z_k − z_j) that the null control integrates (§3) is formalized as a Calogero–Moser particle system for polynomial zeros under heat flow in Cuenca & McSwiggen, "The Rectangular Finite Free Heat Flow" (arXiv:2606.06859, 2026), and explored for random polynomials in Hall & Ho, "Zeros of random polynomials undergoing the heat flow" (arXiv:2308.11685, 2023). The 0.04% PDE-vs-ODE agreement measured here is an empirical instance of their universality on a non-polynomial (DH) case. Both added to references/papers.md §6.

Standing-checklist accounting

• Rival: this hunt runs on the rival — and the rival-as-validation leg (§0) makes the instrument reproduce ζ's sibling module first.
• Decoy/surrogate: the arithmetic-free N-body integration (§3) is the matched null, and it explains the effect — the honest outcome for a quantity that was always geometry.
• Lesion: run (§4); both refusals fire; the blind window is measured.
• Precision response: run (§5), spread zero.

Disposition

Instrument (probe.py: the generic-Φ flow evaluator, contour-moment pair tracker, collision-safe ODE) retained. No claim promoted; no ledger entry. The headline numbers are measurements about a rival function's zero geometry under a classical flow; nothing here is evidence for or against RH (Littlewood, docs/08), and a measured max over nine quadruples is a floor for Λ_DH, never the value. Candidate for the spine, if anyone wants it: zeta/heatflow.py could grow a Phi-parametric entry point (the DH weight differs only in (q, gamma-shift, coefficients)), but that is a zeta/ change and belongs to a session that wants it, not to this hunt.