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.

Correction (2026-10-19): an absolute floor in the DH weight

Added after the original record. Every section after this one is the original text, unchanged; where a number in it is affected, this section says so.

The defect. omega_dh dropped every term n·e^(−πn²x/5) below an absolute floor 10^-(P+8), P the ambient precision, so Φ_DH was exactly 0 wherever e^(−πx/5) fell below that floor: for u > ½·log(5(P+8)·ln 10/π), x = e^(2u). The same construction as hunts/jensen_clock's, where hunt #135's Davenport-Heilbronn gate found it (hunts/jensen_wedge/DH_GATE.md section 1) and its own correction fixed it (PR #315).

The fix. omega_dh now factors out the leading term and stops at 2^-(prec+20) of it, jensen_clock's corrected logic, reimplemented. The original is kept as omega_dh_absolute_floor, for the record and as a planted fault. test_flow_repair.py checks the fixed weight against a cutoff-free 30-term sum past the cut at the pair-1 flow's precision (P = 64, cut at u = 2.788, where the original returns 0) and in the moment ∫u⁸⁰⁰Φ_DH du at dps 75 (the original is off by 1.7e-4 in its log); with the original floor put back, 4 of its 5 tests go red.

What the floor reached here: no quadrature node. DHFlow evaluates the weight at dps + 10 digits, so its floor sits at 10^-(dps+18), and ends its integral where the integrand of its design box (t ≤ 0.12, |Im z| ≤ 1.6) falls to 10^-(dps+12). In every rule a published stage built, that limit stops short of the cut, by 0.0125 in u at dps 40 (validate) closing to 0.0013 at dps 165 (survey pair 9); the default rule would first cross it between dps 250 and 300. So no node ever carried weight 0. The floor did cut the n = 2 term early in a band of u, so the weights at the nodes move with the fix, by at most 5.7e-45 relative (dps 40) down to 1.7e-138 (dps 165), and H and H′ at sampled contour points by at most 9.2e-44 relative (measured with both cutoffs on every published rule).

Re-verification, 2026-10-19. Every stage that evaluates the DH weight was re-run with the fix on GitHub Actions (run 39355357518, RUNS.md) and every field compared with results.json by compare_rerun.py; the report is rerun_2026-10-19.json. 12 stage blocks, 370 fields, 364 identical as printed (only seconds is skipped), 6 changed, all in one figure.

• pair1, precision, lesions, survey pairs 2 to 9: identical in every field. Every t\* in §1 and §5 to its last stored digit, the headline Λ_DH ≥ 0.0576518 (pair 5), the precision spread 0.0, the t = 0 checks of §0 (y₀ to 24 digits, winding defect 8.5e-29, Δ(0) + 4y₀² = 4.7e-29), the slope 9.248 and the lesion rows of §4 stand as printed.
• validate: one published number changes. §0's "relative defect ≤ 4.2e-51 for |u| ≤ 0.5 at dps 50" re-runs as ≤ 7.1e-51 (phi_evenness_defect, relative, at u = 0.1, 0.2, 0.35, 0.5: 2.35e-51, 0.0, 4.15e-51, 3.47e-51 become 7.05e-51, 0.0, 5.53e-51, 6.94e-51; the absolute column moves the same way). Both are rounding, not the floor: at every probe, each side of the check is within 4 units of 2^-169 = 1.3e-51 (the dps-50 rounding unit) of a 120-digit reference, under either cutoff; the fixed sum multiplies by the factored-out leading term, which rounds differently. What it supported, Φ_DH even to working precision, stands. The rest of validate is identical (route agreement 7.2e-41, |c − 1| = 4.2e-42, |a − 1| = 2.0e-18, the ζ sibling 5.4e-42, Φ_DH(0) to 25 digits). results.json's validate block keeps the original values.
• Not re-run: ode and figure. They evaluate no DH weight; they read the polished zeros (from dh_f, no weight) and the PDE t\* values, which re-ran identical, so §3's table stands by construction.

Readers of these numbers: hunts/jensen_clock reads the pair1 and precision blocks; its three stages that were waiting on this were re-run on the re-run blocks and are unchanged (its RESULTS.md, "Correction"). hunts/li_dh_onset reads results.json only. hunts/lambda_dh_bounds builds DHFlow at dps up to 205, also short of the cut, so its flows lose no node either; its own published figures were not re-run here.

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.

7. The v6 census sweep: every off-line pair below height 20,000 (2026-10-30)

Added after the original record; nothing above changes. census_sweep.py lands all 1,268 off-line pairs of the Davenport-Heilbronn v6 census (Nathan Berzai, Zenodo 10.5281/zenodo.23585903, data CC-BY-4.0) with §3's null control, raw numbers in census_sweep.jsonl, pins in test_census_sweep.py. Grade: measured, a float ODE model of zero motion. §3 compared it with the measured flow only up to height 412; above that it is the model alone.

Inputs, no zero search. The window zeros come from the replayed census receipts in data/dh_v6_replay/receipts.tar.gz (25,078 on-line ordinates; window closure exact in all 400 windows, docs/46), the pairs from the census table in the same tarball, both read after the manifest hash check. Completeness is the census's own: docs/46 replayed it by one route, the author's verifier, which stops short of hardened (its section 3). The dynamics are a numpy transcription of probe.ode_null_control, same window ±40, 800 RK4 steps, mirrors, tail and t_end. The full census takes about three minutes on 16 cores (RUNS.md).

Checks. The transcription equals ode_null_control on identical inputs to 1e-13. Pairs 1 to 5 from the receipts reproduce §3's results.json landings to 5e-8 relative with identical zero counts. hunts/lambda_dh_exact landed the same model on windows it built itself (sign scan, located off-line zeros, closed strip counts): its 21 pairs below 10^4 agree to 1.1e-6 and its 19 pairs in [10^4, 2·10^4] to 4.3e-6. Deleting one on-line zero next to pair 5 moves its landing by more than 1e-4, so the reproduction check can fail.

Results. 1,268 of 1,268 pairs land, every one before its isolated y0²/2. Three pairs (v6 ranks 1266 to 1268) are flagged edge: their window runs past the census top, which lacks the zeros above 20,000. All three are shallow, landing near 0.01. The latest landings:

In the ODE model, the pair at height 240.4 lands last of all 1,265 non-edge pairs, so P4's winner stands across the census, extending hunts/lambda_dh_exact's census of the 21 pairs with β > 0.85 below 10^4 (its RESULTS.md section 8.6) to every depth below 20,000. The deepest pair (height 2442.53) has the largest isolated estimate and is shaved 29.4% into second place.

Added at landing, 2026-10-31. This sweep is also the antecedent that hunts/lambda_dh_exact's own P4 ("a complete census to 10^4 raises the best landing") lacked: its RESULTS.md section 8.7 grades it "unresolved, evidence against" because pairs with y0 at most 0.35 were never landed. With every v6 pair below 10^4 landed by the same ODE, the best landing is not raised. That verdict belongs to that hunt and is not changed here.

The deep predictions file. deep_predictions_v1.json (SHA-256 2616116f…) holds a crowding-model refit's predictions for the ten pairs with y0 > 0.37. Its estimand is the flow landing; it is compared here with the ODE landing, which §3 tied to the flow only below height 412.

Corrected at landing, 2026-10-31. This paragraph first called the file frozen and said it was written before any of the ten was landed. The tree supports neither. The file first appears in the commit that compares it, with no timestamp, and neither its source (source_sha256 18010823…) nor the refit's code is committed, so the predictions cannot be reproduced from here. The ODE landings of all ten pairs had been in the tree since 2026-10-26 (hunts/lambda_dh_exact/census_1e4.json and census_runs/full_r2/census_1e5.json), and this sweep reproduces them to 1e-5 (test_census_sweep.py). Read the table as a measured disagreement between a committed file and the ODE, not as a blind test of the model.

All ten predictions are too high and none lands inside its parameter-only 95% band: mean overprediction 9.51%, maximum 29.96%. The bands carry parameter uncertainty only, not model error, so the miss points at the model's form rather than its fitted constant (a reading, not a test).

Scope: one census to height 20,000, a measured maximum over 1,265 pairs, a floor for Λ_DH as in §1 and never its value.

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.