Results: the Jensen clock

Status: probe, complete. Instrument kept; no claim promoted; the headline is a measured dictionary, finite Jensen degree acts on the counterexample's off-line pair as de Bruijn–Newman heat with t_eff = |x₀|/(8d), plus the corollary that textbook-degree Jensen scans are structurally blind to an actual RH violation, and a null control showing the clock reads zero configuration, not arithmetic.

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 theta term below an absolute floor 10^-(dps+8), so the Davenport-Heilbronn weight was exactly 0 wherever e^(-πx/5) fell below that floor: for u > ½·log(5(dps+8)·ln 10/π), which is u > 3.035 at dps 110, 3.113 at dps 130, 3.211 at dps 160 and 3.268 at dps 180. Every DH table this hunt built had zero-weight nodes (815 of the 3072 default nodes at dps 130). A silent zero where the weight is small, not an error. Found by hunt #135's Davenport-Heilbronn gate, hunts/jensen_wedge/DH_GATE.md section 1, which measured it where it bites: at dps 75 (cut at u > 2.859, 1001 of 3072 nodes zeroed) log γ is off by 1.7e-4 at m = 400 and by 832 at m = 3200. DH_GATE.md also checked this hunt's dps-130 and dps-180 tables.

The fix. omega_dh now factors out the leading term and stops at a relative cutoff, 2^-(prec+20) of it: the logic of hunt #135's omega_dh_rel, reimplemented here. The original is kept as omega_dh_absolute_floor, for the record and as a planted fault. test_jensen_clock.py checks the fixed weight against a cutoff-free 30-term sum at dps 75 (log γ(400) agrees to below 1e-65) and goes red on the original floor (1.72e-4 off, 1001 zeroed nodes).

What the floor did to this hunt's tables. The moments used here (j ≤ 400) are carried by u below the cut, so each table moved by a small relative amount, largest at the top index (node sets as (U, segments, Gauss-Legendre level)):

Stages planted and li use no DH weight.

Re-verification, 2026-10-19. Each stage below was re-run with the fix and every field compared with results.json (measured; the comparison script is not committed, the precision re-run is):

• precision (the dps-110 and dps-160 tables DH_GATE.md did not check, plus both dps-130 node sets): identical in every digit. d\* = 20786.973402454 at dps 110 and 20785.132775918 in the other three runs, root Im at d = 30000 to 20 digits, spread 8.86e-5. P7 stands as printed. results.json's precision block is now this re-run; only seconds and a new omega_dh_cutoff field differ.
• dstar, map2: identical in every field. d\* = 20785.13, the 0.097% clock agreement and the 36-order truncation margin stand.
• trajectory, pair2: every root and every number derived from them identical. Only the size of the final Newton step changes, at the working floor (1e-41 to 1e-43 at dps 130, 1e-54 to 1e-56 at dps 180); this page quotes none of them.
• validate: one published number changes. "Two unrelated quadratures agree on the DH γ-table to 2.6e-41" (gamma_two_quadratures_max_rel_defect = 2.63e-41) re-runs as 3.96e-130. The printed figure measured the floor, not the quadrature: the two node sets lose 1.15e-40 and 1.41e-40 of the top moment to it, and 2.6e-41 is the difference. With the fix the two node sets agree at working precision, so the point it supported (the γ-table is quadrature-converged far below anything the instrument reads) holds more strongly than stated. newton_last_step also changes, at the working floor (3.62e-43 to 4.01e-42). The rest of validate is identical: the undamped root against hunt #4's pair to 40 digits, 86.69 cancellation digits, ζ's 1.3e-51. results.json's validate block still holds the original values.
• additivity, shift, additivity2: not re-run (about 27 minutes as recorded). Their flowed tables move by at most 1.9e-39 (t = 0.05 is above every positive flow time they use; negative times shrink it), the same size as the unflowed tables whose roots and d\* re-ran unchanged to 14 to 40 digits, and this page quotes at most 9 digits from them. That is a measured bound on their input, not a re-run of their output. (The bullet as first written; superseded by the next one.)
• additivity, shift, additivity2: re-run 2026-10-19, identical in every field. Re-run on GitHub Actions with both fixes in place, this hunt's omega_dh and hunts/flow_repair's, against flow_repair's pair1 and precision blocks re-run with its corrected weight (themselves identical to the committed ones). Compared field by field with results.json: additivity 13 fields, shift_ladder 20, shift 31, additivity2 7, all 71 identical as printed; only seconds differs. P6's 0.055% and 0.030%, Q1's c₀ = 0.16278, c₁ = 0.18260 and overshoot 3.69, and Q5's defect 1.46e-5 stand as printed, and so does map2, which reads shift's g1. Run and estimate in hunts/flow_repair/RUNS.md, report in rerun_2026-10-19.json.

hunts/flow_repair/probe.py's omega_dh, behind the PDE targets read here, has the same absolute floor; it is not changed or checked here. (Superseded 2026-10-19: fixed there, and every flow_repair stage that evaluates the weight re-run; the targets read here did not change. See hunts/flow_repair/NOTES.md, "Correction".)

Everything below is the accurate regime (mpmath floats, measured cross-route defects). Raw numbers in results.json; predictions were pre-registered in MISSION.md; PDE-side targets are hunt #4's measurements (hunts/flow_repair/results.json), reused, never recomputed here.

The object

For the Davenport–Heilbronn weight Φ_DH, E(x) = Σ γ(n)xⁿ/n! with γ(n) = n!/(2n)!·M₂ₙ puts the completed function's zeros at x = (ρ−1/2)²: negative real when ρ is on the critical line, off the axis otherwise. The degree-d Jensen polynomial obeys the exact identity J^{d,0}(x/d) = Σ γ(j)/j!·Π_{i<j}(1−i/d)·x^j, and the binomial damping Π(1−i/d) ≈ e^{−j²/2d} is a Gaussian coefficient multiplier, de Bruijn's smoothing, applied by the degree itself. Read at the cosh-series saddle j ≈ zu/2 it matches the flow multiplier e^{tu²} at the pair's image x₀ with

t_eff(d) = |x₀| / (8d).

What was measured

• Instrument first (stage validate). The ζ pipeline reproduces zeta.li.xi_taylor_coefficients (itself twice-derived) to 1.3e-51 relative over n ≤ 40; two unrelated quadratures agree on the DH γ-table to 2.6e-41; the undamped Newton root reproduces hunt #4's polished pair 1 to 40 digits. Cancellation at the pair is 86.7 digits, measured, and the reason a dps-15 scan could never see any of this.
• The dictionary, pointwise (P1: holds, with its edge measured). Under t_eff = |x₀|/(8d) the Jensen root trajectory tracks the PDE pair trajectory at 7.8e-8 relative (d = 10⁹) through 4e-4 (d = 4·10⁴, i.e. 52% of the way to landing), degrading to 1.2% at t_eff = 0.0399 and 13% at the last pre-landing rung, both curves are near-vertical there, so a fixed dictionary defect inflates in Im x. The drift constant C = lim d·(Im x₀ − Im x(d)) = 595021 against the flow-side composition 590652: 0.74%, inside P1's 2%.
• The clock adjudicates (P2: holds). The landing degree for pair 1 is d\* = 20785.13, i.e. t\_J = 0.0441690, against the PDE landing t\ = 0.0441263, 0.097%, while the isolated-pair value 0.0475914 is 7.2% away. The Jensen clock reads the interacting flow, neighbors included, not the naive pair formula.
• Scaling across pairs (P3: holds, 10× sharper than asked). Pair 2 (γ ≈ 114.16) lands at d\* = 146365, 7.04× pair 1's degree, and its clock reads 0.0111308 vs the PDE 0.0111296: 0.011%.
• Specificity (P4: holds). ζ through the identical pipeline, no plant: Newton from the DH seed collapses onto the real axis (relative Im 1.2e-200) and the winding box at the pair location counts 1.3e-63.
• The lesion, and the null that explains it (P5: qualitative content holds; the mission's quantitative guess was wrong and is recorded as such). A pair planted into ζ's γ-table at exactly x₀ (an exact three-term recurrence) is detected, with d\* = 21046 → clock 0.0436212. MISSION.md predicted this would read close to the isolated value 0.0476, that prediction failed (8.3% away), because the mission's neighborhood estimate forgot that ζ's ordinates appear doubled in this x-plane, so the plant is not isolated: it has a line neighbor at z-distance 0.96, comparable to DH's own spacing. The correct configuration-only prediction, hunt #4's arithmetic-free N-body dynamics ż = 2Σ1/(z−z′), integrated for the plant's actual zero configuration in collision-safe (x_c, Q) variables, gives 0.0435805, and the Jensen clock read 0.094% away from it. Same x-plane position as DH, different neighbor field: the two clocks separate by 1.1%, and pure zero geometry predicts each. The clock reads configuration, not arithmetic, hunt #4's null result, now on the coefficient side.
• Two clocks add (P6: holds). Flowing the γ-table to time t before damping: t_land(d) + |x₀|/(8d) = 0.0441507 at d = 6·10⁴ (0.055% from t\*) and 0.0441398 at d = 1.2·10⁵ (0.030%).
• Precision response (P7: holds). d\* is identical to the printed 14 digits across dps 130 → 160, series cutoff 320 → 360 and two quadrature geometries; the deliberately underpowered dps-110 run moves it by 8.9e-5 relative. The truncated-series detector's validity margin at the final contour is 36 orders (tail bound 1e-64.6 vs min |F| = 1e-28.5).

The corollary worth keeping

A blind hyperbolicity scan over d ≤ 32, all shifts n ≤ 250, sees nothing on the counterexample, measured here before the targeted instrument was built. The dictionary says why that is structural, not a matter of effort: degree d carries effective flow time |x₀|/(8d), and any d below ~2·10⁴ has already flowed the height-85.7 violation past its landing before the polynomial is inspected. Finite Jensen hyperbolicity at textbook degrees is not a weak detector of off-line zeros; at those degrees it is the wrong side of the landing. Conversely the targeted detector (winding box in the x-plane at the pair image) does separate the rival from ζ at d > d\*, a position-sensitive instrument in the sense of docs/18, and, by the N-body null above, exactly as arithmetic-blind as a position detector should be: it says where zeros sit, not what arithmetic put them there. So nothing here distinguishes ζ structurally, and nothing here is evidence about RH.

Phase 2: the shift axis, and the map that collapsed

Registered in MISSION.md after phase 1 closed; measured afterwards. The shifted Jensen polynomial is the damped n-th derivative, so the shift axis asks what differentiation in x does to the pair, on the same clock. Raw numbers in results.json (shift_ladder, shift, map2).

• Q1 (per-step clock): the direction and configuration claims held; the registered numeric bracket failed and is recorded as such. One differentiation carries the pair past its landing, overshoot factor 3.69: c₀ = 0.16278 of flow time against the violation's entire budget t\* = 0.04413. Measured by letting the heat flow run backward (t < 0) until the pair re-lifts in E′ (t_land = −0.11867) and E″ (−0.30126); the warm-started Newton predicate also tracked the landing's position drift (Re −7344 → −7462 → −7601), which the contour-count predicate could not. The measured saddle (u₀ = 2.1004, j\* = 90) gives the heuristic 1/(2u₀²) = 0.11334, right order, 44% low; the registered wide bracket [0.005, 0.05] for c₀ missed entirely. The per-step clock is also not constant: c₁ = 0.18260, 12% larger, the saddle drifts as the function is differentiated.
• Q3 (the map): all six boundary cells agree with the additive budget rule, and the map degenerates. Detection flips between d = 19000 and 22000 on the n = 0 row, straddling phase 1's d\* = 20785; every tested n = 1 cell (d = 3·10⁴, 10⁶, 10⁸) is blind, exactly as g(1) = 0.163 > t\* predicts, with the window holding a lone real Rolle interlacer where the pair used to be. Since g(1) exceeds every landing time hunt #4 measured (max 0.0577, height-240 pair), the budget rule says all nine known DH off-line pairs are invisible to every shifted Jensen polynomial J^{d,n} with n ≥ 1, at every degree, measured directly here for pair 1, predicted by the clock for the rest. The GORZ direction, fixed degree, growing shift, the direction in which hyperbolicity is a theorem for ζ, is, for detecting this class of violation, the maximally blind direction: the first step of it already erases 3.7× more evidence than the whole violation contains.
• Q2 (three clocks add): tested only in the weak form, stated plainly. The map cells' predictions use the additive budget |x₀|/(8d) + g(n) and all agree with measurement, but g(n) was itself measured through the flow at d = 10⁸, so this is consistency across the degree axis, not the registered independent 1% additivity test. That stronger test was not run; nothing here should be read as having passed it.
• Q4 (specificity): held. ζ through the shifted cells: winding ~1e-95.

Phase 3: the falsifier and the trichotomy

Registered in MISSION.md after phase 2 closed; measured afterwards. Raw numbers in results.json (additivity2, li).

• Q5 (strong additivity: the falsifier passed). Re-measuring t_land(n = 1) at d = 10⁶: the additive budget predicts −0.11957629 from the d = 10⁸ measurement and the degree-budget difference alone, no free parameters, and the measurement gives −0.11959090. Defect 1.46e-5, inside the registered ±1e-4 bar; as a fraction of the budget shift itself it is 1.6%, consistent with the dictionary's percent-level 1/d corrections. Phase 2's Q2 gap is closed: the three knobs (degree, shift, flow) spend one budget.
• Q6 (the trichotomy: measured). The planted symmetric quadruple at ρ = 0.8 + 2.5i drives the Bombieri–Lagarias Li sum negative first at n = 95, inside the registered [80, 160], with exactly the predicted shape: one oscillation trough (n = 95–98, period ≈ 16.1, amplification r = 1.04661 per step), positive again at n = 99, an initial-segment blindness only. One auxiliary statistic failed as coded and is recorded as such: the envelope-crossing indicator compared 2rⁿ against λ_n(ζ) itself, which is degenerate at small n (λ₁ ≈ 0.023), so it fired at n = 1; the registered onset interval came from the amplitude-vs- background estimate and held, but the coded indicator is useless and should not be reused. The same formula pointed at DH pair 1 (r − 1 = 4.2e-5) puts its Li onset at n ≈ 3.3·10⁵, with the stated hedge that the background is the ζ-shaped asymptotic, so this is an order-of-magnitude claim, not a sharp one.
• Q7 (specificity): held. Unplanted, min λ_n(ζ) over n ≤ 200 is +0.0231 (= λ₁), all positive.

The trichotomy, stated. Three coefficient-side hyperbolicity/positivity detectors, one violation, three blindness geometries:

The two Jensen axes destroy the evidence before reading it; the Li axis preserves and amplifies it but pays an exponentially long wait. None of the three blindnesses is a matter of effort, and each has a measured or formula-given boundary. Nothing in this table distinguishes ζ structurally (the null controls read configuration throughout), and nothing in it is evidence about RH.

Prior-art hooks

Gaussian coefficient multipliers and their zero-realifying character are classical (Pólya–Schur multiplier sequences; de Bruijn's e^{−λD²} smoothing; Turán), and Griffin–Ono–Rolen–Zagier obtain hyperbolicity of J^{d,n} for fixed d, n → ∞ through a Hermite/heat limit in the shift.

A short networked pass (2026-08-11) sharpened two hooks:

• The degree/flow/Jensen triangle is qualitatively old. Csordas, Norfolk and Varga (Numer. Math., 1988) obtained Λ ≥ −50 precisely by exhibiting a Jensen polynomial of the flowed function with nonreal zeros, and the successor line (Csordas–Ruttan–Varga and onward, through te Riele and Norfolk–Ruttan) explicitly abandoned Jensen polynomials for direct tracking of zeros of F_λ because the degrees required were impractically large. The dictionary measured here, the degree must outrun the pair's remaining flow distance, d > |x₀|/(8(t\* − t)), is a quantitative law of exactly that documented inefficiency. Whether the constant |x₀|/8 appears anywhere in that literature is not known to this tree; no novelty is claimed, and a proper literature pass is the stated next step for anyone wanting to promote it.
• The shift axis lands in the modern differentiation-as-PDE line. Steinerberger's nonlocal PDE for zeros under repeated differentiation and its rigorous treatments (Hoskins–Kabluchko; Kiselev–Tan) are about real zero densities; phase 2's c₀ is a single-pair, single-step measurement of the same mechanism read against an explicit heat flow, on a function with an actual violation. Same disclaimer.

Disposition

Instrument kept; no ledger entry. The surviving observation, the landing degree is the flow landing time on another dial, and both are configuration geometry, is the null control explaining the quantity, which is a closure of the same kind hunt #4 recorded, not a lead. Nothing here is evidence for or against RH: the nine known pairs bound Λ_DH-type quantities from below and the dictionary adds no new zero knowledge; it re-expresses where known zeros sit. Spine candidate recorded in NOTES.md: a docstring line for zeta/li.py's hyperbolicity scanners stating the finite-degree blindness window (a zeta/ change, not this hunt's).