Question. zeta/li.py's real-rootedness lane asks about Jensen polynomials J^{d,n}; hunt #4 (flow_repair/) measured when the de Bruijn–Newman flow lands the Davenport–Heilbronn off-line pairs. This hunt connects them: at what finite Jensen degree does hyperbolicity actually witness the counterexample's off-line pair, and is the finite degree itself a heat clock?
The observation driving the design (found in a scratch prototype before this mission was written — recorded honestly as such): with E(x) = Σ γ(n) xⁿ/n! and γ(n) = n!/(2n)! · M₂ₙ the moment sequence of the DH weight Φ_DH, the degree-d Jensen polynomial satisfies exactly
J^{d,0}(x/d) = Σ_j γ(j)/j! · Π_{i<j}(1 − i/d) · x^j,
and the damping Π(1 − i/d) ≈ e^{−j²/2d}, read at the cosh-series saddle j ≈ zu/2, acts like the flow multiplier e^{t u²} with an effective time
t_eff(d) = |x₀| / (8 d), x₀ = (β − 1/2 + iγ)² the pair's image.
A prototype trajectory ladder saw the DH height-85.7 pair's image in J^{d,0} drift to the real axis and land between d = 10⁴ and 3·10⁴, with the drift rate agreeing with hunt #4's measured pair dynamics through the dictionary above at the percent level. A naive grid scan (d ≤ 32, all shifts n ≤ 250) sees nothing, which the dictionary explains: those degrees carry t_eff far beyond the pair's landing time, so the degree itself has smoothed the violation away before the polynomial is even inspected.
Pre-registered predictions
Targets below use hunt #4's measured values, read from hunts/flow_repair/results.json: pair 1 (β ≈ 0.8085, γ ≈ 85.699, t\* = 0.044126344551623946, isolated-pair naive value y₀²/2 = 0.047591…), pair 2 (β ≈ 0.6508, γ ≈ 114.163, t\* ≈ 0.0111296).
- P1 (dictionary, pointwise). Under t_eff = |x₀|/(8d), the Jensen pair trajectory Im X(d) matches the PDE trajectory Im X_PDE(t) pointwise within 5% relative over t_eff ∈ [0, 0.8·t\*], and the drift constant C = lim_{d→∞} d·(Im X(∞) − Im X(d)) matches C_pred = |d Im X_PDE/dt|₀ · |x₀|/8 within 2%.
- P2 (the clock adjudicates). The landing degree d\* gives t\_J := |x₀|/(8 d\) inside (0.042, 0.047), and closer to the PDE landing 0.0441263 than to the isolated-pair 0.0475914 — i.e. the Jensen clock feels the same neighbor interaction the PDE flow does.
- P3 (scaling across pairs). Pair 2 lands at d\* ≈ 1.5·10⁵ with t\_J within 8% of its PDE t\ = 0.0111296 — a 7× change in d\* tracked by the same dictionary.
- P4 (specificity). ζ through the identical pipeline, no plant: Newton from x₀ collapses onto the real axis and a winding box at the DH pair location counts zero.
- P5 (lesion, geometry vs arithmetic). A pair planted into ζ's γ-sequence at exactly x₀ (an exact three-term recurrence — multiplication of E by (x − x₀)(x − x̄₀)) is detected, with its own landing degree; since its x-plane neighborhood is ~5× sparser than DH's, its clock should read close to the isolated value 0.0476 rather than DH's 0.0441. Same position, different arithmetic, different neighbor field → different d\*: the clock reads configuration, extending hunt #4's null result to the coefficient side.
- P6 (two clocks add). Flowing the γ-sequence to time t before damping, the landing time t_land(d) satisfies t_land(d) + |x₀|/(8d) = t\* within 5% of t\* at d = 6·10⁴ and 1.2·10⁵.
- P7 (precision response). d\* moves by < 0.1% under dps 110 → 160, series cutoff 280 → 360 and quadrature (U, segments, degree) changes; the undamped Newton root reproduces hunt #4's polished pair to ≥ 30 digits.
Phase 2 — the shift direction (registered 2026-08-11, after phase 1 closed)
Phase 1 fixed the shift n = 0 and measured the degree axis. The shifted Jensen polynomial is exactly the damped n-th derivative, J^{d,n}(x/d) = Σ_j γ(n+j)/j! · Π_{i<j}(1−i/d) · x^j = damped E⁽ⁿ⁾(x), so the shift axis asks: what does differentiation in x do to the off-line pair's image, on the same clock? Differentiation of real entire functions is believed to act on zeros as a smoothing flow (Gauss–Lucas pulls complex pairs toward the real hull; there is a modern PDE literature on repeated differentiation as an erosion/heat-like flow on the zero distribution — a literature pass is part of this phase). Pre-registered questions and predictions, written before any phase-2 measurement:
- Q1 (per-step clock). At d = 10⁸ (degree damping negligible) the pair's image in E⁽ⁿ⁾ descends toward the real axis as n increases, and the per-step effective flow time c_n := t-equivalent of step n → n+1 (read off the PDE trajectory) is of order 1/(2u₀²) with u₀ ≈ 1.3–1.5 the measured cosh-series saddle abscissa at the pair — i.e. c₀ ∈ [0.2, 0.35]·t\*/0.0441 … loosely, c₀ ∈ [0.005, 0.05] is the honest wide bracket; the saddle heuristic is weak and this question is genuinely open. The sharp registered prediction is only the direction (descent, monotone) and that c_n is set by configuration, not arithmetic.
- Q2 (three clocks add). For cells (d, n, t) with all three knobs on, t_land(d, n) + |x₀|/(8d) + g(n) = t\* within 1%, where g(n) = Σ_{k<n} c_k is the measured shift clock — no free parameters once c_k are measured at t = 0.
- Q3 (the map). The detection region for pair 1 in the (d, n) plane is the corner {|x₀|/(8d) + g(n) < t\}: boundary cells flip detected/not-detected as that inequality predicts, including the n-axis end (the largest n at which any degree can see this pair, predicted n_max = max{n : g(n) < t\}).
- Q4 (specificity). ζ through the same shifted cells stays silent.
Phase 2 may also touch only hunts/jensen_clock/.
Phase 3 — the falsifier and the trichotomy (registered 2026-08-11, after
phase 2 closed)
- Q5 (strong additivity, zero free parameters). Phase 2's Q2 was left honestly untested. The test: re-measure t_land(n = 1) at d = 10⁶. The additive picture predicts it moves from the d = 10⁸ value −0.118667407 by exactly the degree-budget difference −|x₀|/8·(10⁻⁶ − 10⁻⁸) = −0.00090888, i.e. t_land(n=1, d=10⁶) = −0.11957629 ± 0.0001. If this fails, phase 2's additive budget rule was curve-fitting and the hunt's map claim is withdrawn.
- Q6 (the trichotomy: Li is an accumulating discriminator). The third coefficient-side detector, Li's criterion, has a kernel 1 − (1 − 1/ρ)ⁿ that amplifies off-line zeros (|1 − 1/ρ| > 1 for β < 1/2) instead of smoothing them. Registered predictions: (i) for a planted symmetric quadruple at ρ_p = 0.8 + 2.5i added to ζ's Li sum (Bombieri–Lagarias form), the sum first goes negative at the envelope crossing of 2·rⁿ against λ_n(ζ), r = |1 − 1/(1 − ρ_p)| ≈ 1.0466 — predicted onset n_Li ∈ [80, 160], with negativity arriving within one oscillation period (≈ 16) of the envelope crossing; (ii) the identical formula applied to DH pair 1 (r − 1 ≈ 4×10⁻⁵) puts its onset beyond 10⁵ — unreachable by any Li computation in this tree; (iii) hence the trichotomy: degree and shift are erasing clocks (their blind sets are cofinal — all small d, all n ≥ 1), Li is an accumulating discriminator (its blind set is an initial segment). Blind in opposite directions; neither blindness is a matter of effort.
- Q7 (specificity). The unplanted λ_n(ζ) stays positive over the whole tested range (already pinned by
tests/test_li.py; recomputed here so the planted run has its own control).
Scope
May touch: hunts/jensen_clock/ only. Reads (never writes) hunts/flow_repair/results.json, zeta.epstein, zeta.heatflow, zeta.li. No changes to zeta/, ontology/, harness/, no cache invalidation, no ledger entry unless something survives the checklist in hunts/README.md.
Everything here is the accurate regime — mpmath floats with measured cross-route defects; the strongest words used are measured and observed. The truncated-series detector reports its own validity margin (tail bound vs minimum modulus on the contour) rather than assuming it.
Prior-art hooks, recorded so nothing is overclaimed: coefficient multiplier sequences of Gaussian type are classical (Pólya–Schur; de Bruijn's e^{−λD²} operators; Turán), and Griffin–Ono–Rolen–Zagier relate large-shift Jensen polynomials to Hermite polynomials through exactly this kind of heat limit. The degree-damping-as-heat reading may well be implicit in that literature; this hunt claims measurements on the counterexample, not novelty.