MISSION: `jensen_wedge`, hunt #135: the polynomial hyperbolicity wedge for Jensen polynomials of xi

Opened 2026-10-18 as a depth hunt by owner decision: the owner asked for one mechanism worked for weeks rather than a one-day forage. The forage stop rule does not apply here. The hunt stops on a kill condition below or at the time box, and an absence of results before a checkpoint is not a kill condition. Time box: through 2026-11-29 (six weeks). Checkpoints on 2026-10-25, 2026-11-08 and 2026-11-22. The owner confirmed this funding and time box on 2026-10-18 ("Fund hunt #135 as scoped"), and the allocation record belongs on fulcrum's roster, not here.

Closed 2026-10-19 by owner decision, after landing in PR #317, about six weeks inside the time box. No kill condition fired. The remaining checkpoints and the rest of the time box are released, and this is an allocation decision, not a mathematical verdict. The open doors are listed in RESULTS.md, "The doors". Reopening them is a new allocation.

This hunt cannot promote its own claims (hunts/README.md). Nothing in it is a statement about RH, and section "What success would mean" says why that holds even if every phase succeeds.

Why this hunt exists, and the target it replaced

It was requested as "explicit Jensen hyperbolicity thresholds at degree 4 and up via the gauss_positivity saddle instrument", following hunts/gauss_positivity section 6 (hunt #134). That target was already met in the literature before this tree proposed it, and the tree's own frontier statements (docs/12 section 8.1, zeta/li.py's module docstring, hunt #134 section 6) read the frontier from GORZ 2019 alone. Checked 2026-10-18 against the primary sources:

• All n, every degree up to about 10^21. Griffin, Ono, Rolen, Thorner, Tripp and Wagner, Jensen polynomials for the Riemann xi-function, Adv. Math. 397 (2022) 108186, arXiv:1910.01227v3. Theorem 1.2: if RH holds to height T then J^{d,n} is hyperbolic for all n >= 0 whenever d <= floor(T)^2. Their Corollary 1.3, with Platt's T = 3.06e10: hyperbolic for all n >= 0 when d <= 9.36e20. Chasse had d <= 2e17 earlier. Applying the same Theorem 1.2 to the Platt-Trudgian height T = 3e12 gives d <= 9e24; that last figure is derived here from their theorem, not quoted from their v3 text.
• Large n, every degree, explicit shape. The same paper, Theorem 1.1: there is a constant c > 0 with J^{d,n} hyperbolic for n >= c e^{d/2} (v3; the published version states n >= c e^d). The constant is not computed.
• The current frontier, two months old. Jonathan Holland, A new hyperbolicity wedge and a joint semicircle limit for Jensen polynomials of Riemann's xi-function, arXiv:2608.08682v1 (9 August 2026). Theorem 1.1: there is an absolute K > 0 such that n^3 log^2(n+2) >= K d^5 implies J^{d,n} has d distinct negative real zeros, i.e. d <~ n^{3/5} up to logarithms. This replaces the exponential threshold by a polynomial one. Its section 12 says plainly that no attempt is made to optimise the eventual absolute constant.

So no finite unresolved set exists at any degree an explicit-threshold computation could reach. The unresolved region of the (d, n) plane begins above d = 9.36e20 and lies below Holland's wedge. Every point in it is far beyond any finite check; it can be reached only by proof.

Frontier, 2026-10-19 (phase 3, from NOVELTY.md): two deposits that predate this hunt reach past the list above. Both are unreviewed, and neither proof was checked here. Savva, Zenodo 10.5281/zenodo.22293642 (3 September 2026), posted the 2/3 wedge n^2 log(n+2) >= K d^3 with K not computed, stronger than the n^4 log^2(n+2) >= K d^6 log^12(d+2) this hunt later derived; it is AI-assisted by its author's account. Sarnowski, Zenodo Papers 134Z and 134AA (20 August 2026), posted the rank-r diagonal Pade spine whose r = 2 case is this hunt's [2/2] Pade point. So Q1 had been answered, in unreviewed form, before this hunt asked it. What the hunt adds with no prior instance in the searched set is Q2: explicit constants, K = 17 900 with GORTTW Cor. 1.3 and K = 9.10e19 standalone for Holland's Theorem 1.1, K' = 7.21e9 and 3.61e30 for the log^12 six-coefficient form, and c = 1 in GORTTW Theorem 1.1 (RESULTS.md section 19; derivation, reviewed once internally, not a theorem). The bullets above are kept as written.

The mechanism this hunt works, as Holland states it

Normalise R_j = gamma(n+j)/gamma(n), so J^{d,n}(X)/gamma(n) = sum_j C(d,j) R_j X^j. Holland builds a real-rooted polynomial in the positive-root class that matches R_0..R_4 exactly, through three nested families: Laguerre (matches R_1, R_2; its error at R_3 has a definite positive sign), Jacobi (absorbs R_3; leaves a defect of the opposite sign at R_4), and a second Jacobi factor joined by finite-free multiplicative convolution (absorbs R_4). His Proposition 2.2, fifth-order multiplier stability, then shows the remaining coefficient multiplier c(j), which equals 1 at five consecutive indices, cannot move the zeros off the axis when sup |c - 1| is small on a neighbourhood of radius r ~ sqrt(nd). The multiplier's fifth derivative is O(1/(n^4 log n)), so the residual error is O((nd)^{5/2} / (n^4 log n)) = O(d^{5/2} / (n^{3/2} log n)), and requiring it small is exactly d^5 <~ n^3 log^2 n. The analytic input is the complex saddle estimate of GORTTW section 3, which is the same object hunts/gauss_positivity section 3 reproduces in real form (pi e^{u_*} = 1/4 + 2n/u_*, hardened against exact quadrature to 4-6 digits at n = 100..3200).

Correction, 2026-10-18 (phase 1, from the PDF): the paraphrase of Proposition 2.2 above is loose. Its threshold is sup over Omega_r of |c - 1| <= eps < 16, on the 2r-neighbourhood of [0, d], and it has a second hypothesis on the model alone, (6): |y^k p^(k)(y)/p(y)| <= r^k at every critical point y of p. The radius enters only through (6), which Holland's Lemma 9.1 verifies with r_F = K_6 sqrt(Bd). RESULTS.md section 9 has the line-by-line check.

The questions, ranked

Q1 (exponent). Can the matching be carried to order m = 6, giving n^4 log^2(n+2) >= K' d^6, i.e. d <~ n^{2/3}?

Heuristic, derived here from Holland's displayed error and not proved: if matching R_0..R_{m-1} succeeds inside the positive-root class and the stability scale stays sqrt(nd), then with g_i = O(n^{1-i}) (GORTTW equation (2.1)) the residual is O(d^{m/2} / (n^{m/2-1} log n)) and the wedge is d <~ n^{1-2/m}. Setting m = 5 recovers Holland's exponent exactly, which is the consistency check that licenses writing the heuristic down. Under the same heuristic the method class stalls at d ~ n, where the terms n (d/n)^i of the log R_j expansion stop decaying. That barrier is a conjecture about this class, recorded so that nobody mistakes it for a theorem.

The concrete open step at m = 6: after Holland's five-coefficient match, does the defect at R_5 have a definite sign across n, and is there a third real-rootedness-preserving factor (Jacobi or Laguerre, joined by finite-free multiplicative convolution, Martinez-Finkelshtein-Morales-Perales IMRN 2024) whose parameters stay in the positive-root class while matching it? Then: does Proposition 2.2 generalise to six-point vanishing with the same radius?

Q2 (effectivity). For whichever wedge stands at the end (Holland's m = 5 or a new m = 6), compute an explicit admissible constant, with every numeric step in the saddle error bounds carried by ball arithmetic (zeta/rigor.py, Arb backend). This is the closest form of the originally requested "explicit threshold" that is still open. Combined with GORTTW's all-n range it would give a fully explicit hyperbolic region.

Plan and checkpoints

1. By 2026-10-25: reproduce before extending. Build R_j(n) and the quotient coordinates q_k = R_{k+1}^2/(R_k R_{k+2}) from the exact gamma(n) table (zeta/li.py, two routes pinned in tests/test_li.py), and the saddle expansion of log R_j through order 7. Reproduce Holland's two sign facts numerically (the Laguerre error at R_3 is positive, the Jacobi defect at R_4 is negative) over n = 50..3200. A failure here means the paper was misread; fix the reading before anything else.
2. By 2026-11-08: the sixth-order step (Q1). Measure the R_5 defect after the five-coefficient match: its sign, and its scaling in n and d, on two independent routes (exact table and saddle). Attempt the third factor. Attempt the six-point stability lemma. Output: a derivation with explicit hypotheses, or an obstruction at that order graded per ALIGNMENT.md section 5.
3. By 2026-11-22: effectivity (Q2). Explicit constants for the standing wedge, ball-arithmetic enclosures for each numeric input, and a planted-fault check that the bound goes red when a constant is deliberately weakened.
4. By 2026-11-29: rival, doors, hand-off. The Davenport-Heilbronn run (below), RESULTS.md with "The doors", the methods entry in docs/37-methods.md, and routing through docs/doors/refute.md. A Lean formalisation is out of scope: finite-free convolution and the saddle machinery are not in Mathlib, and sizing that port is a separate decision.

The rival gate, with its expected outcome stated in advance

The wedge's inputs are the asymptotics of the weight Phi (double-exponential decay e^{-pi e^u}), which Davenport-Heilbronn's weight shares. Prediction: the same construction succeeds for DH at the same (d, n), using the DH gamma table that hunts/jensen_clock/probe.py builds by two quadratures. DH violates RH, and its J^{d,0} stops being hyperbolic near d* = 20785 (hunts/jensen_clock). If the prediction holds, the wedge is shared structure that cannot tell an RH-true function from an RH-false one. That is a measured, in-tree form of Farmer's critique (Adv. Math. 411 (2022) 108781) and of GORTTW's Remark 3. If DH fails where zeta passes, that is unexpected, and it gets the bug, assumption and shared-failure checks of the certainty ladder before anyone reads it as arithmetic content.

What success would mean, and what it would not

Full success is a theorem of the shape "n^4 log^2(n+2) >= K' d^6 with explicit K' implies J^{d,n} hyperbolic", stated as a delta against Holland's arXiv:2608.08682 Theorem 1.1. That would be a real, citable result on the Jensen program, pending external review. It would not be progress on RH. Every wedge of this shape leaves out the corridor of small n and large d, and n = 0 with all d is RH itself. No known mechanism converts partial Jensen hyperbolicity into zeros on the line: Farmer 2022, GORTTW Remark 3 and Holland section 12 all say so, and this tree's jensen_clock P5 says the clock reads zero configuration, not arithmetic. The owner chose depth over breadth; this mission does not claim that this lane is the deep route to RH.

id: jensen_wedge
question: Can Holland's five-coefficient positive-root comparison for J^{d,n} be carried to six coefficients, giving a hyperbolicity wedge n^4 log^2(n+2) >= K d^6, and can the constant of the standing wedge be made explicit?
frontier: All n hyperbolic for d <= 9.36e20 (GORTTW Cor 1.3, arXiv:1910.01227v3, via RH to 3.06e10); n >= c e^{d/2} for every d with c not computed (GORTTW Thm 1.1); wedge n^3 log^2(n+2) >= K d^5 with K absolute and not computed (Holland, arXiv:2608.08682, Thm 1.1, August 2026). Added 2026-10-19: unreviewed deposits predating the hunt claim the 2/3 wedge n^2 log(n+2) >= K d^3 with K not computed (Savva, Zenodo 10.5281/zenodo.22293642, 2026-09-03) and the rank-r Pade spine (Sarnowski, Zenodo Papers 134Z, 134AA, 2026-08-20).
proposed_attack: Reproduce Holland's sign facts at R_3 and R_4 from the exact gamma(n) table and the saddle expansion of log R_j, measure the sign and scaling of the R_5 defect after the five-coefficient match, attempt a third finite-free Jacobi or Laguerre factor and a six-point multiplier stability lemma, then make the constants of the standing wedge explicit with ball arithmetic.
dead_routes:
  - Explicit N(d) at fixed degree d >= 4 plus a finite check; done in the literature (GORTTW Thm 1.1 and 1.2, Chasse), and the open region begins above d = 9.36e20, beyond any finite computation.
  - Newton inequalities for J^{d,n} at higher degree; implied by the d = 2 Turan inequality, proved in hunts/gauss_positivity section 5b (hunt #134).
  - Low-degree hyperbolicity scans as a falsification route; they cannot fail, by the de Bruijn-Newman smoothing t_eff(d) = |x0|/(8d) measured in hunts/jensen_clock and hunts/gauss_positivity section 5c.
  - Positivity of the weight Phi as the real-zeros criterion; refuted by the positive measure delta_0 + delta_1 + delta_3, hunts/gauss_positivity section 1.
required_oracles:
  - the exact gamma(n) table of zeta/li.py, two independent routes pinned in tests/test_li.py
  - mpmath quadrature at two working precisions for every saddle-expansion coefficient
  - Arb ball arithmetic through zeta/rigor.py for every numeric constant stated as explicit
  - exact rational Sturm root counts (zeta.li.is_hyperbolic, method sturm) for spot checks at small d
  - the Davenport-Heilbronn gamma table built by two quadratures in hunts/jensen_clock/probe.py, as the rival
  - the published statements of GORTTW Theorems 1.1 and 1.2 and Holland Theorem 1.1 and Proposition 2.2, as the targets any claim is a delta against
kill_conditions:
  - Holland's sign facts at R_3 (positive) and R_4 (negative) fail to reproduce from the exact table, which means the construction was misread
  - the saddle expansion of log R_j disagrees with the exact gamma table beyond its stated error at any n <= 3200
  - the R_5 defect after the five-coefficient match changes sign within the measured n range, on both routes, which blocks a single sixth-order factor of Holland's type
  - every candidate third factor needs a parameter outside the positive-root class (negative or complex) to match R_5
  - a planted weakening of an explicit constant does not turn the effectivity check red
  - the time box of 2026-11-29 passes, after which the hunt is paused by allocation, which is not a mathematical verdict
agents_may:
  - search
  - derive
  - compute
  - code
  - attack
agents_may_not:
  - declare novelty
  - declare theorem status
  - promote their own claim
  - claim any bearing on RH
  - run heavy work locally

Compute discipline

Phases 0 and 1 are light: the gamma(n) table to a few thousand terms and saddle expansions at symbolic n. Anything heavier runs on GitHub Actions, never locally (AGENTS.md, "Compute discipline"). Time one unit and write the estimate in RUNS.md before any sweep. Every run appends a runmanifest block to RUNS.md, including runs that produced nothing.