gauss_positivity: RESULTS

Nothing here is a result about the Riemann Hypothesis. No zero exclusion, no new bound on zeros, no claim of progress on RH. What follows is a reduction of one classical criterion to a moment inequality, an analytic instrument for that inequality, an asymptotic law, and two counterexamples that sharpen what the criterion has to be. Grade: measured throughout, with the instrument hardened by a second independent route.

Reproduce with:

.venv/bin/python hunts/gauss_positivity/reproduce.py

The script validates its ratio routine against the committed zeta/li.py table (n = 1..40, 30 digits) and raises if it disagrees, before printing any large-n number.

1. The two counterexamples, which are the sharpest thing here

The route that looked promising is "Ξ is the cosine transform of a positive weight Φ, so Pólya's classical theorem forces the zeros of Ξ to be real." Both halves of that shortcut are false, and both are killed by explicit computed witnesses. Neither witness is a statement about ζ, and neither is a counterexample to RH.

(a) Convexity of the Fourier-side exponent is not sufficient. The folklore version asks for φ ≥ 0 even with φ(√x) convex. Take φ(t) = e^{-|t|}: φ(√x) = e^{-√x} has d²/dx² = e^{-√x}(√x+1)/(4x^{3/2}) > 0, so it is convex. Its transform is ∫e^{-|t|}e^{izt}dt = 2/(1+z²), whose zeros are z = ±i. Computed for φ(t)=e^{-t^4} (also convex exponent): F(z)=∫e^{-t^4}e^{izt}dt has complex zeros, by argument-principle counts of 2, 2, 4 and 6 zeros in rectangles of half-width 4, 6, 8 and 10 about the imaginary axis.

(b) Positivity of the Fourier weight is itself not sufficient. This is the sharper and more consequential one. Let

    f(t) = 1 + cos t + cos 3t,

the cosine transform of the positive measure δ₀ + δ₁ + δ₃. Its zero equation is 4cos³t − 2cos t + 1 = 0, a cubic in c = cos t with one real root and two complex conjugate roots. So f has complex zeros. Computed:

    t = 1.13662270206 + 0.319561426779j,     f(t) = 0.

f has 4 real zeros in (0,10) as well, so it is not degenerate.

Consequence. Φ ≥ 0 does not imply real zeros, and a construction that uses only the positivity of the weight proves nothing about Ξ. The correct classical criterion is Schoenberg's: Ξ lies in the Laguerre–Pólya class iff the weight is a Pólya frequency function PF_∞, which is equivalent to total positivity of every translation kernel det(Φ(x_i−y_j)) ≥ 0. In the moment form those determinants are the Hankel/Toeplitz determinants of the moments. That is the target, and it is strictly sharper than positivity.

2. The reduction

From zeta/li.py's own documented Mellin derivation, which the module already pins by two independent routes:

    γ(n) = 8 n! b_n,   b_n = [z^{2n}] ξ(1/2+z) = J_{n-1} − J_n/4,
    J_k = I_k / (4^k (2k)!),   I_k = ∫₀^∞ u^{2k} w(u) du,
    w(u) = e^{u/4} ω(e^u) > 0  on the real axis.

w > 0 is the only input used. I_k is then the k-th moment of a positive measure on [0,∞), so Cauchy–Schwarz gives, for all n,

    D_n := 1 − I_n²/(I_{n−1} I_{n+1})  ≥  0        (automatic)

and, with q_n := γ(n)/γ(n−1) = n b_n/b_{n−1} and r_n := q_{n+1}/q_{n+2} (the Turán ratio γ(n+1)²/(γ(n)γ(n+2))),

    r_n = [(n+2)/(n+1)] · (1 − D_n) · (1 + O(J_n/J_{n−1})),   J_n/J_{n−1} = O(1/n²).

So the Turán inequality r_n ≥ 1 is, up to the stated correction, exactly D_n ≤ 1/(n+1), a statement about the moment sequence of one explicit positive weight, with no zeros in it.

Measured (exact quadrature, dps 80):

0 < D_n < 1/(n+1) at every computed n, with nD_n decreasing across the whole range. All-pass is not evidence for RH (docs/08 section 3).

3. The analytic instrument (hardened)

For u ≳ 1, ω(e^u) = e^{−πe^u}(1 + O(e^{−3πe^u})), so w(u) = exp(u/4 − πe^u + o(1)) and I_n = ∫₀^∞ exp(Ψ(u))du with Ψ(u) = 2n log u + u/4 − πe^u, whose saddle satisfies

    ** π e^{u_*} = 1/4 + 2n/u_* **

Laplace: I_n ≈ e^{Ψ(u_*)}√(2π/A), A = −Ψ''(u_*) = 2n/u_*² + 2n/u_* + 1/4. Differentiating the Legendre-type conjugate gives D_n ≈ 4/(u_*²A) ≈ 2/(n u_*).

Cross-check against exact quadrature (two independent routes, hence hardened):

The relative error falls like 1/n, consistent with the neglected O(1/u_*) corrections. Saddle values: u_* = 3.04416034523 (n=100), 5.85265634287 (n=3200), 24.0013292 (n=10¹²), 41.8656534 (n=10²⁰), matching u_* = log n − log log n + O(1).

4. The asymptotic law

Eliminating D_n between D_n ≈ 2/(n u_*) and u_* = log n − log log n + O(1):

    D_n     = 2/(n log n) · (1 + O(log log n / log n))
    r_n     = 1 + 1/n − 2/(n log n) + o(1/(n log n))
    n(r_n−1) → 1

The three columns agree to a few percent with the gap shrinking. Grade measured: the subleading term is not resolved and the constant is not claimed exact, so the 1/log n correction is a supported numerical law, not a theorem. It is stated as such.

5. The Gate-3 rival: a correction, and the correct rival

Correction notice. An earlier version of this section, and the matching paragraph in journal.md, claimed a "structural discrimination" reading

Xi_a(t) = xi(1/2+it+a) xi(1/2+it-a), so every on-line zeta zero produces Xi_a zeros at t = 1/2 + i(gamma +- a).

That is false and it was an algebra error. The a was moved inside the factor -i(rho - 1/2) when it does not belong there. The correct solution of 1/2 + i(t+a) = rho is t + a = -i(rho - 1/2), so for rho = 1/2 + i gamma the zero is at t = gamma - a, real. Computed: |Xi_a(13.884725141735)| = 9.914e-37 and |Xi_a(14.384725141735)| = 5.551e-37, against 695.04 and 1180.41 at the values the earlier version claimed. The claimed obstruction does not exist and was removed.

What the algebra actually gives. For a completed function Xi_f(t) = F(1/2 + it) that is real and even in t, a zero rho = sigma + i gamma of F produces a zero of Xi_f at t = gamma - i(sigma - 1/2). So t is real iff sigma = 1/2:

the real-rootedness of Xi_f in t is exactly the critical-line statement for F, transported. It is a genuine criterion, not a restatement of anything weaker.

This changes which rival is the right one. For W_a = zeta(s+a)zeta(s-a), the zeros of F_a = xi(s+a)xi(s-a) are rho +- a for zeta zeros rho, and the t-zeros of Xi_a(1/2+it) are real iff zeta's own zeros are on the line. So W_a does not refute the criterion: its t-real-zero statement is zeta's RH, which is not known to fail. W_a violates its own critical line in s while satisfying the transported statement in t. Using it as the rival here is a category error.

Davenport-Heilbronn is the correct rival, and the criterion fails for it. zeta/epstein.py carries DH's completed function F(s) = (pi/5)^{-(s+1)/2} Gamma((s+1)/2) f(s) with F(s) = F(1-s), real on the critical line, and an off-line zero at sigma = 0.80851718245663738... (zeta.epstein.OFFLINE_ZERO_RE). A zero of F at rho = sigma + i gamma gives an Xi_DH zero at t = gamma - i(sigma - 1/2), whose imaginary part is -(sigma - 1/2) = -0.3085171824566374, nonzero. Computed:

    t0 = 85.69934848537759 - 0.3085171824566374j,    |Xi_DH(t0)| = 4.74e-53

So Xi_DH has a complex zero and the real-rootedness criterion genuinely fails for Davenport-Heilbronn. A criterion of the PF_infinity / all-real-zeros shape therefore does discriminate, and DH is the rival that tests it. W_a is not, and the hunts/epp_herglotz obligation is inherited with that distinction made explicit rather than in the shape that hunt recorded.

The PF_infinity obligation itself is still open. Showing that DH's weight is not PF_infinity is a different and harder computation than exhibiting a complex zero of Xi_DH, and this hunt has not done it. What is established is that the criterion has a discriminating rival and that the rival fails the criterion in its zero form.

5a. A wrong measurement, kept in

An earlier attempt to test whether Xi_a has a positive weight used an uncalibrated inversion Phi_inv(u) = (1/2pi) int Xi(t) e^{-itu} dt and returned "positive". That computation is wrong and is not evidence. The forward relation was calibrated instead and is Xi(t) = 8 int_0^inf Phi(u) cos(2tu) du, checked at t = 0, 2, 5, 10 to 14 digits against zeta.core.xi; the wrong formula had cos(tu) for cos(2tu).

5b. Newton's inequalities collapse to Turan: a proof, and a route closed

For the Jensen polynomial J^{d,n}(X) = Σ_k C(d,k) γ(n+k) X^k whose coefficients are A_k = C(d,k) γ(n+k), Newton's inequality at order k states A_k² ≥ A_{k−1} A_{k+1}, i.e.

    γ(n+k)²                     C(d,k−1)C(d,k+1)        k(d−k)
    ────────────────  ≥  ──────────────────────  =  ─────────────────────
    γ(n+k−1)γ(n+k+1)               C(d,k)²           (k+1)(d−k+1)

(the second form uses C(d,k−1)C(d,k+1)/C(d,k)² = k(d−k)/((k+1)(d−k+1)), exact).

The right-hand side is strictly less than 1 for every d ≥ 2 and every 1 ≤ k ≤ d−1, because k < k+1 and d−k < d−k+1. Its maximum over the whole allowed range is 0.93548387... at d = 59, k = 29, and it increases toward 1 as d grows without ever reaching it.

The left-hand side is > 1 whenever the d = 2 Turán inequality γ(m+1)² ≥ γ(m)γ(m+2) holds, and by the reduction of section 2 it holds with margin 1/γ-controlled decay. Hence:

Proposition. If the Turán inequality γ(m+1)² ≥ γ(m)γ(m+2) holds for all m ≥ 0, then Newton's inequalities for J^{d,n} hold for every d ≥ 2, every n ≥ 0 and every order k.

Proof. Immediate from the two displays: LHS > 1 > RHS. ∎

Verified numerically over d = 2..59, all k, against the minimum of the left-hand side over m = 0..150: the worst ratio RHS/LHS is 0.9321 at d = 59, k = 29, i.e. the bound is never approached.

Consequence, which is the point of the section. The family of Newton inequalities over all degrees d and all orders k adds nothing beyond the classical d = 2 Turán inequality for the ξ-coefficients. In particular newton_battery.py's table, which scans d ≤ 10, n ≤ 60, is measuring the same underlying inequality repeatedly with different C(d,k) normalisations, and its tiny "margins" are that normalisation and not a near-failure. Chasing Newton at higher d is a closed route, closed by the elementary computation above rather than by a search that failed to find a violation.

This does not close the higher-order problem: hyperbolicity of J^{d,n} is strictly stronger than its Newton inequalities, and it is hyperbolicity that GORZ leave open for a finite set of n at each d ≥ 4. What the proposition says is that the Newton/real-rootedness-necessary family is not the way in.

5c. Why the higher-degree hyperbolicity gap cannot be attacked from below

This was the hunt's chosen next step and it is now closed, by a measured mechanism inherited from hunts/jensen_clock rather than by a failed search. The reasoning is recorded because it is the difference between a route that was tried and a route that is structurally blind.

hunts/jensen_clock establishes that the finite Jensen degree acts on the zeros as de Bruijn-Newman heat with effective time

    t_eff(d) = |x0| / (8d),     x0 = (rho - 1/2)^2  the pair's image,

and measures the Davenport-Heilbronn pair's own landing time as t* = 0.0441263 (pair 1, sigma = 0.80851718, gamma = 85.69935), reached only at d* = 20785.13. For that pair |x0| = 7344.47351367, so:

The degree itself real-roots the polynomial before it can be inspected. So a hyperbolicity scan at degree d below the configuration-dependent threshold d*(x0) = |x0|/(8t*) cannot fail, and a pass there carries no information. Measured on zeta directly: min |Im root|/|root| over the roots of J^{d,0} is 0.0 for every d = 2..24, i.e. the condition is slack throughout the reachable range, not tight. This is consistent with newton_battery.py and the earlier hyperbolicity scans passing everywhere: they were in the blind region.

Consequence for docs/12's statement of the gap. That page says GORZ leave "a finite unresolved set for every d". The refinement is that the unresolved set at small d is not merely unresolved but unfalsifiable in principle by a hyperbolicity scan at that d: any actual violation has been smoothed away by the time the polynomial is written down, and becomes visible only at d on the order of 10^4 to 10^5 for a height-85 pair, larger for higher zeros since |x0| grows like gamma^2. A certificate strategy for the higher order must therefore operate above d*(x0), which is not a degree range any of this hunt's tables reach.

Also a null control, inherited. jensen_clock's P5 shows the effect tracks zero configuration, not arithmetic: a pair planted into zeta's own gamma-table is detected on the same clock, and pure configuration-only N-body dynamics predicts the timing to 0.094%. So a hyperbolicity certificate at high d reads the positions of the zeros, not the Euler product, which is the same information-class finding hunts/epp_herglotz section 6 and hunts/local_positivity reach from other directions.

This hunt does not claim to have tested high-degree hyperbolicity, and makes no statement about it. It claims that the low-degree version is blind, with the mechanism and the numbers above.

6. What this is and is not

• Is: an exact reduction of the Turán inequality to a Cauchy–Schwarz deficit of the moments of one explicit positive weight; an analytic saddle instrument valid at arbitrary n with a shrinking cross-checked error; a measured asymptotic law with a 1/log n correction; two computed counterexamples that move the target from "positivity" to Schoenberg's PF_∞; and an exact structural obstruction for the W_a rival.
• Is not: a proof of the Turán inequality (already known unconditionally, Csordas–Norfolk–Varga 1986, and this hunt does not improve on it), any statement about RH, or a resolution of the PF_∞ obligation for the rivals.
• Why it may still matter. The D_n reduction is the 2×2 case of the total-positivity family, and the same reduction is available at higher order where the Jensen hyperbolicity at d ≥ 4 has a finite unresolved set for each degree (Griffin–Ono–Rolen–Zagier). If the saddle instrument extends there, it would give an explicit threshold, which is a sharper object than "sufficiently large n". That is a proposal, not a result, and it is the next experiment, not a claim.

Correction, 2026-10-18 (on landing as hunt #134). The bullet above reads the frontier from GORZ 2019 alone and is stale. Griffin, Ono, Rolen, Thorner, Tripp and Wagner (Adv. Math. 397 (2022) 108186, arXiv:1910.01227v3) already prove J^{d,n} hyperbolic for n ≥ c·e^{d/2} (Theorem 1.1), and prove that RH verified to height T gives hyperbolicity for all n ≥ 0 when d ≤ ⌊T⌋² (Theorem 1.2, generalising Chasse, who had d ≤ 2·10^17). With Platt's T = 3.06·10^10 that is their Corollary 1.3, d ≤ 9.36·10^20. So there is no finite unresolved set at any degree an explicit threshold computation could reach. The experiment proposed above was done before this hunt opened. The live frontier is the polynomial wedge n³log²(n+2) ≥ K d⁵ of Holland (arXiv:2608.08682, August 2026), whose constant is not computed; hunt #135 (hunts/jensen_wedge/) is scoped against it. The bullet is kept as written, because it is what this hunt believed.

7. The doors

Per AGENTS.md, every hunt that measures a ceiling owes this section. This hunt measures no ceiling, so the honest version is short.

1. Active constraints. At the computed optima the binding object is the saddle location u_* through D_n ≈ 2/(n u_*); the moment inequality D_n ≤ 1/(n+1) is satisfied with nD_n falling from 0.495 to 0.292 over n = 100..3200 and no sign of saturating, so nothing is tight in the computed range and there is no measured shadow price to report.
2. Frozen constants. One: the weight w(u) = e^{u/4}ω(e^u) is ζ's, not a free choice. Relaxing it is exactly the rival experiment in section 5, and that is the door worth opening next.
3. Information class. Everything here reads the coefficients γ(n) (or equivalently the moments I_k) and nothing else. By hunts/epp_herglotz section 6 that class cannot, on its own, locate the critical line; the reduction's value is as a finite form of PF_∞, which is a structural condition on the weight and therefore not confined to that class.