Gauss-positivity lane: the Fourier structure of Xi and the moment problem

Started 2026-10-03. Route chosen under the 2026-09-12 owner mandate (ALIGNMENT.md section 0), which permits original mathematics toward RH and useful intermediate results. This hunt attempts a contribution and may fail; it does not presume success, and it cannot promote its own claims.

The question

Ξ(t) = ξ(1/2 + it) is the cosine transform of a function Φ that is documented positive on the real axis (zeta/heatflow.Phi, verified in zeta/heatflow.py and pinned by tests/test_heatflow.py). Pólya's classical theorem family says a positivity condition on the Fourier-side function can force the transform's zeros to be real, and RH is exactly the statement that Ξ has only real zeros.

Question. Does the positivity of Φ buy anything, and what is the exact criterion that converts Fourier-side positivity into real zeros?

The naive form of the folklore criterion is false and this hunt says so with a computed witness: φ(t) = e^{-t^4} is positive and even with a convex exponent, and F(z) = ∫ φ(t) e^{izt} dt has complex zeros (argument-principle count, this hunt's journal.md). So convexity of the exponent is not sufficient, and any real criterion is strictly sharper. Finding the sharp criterion and testing whether Φ meets it is the work.

The concrete plan

1. The exact moment table. b_n = ∫_0^∞ Φ(u) u^{2n} du. Nobody in this tree has this table; zeta/li.py has γ(n) (a different normalisation of the same data), and the exact conversion between them is derived and checked here rather than quoted.
2. The Newton / higher-Turán battery. For a function with only real zeros, the normalised coefficient sequence satisfies the Newton inequalities at every order. Each order is a necessary condition for RH and each is separately checkable. The first violation at any order refutes RH; an all-pass table for the orders computed is not evidence for RH (docs/08 section 3, Littlewood).
3. The Gate-3 rival check. Anything proposed is run against W_a = ζ(s+a)ζ(s-a) and Davenport-Heilbronn, per zeta.epstein.battery and the hunts/epp_herglotz finding that the standing rivals are all linear combinations of genuine Euler products.

Compute discipline

Heavy work runs on Modal, never locally, per AGENTS.md "Compute discipline". One unit is measured and multiplied before any sweep. Budget and per-run notes in RUNS.md.

Dead routes carried in

• hunts/epp_herglotz: "Euler-product positivity plus the functional equation implies RH" is refuted by W_a. Any candidate here must survive that rival.
• hunts/local_positivity: place-local norm identities do not globalise; Q_p - ||f||^2 has mixed sign (52/60 positive).
• docs/35: the out-of-band positivity information is worth about +0.0068 in the configuration class but the certificate machinery cannot reach it, for a stated structural reason.
• hunts/family_wall: the n-point pressure family has a configuration ceiling it cannot reach.

id: gauss_positivity
question: Does Fourier-side positivity of Phi, via the sharp Pólya-type criterion, force the zeros of Xi to be real?
frontier: The folklore convexity criterion is false by an explicit computed witness; the sharp criterion and the moment table b_n = int Phi u^{2n} are not established in this tree.
proposed_attack: Build the exact moment table on Modal, test the full Newton/higher-Turan inequality family at each order, and check every candidate criterion against the W_a and Davenport-Heilbronn rivals.
dead_routes:
  - Convexity of the Fourier-side exponent is not sufficient; e^{-t^4} has a convex exponent and its transform has complex zeros, computed by argument principle in this hunt.
  - Euler-product positivity plus the functional equation, refuted by W_a = zeta(s+a)zeta(s-a) in hunts/epp_herglotz.
  - Place-local norm identities do not assemble into a global one, measured in hunts/local_positivity.
required_oracles:
  - mpmath quadrature at two independent precisions and two independent panel families
  - the committed zeta/li.py gamma(n) table, computed by two independent routes already pinned in tests/test_li.py
  - exact integer arithmetic for the Newton identities on rationalised coefficients
  - zeta.epstein.battery for the rival gate
kill_conditions:
  - the moment table disagrees between the two independent quadrature routes beyond the stated error budget
  - a Newton inequality fails, which would refute RH and requires independent confirmation before being reported as such
  - the derived exact relation between b_n and gamma(n) fails to hold numerically
  - a candidate criterion is satisfied by W_a or Davenport-Heilbronn
agents_may:
  - compute
  - derive
  - code
  - attack
  - formalize
agents_may_not:
  - declare novelty
  - declare theorem status
  - promote their own claim
  - run heavy work locally