MISSION: what has to cancel for the Weil form to be positive

Scope: hunts/arch_cover/ only. Nothing outside this directory is modified. Status: exploratory. Nothing here is a result about the Riemann Hypothesis.

The question

Weil's functional, in this repository's own convention (zeta/weil.py),

W(h) = pole + arch + prime, pole = h(i/2) + h(-i/2) arch = (1/2pi) * int h(r) [Re psi(1/4 + ir/2) - log pi] dr prime = -2 * sum_n Lambda(n) n^{-1/2} g(log n) (negative for h > 0)

and W(h) = sum_rho h(gamma_rho). Positivity on the admissible class (h = |f^|^2) is equivalent to RH. Define the cover ratio

C(f) = (pole + arch) / (-prime), so C - 1 = W / (-prime).

C >= 1 for every admissible f is RH, restated with the two computable pieces separated: the pole and archimedean contributions must cover the prime-weighted sum.

A first look (Gaussian, a = 0.05) shows pole = +2.025, arch = -1.909, prime = -0.116: the pole and the archimedean integral cancel to four digits and the prime side is the small leftover. If that is generic, then any numerics-based "enclosed positivity" is a statement about a cancellation, and the number of digits of cancellation is the real difficulty parameter of the method, not the size of W.

What is measured

For each family member: the enclosure of W (Arb, prec_bits), the piecewise enclosures pole, arch, prime, the cover ratio as an interval, and

cancel_digits = log10( max(|pole|, |arch|) / |W| ),

the number of digits by which the large pieces cancel to produce W.

What this hunt may not claim

Not evidence for or against RH. C >= 1 on finitely many f is what it says.
Not a uniform bound: no statement survives outside the listed grid.
Not independent: the archimedean term has no second backend in this tree (enclose_weil_functional reports two_backend_cross_check unavailable), so a failure of the archimedean quadrature would appear here as a cover ratio moving, not as an error.

The door this opens, if it opens

If cancel_digits grows without bound along a family while C stays above 1, then positivity on that family is asymptotically critical and any proof through this functional must control the archimedean integral at that depth, not merely prove a positive lower bound. That would be a statement about the shape of a proof, and it is falsifiable: find a family whose cancel_digits stay bounded and whose C is bounded away from 1.

Instruments

cover.py, which writes results.json. Entry points: scan, row, summary.

HuntSpec

id: arch_cover
question: How do the pole, archimedean and prime pieces of the Weil functional balance, and how deep is their cancellation?
frontier: measured here from 0.27 to 39.1 digits of cancellation on the Gaussian family; the prime term runs from 1e-10 to 1.24 over the same range
proposed_attack: evaluate the three pieces separately under Arb enclosures, compare both sides of the explicit formula, and read the cancellation depth against the zero side
dead_routes:
  - a uniform positive lower bound on W for all admissible h, killed here by the Gaussian family driving W to zero with every piece at O(1)
  - a place by place positivity argument, already recorded as not assembling globally in hunts/local_positivity
required_oracles:
  - Arb ball arithmetic from zeta.rigor.enclose_weil_functional with explicit truncation bounds
  - the verified zero list to height 10^4 for the independent side of the formula
  - the closed form depth a gamma_1 squared over ln 10, checked against every measured row
kill_conditions:
  - the arithmetic side disagrees with the zero side beyond the reported truncation tails
  - the closed form depth fails to reproduce a measured row to the printed digits
  - a sign is reported while the enclosure still straddles zero
agents_may:
  - evaluate
  - measure
  - compare
  - code
agents_may_not:
  - declare a bound
  - promote an observation to a result
  - use the reserved vocabulary of zeta/rigor.py

Note, 2026-10-30 (landing review). The oracle above names the closed form a gamma_1^2 / ln 10. That form omits the pole's e^{a/4}, and against it the second kill condition fired (17.38 measured against 17.35 at a = 0.2). The form that reproduces every row is a (gamma_1^2 + 1/4) / ln 10; RESULTS.md section 4(b) records both. The HuntSpec is left as written.