MISSION: the prime side, place by place, for a compactly supported test function

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

Why this differs from local_positivity

hunts/local_positivity decomposed the prime side into local norms Q_p and found that every place is positive (given Ramanujan-type bounds) and that the local norms do not assemble globally. It could not say where the covering mass sits, because its gate works on Toeplitz sections of local coefficients, not on an L^2 norm of a concretely shifted function.

For a compactly supported f the shift operator is finite: Phi_p f = sum_{m>=0} p^{-m/2} f(. - m log p) has finitely many overlapping terms, so ||Phi_p f||^2 is an exact finite sum of rectangle overlaps. With f = b^{-1/2} 1_{[-b/2, b/2]} and g = f * f (the triangle), every Q_p is an exact finite sum (not rational: it carries p^{-1/2}, log p and b; corrected 2026-10-30), and the per-place energy

cost_p = log p * (Q_p - 1), Q_p = (1 - 1/p) ||Phi_p f||^2

is exact, and is non-negative wherever the shifted copies pile up. Then

W(h) = pole + arch - sum_p cost_p,

and the question this hunt asks is: how is the cost distributed over the primes, and does the distribution move as the test function widens?

Check that must pass first

E computed place by place must equal -prime_term from zeta.weil.explicit_formula_sides, which computes the prime side as a sum over prime powers. If they disagree the decomposition is wrong and nothing else here counts. The zero side gives a second, independent check on W itself.

What this may not claim

• Not evidence for RH. E <= pole + arch on finitely many b is what it says.
• The test functions are a one-parameter family of rectangles; no statement survives outside it.
• Raghavan/Ramanujan-type bounds are not proved here; where Q_p > ||f||^2 the place is reported with a negative e_p rather than hidden.

HuntSpec

id: weil_split
question: How is the prime side of the Weil functional distributed over places, for a compactly supported test function?
frontier: measured here, the per-place cost spreads from 4 places carrying 98 percent in their top three to 429 places carrying 13 percent, while the covering margin falls from 3.0e-2 to 1.6e-4
proposed_attack: exact rectangle-overlap norms of the shift operator, giving each place an exact closed-form cost (not rational: it carries log p and p^(-1/2); corrected 2026-10-30)
dead_routes:
  - finitely many important primes doing the covering, killed here by the falling top three share
required_oracles:
  - exact closed-form rectangle overlap sums
  - equality of the per place sum with the explicit formula prime term to 1e-16
  - the verified zero list for the covering ratio
kill_conditions:
  - the per place sum disagrees with the prime term
  - the covering ratio is quoted from the slow quadrature instead of the zero side
agents_may:
  - decompose
  - measure
  - compare
  - code
agents_may_not:
  - declare a bound
  - promote an observation to a result
  - use the reserved vocabulary of zeta/rigor.py