RESULTS: weil_split

Nothing here is a result about the Riemann Hypothesis. This is an exact place-by-place decomposition of the prime side for one family of compactly supported test functions. Grade: measured, with one exact identity.

1. The decomposition, and the identity that was checked

For f = b^{-1/2} 1_{[-b/2, b/2]} (so ||f||^2 = 1), g = f * f is the triangle of half-width b and h = g^ is (2/b)(1 - cos br)/r^2. Because f is compactly supported, Phi_p f = sum_{m>=0} p^{-m/2} f(. - m log p) has only finitely many overlapping terms:

||Phi_p f||^2 = (1/b) sum_{m,n>=0} p^{-(m+n)/2} (b - |m-n| log p)_+ Q_p = (1 - 1/p) ||Phi_p f||^2 cost_p = log p (Q_p - 1) exact, a finite closed form sum_p cost_p = -prime_term checked to 1e-16 W = pole + arch - sum_p cost_p

Only primes with log p < b contribute, i.e. p <= e^b; the shift series is a geometric sum beyond that. The check sum_p cost_p = -prime_term passes at every b tested, with the residual at the float-sum floor (1e-16), so the decomposition is the same object zeta.weil computes, split by place rather than asserted.

Scope note, 2026-10-30 (landing of the referee check, REFEREE.md). The identity holds place by place by construction. Expanding the overlap sum, Q_p = (1/b) sum_{d in Z} p^{-|d|/2} (b - |d| log p)_+, so cost_p = 2 log p sum_{d >= 1} p^{-d/2} g(d log p), which is exactly the p-part of -prime_term. sum_p cost_p = -prime_term is therefore the prime sum regrouped by prime: the 1e-16 check verifies the implementation, not a mathematical fact, and "one exact identity" in the grade line means this regrouping. test_weil_split.py pins it place by place at 50 digits, together with the referee's sums.

2. Where the cost sits

Two monotone facts, and they point the same way (on these five widths; note, 2026-10-30: hunt #158's 0.25 grid shows the margin rising on 5 of 24 steps, so the second holds from b = 2 to b = 8, not step by step):

• The cost spreads. At b = 2 the prime side is three places (p = 2, 3, 5 carry 98% of it). At b = 8 it is 429 places and the top three carry 13%.
• The cover margin collapses. cover - 1 falls from 3.0e-2 to 1.6e-4 over the same range; the local exponents between rows run from 2.2 to 5.6, and about b^{-3.8} from b = 2 to b = 8 (first written "roughly like b^{-2.5}", which the table does not support; corrected 2026-10-30).

So the archimedean side is asked to cover a sum over an unbounded set of places, with a margin that shrinks as the test function widens. One caution on reading the margin: by construction cover - 1 = W_zeros / sum_p cost_p, and W_zeros = sum h(gamma) >= 0 because h >= 0 and the verified zeros are on the line. So the margin is W normalized by the prime cost, not an independent statement about the archimedean term; its decay says W grows more slowly than the prime side does. There is no finite set of "important" primes doing the work, and no bounded slack.

3. A caveat that had to be handled, not hidden

h here decays only like 1/r^2 (that is what compact support of f costs), so the archimedean quadrature converges slowly. explicit_formula_sides reports agreement between the arithmetic and zero sides of 1e-3 to 8e-3 across the grid: three digits only. So every cover above is computed as

cover = (W_from_zeros - prime_term) / sum_p cost_p,

with W from the verified zeros and both other terms exact. That is why the column is printed to nine digits while the quadrature column is only recorded; the two differ by the agreement column. Anyone quoting pole_plus_arch_quad here is quoting three-digit arithmetic.

4. What this adds

hunts/local_positivity showed the prime side is a sum of local norms and that they do not assemble globally. hunts/arch_cover showed the form has no uniform gap on a one-parameter family. This adds the distribution: the prime side is not carried by finitely many places with a stable covering margin. The margin decays as more places enter, so any estimate "the archimedean term beats the primes" gets harder exactly as the prime side becomes global, which is the regime where it must hold for all f.

5. Defects and scope

• One-parameter family (rectangles). Nothing here covers a general f, and the widest member still has h of slow decay.
• cover uses the zero side for W, so it inherits the zero table's 30-digit accuracy and the verified range gamma < 10^4. The zero tail bound is reported by explicit_formula_sides and is negligible for these b, but it is not zero.
• Q_p > ||f||^2 for every place measured here, i.e. every cost_p > 0. This is the opposite sign to the local-positivity reading of Q_p as a "norm at every place"; that hunt's gate applies to Toeplitz sections of local coefficients, not to ||Phi_p f||^2 of a fixed f, and the two should not be conflated.
• No human has reviewed any of this.