RESULTS: arch_cover

Nothing here is a result about the Riemann Hypothesis. This is a measurement of how the three pieces of the Weil functional balance, on two families of test functions, with enclosures where it is cheap and 30-digit mpmath elsewhere. Grade: measured. No proof, no review.

0. What was actually run

Every command runs from the repository root with the repository .venv and PYTHONPATH unset. (An operator environment note was removed at landing, 2026-10-30: it described one machine, not the record.)

1. The instrument

In this repository's 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),

and W(h) = sum_rho h(gamma_rho) for every admissible h. For h = |f^|^2 this says W >= 0 is RH, and the zero side is then manifestly a sum of non-negative terms. The only non-trivial content of the identity is that the arithmetic side, three terms each of size O(1), produces that non-negative number.

Define the cancellation depth

depth = log10( max(|pole|, |arch|, |prime|) / |W| ),

the number of digits by which the three O(1) pieces cancel to produce W.

2. Two families that probe opposite regimes

The Gaussian h = e^{-a r^2} has Fourier partner g proportional to e^{-x^2/(4a)}, a bump at the origin of width 2 sqrt(a). The prime sum sees g(log n), and log n >= log 2 = 0.693. So widening g (larger a) to reach the primes simultaneously narrows h, which pushes zeros out of the sum. Prime activity and zero activity move together along this family, in opposite directions.

The comb f(u) = sum_j c_j exp(-(u - j log 2)^2 / (2 sigma^2)) in zeta.weil.autocorrelation_pair decouples them in the other direction. Its partner g sits on log 2, 2 log 2, ..., so the prime side is O(1) by construction, while h = |f^|^2 stays broad in r and hundreds of zeros contribute.

The Gaussian does not decouple the two, and at large a it is in both regimes at once: g is wide enough to reach the primes (prime term 1.24 at a = 0.65) while h is narrow enough to leave a single zero. That combination is where the arithmetic side has to produce an exponentially small number.

3. The measurement (dps 30, zero side agrees to 1e-22 or better)

Zero side against arithmetic side: agreement between 1e-22 and 1e-32 on every row, so the three pieces are the true pieces of the identity, not a numerical artefact.

4. What the table says

(a) The prime side is not always negative. For the comb at sigma = 0.01 it is +0.0174: the prime sum helps the inequality. The sign follows the phase-space placement of f, because g may be negative where the zeros of the comb are. So my opening framing, "the archimedean term must cover the prime side", is wrong as a general statement. What has to happen is that the three-term sum comes out non-negative, and the terms can carry either sign.

(b) Depth is set by how far the test function sits from the first zero. For the Gaussian the zero side is W = 2 sum_rho e^{-a gamma^2} = 2 e^{-a gamma_1^2}(1 + O(e^{-a(gamma_2^2 - gamma_1^2)})), and the largest piece is pole = 2 e^{a/4}. So

depth = a * (gamma_1^2 + 1/4) / ln 10 = 86.88 * a,

with gamma_1^2 = 199.79. Measured against this: 4.344 vs 4.344 (a=0.05), 17.375 vs 17.375 (a=0.2), 26.063 vs 26.063 (a=0.3), 39.094 vs 39.094 (a=0.45), to every digit results.json and a direct sum over the zeros carry. The statistic carries no information beyond h(gamma_1) against the pole term; it measures the gap between the test function's scale and the first zero, nothing deeper.

Corrected 2026-10-30, at landing. This paragraph first gave depth = a gamma_1^2 / ln 10 and "predicted" values 17.36, 26.05 and 39.07, which match neither that form (17.354, 26.030, 39.046) nor the measurement. The first form drops the pole's e^{a/4}. Against it, the MISSION kill condition "the closed form depth fails to reproduce a measured row to the printed digits" fired (17.38 measured against 17.35), and that was not recorded. It is recorded here: the condition fired against the form as first written, and the corrected form reproduces every row.

(c) Both difficulties are active at large a, contrary to a first reading of the small-a rows. The prime side grows with a (0.116 at 0.05, 0.603 at 0.2, 1.240 at 0.65) while the zero sum collapses onto gamma_1. So the hard regime is not exotic: it is inside the Gaussian family. There the arithmetic side, three O(1) terms with either sign, has to land on a number of size e^{-a gamma_1^2}. That is the actual shape of the difficulty, and it is a statement about cancellation on the arithmetic side, not about the zeros: the zero side sum h(gamma) is a sum of non-negative terms and never cancels at all.

(d) The inequality has no uniform gap. Along this family W -> 0 while every piece stays O(1), so the bottom of the Weil form over the admissible class is 0 and is approached, not attained away from the zeros. Normalized by ||f||^2 = (1/2 pi) int h = 1/(2 sqrt(pi a)), the ratio W / ||f||^2 is about 4 sqrt(pi a) e^{-a gamma_1^2} and also tends to 0. Consequence: no inequality W(|f^|^2) >= c ||f||^2 with a fixed c > 0 holds over the admissible class. A proof must be an identity, or an inequality whose constant degenerates exactly as the test function concentrates away from the zeros. This is the single most useful constraint the measurement produces, and it is elementary once stated.

(e) The enclosed instrument has a floor at ~33 digits, set by prime truncation. At a = 0.65 the enclosure is [-7.12e-33, +6.36e-34], sign = 0. It was reported identical at 256 and 512 bits; only the 256-bit run is committed (rows.jsonl), so the 512-bit half of that statement is unpinned. The binding term is prime_tail_bound = 7.76e-33 at the n_max cap of 2e6; arithmetic precision is irrelevant. The value from the zeros there is 7.98e-57 (first written "about 4.8e-57"; corrected 2026-10-30). So enclosed positivity cannot be pushed past roughly 33 digits of cancellation by adding bits: it needs a different prime-tail argument. Every UNPROVEN row in rows.jsonl is this floor, not a failure of the math.

5. The door

The measurements above close one route rather than open one: "establish W > 0 numerically and extrapolate" is dead, because the quantity to be enclosed is exponentially small in a parameter while its three pieces are not, and the arithmetic side is where the cancellation happens.

What survives is the structural question. W is linear in h, so for a family f = sum a_j f_j the Gram matrix

M_jk = W(h_jk), h_jk(r) = f^_j(r) f^_k(r), G_jk = <f_j, f_k>

gives W(|f^|^2) = a^T M a. A proof by positivity would amount to exhibiting M = B^T B with B built from arithmetic data alone. The first job of that instrument is negative-control work: reproduce the known degenerate direction (concentrated h, a^T M a -> 0) so that any claimed factorization is tested against the one case where the answer is known to be zero. That is the next hunt, not this one; hunts/arch_cover measures and does not enclose lambda_min.

6. Scope and defects

• Finite families, finite grids. No statement here is uniform in f.
• The archimedean term has no second backend in this tree: enclose_weil_functional reports two_backend_cross_check unavailable because mpmath.iv has no enclosed quadrature. A quadrature error in arch would look like W moving, not like an error. This is the single largest unmitigated risk in every enclosed number produced here.
• The prime tail is truncated at a hard n_max = 2e6, and at narrow h that truncation, not precision, sets the enclosed floor (prime_tail_bound = 7.76e-33 at a = 0.65). Raising prec_bits from 256 to 512 was reported to change the enclosure not at all; that run is not committed. The row that reports UNPROVEN at a = 0.65 is this cap; the arithmetic is fine and the true value is ~e-57.
• cover.py's cover ratio is undefined when the prime side is positive; the code now reports None and prime_positive instead of dividing. The first version asserted positivity of -prime and died mid-scan without writing results.json; scan() now appends to rows.jsonl after every row.
• No human has reviewed any of this.

The doors

This hunt measures a ceiling, the 33-digit floor of the enclosed instrument, so it owes this section. The floor is a measured optimum at one cap, not a proved ceiling: candidate, Gaussian family a <= 0.65, information class the prime sum truncated at n_max, proof status none.

Active constraints at the optimum. One binds: prime_tail_bound at the n_max = 2e6 cap, 7.76e-33 at a = 0.65, against a true value of 7.98e-57. Arithmetic precision does not bind (256 bits leave the enclosure unchanged on the committed run). The zero side does not enter the enclosure.

The frozen-constant inventory.

The information class. Raising n_max or sharpening the tail bound stays inside the data this family reads (primes, the archimedean integral, the pole). Neither reads the zeros. Reaching depth 56 at a = 0.65 enclosed needs the tail of the prime sum to be bounded to about 1e-57, which the current bound does not approach at any affordable n_max.