RESULTS: weil_gram

Nothing here is a result about the Riemann Hypothesis. This measures the bottom of the Weil form on Gaussian subspaces. Grade: measured. No proof, no review, no ball-arithmetic enclosure of the eigenvalues.

1. The instrument, and its check

For a real basis f_1..f_J,

M_jk = W(h_jk), h_jk = f^_j * f^_k, G_jk = <f_j, f_k>, W(|f^|^2) = a^T M a for f = sum a_j f_j.

Gaussian basis, f_j = N(0, sigma_j^2):

h_jk(r) = exp(-a2 r^2 / 2), a2 = sigma_j^2 + sigma_k^2, g_jk(u) = N(0, a2)(u), G_jk = sqrt(2 sigma_j sigma_k / a2).

Every entry computed twice: arithmetic side (explicit_formula_sides) and zero side (2 sum_{gamma>0} h_jk(gamma)). Residuals, 15 unique entries, dps 50:

Two things about this table, both of which constrain what can be claimed:

• The residual floor is ~1e-32 absolute, and it is the zero table's 30-digit precision, not the arithmetic side: raising dps from 50 to 90 changes no eigenvalue at any printed digit, and no residual beyond its 15th digit (results_dps90.json). Corrected 2026-10-30, at landing: the first results_dps90.json was computed at dps 50, because run_base took its dps default from DPS when it was defined and main() never passed it, so the file recorded "dps": 50 for every basis. With the default passed explicitly, the run was repeated at dps 90 (3 min 45 s) and the file replaced; the statement above now rests on that run. So the arithmetic side is more accurate than the check, and the check cannot confirm entries below ~1e-32.
• (0.60,0.60) is the exception that matters: its prime_tail_bound is 1.2e-33, comparable to the value 1.16e-31. That entry is 1% uncertain by truncation, and that uncertainty is what makes the plus2 eigenvalue meaningless (section 3). The uncovered flag written by gram.py compares the residual against the truncation tails only, which for every other entry are e^{-aR^2} with R = 10^4 and therefore absurdly small (1e-434294); the flag fires on the zero-table floor and is uninformative except near the floor.

2. The gap, and what happens when one element is concentrated

The collapse from 3.6e-7 to 3.9e-25 is the single-element degeneracy, not a collective effect: the bottom eigenvector is the coordinate vector of the concentrated element. That is the same direction hunts/arch_cover measured (depth = sigma^2 (gamma_1^2 + 1/4) / ln 10; first written without the + 1/4, corrected 2026-10-30), restated at the level of the form.

plus2 is not a measurement. The (0.60,0.60) entry carries 1% truncation uncertainty, i.e. ±1e-33, and the eigenvalue is 7e-35: the two sides disagree by 25 orders because the eigenvalue is far below the entry uncertainty. What is established is only lambda_min(plus2) < 1e-30.

3. The measured content: the gap is not stable under refinement

The broad span alone, with no concentrated element, refined by adding sigma = 0.05, 0.25, 0.30 in that order (refine_dps50.json):

Every element here is individually healthy: W(h_jj) at sigma = 0.30 is 3.10e-8, and at sigma = 0.05 it is 2.75. Nothing in the basis is close to missing the zeros. Yet the bottom of the form falls by 19 orders from k=2 to k=6, and the arithmetic and zero sides agree at every printed digit.

The mechanism is visible in the eigenvector at k=6: a combination concentrated on sigma = 0.30 with a -0.075 admixture of sigma = 0.25, which cancels F = sum a_j f^_j against the low zeros. At k=6 the eigenvalue is 14 orders below the best single element in the span. A J-dimensional Gaussian span has J parameters with which to make F small at gamma_1, ..., gamma_J, and it uses them: the descent is geometric in the dimension, roughly a factor 1e-6 per added element at this end of the range.

4. What this settles

lambda_min on a span is the minimum of W(|f^|^2) / ||f||^2 over that span, and W(|f^|^2) = sum_rho h(gamma_rho). If every zero is on the line this is sum_gamma |F(gamma)|^2 with F = f^, a measure of how small an element of the span can be made at the zeros relative to its norm (it is not a distance, and reading it as |F|^2 presupposes the zeros are on the line, which the verified table guarantees only to its height). The measurement says this minimum falls geometrically with the dimension of a perfectly ordinary approximation space.

Consequence for strategy, and it is a negative one: a uniform positive gap does not survive enlargement. An argument that needs lambda_min bounded below independently of the subspace cannot work on these spans: the gap is created by the choice of subspace and destroyed by adding dimensions. Nonnegativity on each subspace of an exhausting sequence does pass to the limit; that weaker route is untouched by this measurement. (Scope corrected 2026-10-30: this paragraph first read "establish a positive gap on a growing subspace, then pass to the limit cannot work", and called lambda_min exactly a squared distance.) Any proof through this functional must handle the degeneracy explicitly (an identity, or a decomposition whose constant degenerates in step with the test function), which is the same conclusion hunts/arch_cover reached from the other direction, with a one-parameter family instead of a subspace.

It is not evidence about RH in either direction. lambda_min > 0 at every k is consistent with RH and with its negation on other spaces.

5. Defects and scope

• Eigenvalues are computed in mpmath, not enclosed in ball arithmetic. The (M, G) problem is badly conditioned by construction (entries span 1e-31 to 1), and nothing here is enclosed.
• Entry residuals sit at the zero table's 30-digit floor; the arithmetic side agrees at dps 50 and 90 to all printed digits (the dps-90 run as repeated on 2026-10-30, above), so the dps-90 run is a null test of the arithmetic and a positive test only of the zero table's limit.
• plus2 is unresolved, as stated. Do not quote its 7.1e-35 or 5.8e-60.
• The basis is Gaussian and centred at the origin. Shifts and non-Gaussian elements are not measured here.
• The archimedean term still has no second backend in this tree.
• No human has reviewed any of this.