MISSION: the Gram matrix of the Weil form, and its bottom eigenvalue

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

The object

W is linear in h, so for a real basis f_1..f_J and f = sum a_j f_j,

W(|f^|^2) = a^T M a, M_jk = W(h_jk), h_jk = f^_j * f^_k, G_jk = <f_j, f_k>, lambda_min = min_a a^T M a / a^T G a.

RH says lambda_min >= 0 for every admissible subspace. The useful question is not the sign (the answer is known) but the size: whether the bottom of the form is bounded away from zero on a subspace, or sits at the numerical floor.

Basis used here

Normalised Gaussians f_j(x) = N(x_j, sigma_j^2). Then

h_jk(r) = exp(-(sigma_j^2+sigma_k^2) r^2 / 2) * cos(r (x_j + x_k)), g_jk(u) = N(x_j + x_k, sigma_j^2 + sigma_k^2),

because the Fourier partner of h_jk is the convolution f_j * f_k. Both are closed form and W(h_jk) is the repository's zeta.weil.weil_functional. The sigma_j control the width of h in r: large sigma means h is narrow and misses the zeros, which is the degenerate direction measured in hunts/arch_cover (depth = sigma^2 (gamma_1^2 + 1/4) / ln 10; first written without the + 1/4 from the pole term, corrected 2026-10-30).

What is measured

1. Entrywise agreement. Every M_jk computed twice: from the arithmetic side (weil_functional) and from the zero side (2 sum_gamma h_jk(gamma)). If these disagree the instrument is wrong and nothing else on the page counts.
2. lambda_min on a broad subspace (sigma <= 0.2), where h has mass on many zeros. Expected: bounded away from zero.
3. lambda_min after adding one concentrated element (sigma = 0.45). Expected: collapses to the floor, because the subspace now contains the direction hunts/arch_cover showed to be degenerate. This is the control: an instrument that still reports a positive gap here is lying.

What this may not claim

• Not evidence for RH, and not a proof of anything: lambda_min >= 0 on finitely many subspaces is what it says.
• lambda_min is computed in mpmath at finite precision and is not enclosed in ball arithmetic. The eigenvalues below ~1e-25 are noise, and are labelled so.
• The basis is Gaussian and centred at the origin; no statement survives outside it.

HuntSpec

id: weil_gram
question: Does the bottom eigenvalue of the Weil form stay bounded away from zero as the subspace grows?
frontier: measured here, lambda_min falls from 2.4e-3 to 5.0e-22 as the Gaussian span grows from 2 to 6 elements, with every entry checked against the zero side
proposed_attack: assemble the bilinear Gram matrix of W on a Gaussian basis and solve the generalized eigenproblem against the L2 Gram matrix
dead_routes:
  - establish a gap on a finite subspace and pass to the limit, killed here because the gap is created by the subspace choice and destroyed by enlargement
required_oracles:
  - zero side evaluation of every entry, independent of the arithmetic side
  - closed form Gaussian pair with exact inner products
kill_conditions:
  - an entry disagrees with the zero side beyond the table precision
  - an eigenvalue lies below the entry uncertainty and is nevertheless quoted
  - the bottom eigenvector is not reproducible under refinement
agents_may:
  - assemble
  - diagonalise
  - compare
  - code
agents_may_not:
  - declare a bound
  - promote an observation to a result
  - use the reserved vocabulary of zeta/rigor.py