Spectral Properties of the Truncated Weil Operator & Numerical Verication of Weil Positivit

We study the truncated Weil operator AΛ on L2([− log Λ, log Λ]), following
Connes's programme [3, 4] for the Riemann Hypothesis.
We establish ve unconditional spectral results: a traceenergy identity provid-
ing structural positivity on the Sonin space, J-self-adjointness with respect to the
Kren involution, resolvent convergence at rate O(Λ−1/2), a variational lower bound
ε0(Λ) ≤ −2√Λ + C, and a HilbertSchmidt decomposition separating archimedean
and atomic contributions. We further prove that norm convergence of the trun-
cated operators to the prolate projection is impossible, ruling out the most natural
convergence route.
Our main contribution is a precise structural reduction: the Riemann Hypoth-
esis in this framework is equivalent to asymptotic kernel invisibilitythe property
that Weil test functions become orthogonal to the kernel of the compressed atomic
operator as Λ → ∞. All currently identied nite-scale algebraic obstructions are
resolved; the remaining diculty is concentrated in this single analytic statement,
which we formulate as a concrete, falsiable problem.