Certified resolvent detection of the third Weyl-blind twisting harmonic

Abstract

The leading Weyl profile of a rapidly twisted tube with a fixed connected step section annihilates every odd logarithmic harmonic of inverse twist speed. A preceding heat-trace result detects its fundamental harmonic but leaves the third harmonic unresolved. We give a computer-assisted resolution for that original third harmonic, without altering the section or adding a lower-order perturbation. The proof evaluates a complex Mellin invariant of the connected junction. We derive its half-strip resolvent formula, retain the noncommuting Dirichlet–Dirichlet and Dirichlet–Neumann interval roots, and enclose the infinite aperture inverse through arbitrary trial vectors and complete residual bounds. Sectorial trace estimates permit a wide analytic log strip, reducing the required quadrature to 27 certified complex nodes. Infinite angular and input tails, floating arithmetic, exact-node conversion, aliasing and the two sample tails are included. The resulting Mellin modulus is below 5.653 × 10^(-6), whereas cancellation requires modulus above 1.13 × 10^(-5). The actual third heat coefficient therefore has modulus greater than 5.08 × 10^(-5). The supplied source certificate states its arithmetic assumptions and regenerates its numerical inputs.