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Certified Resolvent Detection of a Third Weyl-Blind Twisting Mode

Scott Schubr

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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 the fundamental harmonic. This paper gives a computer-assisted resolution for the original third harmonic without changing the section or adding a lower-order perturbation. A half-strip resolvent formula and complete residual estimates enclose a complex Mellin invariant of the connected junction. Sectorial trace bounds reduce the certified quadrature to 27 complex nodes while retaining infinite tails, arithmetic error, node conversion and aliasing. The resulting Mellin modulus is below 5.653 × 10⁻⁶, whereas cancellation requires a modulus above 1.13 × 10⁻⁵; the third heat coefficient therefore has modulus greater than 5.08 × 10⁻⁵. The source certificate states its arithmetic assumptions and regenerates its inputs. Higher odd modes and finite-spectrum stability are outside this paper’s scope.

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Edition files

Available editions

  1. Edition 2 · 2026-10-27 · public
  2. Edition 1 · 2026-10-23 · public

Research details

Subjects
Mathematics · Analysis and Spectral Theory
Keywords
twisted tubes, relative heat trace, Mellin transform, Dirichlet-to-Neumann operators, validated numerics, inverse spectrum
Manuscript date
2026-10-22
Reader reuse
No additional permission granted

Scope

Computer-assisted detection of the original third Weyl-blind harmonic in the fixed connected polar-strip geometry. Includes the complete infinite-aperture and arithmetic error bounds under the stated implementation contract. The result establishes ideal asymptotic audibility.

Disclosures

AI tools assisted derivations, implementation, literature checking and independent internal verification. The manuscript and runnable certificate are supplied for public scrutiny.

Reproduction

The complete package contains the manuscript, editable source, certificate code, reader instructions, citation metadata and checksums. Its source regenerates the 27 connected-aperture node disks and scalar acceptance tests. The reference implementation uses CPython 3.12, NumPy 2.3.5 and the explicit IEEE binary64 contract described in the paper and README.

References and related resources

  1. S. Schubr. Heat-Trace Recovery of a Weyl-Blind Twisting Mode. Luminary Research Archive, edition 1 (2026).
  2. M. van den Berg and S. Srisatkunarajah. Heat equation for a region in R² with a polygonal boundary. Journal of the London Mathematical Society 37 (1988), 119–127.
  3. J. Behrndt, F. Gesztesy and S. Nakamura. Spectral shift functions and Dirichlet-to-Neumann maps. arXiv:1609.08292.
  4. L. N. Trefethen and J. A. C. Weideman. The exponentially convergent trapezoidal rule. SIAM Review 56 (2014), 385–458.
  5. NumPy developers. NumPy 2.3.5 source.
  6. E. Anderson et al. LAPACK Users’ Guide, third edition. SIAM, 1999: Error Bounds for Fast Level 3 BLAS.