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Audibility of all odd blind modes in rapidly twisted polar-strip tubes

Scott Schubr

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Abstract

For a fixed connected polar section, the leading Weyl profile of a rapidly twisted Euclidean tube is independent of every odd logarithmic harmonic of inverse twisting speed. We prove that the next heat profile detects the entire odd lattice. A finite positive-root matrix lemma and two controlled aperture limits represent the full connected relative heat trace as the Laplace transform of a positive measure. An analytic mass bound and Euler-product domination then give a strict, frequency-dependent lower bound for every odd Mellin multiplier. Together with the established actual-tube heat expansion, this determines each admissible pure amplitude and every admissible finite odd Fourier profile from the complete spectrum. The argument is analytic; exponential decay of the bound records the conditioning of high modes.

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Edition files

Available editions

  1. Edition 3 · 2026-11-02 · public
  2. Edition 2 · 2026-10-27 · public
  3. Edition 1 · 2026-10-23 · public

Research details

Subjects
Mathematics · Analysis and Spectral Theory
Keywords
twisted tubes, inverse spectrum, relative heat trace, Stieltjes transform, Mellin transform
Manuscript date
2026-10-23
Reader reuse
No additional permission granted

Scope

For the specified connected polar section and one specified end or identical specified two ends, every original odd logarithmic inverse-speed harmonic has a nonzero next-heat multiplier. Complete spectral data determine admissible pure amplitudes and finite odd profiles, with exponential frequency conditioning.

Disclosures

Author-posted preprint. OpenAI ChatGPT and Codex assisted mathematical exploration, literature comparison, drafting and proof and code checks. The author assumes responsibility for all content.

Reproduction

Nine-page primary paper and nine-member package. The full current heat-lift predecessor is supplied as a proof supplement. build_pdf.py gives a reproducible Tectonic build. Two standard-library exact rational scalar checkers run normally and under Python -O; operator and geometric proofs are analytical.

References and related resources

  1. S. Schubr. Heat-Trace Recovery of a Weyl-Blind Twisting Mode. Edition 2, with original theorem from edition 1. Luminary Oddities (2026).
  2. S. Schubr. Certified resolvent detection of the third Weyl-blind twisting harmonic. Luminary Oddities, edition 2 (2026).
  3. S. Schubr. Left-Sector Mellin Certification of a Weyl-Blind Twisting Mode. Luminary Oddities, edition 2 (2026).
  4. M. van den Berg and S. Srisatkunarajah. Heat equation for a region in R^2 with a polygonal boundary. Journal of the London Mathematical Society 37 (1988), 119-127.
  5. J. Behrndt, F. Gesztesy and S. Nakamura. Spectral shift functions and Dirichlet-to-Neumann maps. arXiv:1609.08292v1.
  6. A. Gomilko, M. Haase and Y. Tomilov. Bernstein functions and rates in mean ergodic theorems for operator semigroups. Author preprint (2011), Section 2.
  7. S. Schubr. Finite-spectral recovery of Weyl-blind twisting modes. Luminary Oddities, edition 2 (2026).