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

Scott Schubr

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Abstract

A fixed rapidly twisted Euclidean tube has an infinite lattice of logarithmic inverse-speed harmonics invisible to its leading Weyl profile. This paper proves that every odd harmonic on that lattice is detected by the next heat profile. The connected-versus-cut relative heat trace of the step aperture is represented as a Laplace transform of a positive measure through a finite positive-root matrix lemma, threshold atoms and controlled aperture limits. A noncommuting trace estimate gives an analytic mass bound, and Euler-product domination yields uniform noncancellation at every original odd frequency. Combined with the preceding actual-tube heat expansion, this recovers each admissible pure amplitude and gives injectivity on admissible finite odd Fourier profiles. The argument is analytic.

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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

Ideal spectral identifiability for admissible pure amplitudes and finite odd Fourier profiles in the specified fixed geometry.

Disclosures

Author-posted preprint. AI-assisted internal research and verification were used.

Reproduction

The nine-member package includes editable LaTeX, the unchanged foundational heat-trace paper and two independent exact scalar checkers. With Python 3.10 or later, run python certify.py and python independent_check.py without -O or -OO. Only the standard library is required. These check scalar estimates; the operator and aperture-limit arguments are analytic.

References and related resources

  1. S. Schubr, Heat-Trace Recovery of a Weyl-Blind Twisting Mode , first edition, Luminary Oddities, 2026. https://luminaryoddities.com/researcharchive/ heat-trace-recovery-weyl-blind-twisting-mode/v/1 .
  2. S. Schubr, Certified Resolvent Detection of a Third Weyl-Blind Twisting Mode , companion manuscript, Luminary Oddities, 2026.
  3. S. Schubr, Left-Sector Mellin Certification of a Weyl-Blind Twisting Mode , companion manuscript, Luminary Oddities, 2026.
  4. M. van den Berg and S. Srisatkunarajah, Heat equation for a region in R2 with a polygonal boundary, J. London Math. Soc. (2) 37 (1988), 119–127. doi:10.1112/jlms/s2-37.121.119.
  5. J. Behrndt, F. Gesztesy and S. Nakamura, Spectral shift functions and Dirichlet-to-Neumann maps , arXiv:1609.08292. https://arxiv.org/abs/1609.08292.
  6. A. Gomilko, M. Haase and Y. Tomilov, Bernstein functions and rates in mean ergodic theorems for operator semigroups , author preprint, 2011, Section 2. https://www.math.uni-kiel. de/analysis/en/haase/research/9_GHT2-pre.pdf.