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Research Article · Preprint Mathematics

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

Scott Schubr

Luminary Oddities

  • Mathematics
  • Analysis and Spectral Theory
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
2 · Permanent link
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Overview

Abstract

A fixed rapidly twisted Euclidean tube has an infinite lattice of logarithmic inverse-speed harmonics invisible to its leading Weyl profile. We prove that every odd harmonic on that lattice is detected by the next heat profile. The proof avoids separate oscillatory calculations at successive frequencies. The full connected-versus-cut relative heat trace of the step aperture is a Laplace transform of a positive measure. We establish this fact by a finite positive-root matrix lemma, including threshold atoms, and two controlled aperture limits. A noncommuting trace estimate gives an entirely analytic mass bound. Euler-product domination then proves a uniform noncancellation inequality at all original odd frequencies. Combined with the preceding actual-tube heat expansion, this yields recovery of each admissible pure amplitude, and injectivity on admissible finite odd Fourier profiles. No sampled aperture calculation or floating-point PDE enclosure is needed. The lower bound decays with frequency.

Keywords twisted tubes · inverse spectrum · relative heat trace · Stieltjes transform · Mellin transform

Scope of the work

Simultaneous noncancellation of every original odd inverse-speed harmonic in the next heat profile of the specified rapidly twisted tube, with recovery of admissible pure amplitudes and finite odd Fourier profiles.

Reproduction notes

The paper has 9 pages. The editable-source archive contains 9 files. Compile main.tex. The two standard-library Python scalar checkers verify the stated constants; the included heat-trace predecessor supplies the cited geometric foundation.

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Author disclosures

Author-posted preprint. AI-assisted internal research and verification do not constitute external human peer review or proof-assistant formalization.

References and related publications

  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.

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Cite this edition

Scott Schubr. (2026). Audibility of all odd blind modes in rapidly twisted polar-strip tubes (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/audibility-all-odd-blind-modes-rapidly-twisted-polar-strip-tubes/v/2

BibTeX
@techreport{luminary_audibility_all_odd_blind_modes_rapidly_twisted_polar_strip_tubes_v2,
  title = {Audibility of all odd blind modes in rapidly twisted polar-strip tubes},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/audibility-all-odd-blind-modes-rapidly-twisted-polar-strip-tubes/v/2}
}

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