Research Article · Preprint Mathematics
Certified resolvent detection of the third Weyl-blind twisting harmonic
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
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.
Keywords twisted tubes · relative heat trace · Mellin transform · Dirichlet-to-Neumann operators · validated numerics · inverse spectrum
Scope of the work
A certified connected-junction resolvent/Mellin calculation detects the original third Weyl-blind twisting harmonic, with heat multiplier modulus greater than 5.08 × 10⁻⁵ in the specified tube family.
Reproduction notes
The paper has 12 pages. The editable-source archive contains 10 files. Compile main.tex. certify.py assembles saved balls using the standard library; --recompute regenerates 27 nodes with NumPy 2.3.5 under the stated arithmetic model.
The manuscript
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Author disclosures
AI tools assisted derivations, implementation, literature checking and independent internal verification. Internal checks are not external human peer review; the manuscript and runnable certificate are supplied for public scrutiny.
References and related publications
- S. Schubr. Heat-Trace Recovery of a Weyl-Blind Twisting Mode. Luminary Research Archive, edition 1 (2026).
- 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.
- J. Behrndt, F. Gesztesy and S. Nakamura. Spectral shift functions and Dirichlet-to-Neumann maps. arXiv:1609.08292.
- L. N. Trefethen and J. A. C. Weideman. The exponentially convergent trapezoidal rule. SIAM Review 56 (2014), 385–458.
- NumPy developers. NumPy 2.3.5 source.
- E. Anderson et al. LAPACK Users’ Guide, third edition. SIAM, 1999: Error Bounds for Fast Level 3 BLAS.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Certified resolvent detection of the third Weyl-blind twisting harmonic (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/certified-resolvent-detection-third-weyl-blind-twisting-mode/v/2
BibTeX
@techreport{luminary_certified_resolvent_detection_third_weyl_blind_twisting_mode_v2,
title = {Certified resolvent detection of the third Weyl-blind twisting harmonic},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/certified-resolvent-detection-third-weyl-blind-twisting-mode/v/2}
}Loading discussion…