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Left-Sector Mellin Certification of a Weyl-Blind Twisting Mode

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

We certify a complex Mellin invariant of a fixed connected step junction on a ray with negative real part. Positive-axis exponential flatness is propagated through an exterior sector using a classical Phragmén–Lindelöf multiplier, with explicit constants. A left-sector contour then suppresses the reciprocal-Gamma conditioning of the Mellin formula. The application proves that the original fifth logarithmic harmonic of inverse twist speed, invisible to the leading Weyl profile, is detected by the lower heat profile of the same fixed Euclidean tube. The noncommuting DD/DN aperture inverse is enclosed by arbitrary trial columns, complete residuals and infinite input tails. Sixty-seven complex nodes, a tiny endpoint calculation, every infinite quadrature tail and the stated floating arithmetic errors are included. The certified junction Mellin modulus is below 3.198 × 10^(-9), while cancellation requires modulus above 1.18 × 10^(-8); the fifth heat multiplier has modulus greater than 2.16 × 10^(-7). The general propagation mechanism is classical; the contribution is its explicit connected-junction certification and this original-mode separation. A standalone replay suite accompanies the proof.

Keywords relative heat trace · Mellin transform · sectorial resolvent · twisted tubes · validated computation · inverse spectrum

Scope of the work

An explicit left-sector connected-junction Mellin certificate detects the original fifth Weyl-blind twisting harmonic, with |C₅| > 2.16 × 10⁻⁷ under the stated operator and arithmetic hypotheses.

Reproduction notes

The paper has 12 pages. The editable-source archive contains 11 files. Compile main.tex. certify.py assembles saved aperture disks using the standard library; full 67-node replay requires NumPy 2.3.5 and the stated IEEE model. The optional independent scalar assembly uses python-flint 0.9.0.

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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. 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.
  3. J. Behrndt, F. Gesztesy and S. Nakamura, Spectral shift functions and Dirichlet-to-Neumann maps , arXiv:1609.08292. https://arxiv.org/abs/1609.08292.
  4. L. N. Trefethen and J. A. C. Weideman, The exponentially convergent trapezoidal rule, SIAM Review 56 (2014), 385–458. doi:10.1137/130932132.
  5. NumPy developers, NumPy 2.3.5 source: einsumfunc.py, einsum_sumprod.c.src, and loops_umath_fp.dispatch.c.src. https://github.com/numpy/numpy/tree/v2.3.5/numpy/ _core.
  6. E. Anderson et al., LAPACK Users’ Guide, third edition, SIAM, 1999, “Error Bounds for Fast Level 3 BLAS. ” https://netlib.org/lapack/lug/node108.html.
  7. S. Schubr, Certified Resolvent Detection of a Third Weyl-Blind Twisting Mode , prepared companion manuscript, Luminary Oddities, 2026.
  8. L. G. Rogers, Estimates for the resolvent kernel of the Laplacian on p.c.f. self similar fractals and blowups, arXiv:1009.5334, Section8, Corollary8.2 and its proof. https://arxiv.org/abs/1009. 5334.
  9. T. Coulhon and A. Sikora, Gaussian heat kernel upper bounds via the Phragmén–Lindelöf theorem, Proc. London Math. Soc. 96 (2008), 507–544. doi:10.1112/plms/pdm050.

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Left-Sector Mellin Certification of a Weyl-Blind Twisting Mode (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/left-sector-mellin-certification-weyl-blind-twisting-mode/v/2

BibTeX
@techreport{luminary_left_sector_mellin_certification_weyl_blind_twisting_mode_v2,
  title = {Left-Sector Mellin Certification of a Weyl-Blind Twisting Mode},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/left-sector-mellin-certification-weyl-blind-twisting-mode/v/2}
}

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