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

Scott Schubr

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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.

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Edition files

Available editions

  1. Edition 2 · 2026-10-27 · public
  2. Edition 1 · 2026-10-23 · public

Research details

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

Scope

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.

Disclosures

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

Reproduction

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.

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 .
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  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.
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  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.
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  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.
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