Spectral Admissibility, Harmonic Sums, and the Emergence of Lawful Modes
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2025
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| _version_ | 1866901697310228480 |
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| author | Stanford, Paul Vincent Raymond |
| author_facet | Stanford, Paul Vincent Raymond |
| contents | <p>This paper introduces a real-domain spectral admissibility framework, implemented in the Sc-Rubs rectified Laplacian engine, and demonstrates a structural parallel between this framework and the critical-line constraint of the Riemann zeta function.</p> <p>Rather than attempting to compute or reproduce Riemann zeros, the work focuses on the underlying mechanism that governs their lawfulness: the filtering of an infinite mode spectrum by a hard admissibility constraint that allows only a narrow, coherent set of survivors.</p> <p>The appearance of the half-spectrum constant π² / 12 in the Sc-Rubs system is interpreted as the real-domain analogue of the Riemann critical line Re(s) = 1/2. Both mark the boundary at which spectral growth and decay are balanced, and only lawful modes persist.</p> <p>This contribution is intended as a bridge between analytic number theory and physical spectral systems, offering a concrete operator-based realisation of the admissibility principle that the Riemann Hypothesis encodes abstractly.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18146403 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Spectral Admissibility, Harmonic Sums, and the Emergence of Lawful Modes Stanford, Paul Vincent Raymond spectral admissibility Riemann Hypothesis Riemann zeta function half-spectrum π² / 12 rectified Laplacian Sc-Rubs Hilbert–Pólya eigenmodes spectral filtering analytic continuation harmonic sums zeta regularisation emergent geometry physical operator models <p>This paper introduces a real-domain spectral admissibility framework, implemented in the Sc-Rubs rectified Laplacian engine, and demonstrates a structural parallel between this framework and the critical-line constraint of the Riemann zeta function.</p> <p>Rather than attempting to compute or reproduce Riemann zeros, the work focuses on the underlying mechanism that governs their lawfulness: the filtering of an infinite mode spectrum by a hard admissibility constraint that allows only a narrow, coherent set of survivors.</p> <p>The appearance of the half-spectrum constant π² / 12 in the Sc-Rubs system is interpreted as the real-domain analogue of the Riemann critical line Re(s) = 1/2. Both mark the boundary at which spectral growth and decay are balanced, and only lawful modes persist.</p> <p>This contribution is intended as a bridge between analytic number theory and physical spectral systems, offering a concrete operator-based realisation of the admissibility principle that the Riemann Hypothesis encodes abstractly.</p> |
| title | Spectral Admissibility, Harmonic Sums, and the Emergence of Lawful Modes |
| topic | spectral admissibility Riemann Hypothesis Riemann zeta function half-spectrum π² / 12 rectified Laplacian Sc-Rubs Hilbert–Pólya eigenmodes spectral filtering analytic continuation harmonic sums zeta regularisation emergent geometry physical operator models |
| url | https://doi.org/10.5281/zenodo.18146403 |