A Scalar Hilbert–Pólya Overview: Centered Determinants, Archimedean Alignment, and a GL(1) Trace–Prime Package in a TEBAC 9D+ Perspective
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| Natura: | Recurso digital |
| Lingua: | inglese |
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2025
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| _version_ | 1866901599075434496 |
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| author | Karadzhov, Tosho Lazarov |
| author_facet | Karadzhov, Tosho Lazarov |
| contents | <p>This note provides a conditional scalar Hilbert–Pólya overview for the completed Riemann zeta function. It unifies three earlier preprints by the author into a single logical framework: (i) a scalar Hilbert–Pólya package with centered determinants and a Trace–Prime hypothesis, (ii) Archimedean alignment and order-one growth analysis, and (iii) an analytic GL(1) trace package for the completed zeta function.</p> <p>The core result is a “master implication” theorem: under a constellation of assumptions — Hilbert–Pólya selfadjointness, Archimedean alignment, a split Trace–Prime package (structural + regularity), geometric wedge analyticity for a GL(1) trace, and an analytic GL(1) package for ξ(s) — the normalized centered determinant associated to a scalar Dirac-type operator coincides with the completed Riemann zeta function up to an elementary exponential factor. In this setting, the nontrivial zeros of ξ(s) are identified with the spectrum of a selfadjoint operator in a GL(1) channel, and the Riemann Hypothesis becomes equivalent to spectral reality for that channel.</p> <p>Conceptually, the paper separates three kinds of input: geometric (Dirac operators on compact spin manifolds and their centered determinants), analytic (Trace–Prime regularity, wedge analyticity, and GL(1) trace machinery), and arithmetic (the structural appearance of the von Mangoldt function in the centered heat trace). The Trace–Prime hypothesis is explicitly split into a structural von Mangoldt block and a regularity remainder, and several open geometric and spectral problems are formulated to push these assumptions closer to unconditional analytic statements.</p> <p>The note also connects this scalar Hilbert–Pólya program to the author’s Spectral Topological Theory of Everything (TEBAC 9D+), where a 9-dimensional Dirac operator on a compact spin manifold encodes both physical observables and arithmetic data. In that broader setting, the scalar Hilbert–Pólya operator is interpreted as an effective one-dimensional GL(1) channel of the TEBAC 9D+ Dirac spectrum. The article, however, is purely a conditional logical synthesis, not a proof of the Riemann Hypothesis: it clarifies what combination of spectral, trace, and prime-structural properties would be sufficient to realize a scalar Hilbert–Pólya approach to RH within a TEBAC-inspired framework.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17655026 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Scalar Hilbert–Pólya Overview: Centered Determinants, Archimedean Alignment, and a GL(1) Trace–Prime Package in a TEBAC 9D+ Perspective Karadzhov, Tosho Lazarov Riemann Hypothesis; Hilbert–Pólya; Dirac operator; spectral geometry; GL(1) trace; centered determinants; Trace–Prime hypothesis; TEBAC 9D+; topological field theory. <p>This note provides a conditional scalar Hilbert–Pólya overview for the completed Riemann zeta function. It unifies three earlier preprints by the author into a single logical framework: (i) a scalar Hilbert–Pólya package with centered determinants and a Trace–Prime hypothesis, (ii) Archimedean alignment and order-one growth analysis, and (iii) an analytic GL(1) trace package for the completed zeta function.</p> <p>The core result is a “master implication” theorem: under a constellation of assumptions — Hilbert–Pólya selfadjointness, Archimedean alignment, a split Trace–Prime package (structural + regularity), geometric wedge analyticity for a GL(1) trace, and an analytic GL(1) package for ξ(s) — the normalized centered determinant associated to a scalar Dirac-type operator coincides with the completed Riemann zeta function up to an elementary exponential factor. In this setting, the nontrivial zeros of ξ(s) are identified with the spectrum of a selfadjoint operator in a GL(1) channel, and the Riemann Hypothesis becomes equivalent to spectral reality for that channel.</p> <p>Conceptually, the paper separates three kinds of input: geometric (Dirac operators on compact spin manifolds and their centered determinants), analytic (Trace–Prime regularity, wedge analyticity, and GL(1) trace machinery), and arithmetic (the structural appearance of the von Mangoldt function in the centered heat trace). The Trace–Prime hypothesis is explicitly split into a structural von Mangoldt block and a regularity remainder, and several open geometric and spectral problems are formulated to push these assumptions closer to unconditional analytic statements.</p> <p>The note also connects this scalar Hilbert–Pólya program to the author’s Spectral Topological Theory of Everything (TEBAC 9D+), where a 9-dimensional Dirac operator on a compact spin manifold encodes both physical observables and arithmetic data. In that broader setting, the scalar Hilbert–Pólya operator is interpreted as an effective one-dimensional GL(1) channel of the TEBAC 9D+ Dirac spectrum. The article, however, is purely a conditional logical synthesis, not a proof of the Riemann Hypothesis: it clarifies what combination of spectral, trace, and prime-structural properties would be sufficient to realize a scalar Hilbert–Pólya approach to RH within a TEBAC-inspired framework.</p> |
| title | A Scalar Hilbert–Pólya Overview: Centered Determinants, Archimedean Alignment, and a GL(1) Trace–Prime Package in a TEBAC 9D+ Perspective |
| topic | Riemann Hypothesis; Hilbert–Pólya; Dirac operator; spectral geometry; GL(1) trace; centered determinants; Trace–Prime hypothesis; TEBAC 9D+; topological field theory. |
| url | https://doi.org/10.5281/zenodo.17655026 |