ON THE SPECTRAL REALIZATION OF THE GEOMETRIC SIEVE VIA CENTERED LINEAR PENCILS AND SELF-ADJOINT OPERATORS

Abstract

This paper establishes a rigorous proof of the Riemann Hypothesis by constructing a densely defined, self-adjoint operator ABT on a weighted Hilbert space HBT

whose spectrum is in bijective correspondence with the non-trivial zeroes of the Riemann

Zeta function. By anchoring the algebraic center of inversion at the exact fractional

coordinates C(︁−1/2, −1/2)︁ under a stratified linear pencil of odd slopes km = 2m + 1, we

construct a geometric sieve that maps grid positions p yielding composite odd integers

N = 2p + 1 onto the lattice grid Z2. The prime locations are isolated as a strict topological

complement variety Mprime. Utilizing the U(1) local gauge symmetries of the resulting

Mittag-Leffler meromorphic derivative M(z) and applying the Guinand-Weil trace formula,

we demonstrate that the eigenvalue spectrum satisfies strict spectral completeness against

the von Mangoldt density. The self-adjointness of the core operator forces all non-trivial

eigenvalues to be strictly real, mapping the non-trivial zeroes ρn identically onto the critical

line Re(s) = 1/2.

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BibTeX RIS

BibTeX

@misc{gabbianelli2026spectral,
  title = {{ON THE SPECTRAL REALIZATION OF THE GEOMETRIC SIEVE VIA CENTERED LINEAR PENCILS AND SELF-ADJOINT OPERATORS}},
  author = {Silvio Gabbianelli},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23480087},
  url = {https://zenodo.org/records/23480087},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - ON THE SPECTRAL REALIZATION OF THE GEOMETRIC SIEVE VIA CENTERED LINEAR PENCILS AND SELF-ADJOINT OPERATORS
AU  - Silvio Gabbianelli
PY  - 2026
DA  - 2026/10/19
PB  - Zenodo
DO  - 10.5281/zenodo.23480087
UR  - https://zenodo.org/records/23480087
AB  - This paper establishes a rigorous proof of the Riemann Hypothesis by constructing a densely defined, self-adjoint operator ABT on a weighted Hilbert space HBT  whose spectrum is in bijective correspondence with the non-trivial zeroes of the Riemann  Zeta function. By anchoring the algebraic center of inversion at the exact fractional  coordinates C(︁−1/2, −1/2)︁ under a stratified linear pencil of odd slopes km = 2m + 1, we  construct a geometric sieve that maps grid positions p yielding composite odd integers  N = 2p + 1 onto the lattice grid Z2. The prime locations are isolated as a strict topological  complement variety Mprime. Utilizing the U(1) local gauge symmetries of the resulting  Mittag-Leffler meromorphic derivative M(z) and applying the Guinand-Weil trace formula,  we demonstrate that the eigenvalue spectrum satisfies strict spectral completeness against  the von Mangoldt density. The self-adjointness of the core operator forces all non-trivial  eigenvalues to be strictly real, mapping the non-trivial zeroes &rho;n identically onto the critical  line Re(s) = 1/2.
ER  -

Abstract

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