A Gram-Spectral Autopsy of the BNBD Inversion Problem

The Nyman–Beurling–Báez–Duarte framework converts the Riemann Hypothesis into a Hilbert-space closure problem. This note isolates two distinct components of the finite inversion problem: a rigid Blaschke skeleton associated with hypothetical right-half off-critical zeros, and an arithmetic Gram projection defect arising from the finite fractional-part basis. We prove an orthogonal splitting identity showing that these two components interact with zero cross-term, record the spectral whitening formula for Tikhonov regularization, and present a finite-dimensional audit up to M=60. The diagnostics show that the tested projection is dominated by a low-divisor channel, while high-divisor indices provide only weak correction at substantial conditioning cost. No proof of the Riemann Hypothesis is claimed.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-05-19
DOI
https://doi.org/10.5281/zenodo.20277868
Primary Topic
Mathematical Analysis and Transform Methods
Type
preprint
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preprint

A Gram-Spectral Autopsy of the BNBD Inversion Problem

T.A. Ziad
Zenodo (CERN European Organization for Nuclear Research)
Mathematical Analysis and Transform Methods
preprint

A Gram-Spectral Autopsy of the BNBD Inversion Problem

T.A. Ziad
preprint en

Abstract

The Nyman–Beurling–Báez–Duarte framework converts the Riemann Hypothesis into a Hilbert-space closure problem. This note isolates two distinct components of the finite inversion problem: a rigid Blaschke skeleton associated with hypothetical right-half off-critical zeros, and an arithmetic Gram projection defect arising from the finite fractional-part basis. We prove an orthogonal splitting identity showing that these two components interact with zero cross-term, record the spectral whitening formula for Tikhonov regularization, and present a finite-dimensional audit up to M=60. The diagnostics show that the tested projection is dominated by a low-divisor channel, while high-divisor indices provide only weak correction at substantial conditioning cost. No proof of the Riemann Hypothesis is claimed.

Zenodo (CERN European Organization for Nuclear Research)
Mathematical Analysis and Transform Methods
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