What question did this study set out to answer?

This work aims to provide a comprehensive algebraic and geometric approach to the Riemann Hypothesis through the 59-dimensional framework.

May 11, 2026Open Access

The No-Vacuum Principle and the Riemann Hypothesis: A Proved Algebraic Theorem, the Seifert Matrix Construction from Prime Numbers, and the Hermitian Operator H₅₉

Key Points

This work aims to provide a comprehensive algebraic and geometric approach to the Riemann Hypothesis through the 59-dimensional framework.
Developed a main theorem proving the eigenvalues of certain matrices are real with a specific part.
Constructed the Hermitian operator H₅₉ and verified it through numerical checks and algebraic proof for dimension 59.
Established equivalence between the Riemann Hypothesis and a derived L²-condition.
Proved that any matrix S satisfying the condition has eigenvalues with real part 1/2 and verified this for dimensions 4, 10, 29, and 58.
Showed that all roots of the Alexander polynomial lie on the unit circle, supporting the findings.
Presented numerical evidence suggesting that the conjectured Seifert matrices align with Riemann zeros to within 0.02.

Abstract

This paper presents the most complete synthesis of the59-dimensional approach to the Riemann Hypothesis. WHAT IS PROVED (rigorously): 1. Main Theorem: Any real matrix S satisfying S + Sᵀ = I has all eigenvalues with real part exactly 1/2. Complete algebraic proof. Machine-precision numerical verification for n = 4, 10, 29, 58. 2. Corollary: All roots of the Alexander polynomial det (tS − Sᵀ) lie on the unit circle |t| = 1. 3. Theorem: The candidate operator H₅₉ = DEuler + (29/59) ·iHHilbert + (1/59) ·Hₒsc is Hermitian, constructed entirely from the partition 59 = 29+1+29 with no free parameters. 4. Equivalence: The Riemann Hypothesis is equivalent to an L²-condition derived from the 59-dimensional framework. WHAT IS NEW vs ALL PRIOR APPROACHES: Every prior approach — Berry-Keating (1999), Connes (1999), Lapidus-Herichi (2012) — treats the value 1/2 as amathematical given. This work is the first to providea geometric derivation of why 1/2 is the critical value, arising from the exact arithmetic identity: D (+) / (D (+) + D (-) ) = 29/58 = 1/2within the 59-dimensional structure 59 = 29+1+29. WHAT IS CONJECTURED (not proved): An explicit sequence of Seifert matrices SN constructedfrom the prime numbers satisfies SN + SNᵀ = I byconstruction. The conjecture — equivalent to the RiemannHypothesis — is that the Alexander polynomialsdet (tSN − SNᵀ) converge to 1/ζ (s) under thesubstitution t = exp (2π (s − 1/2) ). Numerical evidence for N = 59 shows agreement with thefirst four Riemann zeros to within 0. 02. WHAT REMAINS OPEN (stated explicitly): Whether Spectrum (H₅₉) = tₙ: ζ (1/2 + itₙ) = 0— the full Hilbert-Pólya conjecture, open since 1914. This paper is an open invitation to number theorists, harmonic analysts, and mathematical physicists to engagewith this derivation programme. Synthesis of: DOI 10. 5281/zenodo. 20060838, 10. 5281/zenodo. 20072650, 10. 5281/zenodo. 20074018, 10. 5281/zenodo. 20076536. Declaration: This work was developed in collaborationwith Claude (Anthropic, Claude Sonnet 4. 6) used foralgebraic verification and numerical computation. All mathematical ideas and responsibility rest withthe author.

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