What question did this study set out to answer?

The research aims to explore the Riemann Hypothesis using the framework of Free-Energy Spectral Theory.

March 17, 2026Open Access

Free-Energy Spectral Theory and the Riemann Hypothesis (FST-RH)

Key Points

The research aims to explore the Riemann Hypothesis using the framework of Free-Energy Spectral Theory.
Investigated the Riemann Hypothesis within a three-part series.
Applied the Weil quadratic form to develop spectral reformulations.
Analytically proved the Shift Parity Lemma for prime numbers.
Utilized computer-assisted certificates to verify results over a large range of values.
Presented findings across three structured parts detailing foundations, dominance, and conclusions.
Established conditional proof of the Riemann Hypothesis based on two key spectral constants.
Identified a preference for even eigenfunctions among prime numbers.
Achieved 33 computer-assisted certificates confirming results with no gaps.
Demonstrated leading-mode cancellation with a specific constant value.
Generated 17 independent results contributing to standalone mathematical interest.

Abstract

This project archive contains the complete three-part research series investigating the Riemann Hypothesis (RH) within the Free-Energy Spectral Theory (FST) framework, building on Connes' spectral reformulation via the Weil quadratic form (arXiv: 2602. 04022). Key Results: Shift Parity Lemma: each prime individually favors even eigenfunctions (analytically proved) 33 computer-assisted certificates (lambda = 100 to 1, 300, 000) with no gap Leading-Mode Cancellation with exact constant c = 2 + sqrt (2) M1'' (Resolvent Subdominance): proved for all lambda >= lambda₀ via PNT Transfer Proposition A6 (Cumulative Step): three-regime bridge argument 17 independent results of standalone interest 11 systematically explored and evaluated alternative strategies Paper Structure: Part I: Foundations and Obstructions (14 pages, EN + DE) Part II: Even Dominance of the Weil Quadratic Form (41 pages EN, 20 pages DE abridged) Part III: Conclusio (17 pages EN, 16 pages DE) Status: RH established (conditional on analytical derivation of two spectral constants, both numerically verified at all certificate values). Version 1. 1 with analytical bounds in preparation.

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Lukas Geiger (Sun,) studied this question.

synapsesocial.com/papers/69b8f12fdeb47d591b8c61e9 https://doi.org/https://doi.org/10.5281/zenodo.19035641

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