What question did this study set out to answer?

The aim is to establish a proof of the Riemann Hypothesis through gauge theory and spectral analysis of eigenfunctions.

March 17, 2026Open Access

The Riemann Hypothesis: A Gauge-Theoretic Proof via Even Dominance and Spectral Stability

Key Points

The aim is to establish a proof of the Riemann Hypothesis through gauge theory and spectral analysis of eigenfunctions.
Developed Lemma B concerning the spectral gap bound for even and odd-index modes.
Reformulated Lemma C to control sector differences without global bounds.
Utilized computer-assisted methods for verifying eigenfunction behavior across a wide spectrum.
Each prime number shows a preference for even eigenfunctions as established in the Shift Parity Lemma.
Proved the Leading-Mode Cancellation phenomenon with defined mathematical constants.
Presented 17 independent results linking the findings across different regimes.

Abstract

Three-part research series establishing the Riemann Hypothesis via Connes' spectral program and the Weil quadratic form. v1.1 Changelog (from v1.0): Lemma B upgraded: Parity-split spectral gap bound (fully analytical). Even-index modes: gap ~ 2√λ. Odd-index modes: gap ~ 4π²(j²-1)√λ/L², with matched coupling suppression. Lemma C reformulated: Sector-difference control replaces the global ||R0K|| < 1 condition. Tail contributions cancel in the even-odd difference: |correction|/D ≤ 1/N0². Status upgrade: "established†" → "proved". No numerical constants remain in the proof of Regime 2. All daggers (†) removed from status tables. Key Results: Shift Parity Lemma: each prime individually favors even eigenfunctions 33 computer-assisted certificates (λ = 100 to 1,300,000) Leading-Mode Cancellation (c = 2 + √2) Proposition A6: three-regime bridge argument (proved) 17 independent results, 11 explored alternatives Papers: Part I (14p), Part II (41p), Part III (17p). English + German.

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Lukas Geiger

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Schwarzwald-Baar Klinikum

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The Riemann Hypothesis: A Gauge-Theoretic Proof via Even Dominance and Spectral Stability

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