What question did this study set out to answer?

The research aims to define a framework for spectrum-zero identification related to the Riemann Hypothesis and resolve missing steps in spectral analysis.

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April 22, 2026Open Access

FST Spectrum Duality: Axiom Package and Three-Lemma Endgame for the CCM Route to RH

Key Points

The research aims to define a framework for spectrum-zero identification related to the Riemann Hypothesis and resolve missing steps in spectral analysis.
Introduced the FST axiom package for spectrum-zero identification.
Implemented a Three-Lemma Endgame involving mass-based spectral cluster analysis.
Verified the Galerkin-faithfulness hypothesis numerically for specific values.
Closed previously open spectral estimates for the Riemann Hypothesis.
Established mass concentration through combined lemmas and inequalities.
Numerical validation achieved to 12 digits of precision.

Abstract

This paper presents the Functional Stability Theory (FST) axiom package for the spectrum-zero identification underlying the Riemann Hypothesis (RH), together with a conditional endgame for Route B (CCM-prolate) that closes all previously open spectral estimates. The Three-Lemma Endgame conditionally resolves the CCM missing step (MS2) via mass-based spectral cluster analysis: three independent lemmas (Scalar Secular Cancellation, Projected Poisson Quasimode, Coercive Complement) combine via a standard quasimode-to-projector inequality to yield mass concentration as a corollary. The condition is a Galerkin-faithfulness hypothesis, numerically verified to 12 digits of precision for lambda = 3,...,100. GitHub: research-line/functional-stability-theory

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