What question did this study set out to answer?

This work aims to provide a computational and theoretical solution to the Riemann Hypothesis through discrete topology.

May 12, 2026Open Access

Computational Simulations of The Spectral Operator for the Riemann Hypothesis: Topological Disintegration, the Analytical Heart of the Proof, and Thermodynamic Convergence

Key Points

This work aims to provide a computational and theoretical solution to the Riemann Hypothesis through discrete topology.
Introduced a self-adjoint Real Symmetric Matrix operator derived from integer partitions under $A_{k-1}$ Weyl reflections.
Developed a 7-step algorithmic benchmark across 22 dimensions, scaling from $N=10$ to $N=2500$.
Mapped discrete eigenvalues using the inverse Cayley transform to analyze spectral stabilization.
As the spectral radius expands to approximately $10^{158}$, eigenvalues exhibit absolute asymptotic stabilization.
Achieved resonance alignment reproducing non-trivial zeros of the Riemann Zeta function on the critical line.
Demonstrated that these zeros act as fundamental topological defects of the partition manifold.

Abstract

This paper presents a definitive computational and theoretical resolution to the Riemann Hypothesis (RH) through discrete topology. We introduce a self-adjoint Real Symmetric Matrix operator derived exclusively from the additive combinatorics of integer partitions under A₊-₁ Weyl reflections (the Kaleidoscopic Filter). The core of this work lies in its unprecedented computational transparency: we provide exhaustive 7-step algorithmic benchmarks across 22 dimensional levels, scaling from a local state at N=10 to a massive thermodynamic limit at N=2500. We demonstrate mathematically and empirically that as the spectral radius of the system expands to quantum magnitudes (10^158), the discrete eigenvalues—mapped via the inverse Cayley transform—achieve absolute asymptotic stabilization. This exact resonance alignment perfectly reproduces the ordinates of the non-trivial zeros of the Riemann Zeta function on the critical line, proving that these zeros are fundamental topological defects of the partition manifold.

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