Zeta Zeros Mirror Random Matrix Eigenvalues, Linking Primes to Geometry — E8 Intelligence Research

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Riemann zeta zeros exhibit statistical correlations identical to the eigenvalues of random Hermitian matrices (GUE), establishing a link between prime number distribution, root-system geometry, and Painlevé transcendents. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22702062

September 11, 2026Open Access

Zeta Zeros Mirror Random Matrix Eigenvalues, Linking Primes to Geometry — E8 Intelligence Research

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Key Points

Characterize the statistical correspondence between the non-trivial zeros of the Riemann zeta function, random matrix spectra, and root-system geometry.
Evaluated the statistical distributions and pair correlation functions of critical-line Riemann zeta zeros against the Montgomery–Odlyzko law.
Modeled spectral spacing and eigenvalue repulsion using the Gaussian Unitary Ensemble (GUE) and Fredholm determinants based on Hermite, Laguerre, and Jacobi polynomial kernels.
Derived connections between eigenvalue distributions and root system Weyl chambers alongside Painlevé V and VI transcendent representations.
Zeta zero spacings demonstrated local level repulsion matching the Gaussian Unitary Ensemble distribution scaling as s² exp(-πs²/4), distinct from Poisson statistics.
Pair correlations strictly aligned with the Montgomery–Odlyzko formulation 1 - (sin(πs)/(πs))².
Extremal eigenvalue distributions across ensembles were directly mapped to Painlevé transcendents and Weyl chamber volume formulas.

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Implication

Mathematical analysis demonstrates statistical equivalence between Riemann zeta zeros and random matrix eigenvalues, highlighting fundamental geometric connections to prime numbers.

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Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6aa7b3440926e14a848b20b6 https://doi.org/10.5281/zenodo.22702062

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Zeta Zeros Mirror Random Matrix Eigenvalues, Linking Primes to Geometry — E8 Intelligence Research

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Zeta Zeros Mirror Random Matrix Eigenvalues, Linking Primes to Geometry — E8 Intelligence Research

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