What question did this study set out to answer?

The primary aim is to address the negative eigenvalue issue in the Lo-Shu Yang-Mills framework.

May 16, 2026Open Access

Diagonal Rotation Correction for Open Problem 1: |Lₛym| Resolves Negative Eigenvalue and Reveals Isotropic Excited Subspace (sqrt (3) *I) in Lo-Shu/Yang-Mills Framework

Key Points

The primary aim is to address the negative eigenvalue issue in the Lo-Shu Yang-Mills framework.
Evaluated the operator absolute value |L_sym| to determine eigenvalues.
Used 100,000 Monte Carlo directions to verify results without violations.
Investigated the relationship between |L_sym| and the bridge operator Phi to the Hilbert space.
|L_sym| has eigenvalues {sqrt(3), sqrt(3), 15} indicating positive semi-definiteness.
|L_sym| restricted to the excited subspace equals exactly sqrt(3)*I2, demonstrating complete isotropy.
|L_sym| >= C_min * P_h^2 with C_min = 8.119 was confirmed statistically.

Abstract

We resolve the negative eigenvalue obstruction in the Lo-Shu Yang-Mills framework (Paper v7, DOI: 10. 5281/zenodo. 19910348). Lₛym has eigenvalues -sqrt (3), +sqrt (3), 15; the negative value arises from negative Cartan coefficients (lambda₃, lambda₈). Taking the operator absolute value |Lₛym| yields eigenvalues sqrt (3), sqrt (3), 15, which is positive semi-definite. Key new result: |Lₛym| restricted to the excited subspace u2, u3 equals exactly sqrt (3) *I2 (scalar matrix), implying complete isotropy — a discrete analogue of gauge invariance. We prove |Lₛym| >= Cₘin * Pₕ² with Cₘin = 8. 119, verified by 100, 000 Monte Carlo directions with zero violations. The remaining gap is the bridge operator Phi: Lo-Shu R³ -> HYM Hilbert space, with a candidate intertwining relation HYM * Phi = Phi * |Lₛym|.

Diagonal Rotation Correction for Open Problem 1: |Lₛym| Resolves Negative Eigenvalue and Reveals Isotropic Excited Subspace (sqrt (3) *I) in Lo-Shu/Yang-Mills Framework

Key Points

Abstract

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