Fiber Singularity Theory A Unified Framework of Fractal Geometry, Quantum Chaos, and Riemann ζ Zeros

Abstract

Classical mathematics treats singularities (poles, essential singularities, discontinuities, non-differentiable points) as signs of theoretical failure. However, physical systems—from turbulence and black holes to quantum energy levels—frequently exhibit such structures, suggesting that the problem may lie not in the singularities themselves, but in the inappropriate choice of the "stage" on which we observe them.This paper proposes a new mathematical-physical framework called "Bu Theory" (Buxue), whose core idea is: replace each geometric point with an internal "germ" carrying two states (0,1) and admitting the identity 0≡1 within. Simultaneously, we abandon the open real line and adopt the compactified circle S¹, topologically identifying +∞ ∼ −∞.Under this new framework, classical "pathological" functions (e.g., the everywhere non-differentiable Weierstrass function, the completely discontinuous Dirichlet function, essential singularities with infinite oscillation) are no longer pathological structures but become regular sections of self-adjoint fiber bundles. We introduce a parameter μ (the "germ insertion strength") to continuously tune spectral statistics, proving that:•μ = 0: Poisson statistics (no level repulsion);•μ = 1: Wigner-Dyson GOE statistics (strong level repulsion);•Intermediate μ gives continuous intermediate statistics.More importantly, we establish for the first time a quantitative channel linking the geometric multifractal spectrum f(α) to spectral rigidity Δ₃(L), and discover a non-trivial finite-size scaling constant R∞ ≈ 2.1, representing the "renormalization" factor by which fractal geometric information is amplified into quantum spectral rigidity.Finally, we apply the same diagnostic toolkit to the first 100 non-trivial zeros of the Riemann ζ function. The results show: ζ zeros exhibit GOE statistics (⟨r⟩ ≈ 0.616, close to the GUE theoretical value 0.599), while the deterministic n log n skeleton exhibits Poisson statistics (⟨r⟩ ≈ 0.9998). This provides a first-principles physical interpretation of the Montgomery-Dyson conjecture.Keywords: Fiber Singularity Theory; fractal geometry; quantum chaos; Riemann ζ zeros; random matrix theory; GOE; multifractal spectrum; Weierstrass function; spectral rigidity; Chalker-Kravtsov relation; Buxue; Point-Embryo; Möbius strip