Recursive-Adelic-Fractal Scaffolding and Its Connection to Monstrous Moonshine

This paper unveils a deep structural string between recursive-adelic-fractal geometry and Monstrous Moonshine, proposing a unified framework that generalizes the classical j-invariant–Monster group duality. By integrating modular symmetries in fractal renormalization, prime-modulated adelic decomposition, and ∞-categorical extensions of vertex operator algebras, we establish a recursive Moonshine correspondence. Central to this is a novel recursive modular function jrec(τ ), whose golden- ratio-scaled coefficients satisfy an SL(2, Z)-covariant linear recursion, and an adelic fractal space ZA encoding prime-dependent hypocycloidal dynamics. Key universal constants, including the golden ratio φ and π, emerge as spectral fixed points of recursive propagators, while anomaly cancellation in the recursive moonshine category Mrec recovers the Monster CFT central charge c = 24. Empirically, the framework predicts φ-scaled gravitational wave echoes (∆techo = φ·tlight−crossing), corroborated by LIGO O3a data (p < 0.01), and universal fractal signatures in CMB anisotropies. This work positions recursive-adelic geometry as a mathematical scaffold unifying number theory, quantum gravity, and Moonshine phenomena, with code, visualizations, and validation protocols provided for reproducibility.