APURⁿᵃᵈ: 8th Order Resonance Completion – A Fractal Scaling Framework for Gravitational Dynamics

Will be updated soon. See Information: https://doi.org/10.5281/zenodo.15556420 This work presents APURⁿᵃᵈ, a closed cosmological framework that replaces the speculative constructs of ΛCDM with a resonance-based model grounded in five measurable parameters: mass, length, the speed of light (c), the Hubble constant (H₀), and the resonance coefficient (β₍Res₎ ≈ 1.173). At its core, the framework redefines dark energy as a scale-dependent resonance effect and dark matter as gravitationally active unresolved mass-energy, unifying both through a fractal scaling operator. Key innovations: 8th Order Resonance Structure: Derived from 2³ combinatorial configurations of mass, length, and time/information, ensuring mathematical completeness. Core Equation:(APUR)n⋅a^⋅d=(1+H0Lc)n⋅a^⋅d(APUR)n⋅a^⋅d=(1+cH0L)n⋅a^⋅dwhere n (mass-scale exponent), â (active resonance operator), and d (fractal dimension) govern gravitational interactions. Empirical Validation: Strong correlation with observed phenomena (Bootes Void underdensity, galaxy rotation curves, SMBH mergers). Applications: Resolves the Hubble tension via β₍Res₎-scaled resonance. Predicts gravitational wave modulation (e.g., Δf/f ≈ 1.7 × 10⁻⁵ for LISA). Modular design allows integration with astrophysical, quantum, and interdisciplinary systems. Significance:APURⁿᵃᵈ shifts cosmology from unobservable "dark" entities to testable scale-resonant dynamics, offering a laptop-compatible alternative to ΛCDM’s supercomputer demands. Related Works: Predecessor frameworks: 10.5281/zenodo.15556420 Modular extensions: 10.5281/zenodo.15605970 Audience:Cosmologists, theoretical physicists, and researchers in gravitation, fractals, or computational astrophysics. This Zenodo entry is self-contained, with no esoteric or speculative claims – purely mathematical rigor and empirical alignment.