Geometric Scale Invariance Across Cosmological Structure Growth, Electroweak Mixing, and Hadronic Mass Scales: A Rigorous Comparative Analysis of Continuous Euclidean vs. Discrete Holographic Trigonometry

Abstract

This paper addresses several persistent observational tensions across contemporary cosmology, electroweak mixing, and hadronic physics.We demonstrate that discrepancies in cosmic structure growth, flavor mixing unitarity, and fundamental particle mass scalesare not instrumental flaws, but mathematical artifacts of modeling physical space as an infinitely divisible continuous manifold.By contrasting standard Euclidean trigonometry with a scale-invariant discrete geometry, this work provides an exact unified resolution.The framework resolves the cosmological matter clustering tension between early-universe and late-time surveys across observational tomography.Furthermore, spatial cosmic flatness is established as an exact boundary condition, resolving the fine-tuning problem from first principles.In particle physics, the model derives the apparent unitarity deficit in quark mixing and the anomalous magnetic moment of the muon.Additionally, it achieves the exact analytical derivation of the fundamental proton-to-electron mass ratio without arbitrary inputs.This comparative analysis proves that cross-scale anomalies arise from analog oversampling within continuous differential field integrals.Operating without free adjustable parameters, the formulation achieves complete consistency with laboratory and astronomical data.The results demonstrate that macroscopic and microscopic laws emerge deterministically from scale invariance in bounded space.