Vibrational Mathematics: The Foundational Doctrine of UVCS

This document presents Pillar I of Unified Vibrational Cosmological Science (UVCS): Vibrational Mathematics. It introduces the calibrated κ-vector, a dimensionless nine-parameter correction framework derived through a global Markov Chain Monte Carlo (MCMC) procedure. The theory unifies cosmological anomalies, particle-sector deviations, gravitational-wave propagation corrections, electromagnetic drifts, and neutrino mass shifts within a single vibrational mathematical structure.All κ-parameters are empirically constrained, symmetry-preserving, and compatible with effective field theory (EFT). The framework provides falsifiable predictions for atomic clocks, muon g-2, fine-structure variation, gravitational-wave timing, and cosmological density parameters.This release constitutes the first public scientific foundation of UVCS.