MUT v5.8.13: Supplement X: Determining the Age of the Universe through Background Scalar Field Frequency Evolution

This supplement extends the MUT v5.8 scalar-impedance framework to derive the age of the universe from first principles. Rather than relying on the Hubble constant alone, cosmic time is formulated as the accumulation of phase cycles of the Background Scalar Field (ΦA).In this framework, the temporal duration of the universe is determined by the inverse of the scalar field frequency, integrated over the cosmic scaling parameter z. Specifically, the total age is obtained by integrating the reciprocal of the frequency from the Planck-scale regime to the present-day scaling factor.The frequency decay of the Background Scalar Field, as the universe expands, naturally produces a forward flow of time and a deterministic measure of elapsed cosmic time. Numerical evaluation of this integral yields an age of approximately 13.8 billion years, fully consistent with Lambda-CDM observations, but derived purely from the evolution of the fundamental scalar field.This work completes the temporal sector of the MUT v5.8 framework, linking cosmological expansion, frequency decay, and measurable cosmic age within a unified Background Scalar Field model.Series Note:This is part of the MUT v5.8 Temporal Series, following Supplement VIII on the Arrow of Time and Supplement IX on Consciousness, forming a cohesive framework for scalar-field-based temporal physics.