V1.30 Cosmological Redshift from Homogeneous Dilation

V1.30 Cosmological Redshift from Homogeneous Dilation in Unified Spacetime Theory (companion paper for Part 6 - History paper: https://doi.org/10.5281/zenodo.18102268 ) Abstract Unified Spacetime Theory treats the vacuum as a real physical medium whose state is described by a single configuration field. In that framework, cosmological “expansion” is not introduced as a metric postulate or an added cosmic fluid. It is the time-dependence of a homogeneous dilation of the medium itself. This paper shows that when the medium undergoes homogeneous isotropic dilation, the spacing of crests in a propagating shear-wave signal is stretched by the same dilation factor. As a result, an observer naturally measures a redshift that matches the standard wavelength-stretching law, without importing FRW geometry, continuity equations, or additional cosmological ingredients. The derivation is presented as a strict, reproducible chain: (i) a kinematic result from the homogeneous configuration map that fixes how physical wavevectors and wavelengths scale, and (ii) two explicit transport gates that control when the observed frequency tracks the instantaneous shear-wave dispersion with effectively constant propagation speed. Under these gates, the familiar frequency scaling follows and the redshift law is obtained directly from the medium’s deformation. As an internal consistency closure, the same chain implies the standard radiation bookkeeping scaling for shear-wave energy density. Vault-locked constants are held fixed throughout, the SI firewall is maintained, and all assumptions are stated explicitly as validity conditions rather than hidden adjustments. The 10 Part Series 1 & 2: Foundations paper - Hooke's Law/USEMP and Lorentz Invariance papers:https://doi.org/10.5281/zenodo.17887509 3: Spin 1/2 paper https://doi.org/10.5281/zenodo.17795865 4: Black Hole and Quantum Gravity paperhttps://doi.org/10.5281/zenodo.17804079 5: Dark Mass https://doi.org/10.5281/zenodo.18474582 6: Mechanical History of the Universe https://doi.org/10.5281/zenodo.18102268 Mechanical Strong Force in Unified Spacetime Theory: Flux Tubes and Confinement from Torsional Saturation https://doi.org/10.5281/zenodo.18602945 The foundational series “The Mechanical Foundations of a Unified Spacetime” from UST is organized as a ten-paper series, all built on the same mechanical medium developed here. Beginning with the foundations (USEMP) and the Lorentz-invariant paper, followed by Spin 1/2 paper, Black Holes and Quantum Gravity: The Mechanical Origin of Hawking Radiation, Dark Mass, History of the Universe, The Strong Force, The Weak Force, Emergence of Quantum Dynamics: Deriving the Schrödinger Equation from Continuum Mechanics, and finally The Particle Zoo: Deriving the Mass Spectrum from Solitonic Resonance. Together they extend the same underlying medium to black holes and Hawking radiation, dark mass, cosmology, the strong and weak interactions, the emergence of quantum dynamics, and the full particle zoo, always with the goal of giving a minimal, reproducible and testable mechanical account of observed physics rather than stitching together separate theories by hand. Contact: jared@ustphysics.org