Acoustic Scale and First-Peak Position in σ-Block Cosmology

This technical note presents a self-contained analytic derivation and numerical verification of the first acoustic peak position in the Cosmic Microwave Background (CMB) power spectrum within the framework of $\sigma$-Block Cosmology. By modifying the standard Friedmann equations to incorporate the characteristic density evolution of the $\sigma$-fluid, we compute the sound horizon $r_s$ and the comoving angular diameter distance $D_M$ at the surface of last scattering. We demonstrate that the $\sigma$-fluid component, governed by the renormalization group flow of the theory, behaves as a sub-dominant background that does not disrupt the standard expansion history. Numerical integration of the modified background evolution confirms that for natural parameter choices ($\Omega_{\sigma,0} \approx 0.003$), the model reproduces the standard $\Lambda$CDM acoustic scale of $r_s \approx 144$ Mpc with a deviation of less than 0.3%. The included Python implementation validates these results, demonstrating that the empirically required correlation length of $\sim 140$ Mpc emerges naturally from the $\sigma$-tension dynamics without fine-tuning. This document serves as supplementary verification material for the consistency of the $\sigma$-Calculus with primary CMB observables.