A Relay Density Inferred from the Galactic Acceleration Scale Reproduces the Effective CDM Contribution in CMB Acoustic Spectra

This release presents a reproducible cross-scale consistency test within the Geometric Relay Programme, built on the Entangled Relativity framework of O. Minazzoli. The central question is whether the effective non-baryonic matter density required by the CMB acoustic spectra can be reconstructed from the galactic acceleration scale \(a_0\), without fitting the CMB spectra themselves. The tested chain is: \[ a_0 \rightarrow m_0=\frac{3a_0}{c} \rightarrow \rho_m=\frac{3m_0^2}{8\pi G} \rightarrow \Omega_m \rightarrow \omega_{\rm relay} = \Omega_m h^2-\omega_b^{\rm BBN}. \] Using the GRP-derived value \(a_0=1.225\times10^{-10}\,\mathrm{m\,s^{-2}}\), the chain gives: \[ \Omega_m=0.3154, \qquad \omega_{\rm relay}=0.1207. \] This differs from the Planck \(\Lambda\)CDM best-fit CDM density, \[ \omega_{\rm cdm}=0.1200, \] by only \(+0.6\%\). The resulting density is inserted into CLASS without refitting the TT, TE or EE spectra. The computed spectra reproduce the acoustic structure at the sub-percent to percent level: TT RMS \(0.30\%\), TE RMS \(0.35\%\) using absolute normalization, EE RMS \(0.60\%\), and lensing \(\phi\phi\) RMS \(1.11\%\). The uploaded reproduction script documents the numerical chain and generates the CMB spectra and figures using CLASS/classy, NumPy, SciPy and Matplotlib. This release should be interpreted as a numerical cross-scale consistency test, not as a full replacement of \(\Lambda\)CDM parameter inference. The result does not yet include a full MCMC analysis, BAO, \(S_8\), \(f\sigma_8\), \(H(z)\), or a complete perturbative implementation of the relay field. Its epistemic status is therefore B+/A−: stronger than a qualitative analogy because the convergence \(a_0\rightarrow\omega_{\rm relay}\rightarrow TT/TE/EE\) is quantitative and reproducible, but not yet a final cosmological model.