Temporal Equivalence Principle: hi_class Background Implementation and CMB Acoustic Peak Preservation

Standard cosmology relies on spatial expansion and a Big Bang singularity to explain the Cosmic Microwave Background (CMB) acoustic peaks, the pre-recombination sound horizon, and the thermal scaling relevant to Big Bang Nucleosynthesis (BBN). This paper demonstrates that these early-universe successes are natively preserved under a static conformal temporal-transport geometry governed by the Temporal Equivalence Principle (TEP). The TEP addresses late-universe anomalies via a dynamical proper-time field, governed by environment-dependent Gradient Screening with saturation scale rho_T ~ 20 g/cm^3 (Paper 0, §7). The scalar field is frozen during radiation domination (T^mu_mu ~ 0), and the current numerical analysis isolates the background-only realization as the baseline implementation. This paper implements the native TEP background-only modification directly in hi_class via the transition function f_T(z) = ln(1+z) * exp(-(z/z_T)^n_T), with the exact Jordan-frame conformal factor M(z) = A/(1-alpha_A) giving H_TEP(z) = H_LCDM(z) * M(z) while preserving standard General Relativistic perturbations. The functional form is the shared TEP-C0 implementation (Paper 26): the exp(-(z/z_T)^n_T) factor enforces early-time freezing, so the temporal-shear field is suppressed for z >> z_T. Direct Boltzmann integration verifies this phenomenological boundary condition, confirming that the code preserves the pre-recombination sound horizon to parts-per-million (r_s^TEP / r_s^LCDM = 0.999994) and leaves the acoustic-peak morphology untouched. The exact geometric relation M = A/(1-alpha_A) is evaluated directly in the production implementation (Section 5.4 and Appendix A.3). The only CMB-level effect of a non-zero epsilon_T is a late-time angular-diameter-distance projection that rigidly rescales the angular acoustic scale theta_s (+0.185% at the fiducial epsilon_T = 0.0066, z_T = 5, n_T = 2) and is largely degenerate with H_0. A joint hi_class Cobaya MCMC (Planck 2018 low-l TT/EE + lensing + BAO + Pantheon+) yields epsilon_T = 0.0056 +/- 0.0043 and H_0 = 66.63 +/- 1.70 km/s/Mpc; TEP-C0 (Paper 26) with full Planck TTTEEE drives the homogeneous amplitude to epsilon_T = (6.75 +/- 0.24) x 10^-6. Pantheon+ nested sampling in TEP-C0 gives strong Bayesian preference for the TEP geometry (Bayes factor 131.6 vs Lambda CDM for z_T = 5). Within the TEP framework, the Hubble tension is interpreted as a late-time, environment-dependent clock-transport effect (Paper 11) rather than through a crisis in early-universe physics. Keywords: cosmology theory, cosmic microwave background, static universe, scalar-tensor theories, conformal gravity, Hubble tension, modified gravity, effective field theory of dark energy, hi_class, Horndeski, temporal equivalence principle, proper time, Cobaya, Planck 2018 Website: https://mlsmawfield.com/tep/hcRepository: https://github.com/matthewsmawfield/TEP-HC DOI: 10.5281/zenodo.20572722 Open Science Statement: This work is a preprint and is open to community review, ideas, and collaboration. All analysis code, configuration files, and manuscripts are open source. Feedback and contributions to further test these results are welcome.