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

General Relativity is extensively validated in the deeply screened, dense regime of the Solar System, but cosmological tensions—specifically the Hubble discrepancy and galactic mass anomalies—motivate testing a scale-dependent breakdown of the isochrony axiom. The Temporal Equivalence Principle (TEP) addresses these 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). Previous analytical approximations and CAMB integrations suggest this scalar field is frozen during radiation domination (T^mu_mu ~ 0). Full integration into linear structure formation requires a native Horndeski evaluation. This paper maps the TEP bi-metric framework—defined by a conformal factor A(phi) and a disformal deformation B(phi)—onto the Bellini-Sawicki Effective Field Theory (EFT) of Dark Energy to provide the theoretical blueprint for full linear structure mapping. To isolate the fundamental TEP effect and avoid Horndeski-class instabilities, the current numerical analysis isolates the background-only realization as the baseline numerical 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), applied through the Jordan-frame conformal factor M(z) = A/(1-alpha_A) as 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. A direct exact-frame-factor check confirms that replacing the implemented first-order factor A/(1-alpha_A) with the exact form A(1+alpha_A) leaves r_s unchanged to quoted precision in the screened regime (Section 5.4 and Appendix A.3a). 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 (a +0.185% shift 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 thereby reinterpreted as a late-time, environment-dependent clock-transport effect (Paper 11) rather than through a modified homogeneous expansion history at recombination. Keywords: cosmology theory, cosmic microwave background, Hubble tension, scalar-tensor theories, 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.20572723 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.