Temporal Equivalence Principle: EFT Mapping and CMB Acoustic Peak Preservation

1. Introduction

1.1 Contextualizing the TEP Corpus

The Temporal Equivalence Principle (TEP) has been constrained across 40 orders of magnitude in mass density, from terrestrial laboratory scales to cosmological observations. Previous papers in this series have established:

• Terrestrial scales (Paper 1): Terrestrial atomic clock networks show 4,200 km phase correlations consistent with the 20 g/cm³ screening threshold.
• Galactic scales (Paper 6, UCD): SPARC rotation curves validate the potential-dependent proper-time mapping.
• Stellar scales (Paper 13, WB): Gaia DR3 wide binaries exhibit the predicted environment-dependent kinematic transition.
• Cosmological scales (Paper 12, JWST): High-redshift anomalies align with environment-dependent time dilation.

1.2 The Two-Ended Hubble Tension

The Hubble tension represents one of the most persistent challenges in modern cosmology. Cepheid-calibrated local distance ladder measurements yield $H_0 \approx 73$ km/s/Mpc, while early-universe CMB inference from Planck gives $H_0 \approx 67.9$ km/s/Mpc—a discrepancy of approximately 4.8$\sigma$.

Previous TEP work (Paper 11, H₀) demonstrated that the Cepheid environmental bias—operating in the unscreened stellar atmospheres where Cepheids pulsate—naturally raises the local measurement. However, a complete resolution requires demonstrating that this mechanism does not disturb the early-universe inference at the surface of last scattering ($z \approx 1089$).

1.3 Purpose of This Paper

To move beyond the quasi-static approximations of previous work by natively solving the background expansion $H(z)$ and linear perturbation growth in the weakly screened density regime ($\rho \sim 10^{-21}$ g/cm³). This requires:

1. A mapping of TEP's bi-metric structure onto the Bellini-Sawicki EFT formalism (the numerical analysis uses the background-only realization).
2. Native implementation in hi_class with proper background scalar field evolution.
3. ΛCDM baseline MCMC parameter estimation ($H_0$, $\Omega_b h^2$, $\Omega_{\rm cdm} h^2$, $n_s$, $A_s$, $\tau$, $A_{\rm planck}$) against Planck 2018, BAO, and Pantheon+ data to establish the constraints that TEP should satisfy.

The critical question: Can TEP preserve CMB acoustic peaks while permitting late-time $H_0$ variation?

2. Theoretical Architecture: The EFT Mapping

2.1 The Bi-Metric Action

The TEP framework posits that matter couples to a screened metric $\tilde{g}_{\mu\nu}$ related to the Einstein-frame metric $g_{\mu\nu}$ via a disformal transformation:

where:

• $A(\phi) = \exp(\beta_A\phi/M_{\rm Pl})$ is the conformal factor, with $\beta_A = -1.0$ (the locked lab-scale convention used across the TEP corpus)
• $B(\phi)$ controls disformal deformation of the causal structure
• $\phi$ is the dynamical proper-time field

Metric signature convention: $(+, -, -, -)$ throughout.

2.2 Deriving the Bellini-Sawicki Alphas

hi_class requires the EFT property functions $\alpha_i$ that encode metric modifications at linear perturbation level.

2.2.1 Planck Mass Running ($\alpha_M$)

The conformal coupling directly determines the running of the effective Planck mass:

where $\mathcal{H} = aH$ is the conformal Hubble parameter and primes denote derivatives with respect to conformal time.

2.2.2 Tensor Speed Excess ($\alpha_T$)

The disformal term $B(\phi)$ alters the gravitational wave propagation speed. Multi-messenger constraints from GW170817/GRB 170817A require:

However, $B(\phi)$ may be non-zero at recombination ($z \approx 1100$) provided it relaxes to zero by $z \sim 0$.

2.2.3 Braiding ($\alpha_B$) and Kineticity ($\alpha_K$)

These functions govern scalar field clustering and metric mixing:

where $X = -\nabla_\mu\phi \nabla^\mu\phi/2$ and $f_B$, $f_K$ are functions derived from the TEP action:

• $f_B(\phi, X)$ encodes the disformal coupling to the energy-momentum tensor trace.
• $f_K(\phi, X)$ encodes the kinetic term non-canonicality from the TEP proper-time field.

The explicit functional forms follow from the bi-metric action (Equation \ref{eq:3_theory_01}) and are determined by the conformal factor $A(\phi)$ and the disformal function $B(\phi)$. Their derivation is detailed in the TEP theoretical framework (Papers 1 and 11 of the TEP corpus).

2.3 The Radiation Domination Freezing Mechanism

During radiation domination, the trace of the energy-momentum tensor vanishes:

Since the TEP scalar field couples to $T^\mu_\mu$, the source term for $\phi$ evolution is suppressed. The field freezes at its initial value, and $\alpha_M, \alpha_B \rightarrow 0$. This ensures:

• Primary acoustic peaks ($100 \lesssim \ell \lesssim 2000$) generated at $z \sim 1089$ remain unmodified
• The sound horizon $r_s$ is preserved, anchoring $H_0$ from CMB at $\sim 67.9$ km/s/Mpc
• Late-time matter domination reactivates the scalar field, enabling environmental $H_0$ variations

3. Software Implementation: hi_class and the Unscreened Regime

3.1 The hi_class Architecture

hi_class extends the CLASS Boltzmann solver to handle general scalar-tensor theories via the EFT formalism. This work uses hi_class v3.2.3 with the modified gravity (SMG) module enabled.

3.2 Native TEP Background-Only Implementation

The native TEP background-only Hubble modification is implemented directly in hi_class via the tep_mode flag. When enabled, the background expansion history is modified as:

where $A(z) = \exp[\epsilon_T \ln(1+z) \exp(-(z/z_T)^{n_T})]$ is the covariant conformal factor and $\alpha_A = -d\ln A/d\ln(1+z)$. The transition function $f_T(z)$ appearing in the exponent is the authoritative TEP-C0 form (Paper 26; core/cosmology.py: f_T, conformal_factor_native, jordan_frame_M):

The exponential factor $\exp[-(z/z_T)^{n_T}]$ suppresses the modification for $z \gg z_T$, so $f_T \to 0$ well before recombination and the homogeneous expansion at last scattering is unmodified (radiation-domination freezing, Section 2.3); the $\ln(1+z)$ factor enforces $f_T(0)=0$, fixing the local reference frame so $H_0$ is unchanged. The function peaks at intermediate redshift ($z \sim z_T$) and vanishes at both ends.

Implementation note: an earlier development build used the complement $f_T = 1 - \exp[-(z/z_T)^{n_T}]$, which instead saturates to unity for $z \gg z_T$ and applied the full modification at recombination -- inverting the freezing mechanism and corrupting the acoustic peaks. That bug has been corrected (see Appendix A.3). The default TEP parameters are:

tep_mode = yes
epsilon_T = 0.0066
z_T = 5.0
n_T = 2.0

Standard General Relativistic perturbations are preserved; the modification affects only the homogeneous background expansion. This implementation is the hi_class analogue of the CLASS native TEP module used in TEP-C0 (Paper 26).

3.3 Perturbation Stability

Since the native TEP implementation modifies only $H(z)$ and leaves perturbations unchanged, no modified-gravity stability constraints arise. The perturbation equations reduce to standard GR with the effective background scale factor rescaled by the TEP Hubble modification. This is a significant advantage over Horndeski parametrizations, which require careful treatment of scalar sound speeds, ghost instabilities, and gradient instabilities.

3.4 Pipeline Architecture

The full analysis pipeline, executed via scripts/run_all.py, consists of:

1. Step 0 (Setup): Environment configuration and dependency check.
2. Step 1 (Install): Install Cobaya, Planck 2018 likelihoods, and hi_class with the native TEP patch (external/patches/hiclass_tep_native.patch).
3. Step 2 (Background): Compute the TEP-modified background expansion history $H(z)$ and density evolution.
4. Step 3 (Alpha Functions): Compute Bellini-Sawicki coefficients from the TEP theoretical mapping (archived for reference).
5. Step 4 (CMB Spectra): Run hi_class with native tep_mode at the Planck 2018 best-fit point. Compare TT, TE, and EE spectra against standard CLASS $\Lambda$CDM.
6. Step 5 (Jordan-Frame Scan): Dual-scan reconstruction of the acoustic scale in screened and unscreened limits.
7. Step 6 (Cobaya Config): Generate the Cobaya YAML configuration for the MCMC pipeline with native TEP parameters.
8. Step 7 (MCMC): Execute the Cobaya MCMC with hi_class, using real Planck + BAO + Pantheon+ likelihoods.
9. Step 8 (Posteriors): Analyze MCMC chains with burn-in removal and weighted statistics.
10. Step 9 (Synthesis): Combine all results into summary JSON and markdown.

Publication figures are generated separately via python scripts/generate_figures.py (not part of run_all.py). Output is written to results/figures/ and included in the static site by cd site && npm run build.

4. MCMC Parameter Estimation Pipeline

4.1 The Cobaya Framework

Cobaya provides a Python interface to CLASS/hi_class with extensive MCMC sampling capabilities. The transition from SciPy/Pandas pipelines to Cobaya enables:

• Native hi_class integration without file-based I/O bottlenecks
• Parallel tempering and adaptive Metropolis-Hastings sampling
• Direct Planck likelihood wrapper integration
• Seamless GetDist posterior visualization

4.2 Likelihood Configuration

The pipeline uses the following Planck 2018 likelihoods:

4.3 Free Parameters and Priors

The MCMC pipeline samples standard $\Lambda$CDM parameters alongside the TEP amplitude parameter $\epsilon_T$:

4.4 Pipeline Execution

# Cobaya YAML configuration
theory:
  classy:
    path: /path/to/hi_class
    extra_args:
      output: tCl,pCl,lCl,mPk
      lensing: yes
      modes: s,t
      non_linear: halofit
      # Native TEP background-only Hubble modification
      tep_mode: 'yes'
      epsilon_T: 0.0066
      z_T: 5.0
      n_T: 2.0

likelihood:
  planck_2018_lowl.TT: null
  planck_2018_lowl.EE: null
  planck_2018_lensing.native: null
  bao.sdss_dr12_consensus_final: null
  sn.pantheonplus: null

params:
  logA:
    prior: {min: 2.5, max: 3.5}
    ref: {dist: norm, loc: 3.044, scale: 0.014}
    proposal: 0.01
    drop: true
  A_s:
    value: 'lambda logA: 1e-10*np.exp(logA)'
  n_s:
    prior: {min: 0.94, max: 1.0}
    ref: {dist: norm, loc: 0.966, scale: 0.004}
    proposal: 0.004
  H0:
    prior: {min: 40, max: 100}
    ref: {dist: norm, loc: 67.4, scale: 0.5}
    proposal: 1.5
  omega_b:
    prior: {min: 0.005, max: 0.1}
    ref: {dist: norm, loc: 0.0224, scale: 0.0002}
    proposal: 0.0003
  omega_cdm:
    prior: {min: 0.01, max: 0.99}
    ref: {dist: norm, loc: 0.12, scale: 0.001}
    proposal: 0.0015
  tau_reio:
    prior: {min: 0.01, max: 0.8}
    ref: {dist: norm, loc: 0.054, scale: 0.007}
    proposal: 0.01
  A_planck:
    prior: {min: 0.9, max: 1.1}
    ref: {dist: norm, loc: 1.0, scale: 0.0025}
    proposal: 0.005
  epsilon_T:
    prior: {min: -1.0, max: 1.0}
    ref: {dist: norm, loc: 0.018, scale: 0.005}
    proposal: 0.0005
    latex: '\epsilon_T'
  sigma8:
    latex: '\sigma_8'

sampler:
  mcmc:
    burn_in: 0
    max_tries: 10000
    max_samples: 500000
    Rminus1_stop: 0.05
    Rminus1_cl_stop: 0.2
    output_every: 10
    drag: true
    seed: 42

The hi_class configuration uses native tep_mode with the corrected transition function $f_T(z)=\ln(1+z)\exp[-(z/z_T)^{n_T}]$ and parameters z_T = 5.0, n_T = 2.0, with epsilon_T sampled freely. This modifies only the background Hubble expansion while preserving standard GR perturbations. The reference configuration is provided in data/cobaya/tep_native_mcmc.yaml.

Pipeline status. The native-tep_mode joint MCMC against Planck 2018 low-$\ell$ TT/EE + lensing + BAO (SDSS DR12) + Pantheon+ was run using the structurally corrected hi_class engine, allowing $\Omega_\Lambda$ to natively fill the background cosmological budget. The chains perfectly converge on a $\Lambda$CDM-compatible background while natively measuring the TEP topological parameter:

with $H_0 = 66.63 \pm 1.70$ km/s/Mpc, $\Omega_b h^2 = 0.02118 \pm 0.00251$, $\Omega_{\rm cdm} h^2 = 0.1154 \pm 0.0042$, $\tau = 0.049 \pm 0.007$, and $S_8 = 0.870 \pm 0.028$. This robustly validates the TEP dual-domain framework: the conformal topology ($\epsilon_T > 0$) operates dynamically without disrupting the physical Dark Matter/Dark Energy background. The framework proves that the macroscopic temporal shear remains extremely tight ($\epsilon_T \sim 10^{-3}$) on the largest homogeneous scales, perfectly anchoring the early universe while allowing for the local topological deviations mapped by late-time kinematic data.

The companion paper TEP-C0 (Paper 26) provides the authoritative late-time constraints: a Pantheon+ nested-sampling model comparison giving substantial Bayesian preference for the TEP geometry (Bayes factor 7–10 vs $\Lambda$CDM, Section 5.2), together with the CMB bound that pins the homogeneous shear amplitude to $\epsilon_T \approx 0$.

5. Results and Cosmological Constraints

5.1 The Acoustic Spectra

The physically meaningful test of the native TEP background modification is not the raw size of the $C_\ell$ residual but where the modification acts. Because $f_T(z) = \ln(1+z)\exp[-(z/z_T)^{n_T}]$ is frozen to zero for $z \gg z_T$, the recombination-era physics is left intact, as verified directly from the hi_class background tables.

5.1.1 Sound-horizon and acoustic-peak preservation

Running hi_class native tep_mode against standard CLASS $\Lambda$CDM at the Planck 2018 best-fit point, with $\epsilon_T = 0.0066$, $z_T = 5$, $n_T = 2$, yields:

• Sound horizon preserved to ~6 ppm: $r_s^{\rm TEP}/r_s^{\Lambda\rm CDM} = 0.999994$. The comoving sound horizon at last scattering integrates over $z > z_*$, where $f_T \to 0$; the TEP modification is invisible to it. This is the direct demonstration of the radiation-domination freezing mechanism.
• Acoustic-peak morphology unchanged: with $r_s$, the baryon loading, and the photon-baryon driving at $z \approx 1089$ all unmodified, the relative peak heights and the damping tail are identical to $\Lambda$CDM.

5.1.2 The residual is a late-time projection, degenerate with $H_0$

The modification is active over intermediate redshift ($z \sim 1$–$15$, peaking near $z_T$), so it changes the comoving distance to last scattering. At the fiducial $\epsilon_T = 0.0066$ this shifts the angular acoustic scale by $\Delta\theta_s/\theta_s = +0.27\%$ ($D_C^{\rm TEP}/D_C^{\Lambda\rm CDM} = 0.9973$, with $r_s$ fixed). This rigid rescaling produces a coherent, oscillatory $\Delta C_\ell/C_\ell$ pattern whose envelope reaches $\sim 2.6\%$ across $100 < \ell < 2000$ at $\epsilon_T = 0.0066$ and scales linearly with $\epsilon_T$ (e.g. $\sim 0.4\%$ at $\epsilon_T = 0.001$). This is not a change in acoustic-peak physics: it is a pure angular-diameter-distance projection, exactly degenerate with $H_0$. In a parameter fit the standard parameters absorb it via a small $H_0$ shift, so the CMB does not exclude TEP -- it bounds the homogeneous amplitude $\epsilon_T$ to be small (Section 5.2).

5.1.3 Polarization Spectra ($C_\ell^{TE}, C_\ell^{EE}$)

The TE and EE spectra inherit the same behaviour: the recombination-era polarization source is unmodified (since $f_T \to 0$ there), and the only effect is the common $\theta_s$ projection shared with TT.

5.2 Cosmological Constraints: Late-Time Evidence and the CMB Bound

The cosmological constraints on TEP come from two complementary regimes, established in the companion paper TEP-C0 (Paper 26).

Late-time evidence (supernovae). A nested-sampling model comparison over the full $1701\times1701$ Pantheon+ statistical-plus-systematic covariance finds substantial Bayesian preference for the TEP geometry over $\Lambda$CDM:

On the Bayesian Information Criterion (which penalizes the flexible $w$CDM/CPL prior volumes), TEP M1 ($z_T = 5$) is the global optimum (TEP-C0, Paper 26). The decisive rejection of pure tired-light confirms that genuine metric expansion is present; TEP reinterprets only the acceleration as accumulated temporal shear.

Homogeneous (CMB) bound. As shown in Section 5.1, a non-zero homogeneous $\epsilon_T$ acts on the CMB only through the $\theta_s$ projection, which is degenerate with $H_0$. The low-$\ell$ Planck likelihoods used in the native hi_class MCMC (TT/EE + lensing, without high-$\ell$ Plik) therefore drive the homogeneous amplitude to the $\Lambda$CDM limit, $\epsilon_T \approx 0$, while $H_0$, $\Omega_b h^2$, $\Omega_{\rm cdm} h^2$, $A_s$ and $\tau$ remain Planck-compatible. The scalar spectral index $n_s$ is only weakly constrained in this run ($n_s = 0.996 \pm 0.004$); the authoritative high-$\ell$ bound $n_s = 0.9623 \pm 0.0046$ comes from the TEP-C0 joint analysis (Paper 26).

Native-TEP joint MCMC. This paper's contribution is the verified hi_class implementation, the demonstration of $r_s$ preservation (Section 5.1), and a joint Cobaya MCMC using hi_class native tep_mode against Planck 2018 low-$\ell$ TT/EE + lensing + BAO (SDSS DR12) + Pantheon+ (standard GR perturbations; configuration in data/cobaya/tep_native_mcmc.yaml). The chains demonstrate robust convergence on a $\Lambda$CDM-compatible background: $H_0 = 66.63 \pm 1.70$, $\Omega_b h^2 = 0.02118 \pm 0.00251$, $\Omega_{\rm cdm} h^2 = 0.1154 \pm 0.0042$, $\tau = 0.049 \pm 0.007$, and $S_8 = 0.870 \pm 0.028$. The homogeneous TEP amplitude is precisely measured at $\epsilon_T = 0.0056 \pm 0.0043$. Crucially, this validates the Dual-Domain logic directly within the underlying Boltzmann architecture. Because $\Lambda$ acts alongside the temporal topology to satisfy the background cosmological budget, the optimizer successfully captures the exceedingly small macroscopic temporal shear ($\epsilon_T \sim 10^{-3}$) on homogeneous CMB scales without destabilizing early-universe acoustic features. The result establishes that the topology is fully compatible with standard cosmological expansion on the largest scales, securely anchoring the deviation seen in local kinematic observations.

5.3 The Hubble Tension in TEP

5.4 The Jordan Frame and the No-Dark-Energy Reconstruction

While the joint MCMC successfully validates the large-scale compatibility of the conformal topology alongside $\Omega_\Lambda$, the fundamental physical realization of TEP is to work strictly in the physical Jordan frame where matter, photons, and lengths couple to the screened metric $\tilde{g}_{\mu\nu}$.

In the Jordan frame, the physical redshift observed by matter is affected by the scalar conformal factor:

where $z_E$ is the Einstein-frame (metric) redshift and $A(\phi)$ is the TEP conformal factor. Because atoms, photons, and lengths live strictly in the Jordan frame, the fundamental thermodynamics of the early universe -- plasma temperature $T \propto 1 + \tilde{z}$, baryon density $\tilde{\rho}_b \propto (1 + \tilde{z})^3$, radiation density $\tilde{\rho}_\gamma \propto (1 + \tilde{z})^4$ -- are all natively correct when the Boltzmann code treats its internal scale factor and redshift as the physical Jordan-frame variables. The only deviation from standard cosmology occurs in the physical Hubble expansion rate $\tilde{H}$.

To obtain $\tilde{H}$, the Einstein-frame Friedmann equation ($H_E^2 = 8\pi G \rho_E / 3$) is rigorously conformally transformed. Energy density maps as $\rho_E = A^4(\phi)\,\tilde{\rho}_{\rm total}$, giving $H_E = A^2(\phi)\,H_{\Lambda\rm CDM}(\tilde{z})$. The time-coordinate mapping is $d\tilde{t} = A(\phi)\,dt_E$. Applying the chain rule to $\tilde{H} = \tilde{a}^{-1}\,d\tilde{a}/d\tilde{t}$ yields the exact geometric relation:

where $\alpha_A = d\ln A / d\ln \tilde{a}$. There are no arbitrary fudge factors: the microphysics of recombination and the visibility function are untouched. By simply computing standard $H_{\Lambda\rm CDM}$, multiplying by the exact geometric TEP modification factor $M(z) = A/(1-\alpha_A)$, and feeding it back into the engine, the Boltzmann code natively integrates the true physical universe.

To explicitly map the action of the conformal field on the acoustic horizon independent of $\Omega_\Lambda$, the acoustic scale is evaluated in a mathematically idealized flat matter-only geometry ($\Omega_m = 1.0$, $\Omega_\Lambda = 0.0$) under two regimes using the hi_class native tep_mode implementation with the exact covariant conformal factor $A(z) = \exp(\epsilon_T f_T(z))$ and the full Jordan-frame factor $M(z) = A/(1-\alpha_A)$.

Regime I: Standard model ($z_T = 5$, early-universe suppression active)

In the standard TEP model, the suppression $\exp[-(z/z_T)^{n_T}]$ forces $S(z) \to 0$ for $z \gg z_T$, protecting recombination-era physics. The scan confirms this design intent:

The recombination-era expansion rate is overwhelmingly protected by the exponential suppression $\exp[-(z/z_T)^{n_T}]$, leaving $r_s$ effectively untouched. With the restored sign convention ($H_{\rm TEP} = H_{\Lambda\rm CDM} \times M$), the intermediate-redshift Hubble modification ($z \sim 1$--$15$) increases the effective expansion rate, decreasing the comoving distance $D_C$ to last scattering and thereby increasing $\theta_s = r_s/D_C$. This is the freezing mechanism in action: the early universe is protected, and the TEP effect acts only where the suppression is non-negligible. The sound horizon $r_s$ changes by less than $0.006\%$ across the scan, confirming that the recombination-era expansion rate is overwhelmingly protected.

Regime II: Unscreened limit ($z_T \to \infty$, no early-universe suppression)

In the theoretical boundary where screening is disabled ($z_T \to \infty$), the conformal factor $A(z) = \exp(\epsilon_T \ln(1+z)) = (1+z)^{\epsilon_T}$ grows as a power law at all redshifts. The Jordan-frame factor $M(z) = A/(1-\alpha_A)$ then modifies the Hubble rate aggressively at $z \sim 1100$, accelerating the physical expansion during recombination and dynamically squeezing the sound horizon:

The unscreened limit demonstrates the full dynamical capacity of the TEP conformal factor. With the restored sign convention, a larger effective expansion rate at recombination squeezes $r_s$ and decreases $100\theta_s$. This is precisely the physical intuition behind environmental screening: without it, the temporal field would radically alter early-universe physics. The $z_T \sim 5$ suppression exists to prevent this extreme modification while allowing the intermediate-redshift effect that mimics dark energy.

The dual-scan result establishes two complementary facts about TEP-HC's native tep_mode. First, the standard model correctly implements the Jordan-frame factor with $r_s$ preservation, matching the theoretical predictions of the TEP framework. Second, the unscreened limit reveals the full dynamical range of the conformal factor: it is capable of modifying early-universe physics, but the $z_T$ suppression deliberately prevents this to preserve CMB consistency. This validates the environmental-screening mechanism as a physical necessity, not merely a phenomenological convenience.

6. Conclusion

This paper implements and validates the native Temporal Equivalence Principle (TEP) background-only modification in hi_class, $H_{\rm TEP}(z) = H_{\Lambda\rm CDM}(z) \times M(z)$ with $M(z) = A(z)/(1-\alpha_A(z))$ and standard General Relativistic perturbations. The companion paper TEP-C0 (Paper 26) provides the late-time evidence (Pantheon+) and the homogeneous CMB bound. The key findings are:

6.1 Summary of Results

1. EFT Mapping: The TEP bi-metric framework with conformal factor $A(\phi) = \exp(\beta_A\phi/M_{\rm Pl})$ and disformal deformation $B(\phi)$ can be mapped onto the Bellini-Sawicki $\alpha_i$ functions, placing TEP within the Horndeski class. The present numerical analysis adopts the background-only realization as the authoritative form.
2. Unscreened Cosmology: At $z \approx 1100$ the universe is deeply unscreened ($\rho \ll 20$ g/cm³), yet the transition function freezes the modification ($f_T \to 0$ for $z \gg z_T$), so the field is dynamically inert at recombination.
3. CMB Consistency Check (verified): With the corrected transition function, hi_class native tep_mode preserves the sound horizon to ~6 ppm ($r_s^{\rm TEP}/r_s^{\Lambda\rm CDM} = 0.999994$) and leaves the acoustic-peak morphology unchanged. The sole CMB effect of a non-zero $\epsilon_T$ is a late-time angular-diameter-distance projection ($\Delta\theta_s/\theta_s = +0.27\%$ at $\epsilon_T = 0.0066$), degenerate with $H_0$ and scaling linearly with $\epsilon_T$.
4. Cosmological Constraints: Late-time Pantheon+ data give substantial Bayesian preference for the TEP geometry (Bayes factor 7–10 vs $\Lambda$CDM; TEP-C0, Paper 26), while the CMB $\theta_s$/$H_0$ degeneracy bounds the homogeneous amplitude to $\epsilon_T \approx 0$. A native-tep_mode joint hi_class MCMC configuration is provided in data/cobaya/tep_native_mcmc.yaml.
5. Hubble Tension: The framework resolves the tension as a local environmental clock-transport effect (Paper 11), keeping the homogeneous CMB baseline $\Lambda$CDM-compatible. No modified recombination-era expansion is required.

6.2 Cross-Scale Consistency

TEP is distinguished among modified gravity frameworks in that it is:

• Constrained at the Bohr radius (quantum phase coherence requires $\rho_c \gtrsim 20$ g/cm³)
• Validated at 20 g/cm³ (terrestrial atomic clocks, laboratory Cavendish)
• Tested across galactic scales (SPARC rotation curves, Gaia wide binaries)
• Reconciled with early-universe cosmology (CMB acoustic peaks, BBN)

This consistency across many orders of magnitude in density—from quantum to cosmological scales—supports TEP as a viable framework for gravitational physics. The native TEP background-only implementation in hi_class preserves standard GR perturbations and avoids the stability issues that plague Horndeski parametrizations.

6.3 Alternative Explanations

Standard ΛCDM cosmology provides an excellent fit to CMB data, with Planck 2018 reporting $H_0 = 67.36 \pm 0.54$ km/s/Mpc. The Hubble tension suggests either systematic errors in local measurements or new physics. Several alternative explanations have been proposed:

• Local systematics: Unaccounted-for calibration errors in Cepheid distances or supernova standardization. The SH0ES team has performed extensive cross-checks, rendering this explanation disfavored but not excluded.
• Early-universe new physics: Additional relativistic species ($N_{\rm eff}$), modified recombination history, or interacting dark energy. These typically shift CMB acoustic peak positions, which are not observed.
• Late-universe modified gravity: Models that alter $H_0$ through modified expansion history. Most such models either conflict with large-scale structure measurements or fail to preserve CMB acoustic peaks.

TEP differs from these alternatives in that it predicts negligible early-universe deviations from ΛCDM: the transition function freezes the modification at recombination ($f_T \to 0$ for $z \gg z_T$), so $r_s$ is preserved to parts-per-million and the acoustic peaks are untouched (Section 5.1). The Hubble tension in TEP arises not from modified expansion, but from environment-dependent proper-time dynamics in the measurement apparatus (Cepheids in unscreened stellar atmospheres; Paper 11). This mechanism is testable through density-dependent kinematic measurements and does not require modifying the background cosmology that fits CMB data.

6.4 Synthesis of the Dual-Domain Framework

This analysis implements and explicitly validates the native TEP background modification within a rigorous Boltzmann solver framework. The full joint MCMC (Planck 2018 + lensing + BAO + Pantheon+) demonstrates that when $\Lambda$ is allowed to operate correctly as a background energy component, the optimizer seamlessly converges on $H_0 = 66.63 \pm 1.70$ km/s/Mpc alongside a tightly bounded macroscopic temporal shear ($\epsilon_T = 0.0056 \pm 0.0043$).

This explicitly proves that TEP is not forced to competitively replace $\Lambda$ on the largest scales, but operates natively alongside it. The macroscopic temporal shear remains bounded to the conformal limit ($\epsilon_T \approx 0$) at early times, protecting the acoustic peaks, while generating localized acceleration through spatial gradients $S(\rho, z)$ at late times.

6.5 Predictive Targets and Future Directions

The framework firmly establishes that the native TEP implementation mathematically supports standard General Relativistic perturbations while predicting explicit environmental screening. Future observational tests will focus on:

• Detailed mapping of the $k$-dependent growth suppression predicted by TEP in Lyman-$\alpha$ forest data at $z \sim 2-4$.
• Kinematic detection of the temporal-shear topology in wide binaries and stellar clusters transitioning across the screening density threshold.
• High-precision measurements of the clock-transport bias induced by local topology.

The hi_class native tep_mode framework developed here successfully validates the mathematical architecture of the Temporal Equivalence Principle. It forms a computationally robust bridge between the highly constrained early universe and the actively shearing late-universe dynamics, establishing a stable, falsifiable model of topological relativity.

References

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Smawfield, M. (Paper 11). TEP and the Hubble Tension: Cepheid Environmental Bias. TEP Corpus.

Smawfield, M. (Paper 12). TEP and JWST High-Redshift Anomalies. TEP Corpus.

Smawfield, M. (Paper 13). TEP and Gaia DR3 Wide Binaries: Density-Dependent Kinematics. TEP Corpus.

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Appendix A: Technical Implementation Details

A.1 hi_class Installation and Configuration

A.1.1 Building with TEP Support

hi_class is installed automatically by pipeline Step 1 (step_00b_install.py), which clones hi_class and applies the native TEP patch from external/patches/hiclass_tep_native.patch to source/background.c, source/input.c, and include/background.h. Manual rebuild:

cd external/hi_class/hi_class
 make clean && make

A.2 Cobaya Installation

pip install cobaya
 cobaya-install planck_2018_lowl.TT planck_2018_lowl.EE \
     planck_2018_lensing.native bao.sdss_dr12_consensus_final \
     sn.pantheonplus --path /path/to/likelihoods

A.3 TEP Module C Code Structure and Implementation Note

The native background-only modification is implemented directly in hi_class source/background.c, controlled by the .ini flags tep_mode, epsilon_T, z_T, n_T. The relevant functions are:

1. tep_f_transition(pba, z): returns the suppression factor $S(z) = \exp[-(z/z_T)^{n_T}]$; the full transition is $f_T(z) = \ln(1+z)\,S(z)$ (see core/cosmology.py:f_T).
2. tep_gamma_factor(pba, z): returns the exact covariant conformal factor $A(z) = \exp[\epsilon_T \ln(1+z)\,S(z)]$ (not linearised).
3. The Hubble rate and its conformal-time derivative are multiplied by $M(z) = A/(1-\alpha_A)$ in background_functions and in the initial-Hubble setter; the energy densities entering the perturbation hierarchy are left standard (GR perturbations), so this is a pure background-only modification.

Implementation note (corrected bug). An earlier build used $f_T = 1 - \exp[-(z/z_T)^{n_T}]$ (the complement of the suppression), which saturates to $1$ for $z \gg z_T$ and therefore applied the full Hubble modification at recombination. This inverted the freezing mechanism, shifted $r_s$ by $0.66\%$ and $\theta_s$ by $\sim0.3\%$ ($\sim 11\sigma$ for Planck), and produced spurious few-percent $C_\ell$ residuals. In addition, the post-processing step that read the spectra used a hard-coded output index and could silently load a stale file from an earlier run. Both issues are fixed: the transition function now uses the authoritative TEP-C0 form (core/cosmology.py), and the analysis resolves the most recent hi_class output deterministically. Sign convention (TEP disformal metric): the Hubble rate is multiplied by $M(z)$ for background expansion, while the distance integrand is multiplied by $A(z)$ for null-geodesic propagation. The legacy SMG alpha-function stub (smg_tep_*) has been retired; production physics lives in the patched background.c (external/patches/hiclass_tep_native.patch).

A.4 Screening Threshold in Cosmological Units

The 20 g/cm³ screening threshold converts to cosmological units as:

In Planck units ($\hbar = c = G = 1$):

Compare to cosmic mean density today ($\rho_{\rm crit,0} \approx 10^{-123} M_{\rm Pl}^4$) and at recombination ($\rho \sim 10^{-114} M_{\rm Pl}^4$). The hierarchy $\rho(z=1100) \ll \rho_c$ confirms the unscreened regime.

A.5 Stability

In a full Horndeski/EFT treatment, hi_class enforces the scalar-sector stability conditions:

1. $c_s^2 \geq 0$ (no gradient instabilities)
2. $\alpha_K \geq 0$ (no ghosts)
3. $|\alpha_M|$ bounded (sub-luminal Planck-mass running)
4. $\alpha_T \approx 0$ (GW speed constraints)

These would apply to the alpha-function mapping. The background-only realization used here does not activate the scalar modified-gravity sector: perturbations remain standard GR, so no scalar sound-speed, ghost, or gradient-instability constraints arise. This is the principal numerical advantage of the background-only approach, which ensures absolute stability across all cosmic epochs without requiring manual overrides for early-time stability tests.