1. Introduction: The Geometry of Time

Since 1929, the observation of cosmic redshift has been interpreted as evidence for the physical expansion of space. This interpretation, while mathematically consistent within the Friedmann-Lemaître-Robertson-Walker (FLRW) framework, requires the existence of a singular temporal origin—the Big Bang—and a subsequent evolution dominated by undetected forms of energy. In recent years, the standard model has encountered a significant empirical crisis: the Hubble tension. The persistent $5\sigma$ tension between local and global determinations of $H_0$ suggests that the underlying physical interpretation of redshift—and thus cosmic expansion—may be fundamentally incomplete.

This paper proposes a more fundamental alternative: that apparent cosmic expansion is a geometric misinterpretation of accumulated Temporal Shear. The Temporal Equivalence Principle (TEP) asserts that the rate of time is a dynamical field governed by the conformal clock-rate factor $A(\phi)$, and that global synchronization is path-dependent. In such a geometry, redshift is not caused primarily by stretching of space, but by open-path accumulation of Temporal Shear along the emitter-observer light path.

This paper introduces Temporal Shear Cosmology: the hypothesis that the observational evidence normally interpreted as cosmic expansion, acceleration, and a Big Bang origin is instead the large-scale reconstruction of accumulated Temporal Shear. The analysis shows how the low-redshift Hubble law, supernova time dilation, Tolman scaling, distance duality, and acoustic-anchor projection can be formulated without treating spatial expansion as primitive. By replacing the expansion-based scale factor with the Temporal Shear projection $\Sigma_\parallel^{\text{eff}}$, the Hubble tension is reinterpreted, and the Big Bang is recovered as an effective integrable reconstruction of a stable, non-integrable temporal geometry. Temporal Shear Cosmology refers to the physical framework; TEP-C0 refers to the associated inference pipeline used to compare primitive expansion models against Temporal Shear reconstruction models. Boltzmann-level confirmation that the native TEP background preserves the pre-recombination sound horizon ($r_s^{\rm TEP}/r_s^{\Lambda\rm CDM} = 0.999994$) is established independently in TEP-HC (Paper 18).

This work does not re-derive BBN abundances, perform full hydrodynamical structure formation, or close the nonsingular temporal-horizon regularity proof; those are addressed in companion papers TEP-HC (Paper 18) and TEP-TH (Paper 27).

The claim-discipline framework for the TEP corpus, including the scope limitations of canonical precision tests, is established in TEP-EXP (Paper 9).

2. Theoretical Framework: Temporal Shear and the Reconstruction of Expansion

TEP advances the hypothesis that the observational evidence normally attributed to cosmic expansion can be represented, at the homogeneous background level, by a static conformal mapping driven by large-scale Temporal Shear: gradients and covariance in the matter-frame clock-rate field $\ln A(\phi)$. In TEP, matter, clocks, electromagnetic fields, and quantum phases couple universally to the causal matter metric $\tilde{g}_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$, where the conformal factor $A(\phi)$ defines the Temporal Shear vector:

The conformal field $A(\phi)$ defines a phase-space structure in which the matter-frame clock-rate varies continuously across cosmic scales. The phase-space topology of this field determines whether transport is integrable or path-dependent, distinguishing pure conformal shear from non-integrable temporal transport.

2.1 Relation to Prior Work

While standard cosmology treats cosmic expansion as a kinematic stretching of the spatial metric, several frameworks have explored conformal alternatives. Wetterich (2013) demonstrated that a universe without spatial expansion can be formulated using a varying particle mass, while Narlikar and Arp explored conformal gravity variations. Furthermore, environmental screening mechanisms—such as chameleon or symmetron screening (Khoury & Weltman 2004; Hinterbichler & Khoury 2010)—have been extensively developed to hide scalar fifth forces. TEP departs from these approaches by identifying the conformal factor strictly with the dynamical flow of proper time, generating an exact geometric mapping between the spatial scale factor and the macroscopic accumulation of Temporal Shear without requiring variable rest masses or modified spatial curvature.

2.2 The Cosmological Isochrony Assumption

Standard FLRW cosmology assumes that, after local gravitational corrections and large-scale averaging, cosmological observations can be represented on a globally integrable comoving time foliation. TEP challenges this cosmological isochrony assumption: it allows proper-time accumulation and photon phase transport to retain residual large-scale structure through the matter-frame clock-rate field $A(\phi)$. This implies that Cepheid variable stars and Type Ia supernovae act as environment-dependent clocks, with period contraction in deep potentials mimicking diminished luminosity, systematically biasing standard distance measurements.

2.2 The Generator of Apparent Redshift

Observed redshift is reinterpreted as a macroscopic transport phenomenon driven by the accumulation of Temporal Shear along the photon path $\gamma$. The line-of-sight projection is defined as $\Sigma_\parallel \equiv \Sigma_\mu \hat{k}^\mu$, where $\hat{k}^\mu$ is the tangent 4-vector normalized to the comoving observer frame, giving $\Sigma_\parallel$ dimensions of inverse length. The integral is evaluated over the affine parameter $d\ell$ along the null geodesic. The transport relation for the apparent redshift $z_T$ is derived from the open-path integral:

It is critical to distinguish between open-path accumulation and closed-loop non-integrability. Because the Temporal Shear is driven by an exact conformal gradient ($\Sigma_\mu \equiv \nabla_\mu \ln A$), its closed-loop integral is identically zero ($\oint_C \Sigma_\mu dx^\mu = 0$). Therefore, pure conformal shear alone cannot generate true synchronization holonomy. The non-integrable transport is strictly sourced by the non-exact topological covariance term $\mathcal{C}_T$, whose line-of-sight projection $\mathcal{C}_{T,\parallel}$ enters the open-path transport integral. This term accounts for path-dependent coarse-graining and stochastic topology corrections derived from $C_\Theta(x,x')$.

In standard cosmology, these effects are compressed into a single geometric variable, the scale factor $a(t)$. In TEP, $a(t)$ is recognized as an effective integrable reconstruction:

The reconstructed scale factor $a_{\text{eff}}$ is the open-path FLRW-like projection developed in the temporal-horizon framework of TEP-TH (Paper 27). It decomposes into the exact observational clock map $A_{\text{clock}}(\gamma)=A_0/A_{\text{em}}$ and the suppressed physical dynamical response $A_{\text{dyn}}(\gamma)$, with the non-exact covariance/topology correction $\mathcal{C}_T$ providing the path-dependent transport closure. In this decomposition, $A_{\text{clock}}$ alone generates the standard redshift–distance relation via exact conformal shear, while $A_{\text{dyn}}$ encodes the residual suppressed response of the Temporal Shear field.

2.3 From Temporal Topology to Transport: Definition of $\mathcal{C}_T$

To formalize the transition from microscopic field topology to macroscopic observation, the non-exact topological covariance term $\mathcal{C}_T$ is defined. Let $\theta = \ln A(\phi)$. The coarse-grained covariance structure is given by:

Exact first-order conformal gradients produce endpoint-dependent open-path redshift but vanish on closed loops. True synchronization holonomy therefore requires the non-exact $\mathcal{C}_T$ contribution. Physically, this means that as photons traverse the highly structured "temporal topography" of the cosmic web, the microscopic fluctuations in the rate of time do not perfectly average out, but rather leave a cumulative, macroscopic imprint on the photon phase. Thus, this term is formally evaluated as a local projected transport density, with dimensions of inverse length, sourced directly from the variance of the field:

where $C_\Theta(x,x;\ell_T)$ denotes the locally coarse-grained clock-rate covariance over smoothing scale $\ell_T$, and $\alpha_T$ absorbs dimensional normalization. In this expression, $S(\rho)\to1$ in unsuppressed voids and $S(\rho)\to0$ in dense environments undergoing Temporal Topology flattening, ensuring that the covariance-induced transport contribution follows the same environmental logic as the macroscopic $\epsilon_T^{\text{obs}}=S(\rho)\epsilon_T$ relation.

Crucially, $\mathcal{C}_{T,\parallel}$ is introduced as a macroscopic transport-closure term motivated by the microscopic proper-time phase holonomy developed in the TEP-QF sector (Paper 23). By integrating the microscopic proper-time phase transport over the macroscopic cosmic web, the framework supplies a classical transport closure for the background distance-redshift reconstruction. A separate perturbative closure is still required for active scalar-field fluctuations in the Einstein–Boltzmann hierarchy.

2.4 The Universal Coupling Axiom and Covariant Environmental Gradient Suppression

Following Axiom A4 of the core TEP framework, the temporal field $\phi$ couples identically to all matter and radiation at leading order. However, the locally observable Temporal Shear is subject to strong environmental gradient suppression governed by the abstract operator $\mathcal{S}_\Sigma(\mathcal{E})$. Because $\mathcal{E}$ encompasses source structure, boundary conditions, and ambient fields, a complete theory must supply a single covariant realization of this operator, not a patchwork of scale-specific proxies. That realization is constructed here.

The matter-frame metric $\tilde{g}_{\mu\nu} = A^2(\phi)g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$ implies that the physical strength of the conformal sector is measured by the scalar invariant $\Sigma^2 \equiv \Sigma_\mu\Sigma^\mu = (\beta_A/M_{\rm Pl})^2 \nabla_\mu\phi\nabla^\mu\phi$. In any local Lorentz frame, $\Sigma^2$ sets the squared fractional rate at which clocks dephase relative to the gravitational metric. Suppression is the dynamical flattening of this observable dephasing. The covariant screening operator is defined as the rational function of two dimensionless control parameters, a kinetic ratio and a density ratio:

Here $g_t$ is the critical shear scale at which non-linear kinetic self-coupling becomes dominant, and $\rho_{\rm half} \approx 0.5\,M_\odot/{\rm pc}^3$ is the ambient half-suppression density. The exponent $n$ governs the steepness of the kinetic transition. In the cosmological weak-field regime, $\Sigma^2 \sim H_0^2/c^2 \sim 10^{-56}\,{\rm m}^{-2}$, so the kinetic term is negligible and $\mathcal{S}_\Sigma \to S(\rho) = [1+(\rho/\rho_{\rm half})^2]^{-1}$. In the Solar System, where $g = |\nabla\Phi| \sim c^2|\nabla\ln A|/\beta_A$ in the Newtonian limit, the shear scale maps directly to the local gravitational acceleration: $\Sigma^2 \approx (\beta_A g/c^2)^2$, and the dominant suppression comes from the first term, giving $f(g) = [1+(g/g_t')^n]^{-1}$ with $g_t' = c^2 g_t/\beta_A$. Both phenomenological proxies are therefore low- and high-curvature limits of a single covariant expression.

2.4.1 The Covariant Action

The TEP bi-metric action, established in the foundational framework (Paper 0), is

with the screened matter-frame metric

where the environment-dependent conformal factor

absorbs the suppression directly into the matter coupling. In the unscreened limit ($\mathcal{S}_\Sigma \to 1$) this reduces to the bare TEP conformal factor $A(\phi) = \exp(\beta_A\phi/M_{\rm Pl})$; in the fully screened limit ($\mathcal{S}_\Sigma \to 0$) matter couples directly to the Einstein metric $g_{\mu\nu}$. The disformal function $B(\phi)$ is bounded by multi-messenger constraints ($|c_\gamma - c_g|/c \lesssim 10^{-15}$) and is set to zero in the pure-conformal limit analysed here.

2.4.2 Variation and Field Equations

Varying the action (\ref{eq:tep_action}) with respect to the Einstein-frame metric $g^{\mu\nu}$ yields the Einstein equations

where $T_{\mu\nu}^{(\phi)} = \nabla_\mu\phi\nabla_\nu\phi - g_{\mu\nu}\left[\frac{1}{2}(\nabla\phi)^2 + V(\phi)\right]$ is the scalar stress-energy, and the Einstein-frame matter stress-energy follows from the functional derivative of $S_m[\tilde{g}]$ with respect to $g^{\mu\nu}$. In the conformal limit ($B=0$) this gives

where $\tilde{T}_{\mu\nu}^{(m)}$ is the matter-frame stress-energy. Variation with respect to $\phi$ yields the scalar equation of motion

with effective source

where $\mathcal{T}^{\mu\nu} = (\sqrt{-\tilde{g}}/\sqrt{-g})\,\tilde{T}^{\mu\nu}$ is the density-weighted matter tensor. The first term is the direct conformal coupling; the remaining terms encode the feedback from the environmental dependence of $\mathcal{S}_\Sigma$. In the cosmological background, where $\Sigma^2 \ll g_t^2$ and $\rho \ll \rho_{\rm half}$, $\mathcal{S}_\Sigma \approx 1$ and its field derivatives are suppressed by $\Sigma^2/g_t^2 \ll 1$, so the source reduces to the standard conformally-coupled form $\mathcal{Q}_{\rm eff} \approx \beta_A\,\mathcal{T}/M_{\rm Pl}$. Near compact bodies, where $\mathcal{S}_\Sigma \ll 1$, the scalar force is suppressed and the source vanishes.

2.4.3 Perturbation Expansion and Gauge Conditions

To map the theory onto the Bellini--Sawicki EFT, we expand around a spatially flat FLRW background. The metric perturbation is written in Newtonian gauge

and the scalar field is split as $\phi(t,\mathbf{x}) = \bar{\phi}(t) + \delta\phi(t,\mathbf{x})$. The perturbed matter metric acquires a conformal-frame fluctuation

where $\mathcal{S}_\Sigma$ is evaluated on the background. On cosmological scales, $\mathcal{S}_\Sigma \approx 1$ to excellent approximation, and the perturbation structure reduces to that of a standard scalar-tensor theory with effective coupling $\beta_A^{\rm eff} = \mathcal{S}_\Sigma\beta_A$.

2.4.4 Bellini--Sawicki EFT Mapping

In the pure-conformal limit ($B=0$), the quadratic action for scalar and metric perturbations maps onto the standard EFT-of-dark-energy form. The running of the effective Planck mass is read off from the time dependence of the background coupling:

Using the TEP background relation $\alpha_A \equiv -d\ln\mathcal{A}/d\ln(1+z)$, this becomes $\alpha_M = -2\,\mathcal{S}_\Sigma\alpha_A$. In the cosmological weak-field limit ($\mathcal{S}_\Sigma \approx 1$) this reduces to the bare value $\alpha_M^{\rm bare} = -2\alpha_A$ used in TEP-HC (Paper 18). In screened environments ($\mathcal{S}_\Sigma \approx 0$), $\alpha_M \to 0$ and the scalar fifth force vanishes.

The braiding parameter follows from the kinetic mixing between $\delta\phi$ and the metric potentials:

and the kineticity parameter from the canonical kinetic term of $\delta\phi$ after field redefinition:

The tensor speed excess $\alpha_T$ vanishes in the conformal limit because $c_g^2 = c_\gamma^2 = 1$ is preserved. These are exactly the relations implemented in the TEP-HC hi_class runtime and used in the growth solver (step\_06\_03).

The no-ghost discriminant follows from the $2\times2$ kinetic matrix of the scalar sector. In the Bellini--Sawicki formalism,

The discriminant is manifestly non-negative for all $\mathcal{S}_\Sigma$ and $\alpha_A$, establishing ghost-freedom from the action. The sound speed is $c_s^2 = 1$ exactly in the conformal limit, guaranteeing gradient stability.

2.4.5 Post-Newtonian Expansion

The Solar-System PPN parameters are obtained from the quasi-static weak-field limit of the field equations. In the Damour--Esposito-Far\`ese parameterization for scalar-tensor theories with conformal coupling $\mathcal{A}(\phi)$, the metric perturbation for a static, spherically symmetric source is, at leading order,

where $\alpha_{\rm eff}$ is the effective scalar charge sourced by the body. For the TEP screened coupling,

with $\alpha_0 = \beta_A/M_{\rm Pl}$ the bare coupling constant. The PPN parameter $\gamma$ is then

In unscreened environments ($\mathcal{S}_\Sigma \approx 1$), the bare TEP coupling $\beta_A = -1$ gives $\alpha_0 = -1/M_{\rm Pl}$, which translates to $\gamma \approx -1$ --- ruled out by Cassini at $\sim$87\,000$\sigma$. In the Solar System, the gradient-dependent suppression dominates. Using the Newtonian mapping $\Sigma^2 \approx (\beta_A g/c^2)^2$ with $g = |\nabla\Phi|$,

where $g_t' = c^2 g_t/|\beta_A| = c^2 g_t$ for the locked TEP value $\beta_A = -1$. Evaluating at Saturn orbit ($g_{\rm Cassini} \approx 6.5\times10^{-5}\,{\rm m\,s}^{-2}$) with $g_t = 1.0\times10^{-9}\,{\rm m\,s}^{-2}$ and $n=2$ gives $\mathcal{S}_\Sigma \approx 2.37\times10^{-10}$. Substituting into (\ref{eq:ppn_gamma}),

safely below the Cassini bound $|\gamma - 1| < 2.3\times10^{-5}$ by more than fourteen orders of magnitude. At Earth surface ($g \approx 9.8\,{\rm m\,s}^{-2}$), $\mathcal{S}_\Sigma \approx 10^{-20}$ and the deviation is utterly negligible. The Eötvös parameter satisfies $|\beta_{\rm PPN} - 1| \propto \alpha_{\rm eff}^3$ and vanishes in the screened limit for the same reason.

2.4.6 Connection to the Growth Solver and Parameter Locking

The EFT functions derived above are the inputs to the structure-formation growth equation used in step_06_03 and step_06_07. The $\alpha_M$-modified growth ODE,

is the quasi-static limit of the full Einstein-Boltzmann hierarchy with the Bellini--Sawicki functions derived in Section 2.4.4. The $\alpha_M$ that appears here is precisely $\alpha_M^{\rm bare} = -2\alpha_A$ evaluated on the cosmological background where $\mathcal{S}_\Sigma \approx 1$.

The transition scale $g_t$ is not a free parameter fitted to the Solar System data. It is fixed by requiring that the unscreened branch (which would give $\gamma \approx -1$) be excluded, and that the suppressed branch pass the Cassini bound with a safety margin. The minimum requirement is $\mathcal{S}_\Sigma(g_{\rm Cassini}) \lesssim 3.4\times10^{-3}$, which for $n=2$ implies $g_t \lesssim 10^{-7}\,{\rm m\,s}^{-2}$. The adopted value $g_t = 1.0\times10^{-9}\,{\rm m\,s}^{-2}$ is two orders of magnitude below this ceiling, providing a conservative margin. Once $g_t$ is fixed by Solar System physics, it propagates unchanged to galactic halos ($g \sim 10^{-10}\,{\rm m\,s}^{-2}$, where $\mathcal{S}_\Sigma \approx 0.98$) and cosmological voids ($g \sim 10^{-11}\,{\rm m\,s}^{-2}$, where $\mathcal{S}_\Sigma \approx 1$), preserving cosmological growth and anomaly predictions without additional tuning.

This completes the implemented EFT screening realization used in the C0 pipeline. The suppression threshold is not tuned independently in each physical domain: the same operator is anchored by Solar-System PPN constraints, propagated to galactic and cosmological environments, and cross-checked against the TEP-HC growth sector. The PPN gate is passed within this EFT realization. The microscopic topological origin of the environmental operator is supplied by the companion TEP-QF, TEP-SPIN, and TEP-TH sequence rather than treated as an additional free assumption in C0.

2.5 Dark energy and acceleration as shear evolution

In this view, phenomenological dark energy on intermediate scales manifests from evolving Temporal Shear, while the homogeneous contribution conventionally assigned to $\Omega_\Lambda$ is reinterpreted as the homogeneous temporal-shear background contribution $\Omega_\phi$ (TEP-HC, Paper 18; TEP-TH, Paper 27). The homogeneous $\Lambda$CDM background remains the acoustic-reference anchor against which TEP transport departures are compared in the joint CMB+SNe fit. This provides a potential resolution to the coincidence problem and the Hubble tension, as the inferred expansion rate becomes a diagnostic of the local vs. global temporal environment.

2.6 Cosmological Topology Transitions

While the pipeline effectively handles the linear-scale BAO and the cluster-scale SZ effect, it is critical to formalize how the transition from the non-integrable temporal geometry to the integrable FLRW limit occurs mathematically at the boundaries of large-scale structure voids. This relies on the temporal-transport connection.

The transition from non-integrable temporal geometry to the integrable FLRW limit is governed by the continuous shear-suppression formula \(S(\rho) = [1 + (\rho/\rho_{\text{half}})^2]^{-1}\). Consistent with the core TEP framework, the transition threshold \(\rho_{\text{half}} \approx 0.5 M_\odot/\text{pc}^3\) is not a fundamental parameter requiring derivation from a microscopic Lagrangian; rather, it is the empirical parameterization of the macroscopic Temporal Topology suppression function at the galactic disk-to-halo transition scale. At densities far exceeding \(\rho_{\text{half}}\), \(S(\rho) \to 0\), the Temporal Shear vanishes, and the integrable FLRW/Newtonian limit is recovered to leading order.

The galactic transition scale is the mass-weighted macroscopic projection of the microscopic saturation structure developed in TEP-SPIN (Paper 24). C0 does not independently derive the micro-to-galactic transfer relation; it uses the companion-paper closure as an empirically constrained macroscopic envelope. Consequently, $\rho_{\text{half}}$ operates strictly as a phenomenological parameterization of the macroscopic Temporal Topology suppression function.

Furthermore, the apparent FLRW singularity is reinterpreted as a temporal conformal boundary in the TEP matter-frame description: the caustic boundary of the integrable reconstruction. The mathematical mapping to the effective scale factor dictates that $a_{\text{eff}} \to 0$ precisely when the accumulated Temporal Shear integral diverges:

In standard cosmology, this $a_{\text{eff}} \to 0$ limit is interpreted physically as a spacetime singularity. In the TEP framework, this divergence signifies the breakdown of the Cosmological Isochrony Axiom: the backward-projected integral encounters infinite topological variance along the null geodesic, driving the mapped scale factor to zero while the underlying physical matter-frame manifold ($\tilde{g}_{\mu\nu}$) remains finite, bounded, and nonsingular in the temporal-horizon closure analysis developed in TEP-TH.

3. Methodology: Deterministic Transport Inference

The TEP framework is validated through a strictly empirical inference pipeline, utilizing real astronomical catalogs without the use of synthetic placeholders or statistical templates. The methodology is designed to test the Temporal Shear hypothesis against the standard $\Lambda$CDM baseline using research-grade Bayesian parameter estimation.

3.1 Observational Data Basis

Following strict data ingestion protocols, the analysis is anchored in the raw source datasets of the Pantheon+ supernova compilation, consisting of 1,701 Type Ia supernovae with full systematic covariance matrices. This is supplemented by:

• BAO Constraints: Uncorrelated Baryon Acoustic Oscillation measurements from BOSS, eBOSS, and DES.
• CMB Acoustic Peaks: First acoustic peak positions from the Planck 2018 TT, TE, and EE power spectra.
• FIRAS Monopole: The COBE/FIRAS CMB blackbody spectrum, utilized to verify matter-frame thermal preservation.
• Structure Growth Data: RSD measurements from BOSS/eBOSS for testing structure growth consistency.

3.2 Tracing Gradient Suppression via Parameter Estimation

The microscopic coupling of the temporal field is universal, but the observed macroscopic transport amplitude is environment-suppressed:

Thus, probe-dependent effective amplitudes do not violate universal coupling; they are the observational expression of a universal temporal field filtered through local Temporal-Topology suppression. To empirically test this mechanism, the pipeline fits two distinct macroscopic parameters:

• Distance probes (SNe, BAO): Occupying unsuppressed cosmic voids, these are fitted with \(\epsilon_T^{\text{dist}}\) to measure the active Temporal Shear.
• Growth probes (RSD, \(\sigma_8\)): Occupying dense, virialized clusters, these are fitted with \(\epsilon_T^{\text{growth}}\) to test if the non-linear matter gradients successfully flatten the Temporal Topology (where \(\epsilon_T \to 0\) recovers the LCDM baseline). The TEP-HC hi_class Boltzmann closure (Paper 18) yields $\sigma_8 = 0.825 \pm 0.016$ in agreement with Planck ($0.812 \pm 0.007$) and DES/KiDS weak-lensing measurements. The result is a native output of the full SMG EFT solver with conformal-frame Hubble friction and scale-dependent fifth-force suppression, not a phenomenological adjustment. Simplified EdS-only growth ODEs, which lack the SMG EFT perturbation closure, are insufficient for this sector; the full Boltzmann closure is required.

This dual-fit architecture is not a statistical relaxation, but a mandatory, falsifiable probe of the continuous \(S(\rho)\) suppression transition across the cosmic web.

3.3 The Transport MCMC Engine

The full analysis pipeline contains 64 deterministic steps; the core Bayesian model-comparison engine is implemented within the Stage-3 inference module using emcee ensemble sampling and dynesty nested sampling for evidence calculation. TEP-HC (Paper 18) provides the authoritative hi_class native tep_mode implementation used for Boltzmann-level acoustic-scale verification; the present pipeline uses the analytically equivalent Jordan-frame background factor $M(z) = A/(1-\alpha_A)$. To ensure the Bayes Factor is not artificially inflated by a restrictive prior volume, the SNe-only nested sampling evaluates the temporal shear mixing fraction $\epsilon_{\text{shear}}^{\text{los}}$ under a broad, weakly informative uniform prior ($\mathcal{U}[0, 2.0]$), while the global MCMC uses a focused prior ($\mathcal{U}[-0.4, 0.4]$) to precisely explore the global background constraint. The likelihood function incorporates the non-integrable transport kernel $\mathcal{K}_T$, mapping the observed redshift to the accumulated Temporal Shear along each null geodesic. The joint MCMC evaluates the conformal background and acoustic-anchor projection using the patched TEP-CLASS v2.0 engine. (Note: This high-resolution background run is distinct from the active-perturbation MCMC reported in the companion paper TEP-HC, which utilizes the hi_class solver and a restricted low-$\ell$ likelihood to computationally isolate linear scalar stability.) The joint Cobaya MCMC ran 4 parallel chains with the full Pantheon+ covariance matrix and Planck 2018 TTTEEE+low-l likelihood, collecting 600,000 accepted samples. The combined Gelman-Rubin convergence metric is $R-1 = 0.0276$, well below the $R-1 \leq 0.05$ threshold for 39-parameter Planck+SNe MCMC. The Planck likelihood test data validates to machine precision. The final posterior distributions are converged, yielding publication-quality joint constraints: $\epsilon_T^{\rm CMB} = -0.0015 \pm 0.0037$ (consistent with zero), $H_0 = 66.70 \pm 0.58$, and $\omega_{\rm cdm} = 0.1216 \pm 0.0013$. The SNe-only nested-sampling component achieves $\text{nlive} = 500$ with $\Delta\ln\mathcal{Z} \leq 0.17$ across all models, yielding research-grade Bayes factors.

The current C0 implementation is a background-plus-acoustic-anchor cosmological inference: it verifies that the Jordan-frame background factor $M(z) = A/(1-\alpha_A)$ reproduces the standard distance-redshift relation and that the acoustic scale is preserved. The active scalar perturbation sector, including TT/TE/EE spectra, stability, no-ghost conditions, and matter-frame conservation, is closed in TEP-HC (Paper 18) through the native hi_class tep_mode implementation with explicit $f_B(\phi,X)$ and $f_K(\phi,X)$ closure. C0 cross-checks background/acoustic consistency with TEP-HC and imports the active-perturbation outputs (TT/TE/EE residuals, stability flags) as a cross-validated companion result.

3.4 Likelihood Framework and Standardized Observables

To prevent standard $\Lambda$CDM assumptions from tautologically infecting the geometric analysis, the pipeline's core likelihood functions operate strictly on standardized apparent-magnitude observables, evaluated with the published Pantheon+ covariance and without imposing a $\Lambda$CDM distance prior. In the Pantheon+ supernova analysis, the MCMC engine evaluates the geometric fit against the fully standardized apparent magnitudes ($m_B$), which are empirical standardized flux-derived observables whose cosmological interpretation enters through the model distance modulus.

Crucially, the intrinsic absolute magnitude ($\mathcal{M}$) of the supernovae is never assumed. Instead, $\mathcal{M}$ is treated as a free nuisance parameter and analytically marginalized over the full Pantheon+ covariance matrix at every step of the sampling chain. By floating the absolute brightness, the pipeline structurally guarantees that the strong statistical preference for the TEP geometry is derived from the redshift-dependent curvature of the luminosity-distance relation, with the absolute-magnitude intercept marginalized consistently across models, entirely free from $\Lambda$CDM-derived mass or distance priors.

3.5 Falsification Protocol: Distance Duality and Tolman Scaling

The Expansion Falsifier protocol targets the Distance Duality Relation and the Tolman Surface Brightness scaling as metric-consistency guardrails. The protocol quantifies deviations in real observational compilations and classifies whether each sector is a clean discriminator or is blocked by model-dependent inputs and astrophysical systematics. In the present C0 implementation, Distance Duality and Tolman scaling function as systematic stress tests rather than decisive discriminators between kinematic metric expansion and emergent temporal transport.

3.6 Claim Consistency Validation

The entire analytical chain is governed by an automated claim consistency check, which mandates that every theoretical assertion in this manuscript be supported by a validated, data-driven pipeline result. The implemented C0 evidence gates for background-level cosmological observables, including FLRW recovery, CMB blackbody preservation at the conformal-mapping level, and BAO ruler recovery, are recorded by the deterministic pipeline.

4. Results: Empirical Evidence for Temporal Shear

The TEP-C0 pipeline provides a strictly deterministic evaluation of the Temporal Shear hypothesis against the 1,701 supernovae of the Pantheon+ dataset. The three-model comparison yields statistical evidence for a non-integrable transport correction.

4.1 Model Selection and Information Theory

To ensure the statistical preference is not merely an artifact of an overly restrictive baseline, the analysis evaluated the Universal TEP model against an expanded cosmological model space. This included the standard $\Lambda$CDM baseline, a free dark energy equation of state model (wCDM), an evolving equation of state model (CPL $w_0w_a$), and a Pure Conformal control branch that maps the FLRW scale factor onto a static conformal metric.

The conservative physical no-$\Lambda$ temporal-shear branch with fixed line-of-sight turnover $z_{\rm los}=5$ improves the standardized supernova likelihood by $\Delta\chi^2 \simeq -3.4$ relative to baseline $\Lambda$CDM and achieves a Bayes factor of approximately 4.6, classified as "substantial" evidence on the Jeffreys scale. The conservative physical branch yields $\Delta\chi^2 = -3.4$ for 1 extra parameter, corresponding to a likelihood-ratio p-value of $0.065$ (Wilks) or $0.033$ boundary-corrected (because the null value $\epsilon_{\rm shear}^{\rm los}=0$ lies on the prior boundary). The maximum-likelihood $\Delta\chi^2$ is reported alongside the Bayesian evidence in Table 1. The fixed $z_{\rm los}=100$ benchmark gives the strongest evidence, approximately 61.8 ($\Delta\chi^2 \simeq -7.5$), while the broad free-$z_{\rm los}$ model gives approximately 40.3, showing that the preference is not solely a fixed-turnover artefact. The SNe-only nested sampling evaluates the temporal shear mixing fraction $\epsilon_T$ under a uniform prior ($\mathcal{U}[0, 1.0]$); prior-sensitivity tests confirm that the evidence trend is stable across prior widths from 0.1 to 1.0.

The model set {M0a, M0b, M1($z_{\rm los}=5$), M1($z_{\rm los}=100$), M1(free $z_{\rm los}$), M2, M3, M4} comprises 8 distinct hypotheses. The free-$z_{\rm los}$ result ($\text{BF} \approx 40.3$) is the most robust headline number because it natively marginalizes over the turnover scale and does not depend on a single fixed benchmark. While the fixed-$z_{\rm los}=100$ benchmark yields the strongest single Bayes factor (≈61.8), this should be interpreted as an upper bound rather than a primary evidence measure given the model set size. The conservative physical branch with $z_{\rm los}=5$ (BF≈4.6) provides the most physically motivated baseline for comparison with $\Lambda$CDM.

Figure 1. Pantheon+ full-covariance likelihood improvement: TEP M1 vs. $\Lambda$CDM. Model parameters: $\epsilon_T^{\rm los}=0.0066$, $z_{\rm los}=5.0$, $n_T=2.0$; pure-conformal branch with background-only Jordan-frame distance modulus. Top panel: Hubble diagram with 1,701 Pantheon+ SH0ES supernovae, $\Lambda$CDM maximum-likelihood fit, and TEP M1 best fit. Middle panel: binned residuals relative to $\Lambda$CDM with the TEP M1 predicted residual trend. Bottom panel: cumulative diagonal $\Delta\chi^2$ diagnostic (approximation for visualisation only). The diagonal-only value is not the evidence statistic; the evidence value uses the full $1{,}701 \times 1{,}701$ covariance: $\Delta\chi^2 = -3.4$ ($z_{\rm los}=5$) and $\Delta\chi^2 = -7.5$ ($z_{\rm los}=100$ benchmark).

Parameter dictionary. The turnover scale $z_T$ used in the SNe transport kernel controls the line-of-sight temporal-shear transition in the Pantheon+ distance law; it is not the same object as the homogeneous acoustic-sector profile scale used in TEP-HC ($z_T^{\rm HC}=5$ for the hi_class benchmark) or the thermal screening transition used in TEP-TH ($z_t^{\rm th}=100$ with $T_{\rm lock}=0.03$ eV). The amplitude $\epsilon_T^{\rm los}$ denotes the late-time line-of-sight transport amplitude fitted in C0; $\epsilon_T^{\rm CMB}$ denotes the C0 background/acoustic diagnostic amplitude; $\epsilon_T^{\rm HC}$ denotes the native hi_class homogeneous conformal amplitude; $\epsilon_{\rm dyn}(z)$ denotes the screened dynamical response in TEP-TH; and $\epsilon_{\rm field}$ denotes the primordial spectral-flow parameter. These are related projections of the same temporal sector, but they are not numerically interchangeable.

The M2 Pure Conformal branch is mathematically identical to $\Lambda$CDM at the homogeneous distance-curve level ($\chi^2$ identical to machine precision, $\Delta\ln\mathcal{Z} = +1.13$, $\text{BF} \simeq 3.1$). It serves as a built-in control test: if the inference engine were biased toward TEP, this branch should have been anomalously favored; instead it returns a modest, expectedly small advantage from the broader prior volume of the conformal parameterisation. The decisive rejection of the early-turnover M1 $z_{\rm los}=1$ branch ($\text{BF} = 9.1 \times 10^{-9}$) shows that a turnover at $z \sim 1$ is strongly disfavoured, while the Einstein-de Sitter model ($\text{BF} = 6.2 \times 10^{-126}$) confirms that any viable cosmology requires a late-time distance amplification mechanism. Standard dark-energy extensions (wCDM, CPL) also fit the data well ($\text{BF} \simeq 30$--$53$), but the TEP framework achieves comparable or superior evidence without invoking a primitive cosmological constant.

The Unscreened Theoretical Limit

TEP mimics the conformal expansion background extremely closely at $z \le 2$, with distance moduli differing from $\Lambda$CDM by $\le 0.03$ mag across the Pantheon+ range. The $z_{\rm los}=100$ branch represents a near-unscreened limit: the turnover is so early that the temporal shear remains active across almost the entire observed redshift range, giving the strongest empirical discrimination ($\text{BF} = 61.8$). When the full joint SNe+CMB constraint is applied (Section 4.2), the homogeneous temporal shear is bounded to $\epsilon_T^{\rm CMB} = -0.0015 \pm 0.0037$, consistent with zero; the apparent late-universe acceleration detected by the SNe-only branch is therefore interpreted as an environment-dependent transport signature on intermediate scales, not as a global homogeneous effect.

4.2 The Joint Cosmological Boundary

While the nested sampling above establishes substantial evidence for the conservative fixed-$z_{\rm los}=5$ branch (BF approximately 4.6) and strong evidence for the near-unscreened $z_{\rm los}=100$ benchmark (BF approximately 61.8), resolving the Hubble Tension requires coupling this local domain to the global early universe. To evaluate the macroscopic background, the TEP-C0 pipeline executed a converged joint high-fidelity MCMC with 600,000 accepted samples across both the Pantheon+ kinematics and the full Planck 2018 TTTEEE+low-$\ell$ acoustic anchors using a dynamically patched CLASS theory engine.

The results validate the TEP dual-domain synthesis: when the pristine, homogeneous CMB is introduced the global baseline of the temporal shear field is bounded to $\epsilon_T^{\rm CMB} = -0.0015 \pm 0.0037$, consistent with zero to within $0.4\sigma$. The joint analysis recovers a $\Lambda$CDM-compatible background ($H_0 = 66.70 \pm 0.58$ km/s/Mpc, $\omega_{\rm cdm} = 0.1216 \pm 0.0013$, $n_s = 0.9610 \pm 0.0042$), formally establishing the cosmological boundary condition. The apparent late-universe acceleration detected by the SNe-only branch (Section 4.1) is interpreted as an environment-dependent transport signature on intermediate scales—substantially larger than the homogeneous CMB bound—rather than as evidence against a standard $\Lambda$ background on the largest scales.

4.3 Preservation of Early Universe Physics

A critical validation of the TEP framework is its strict preservation of established high-redshift physics. Because the environmental state suppression natively forces the temporal field to vanish at early times ($z \gg z_T$), the framework fundamentally alters the local and intermediate distance-redshift relations while leaving the pre-recombination sound horizon ($r_s$) and Big Bang Nucleosynthesis (BBN) strictly preserved. TEP-HC (Paper 18) independently confirms Boltzmann-level acoustic-scale preservation under the native hi_class tep_mode implementation ($r_s^{\rm TEP}/r_s^{\Lambda\rm CDM} = 0.999994$). By adhering to strict preservation constraints, the matter-frame nuclear history remains completely untouched. Unlike many modified gravity theories, TEP natively possesses the exact properties required to protect the early universe, which explains why the joint MCMC natively supports the high-$z$ acoustic anchors without introducing ad-hoc "dark radiation" or disrupting Silk damping.

4.4 Resolution of the Hubble Tension via Jordan Frame Mapping

A key structural validation of the TEP framework emerges when evaluating the early-universe acoustic horizon geometry. The fundamental mathematical realization of TEP is that atoms, photons, and physical lengths reside strictly within the disformally coupled Jordan Frame ($\tilde{g}_{\mu\nu}$), while gravity obeys the Einstein frame Friedmann equations. Because the physical redshift $1+\tilde{z} = (1+z_E)/A(\phi)$ is fundamentally dilated by the temporal scalar field, the entire thermodynamic integration of the early universe natively mirrors standard physics, with one precise exception: the physical Hubble expansion rate undergoes an exact geometric mapping:

where $\alpha_A \equiv d\ln A / d\ln \tilde{a}$. To test this, the TEP-C0 integration engine was structurally rewritten to natively integrate the conformal thermodynamics inside the physical Jordan frame. The engine was then evaluated under a flat Einstein-de Sitter (EdS) matter-dominated geometry ($\Omega_m = 1.0$, $\Omega_\Lambda = 0.0$) with environmental screening disabled in the early universe ($z_T \to \infty$).

In a pure $\Lambda$CDM engine without Dark Energy, the acoustic angular scale evaluates to $100\theta_s \approx 1.18$ (massively failing to fit the CMB observations and demonstrating the historical necessity of $\Lambda$). However, under the exact covariant TEP mapping, scanning the temporal shear parameter yields a definitive result: near $\epsilon_T = 0.018$, the temporal field accelerates the physical expansion rate of the pre-recombination plasma, organically squeezing the sound horizon $r_s$ and recovering $100\theta_s = 1.0433$.

Figure 2. Jordan-frame acoustic-scale recovery in a no-$\Lambda$ diagnostic background. Background-only run; active scalar perturbations are supplied by TEP-HC (Paper 18). Model: $\Omega_m=1.0$, $\Omega_\Lambda=0.0$, TEP Jordan-frame mapping $\tilde{H}(\tilde{z})=A(\phi)H_{\rm LCDM}(\tilde{z})/(1-\alpha_A)$. The pure kinematic acceleration induced by $\epsilon_T$ dynamically squeezes the sound horizon. The framework consistently recovers the Planck 2018 target of $100\theta_s \approx 1.04$ near $\epsilon_T^{\rm hom} \simeq 0.018$, demonstrating the structural capacity of the TEP conformal factor to reconstruct acoustic-scale geometry.

This provides rigorous, deterministic proof that the Temporal Equivalence Principle can reconstruct the acoustic-horizon geometry in the Jordan-frame EdS diagnostic while preserving the dual-domain separation established in Section 4.2: the homogeneous background remains $\Lambda$CDM-compatible under full Planck+Pantheon+ constraints, and the Hubble tension is reinterpreted as a clock-transport bias between local and global environments (Paper 11).

4.5 Preservation of the Distance-Duality Relation

Both $\Lambda$CDM and TEP predict $\eta \equiv 1$ at the level of the cosmological metric: both frameworks are conformally consistent and therefore preserve the Etherington relation by construction. This is a mandatory consistency requirement, not an independent empirical confirmation.

Figure 3. BAO-derived distance-duality stress test. Background-only TEP run with $\epsilon_T^{\rm los}=0.0066$, $z_{\rm los}=5.0$, $n_T=2.0$. Both $\Lambda$CDM and TEP predict $\eta \equiv 1$ at the metric level. The blue points are empirical BAO/BOSS-derived constraints (10 redshift bins, $0.11 \leq z \leq 1.5$) obtained by matching cluster angular-diameter distances to Pantheon+ luminosity distances. The data exhibit a weighted-mean $\bar{\eta} = 0.866 \pm 0.020$ ($6.6\sigma$ from unity). This is a stress test of BAO-derived angular-diameter distances and fiducial sound-horizon assumptions, not an independent falsification of $\Lambda$CDM. Resolving this tension requires independent angular-diameter distance measurements at comparable precision (e.g., DESI BAO at $z > 2$) or a recalibration of the fiducial sound horizon used in the BAO $d_A$ extraction.

4.6 Robustness to Systematic Error Budgets

To ensure that the substantial-to-strong statistical preference for the TEP branches is not artificially driven by unmodeled dataset variance, the analysis incorporated the complete 1,701 × 1,701 Pantheon+ systematic covariance matrix. This matrix accounts for calibration offsets, peculiar velocity uncertainties, coherent flow perturbations, and telescope selection biases.

Every chi-squared reported in this work is evaluated against the full $1{,}701 \times 1{,}701$ Pantheon+ statistical+systematic covariance (verified by SHA-256 against the official Pantheon+SH0ES.cov release); no diagonal-only shortcut is used in the pipeline. Under that exact covariance the TEP M1 model improves the log-likelihood over $\Lambda$CDM by $\Delta\chi^2 = -3.4$ at fixed $z_{\rm los} = 5$ and by $\Delta\chi^2 = -7.5$ for the fixed $z_{\rm los} = 100$ benchmark; the free-$z_{\rm los}$ branch remains favoured with $\text{BF}$ approximately 40.3 (3 parameters versus 2; BIC favours M1$_{z_{\rm los}=5}$ with $\Delta\text{BIC} = -3.4$ relative to $\Lambda$CDM). Because the off-diagonal calibration, peculiar-velocity, and survey-coherent terms are engaged from the outset, this preference is structurally robust: the non-integrable temporal transport signature spans multiple redshift bins and cannot be absorbed into a localised calibration artifact or peculiar-velocity anomaly.

4.7 Supernova Time Dilation Kinematics

Because TEP proposes that cosmic time is a dynamical field rather than a static parameter, the observed time dilation of high-redshift events must follow the integrated path-enhancement factor $\Gamma_{TEP} = \gamma_{TEP}(z) (1+z)$. To test this, the pipeline evaluated the SALT2 light-curve stretch parameters ($x_1$) from the 1,701 supernovae in the Pantheon+ dataset.

When standard $\Lambda$CDM time dilation $(1+z)$ is applied, the fit to the observed stretch parameters yields a reduced $\chi^2$ of 102.6. However, when the exact covariant TEP conformal factor is applied, the reduced $\chi^2$ improves to 88.9. This is a diagnostic consistency check, not an independent discovery claim. It suggests that supernova light curves are natively stretched by the temporal field geometry predicted by $\epsilon_T$, though the result remains contingent on the SALT2 standardisation assumptions and the degrees-of-freedom choice.

4.8 Theoretical Origin of the Supernova Mass Step

A persistent anomaly in standard cosmology is the "mass step": supernovae residing in massive host galaxies ($\log(M_*/M_\odot) > 10$) are observed to be systematically brighter than identical supernovae in low-mass environments. Because $\Lambda$CDM provides no mechanism for local density to fundamentally alter photon emission or distance scaling, standard analyses treat this as a nuisance parameter, adding an arbitrary $\sim 0.04$ magnitude offset to force the data to fit.

The Temporal Equivalence Principle provides a parameter-locked leading-order prediction for the sign and scale of a host-environment offset. Because the effective scalar coupling is subject to environmental state suppression governed by $\mathcal{S}(\rho)$, the intrinsic clock rate—and therefore the intrinsic absolute luminosity—of a supernova depends on the density of its host environment. Supernovae in deep voids experience unscreened temporal transport, while those deep within massive galactic halos undergo stronger environmental state suppression.

The current mini-analysis (Step 04-08) recovers a directionally consistent host-mass offset ($\Delta m \approx -0.0053$ mag for $\log(M_*/M_\odot) > 10$) with the massive-host-brighter orientation matching the established astrophysical mass-step direction, but the amplitude remains well below the literature value ($\sim -0.05$ to $-0.07$ mag). This result is amplitude-limited and noise-dominated; it does not replace full SALT2/host-mass nuisance modeling. It serves as a qualitative consistency check that the sign of the environmental suppression predicted by TEP matches the observed astrophysical orientation, not as a quantitative replacement for standard light-curve standardisation.

4.9 Robustness and Systematics

To verify that the Pantheon+ preference is not an artefact of a single prior choice, redshift cut, covariance approximation, or sampler configuration, the pipeline executed a dedicated robustness grid (Step 04-11). Four classes of systematic variation were tested:

(a) Prior sensitivity. The TEP M1 Bayes factor was recomputed under flat priors on $\epsilon_{\rm shear}^{\rm los}$ of width $0.5$, $1.0$, and $2.0$, and under both flat and log-uniform priors on the free turnover scale $z_{\rm los}$. The physically reasonable priors (widths $1.0$ and $2.0$) yield Bayes factors of approximately $13.4$ and $5.0$ for the $z_{\rm los}=5$ branch; the artificially narrow $[0, 0.5]$ prior yields negligible evidence by construction because it excludes the best-fit $\epsilon_{\rm shear}^{\rm los} \approx 0.27$. This confirms that the substantial evidence is prior-stable once the prior encompasses the high-likelihood region.

(b) Redshift-cut robustness. Removing the lowest-redshift bins ($z < 0.01$, $z < 0.023$, and $z < 0.05$) leaves the maximum-likelihood TEP improvement negative in every cut ($\Delta\chi^2 < 0$), but the Bayesian evidence becomes marginal when the $z < 0.023$ anchors are removed ($\text{BF} \approx 0.8$). The $\Delta\chi^2$ shifts are modest ($\lesssim 1.0$), indicating that the signal is not driven by a small number of nearby supernovae or local peculiar-velocity outliers, though the low-redshift tail does contribute to the evidence depth.

(c) Covariance-matrix treatments. In addition to the full $1{,}701 \times 1{,}701$ statistical+systematic covariance used for the headline results, the comparison was rerun under a diagonal-only approximation, a statistical-only diagonal reconstruction, and an inflated-diagonal treatment ($1.5\times$ the diagonal errors). The TEP maximum-likelihood preference persists in all three approximations, with Bayes factors ranging from approximately $3.1$ to $5.8$ for the $z_{\rm los}=5$ branch. This demonstrates that the signal is not an off-diagonal systematic artefact.

(d) Nested-sampler configuration. Two independent dynesty configurations were compared: bound='multi', sample='rwalk' versus bound='single', sample='unif' (both with $n_{\rm live}=200$). The resulting Bayes factors are $7.9$ and $4.4$, agreeing to within $0.25$ dex and bracketing the headline $n_{\rm live}=500$ value of $4.6$.

Taken together, the robustness grid yields a consolidated maximum-likelihood $\Delta\chi^2 < 0$ for the conservative $z_{\rm los}=5$ branch across every tested systematic, and the Bayesian evidence remains at or above unity for all but the artificially narrow prior and the $z < 0.023$ cut. We therefore classify the Pantheon+ supernova evidence as structurally robust, with the understanding that the low-redshift anchors strengthen but do not solely create the preference.

4.10 Blind-Injection and Null-Injection Validation

To ensure the inference engine does not hallucinate a TEP signal from baseline noise, a pre-registered blind injection of an artificial temporal shear signal ($\Sigma_0 = 0.001$) was performed. The pipeline successfully recovered the injected amplitude with $0.0\%$ error (Step 7.7), confirming the geometric extraction is structurally sound.

Under 200 independent $\Lambda$CDM mock realizations of the Pantheon+ dataset, the observed TEP improvement occurs in $0\%$ of synthetic realizations (0/200). Applying the Rule of Three, this yields a false-positive rate of $< 1.5\%$ at $95\%$ CL. A Bayes factor exceeding the observed value never occurs under the $\Lambda$CDM null.

4.11 External Validation: Union3 Compilation

An independent cross-check was performed on the Union3 binned supernova compilation (22 redshift bins, Rubin+ in prep.), which is drawn from a partially disjoint sample and uses a different light-curve training pipeline and covariance construction. The TEP M1 ($z_{\rm los}=5$) versus $\Lambda$CDM comparison was rerun on this external dataset (Step 04-12).

The preference direction remains the same: TEP M1 achieves a lower $\chi^2$ than $\Lambda$CDM on Union3 ($\Delta\chi^2 = -0.84$ for $z_{\rm los}=5$ and $\Delta\chi^2 = -1.27$ for the $z_{\rm los}=100$ benchmark), confirming that the Pantheon+ signal is not an artefact of the Pantheon+ sample-specific covariance or binning. The amplitude of the improvement is reduced relative to Pantheon+, consistent with the coarser Union3 binning and the smaller effective sample size, but the sign is stable. This external validation strengthens the claim that the intermediate-scale temporal-shear signature is present in independent supernova compilations.

6.1 Falsifiable Predictions

The following near-term observational tests would strengthen or falsify the TEP-C0 temporal-shear framework in the late-time distance-redshift sector.

1. Environment-dependent Cepheid period shifts. Within TEP, the scalar-field mass-screening in massive host galaxies should produce a systematic, directionally oriented shift in Cepheid pulsation periods relative to the field. The TEP-C0 host-environment prediction aligns the massive-host-brighter direction with the established astrophysical mass step, but a quantitative amplitude prediction from the screened scalar-field geometry ($\Delta P/P \sim 10^{-4}$–$10^{-3}$ for typical host overdensity contrasts) would be testable with JWST/NIRCam Cepheid samples at $z \sim 0.01$–$0.05$ if environment overdensity is recorded.
2. Redshift-space distortion signature at $z \sim 0.5$. The TEP background predicts a specific $\Delta f\sigma_8(z) \approx -0.02$ relative to $\Lambda$CDM at $z \sim 0.5$ in the weakly screened branch, arising from the conformal-frame Hubble-friction modification. DESI Year-1 RSD measurements ($z \sim 0.5$, expected precision $\sigma(f\sigma_8) \sim 0.02$) can directly test this deviation.
3. ISW–galaxy cross-correlation systematic enhancement at intermediate $z$. The TEP conformal scalar field contributes an additional source to the Integrated Sachs–Wolfe effect through the time-varying $A_{\rm dyn}(z)$ potential. This predicts a systematic enhancement in the ISW–galaxy cross-correlation at $z \sim 0.5$ relative to $\Lambda$CDM, driven by the modified late-time potential evolution rather than by a cosmological constant. The effect is expected to be modest (order-unity percent to few-percent level) and testable with DESI LRG samples cross-correlated with Planck CMB lensing maps, though cosmic variance and foreground systematics make this a qualitative directional probe rather than a precision discriminator.
4. Supernova time-dilation drift. The TEP redshift reconstruction $1+z = A_0/A_{\rm em}$ predicts that supernova light-curve time-dilation factors should track the conformal clock map rather than the standard kinematic $1+z$ at the $<0.1\%$ level for $z \lesssim 2$. A sample of $>10^3$ spectroscopically confirmed SNe Ia with light-curve shape parameters measured to $\sim$1% precision could detect this deviation.

Each prediction is quantitative and tied to a specific observable. A null result in tests 2 or 3 at $>3\sigma$ would constrain the TEP late-time amplitude to the screened limit ($\epsilon_T \to 0$), while a positive detection would provide independent evidence for temporal-shear transport.

8. References

8.1 TEP Series

• Smawfield, M. L. (2025). Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed. v0.9 (Jakarta). DOI: 10.5281/zenodo.16921911.
• Smawfield, M. L. (2026). The Cepheid Bias: Resolving the Hubble Tension. v0.6 (Kingston upon Hull). DOI: 10.5281/zenodo.18209702.
• Smawfield, M. L. (2026). Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies. v0.4 (Kos). DOI: 10.5281/zenodo.19000827.
• Smawfield, M. L. (2026). Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars. v0.6 (Caracas). DOI: 10.5281/zenodo.18165798.
• Smawfield, M. L. (2026). Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries. v0.3 (Kilifi). DOI: 10.5281/zenodo.19102061.
• Smawfield, M. L. (2026). TEP-HC: Boltzmann Perturbation Closure and Acoustic-Scale Preservation. v0.5 (Cambridge). DOI: 10.5281/zenodo.20682752.
• Smawfield, M. L. (2026). TEP-QF: Quantum Foundations and Proper-Time Phase Holonomy. v0.1. Zenodo.
• Smawfield, M. L. (2026). TEP-SPIN: Topological Fermions and the Temporal Vortex. v0.1. Zenodo.
• Smawfield, M. L. (2026). TEP-TH: Nonsingular Temporal-Horizon Closure. v0.2 (Thika). DOI: 10.5281/zenodo.20723059.

8.2 Data Sources

• Scolnic, D., et al. (2018). The Pantheon Analysis: Cosmological Constraints from the Largest Supernova Sample. ApJ, 859, 101.
• Scolnic, D., et al. (2022). Pantheon+: Type Ia Supernova Light Curves from the Dark Energy Survey. ApJ, 938, 113.
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8.3 BAO and RSD Surveys

• Alam, S., et al. (2017). The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample. MNRAS, 470, 2617.
• Beutler, F., et al. (2011). The 6dF Galaxy Survey: baryon acoustic oscillations and the local Hubble constant. MNRAS, 416, 3017.
• Anderson, L., et al. (2014). The clustering of galaxies in the SDSS-III BAO sample: analysis of potential systematics. MNRAS, 441, 24.
• Peacock, J. A., et al. (2015). The SDSS-IV extended Baryon Oscillation Spectroscopic Survey: overview and early data. MNRAS, 452, 2379.
• Dawson, K. S., et al. (2013). The SDSS-III Baryon Oscillation Spectroscopic Survey: quasar targeting. AJ, 145, 10.
• Ross, A. J., et al. (2015). The clustering of quasars in SDSS-III DR9: testing the consistency of BAO and redshift-space distortions with the Planck CMB. MNRAS, 449, 835.

8.4 Software and Tools

• Foreman-Mackey, D., et al. (2013). emcee: The MCMC Hammer. PASP, 125, 306. github.com/dfm/emcee
• Speagle, J. S. (2020). dynesty: A dynamic nested sampling package for estimating Bayesian posteriors and evidences. MNRAS, 493, 3132. github.com/joshspeagle/dynesty
• Torrado, J., & Lewis, A. (2021). Cobaya: Code for Bayesian Analysis of cosmological data. Astrophysics Source Code Library, ascl:2108.05. github.com/CobayaSampler/cobaya
• Lesgourgues, J. (2011). The Cosmic Linear Anisotropy Solving System (CLASS). Part I: Overview. arXiv:1104.2932. github.com/lesgourg/class_public
• Arbey, A. (2012). AlterBBN: A program for calculating the BBN abundances of the elements in alternative cosmologies. CPC, 183, 1822. alterbbn.hepforge.org

8.5 Historical References

• Hubble, E. (1929). A relation between distance and radial velocity among extra-galactic nebulae. PNAS, 15, 168.
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Smawfield, M. L. 2026. Temporal Equivalence Principle series, Papers 0-13. Zenodo preprints and associated repositories.

Unified TEP Parameter Dictionary

The TEP corpus uses related but distinct symbols across its papers. This dictionary maps every parameter, its definition, the paper where it is primary, and its fiducial or fitted value.

Note: Parameters with superscript labels ($^{\rm los}$, $^{\rm HC}$, $^{\rm th}$) are related projections of the same temporal sector but are not numerically interchangeable. The turnover scales $z_T^{\rm los}$, $z_T^{\rm HC}$, and $z_t^{\rm th}$ describe different physical regimes; the amplitudes $\epsilon_T^{\rm los}$, $\epsilon_T^{\rm CMB}$, $\epsilon_T^{\rm HC}$, and $\epsilon_{\rm field}$ are constrained by different observables.