Temporal Equivalence Principle: Dynamical Proper Time and the Illusion of Primordial Deuterium

1. Introduction

The prevailing cosmological paradigm interprets the Hubble redshift as the kinematic expansion of a spatial volume, directly linking high redshifts to a dense, ultra-hot spatial singularity—the Big Bang. Within this framework, the measurement of light element abundances, particularly deuterium (D/H) in high-redshift Lyman-limit systems such as Q1009+2956, serves as a crucial anchor for Big Bang Nucleosynthesis (BBN) [22], [45]. However, this standard inference assumes that cosmological redshift is intrinsically geometric and that the spectroscopic structure of deuterium is uniquely distinguishable from contaminating intergalactic hydrogen.

Tensions in precision cosmology—such as the Hubble tension [17], [18] and the $S_8$ growth tension—have motivated numerous theoretical extensions to the $\Lambda$CDM framework [16]. Recent literature has extensively explored modified gravity (e.g., $f(R)$ or scalar-tensor theories [30], [42]), non-standard recombination histories, and Early Dark Energy (EDE) to resolve these anomalies. While these approaches introduce new dynamical parameters to accommodate the standard hot-thermal history, they generally preserve the foundational assumption that cosmological redshift equates to geometric spatial expansion.

The standard hot-BBN inference is challenged on two fronts within the TEP framework [1], [2]. First, an algorithmically controlled analysis of Q1009+2956 shows that the canonical deuterium interpretation is not uniquely identifiable against the ordinary-H alternative at the available resolution. Second, the temporal-transport framework is developed to show how the associated cosmological observables can be represented without a primordial spatial singularity.

2. Spectroscopic Analysis

The spectroscopic uniqueness of the deuterium identification in high-redshift quasar absorption systems is examined under nested model comparison with Monte-Carlo calibrated significance. The primary finding is that the statistical preference for an unrestricted hydrogen model over a true-D model is robust across sightlines, under both the canonical-parent and parent-reassignment statistics. The parent-reassignment statistic ($T_{\rm parent}$, $p_{\rm parent}$)—which gives D maximum freedom to select the most favorable parent—is the more conservative and informative number, and is reported as the primary result throughout. The canonical-parent (standard) statistic is reported as a secondary comparison. A second key finding is that isotope identifiability is strictly regime-dependent: it must be assessed per absorber rather than assumed, as the distinguishability of H and D Voigt profiles depends critically on the thermal-to-turbulent Doppler ratio.

The analysis is performed on the high-resolution Keck/HIRES spectrum of the benchmark Q1009+2956 absorption system, from Zavarygin et al. 2018 [44] (22 HIRES exposures co-added into four setups using the wspectrum software). The published kinematic architecture and component parameters from Zavarygin et al. 2018 [44] are used as the structural prior for the Voigt component family, fitted with the VPFIT framework [50]. The spectral data comprise four coadds at two instrumental settings: C1 ($R\approx49\,000$, $\sigma_{\rm inst}=2.6$\,km/s) and C5 ($R\approx37\,000$, $\sigma_{\rm inst}=3.4$\,km/s), with S/N $\approx 47$–$78$ per pixel near Ly$\alpha$. The system was originally measured for primordial D/H by Burles & Tytler [46] and subsequently by Zavarygin et al. [44], with independent measurements from other quasar sightlines contributing to the primordial D/H estimate [47], [48]. Two additional benchmark sightlines—PKS\,1937$-$101 (ESPRESSO, $z=3.572$) and J1332$+$0052 (UVES+HIRES, $z=3.421$)—are analysed with the identical framework in Section 2.4.

2.1 Isotope Identifiability Limits in Q1009+2956

2.2 Likelihood Nesting and Significance Testing

A complete structural analysis of the Q1009+2956 spectrum is performed under a rigorously nested model hierarchy. By anchoring the fits to a SHA-256-validated data manifest, the likelihood surfaces of the standard D-interpretation ($M_D$), an unrestricted hydrogen interpretation ($M_{H,\rm free}$), and the joint space ($M_{D+H}$) are mapped. The 141 shared parameters (43 H I Voigt component parameters: centres, $b$-values, and column densities, plus continuum basis coefficients for each data block) are held at the published literature architecture [44], identically for every hypothesis. This fixed scaffold is part of the definition of the test statistic rather than an uncontrolled nuisance choice: the significance of $T$ is not read off an asymptotic $\chi^2$ distribution but calibrated by parametric bootstrap in Section 2.3, where true-D spectra are regenerated on the same scaffold and the identical statistic is recomputed. Because the scaffold enters the observed and simulated statistics in exactly the same way, the resulting $p$-value is valid without requiring the shared architecture to be re-optimised.

The candidate component parameter structure is made fully explicit. Each model differs only in the candidate component; the shared scaffold is identical:

The parameter count difference is $\Delta k = k(M_{H,\rm free}) - k(M_D) = 4 - 3 = 1$: the unrestricted-H model gains one free parameter (the velocity $v_H$, which in $M_D$ is tied to the parent velocity via $v_D = v_{\rm parent} + v_{\rm iso}$). The column density, temperature, and turbulent broadening are free in both models. The $\Delta k = 1$ therefore reflects exactly the velocity-tying difference between the isotope-D and free-H hypotheses — the single physical degree of freedom that distinguishes them.

2.3 Leave-One-Out and Parent Reassignment Robustness

The component misattribution vulnerability was exhaustively tested by tying the candidate D velocity to every single available H component in the model family (43 distinct parent candidate structures). The maximum alternative-parent test statistic is defined as:

Against the most advantageous alternative parent assignment, the free-H interpretation is preferred for Q1009+2956 with $T_{\rm parent} = 161.03$, equal to $T = 161.03$ for the canonical parent. The discrimination is therefore not specific to the published parent assignment, and the parent-reassignment statistic is the conservative quantity to calibrate. When calibrated inside the true-D Monte Carlo loop—utilizing bounded L-BFGS-B optimization with a multi-start grid spanning the full velocity range $[-160, +50]\text{ km/s}$, $\log N_{\rm D} \in [0, 20]$, $T_K \in [100, 4\times10^4]$\,K, and $b_{\rm turb} \in [1, 50]$\,km/s, with the best likelihood retained across all starts to prevent local minima from artificially widening the null distribution—the empirical $p_{\rm parent}$ is reported below.

Parent-reassignment window. Two parent-reassignment windows are reported. The conservative window ($\pm 1000\text{ km/s}$) gives the deuterium hypothesis maximum freedom to find a favorable parent, establishing a lower bound for the $T_{\rm parent}$ statistic. Of the 43 H I components in the Q1009 model, 14 fall within this window; none are DLA-class ($\log N_{\rm HI} > 20.3$), and components from unrelated absorber systems at $v \sim -47\,000$ to $-74\,000\text{ km/s}$ are correctly excluded. The physical window ($\pm 3b_{\rm max}$, where $b_{\rm max} = 56.7$\,km/s is the largest Doppler parameter of the main complex) restricts eligible parents to those within the kinematic structure of the absorber: 9 H I components fall within the velocity window, of which 4 satisfy the column-density eligibility cut ($\log N_{\rm HI} \ge \log N_{\rm max} - 2$\,dex). The physical window is reported as the headline statistic because it is motivated by the absorber's own velocity structure rather than an arbitrary fixed range. Under the physical window, the conclusions are essentially unchanged from the conservative window (see robustness check C1 in Section 5.5), confirming that the $\pm 1000\text{ km/s}$ conservative bound was not driving the results.

A critical interpretation issue arises in the parent-reassignment test. For Q1009+2956, $T_{\rm parent} = T_{\rm obs} = 161.03$, meaning that allowing reassignment does not improve the D fit over the canonical parent—the discrimination is not specific to the published parent assignment. For PKS\,1937$-$101, $T_{\rm parent} = 59.00 < T_{\rm obs} = 161.31$, so the best alternative parent substantially reduces the statistic, but the result remains significant. For J1332$+$0052, $T_{\rm parent} = 135.76 < T_{\rm obs} = 136.97$, so the best alternative parent slightly reduces the statistic. In all three cases, the free-H model is significantly preferred even under the most favorable parent assignment for D. The parent-reassignment statistic is therefore the conservative quantity to calibrate, and the $p_{\rm parent}$ values reported below should be interpreted as the significance under the most favorable parent assignment for D.

Parent-identity diagnostic (Gate 7). Under the isotope-D interpretation, the best alternative parent $j^*$ (the H I component maximizing $\ln L(M_D \mid j)$) should be the main narrow thermal component — D is tied to the gas it formed beside. Under TEP, $j^*$ might be a diffuse, low-column edge component — the edge gas anchors to its own local structure. For all three sightlines, the best non-canonical parent within the eligible window ($\log N_{\rm HI} \ge \log N_{\rm max} - 2$\,dex, $|v| \le 1000$\,km/s) is a core component: Q1009+2956 parent 1 ($v = -43.7$\,km/s, $\log N_{\rm HI} = 16.80$, $b = 17.8$\,km/s), PKS\,1937$-$101 parent 20 ($v = +17.9$\,km/s, $\log N_{\rm HI} = 17.38$, $b = 17.1$\,km/s), and J1332$+$0052 parent 2 ($v = +8.7$\,km/s, $\log N_{\rm HI} = 18.39$, $b = 15.8$\,km/s). This is consistent with the isotope-D interpretation and does not support the TEP edge-parent prediction. The parent-identity test is presently treated as a descriptive diagnostic rather than a TEP-specific discriminator. Under TEP the feature is ordinary edge H, not D tied to any parent; the identity of the parent that minimizes a mis-specified D fit may be a fitting artefact rather than a physical TEP observable. A derivation showing why a displaced edge-H feature mathematically causes the false-D likelihood to maximize on diffuse edge components is required before Gate 7 can be elevated to a TEP-specific prediction.

Offset-exactness test (Gate 4E). Under the isotope-D interpretation, the free-H component should sit at the exact isotope-shifted velocity $v_{\rm iso} = -81.6$\,km/s relative to its parent. Under TEP, the edge-gas feature need not sit at the exact isotope shift — the observed offset depends on the local shear amplitude, which varies with core-edge geometry. The fitted $M_{H,\rm free}$ velocities deviate from the isotope-shifted position by:

Q1009+2956 sits $2.7$\,km/s from the isotope shift — consistent with isotope-D to within the spectral resolution ($2.6$\,km/s), but highly inconsistent when calibrated against the true-D simulation distribution ($\pm 0.2$\,km/s, $p < 0.005$, Section 5.6). PKS\,1937$-$101 deviates by $5.7$\,km/s — also highly significant after calibration ($p < 0.005$). J1332$+$0052 deviates by $-36.6$\,km/s — marginal after calibration ($p = 0.09$). The cross-sightline pattern — increasing offset deviation with increasing D column — is quantitatively testable: if the feature is isotope-D, all three should sit at $v_{\rm iso}$ within spectral resolution; under TEP, the offset varies with core-edge geometry. The J1332 feature shows the largest departure from the isotope-shifted position, and it is also the sightline with the highest D column (saturated, $\log N_{\rm D} \approx 14.65$), where the edge-gas interpretation is most physically motivated.

Finally, performing a transition-level leave-one-out (LOO) test reveals where the empirical discrimination resides. The full LOO vector for Q1009+2956 (10 entries: 4 coadd deletions + 6 transition deletions):

The full LOO vector reveals that the discrimination is strongly concentrated in Ly$\alpha$: removing Ly$\alpha$ decreases $T$ from 161.03 to 34.15, indicating that Ly$\alpha$ provides the bulk of the isotope discrimination. This is physically expected: the isotope velocity shift ($-81.6$\,km/s) is the same across all Lyman transitions, but Ly$\alpha$ has the highest oscillator strength and signal-to-noise ratio, yielding the deepest absorption profile and the largest number of informative pixels over which the mass-scaled Doppler difference between H and D accumulates. Ly$\beta$ is the second most important transition ($T_{-\mathrm{Ly}\beta} = 148.64$, $\Delta T = -12.39$), followed by Ly$\gamma$ ($\Delta T = -4.86$). The high-order transitions (Ly6, Ly13, Ly14) contribute negligibly ($|\Delta T| < 0.5$). The coadd deletions show that the C5x1 setting contributes the most discrimination ($\Delta T = -83.71$), followed by C1x1 ($\Delta T = -35.98$). All coadd deletions decrease $T$, confirming that the discrimination is distributed across all four coadd settings rather than being driven by a single dataset. The result demonstrates that a benchmark high-redshift D/H system is not spectroscopically self-authenticating once the displaced-H model class is admitted.

Velocity-selection theorem (Gate 6). The strongest objection to the TEP interpretation is that the observed velocity offset matches the exact reduced-mass isotope shift ($-81.6$\,km/s) in all three sightlines. The chance probability of three independent features landing within $\pm 5$\,km/s of the isotope shift is $\sim 10^{-6}$ (assuming a $\pm 500$\,km/s prior range of narrow-component velocities). However, this coincidence is a property of the selection function, not of the gas: the D-identification protocol is a filter $|\Delta v - v_{\rm iso}| < \epsilon$ — VPFIT initializes D components at the isotope-shifted velocity, and analysts search for D at that specific offset. Any shear feature at $-40$\,km/s or $-120$\,km/s relative to the core is fitted as an ordinary H I component and never flagged as D. The universality of $-81.6$\,km/s therefore cannot be treated as independent evidence for isotope identity within a sample selected through the isotope-shift criterion; it is true by construction of the sample. The falsifiable prediction is that edge-shear features at other offsets exist and are being absorbed into H I component structure: there should be a population of anomalously narrow "orphan" components in high-S/N LLS spectra whose velocity distribution is broader than the isotope shift and whose columns track the edge-gas prediction.

Quantitative population forecast (Gate 6-II). The observed cross-sightline deviations from the isotope shift ($+2.7$, $+5.7$, $-36.6$\,km/s for Q1009, PKS\,1937, J1332 respectively) imply a shear spread $\sigma_{\rm shear} \approx 19$\,km/s. The D-search selection function $|\Delta v - v_{\rm iso}| < 5$\,km/s captures only $\sim 21\%$ of edge-shear features; the remaining $\sim 79\%$ are orphans absorbed into the H I component structure. In a sample of $\sim 80$ high-S/N LLS sightlines with full Lyman coverage (SQUAD DR1, UVES SLS, ESPRESSO D/H survey), $\sim 24$ shear features are expected, of which $\sim 5$ are identified as D and $\sim 19$ are orphans. The predicted orphan surface density is $\sim 2.4$ per 10 sightlines at $|\Delta v - v_{\rm iso}| > 5$\,km/s, falling to $\sim 1.8$ per 10 sightlines at $> 10$\,km/s. Orphan features are predicted to have narrow Doppler parameters ($b < 10$\,km/s, cool edge gas), low columns ($\log N_{\rm HI} \sim 12$--$14$), and thermal Doppler regimes — distinguishable from the normal H I population by an excess of narrow, low-column components at non-isotope offsets. Under the isotope-D null, no such excess is expected. This is a targetable observational program using existing archival data.

The embedding pipeline has been generalised to support multiple sightlines via a sightline-configuration system. Two additional benchmark D/H quasar sightlines have now been analysed with the identical nested-hypothesis framework, providing the first cross-sightline application of the TEP spectroscopic methodology. The consistent outcome across all three sightlines is qualitatively consistent with TEP's prediction that temporal shear depends on absorber-specific core-edge geometry; an a priori amplitude ranking remains to be calculated.

2.4 Cross-Sightline Validation

The identical Gate 2/3 pipeline was applied to PKS 1937$-$101 (Cooke et al. 2024 [66]; ESPRESSO, $R=70\,000$, $z_{\rm abs}=3.572$) and J1332$+$0052 (Kislitsyn et al. 2024 [67]; joint UVES+HIRES, $R\approx50\,800$ and $49\,200$, $z_{\rm abs}=3.421$). Both sightlines were analysed with all available Lyman transitions covered by their spectral data (13 for PKS 1937$-$101, 12 for J1332$+$0052 which lacks Ly$\alpha$ coverage), the same noise-model calibration, and the same 1000-realization Monte Carlo protocol.

$^{\dagger}$ For J1332$+$0052, the canonical-parent standard statistic is reported as the primary $p_{\rm std}$ for consistency with Q1009 and PKS\,1937. Both definitions yield significant results for J1332$+$0052; the canonical-parent value is reported in the table for cross-sightline uniformity.

Under the conservative parent-reassignment statistic, the results are: for Q1009+2956, $p_{\rm parent} = 0.001$ (0/1000 exceedances), significant—free-H is preferred even under the most favorable parent assignment. For PKS\,1937$-$101, $p_{\rm parent} = 0.001$ (0/1000 exceedances), significant—free-H is preferred even under the most favorable parent assignment. For J1332$+$0052, $p_{\rm parent} = 0.001$ (0/1000 exceedances), significant—free-H is preferred even under the most favorable parent assignment. The standard (canonical-parent) statistic yields $p_{\rm std} = 0.001$ (0/1000) for all three sightlines. The consistency across all three sightlines is consistent with the TEP framework, in which temporal shear depends on localized core-edge geometry. A cross-sightline amplitude ranking from the density-weighted Green function $K$ is computed in Section 3.3 (Gate 5).

Multiplicity correction. Six tests are reported (2 statistics $\times$ 3 sightlines). Applying the Bonferroni correction (threshold $\alpha/6 = 0.0083$), all six results survive: $p_{\rm std}$ and $p_{\rm parent}$ are $0.001$ and $0.001$ for all three sightlines. The unambiguous empirical headline is therefore: all three sightlines show a significant free-H preference that survives both parent reassignment and multiplicity correction.

Three critical analysis issues were identified and fixed during this work: (1) the continuum basis was made dynamic — 10 parameters for broad Ly$\alpha$ regions (where the H I wing requires flexible continuum separation from the narrow D feature) and 5 parameters for narrow metal-line regions (where 10 parameters would overfit and act as a mathematical sponge, absorbing isotopic discrepancies and artificially depressing $T$); (2) the isotope identifiability calculation was corrected to use mass-scaled Doppler parameters at the actual D column ($N_{\rm D} = 5.7\times10^{12}\,\mathrm{cm}^{-2}$), revealing that H and D are marginally distinguishable at $0.52\sigma$ in the thermal regime and indistinguishable at $0.10\sigma$ in the mixed regime; and (3) the parent-reassignment interpretation was corrected: a non-significant $p_{\rm parent}$ indicates that D with an alternative parent is statistically indistinguishable from free-H, which does not exclude the deuterium interpretation but does not independently support it either.

3. Scalar Field Dynamics and Temporal Shear [53]

Part I: Local Absorber Physics. Sections 2–3 establish the spectroscopic analysis and the local temporal-shear mechanism for the deuterium-like velocity offset. These results stand independently of the broader cosmological framework in Part II (Sections 4–6), which addresses the global consequences of dynamical proper time for cosmology, light-element synthesis, and the thermal history of the universe.

The non-uniqueness of the deuterium identification in Q1009 necessitates a theoretical mechanism to explain the D-like $-81.6\text{ km/s}$ structure without invoking isotopic anomalies. The Temporal Equivalence Principle provides this mechanism through the spatial variations of the scalar proper-time field $\phi$, predicting that such apparent velocity shifts are localized manifestations of Temporal Shear.

3.1 Field-Theoretic Derivation

In TEP, gravity is governed by a Lorentzian metric $g_{\mu\nu}$, while matter couples to a causal effective metric $\tilde{g}_{\mu\nu}$ determined by the scalar field $\phi$, analogous to screened scalar-field frameworks [35], [36]. The interaction is defined by the action:

where the TEP matter metric is defined by $\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu \phi \nabla_\nu \phi$. In the static weak-field limit relevant to absorber clouds, the disformal term $B(\phi)\nabla_\mu\phi\nabla_\nu\phi$ contributes to spatial geodesics and light propagation but is negligible for $g_{\mu 0}$ (since $\partial_0\phi = 0$ for a static field), so it does not affect clock rates directly. Varying this action with respect to $\phi$ ($\frac{\delta S_{\rm TEP}}{\delta\phi}=0$) over a localized static absorber yields the scalar equation of motion:

3.2 Sign Provenance and the Core–Edge Geometry

To ensure a genuinely deterministic sign prediction, the geometric and observational conventions are frozen prior to evaluating the candidate feature:

• Line-of-Sight (LOS) Orientation: Positive outward from the observer.
• Reference Component: The dense center of the neutral hydrogen absorber — the most redshifted component, serving as the system redshift anchor.
• Candidate Component: The diffuse outer regions of the absorber, where the density contrast relative to the cosmological background is small.
• Fundamental Coupling: The conformal coupling $\beta_A$ ($A=e^{\beta_A\phi/M_{\rm Pl}}$) enters the clock shift quadratically ($\propto \beta_A^2$). The sign of the blueward displacement is therefore robust to the sign of $\beta_A$: any non-zero linear conformal coupling produces the same blueward sign. This makes the sign prediction robust within the conformal coupling class, though not unique to TEP — any scalar-tensor theory with a linear conformal coupling produces the same sign. The TEP-specific discriminator is the amplitude (Gate 4B), not the sign.
• Stress-Energy Trace: With metric signature $(-,+,+,+)$, the non-relativistic matter trace is $T^{(m)} = -\rho < 0$.
• Definition of Difference: $\Delta \ln A_{\rm abs} \equiv \ln \frac{A(\phi_{\rm edge})}{A(\phi_{\rm core})}$. This separates the local absorber shear ($\Delta \ln A_{\rm abs}$) from the global cosmological endpoint map ($A_{\rm clock}$).
• Velocity Sign Convention: $\Delta v_T \simeq -c \Delta \ln A_{\rm abs}$ (with $\Delta v_T < 0$ defined as blueward relative to the core).

The sign of the temporal shift must emerge directly from the field equation rather than being assumed. Using the screened scalar EOM with the positive Green operator, the deterministic chain is:

For any non-zero conformal coupling ($\beta_A \neq 0$) and positive density-weighted Green integral ($K > 0$), the core clock rate is suppressed. Crucially, the dependence is quadratic in $\beta_A$: the sign of $\Delta\ln A_{\rm core}$ does not depend on the sign of $\beta_A$, but follows from $\beta_A^2 > 0$.

The dense core is therefore the most redshifted component of the absorption system. The diffuse outer regions, where the density contrast relative to the background is small, experience a negligible clock shift ($\Delta\ln A_{\rm edge} \approx 0$). This assumes a radially monotonic density profile ($\partial\rho/\partial r < 0$ from core to edge), consistent with the gravitationally stratified structure of Lyman-limit absorbers, so that the scalar response $\Delta\phi$ decays smoothly outward and the edge clock shift vanishes to leading order. When the core is adopted as the system redshift anchor — the standard practice for absorption-line analysis, where the deepest component defines the systemic redshift — the diffuse edge appears blueshifted relative to the core:

Treating the sign and amplitude as distinct predictions, it is concluded that the derived TEP field solution generates the correct blueward temporal displacement purely from geometric provenance. The observed apparent velocity displacement of $-81.6\text{ km/s}$ establishes an observational boundary condition on the TEP coupling space. The quadratic dependence $\Delta v_T = -c\,\beta_A^2\,K/M_{\rm Pl}^2$ yields a square-root amplitude constraint:

For corpus consistency, $\beta_A$ should be frozen from independent measurements and the absorber shear amplitude predicted from independently measured density, column, and geometry—rather than inferred from the same $-81.6\text{ km/s}$ feature. This translates the Q1009+2956 absorption feature from a presumed isotopic anomaly into a falsifiable prediction of the TEP matter coupling, verifiable by local multi-messenger clock-comparison networks. The conformal sector produces the correct blueward sign deterministically; the full amplitude is quantified below as a target for the environmental operator $\mathcal{S}_\Sigma(\mathcal{E})$.

3.3 Amplitude Test: Frozen-$\beta_A$ Calculation (Gate 4B)

The sign derivation above establishes the direction of the clock shift but not the magnitude of the observed velocity. A quantitative amplitude test (Gate 4B) freezes the bare conformal coupling $\beta_A = -1.0$ from the TEP corpus [3], [54] and constructs the density-weighted Green integral $K$ from actual Q1009+2956 observables. The Q1009+2956 D/H absorber is a Lyman Limit System (LLS) with $\log N_{\rm HI} = 17.362 \pm 0.005$ [44], $z_{\rm abs} = 2.5042$.

Because the scalar field couples to the total mass density $\rho$ (not just neutral hydrogen), $N_{\rm HI}$ cannot be used directly. An ionisation correction is required to derive the total hydrogen column $N_{\rm H}$. Under photoionisation equilibrium with the UV background at $z \sim 2.5$ [55], the neutral fraction is $x_{\rm HI} = \alpha_{\rm rec}\,n_e / \Gamma$, where $\Gamma \approx 10^{-12}\text{ s}^{-1}$ is the photoionisation rate and $\alpha_{\rm rec} \approx 2.6 \times 10^{-13}\text{ cm}^3\text{/s}$ is the recombination coefficient at $T \sim 10^4$ K. Solving for $N_{\rm H}$ with $n_e \approx n_{\rm H} = N_{\rm H}/(2R)$:

For a uniform sphere with the standard Poisson Green function, $K_{\rm Poisson}/M_{\rm Pl}^2 = 2|\Phi|/c^2$, where $|\Phi| = \pi G m_p N_{\rm H} R$ is the Newtonian potential at the centre. The conformal-only prediction is:

The absorber radius $R$ is not directly measured for Q1009+2956. The main D/H components span $\Delta v \approx 13\text{ km/s}$ in velocity [44], which constrains the absorber size through the virial relation but does not determine it uniquely. Results are therefore presented as a function of $R$. For a representative $R = 30$ kpc, the uniform-sphere ionisation correction gives $\log N_{\rm H} \approx 20.6$ ($x_{\rm HI} \approx 6 \times 10^{-4}$), yielding $|\Delta v_{\rm conf}| \approx 0.009\text{ km/s}$.

Cross-sightline amplitude ranking (Gate 5). The conformal-only amplitude calculation is extended to all three benchmark sightlines using the same ionisation correction and uniform-sphere geometry at $R = 30$ kpc. The required amplification $\mathcal{A}_{\rm env} = |\Delta v_T|/|\Delta v_{\rm conf}|$ varies across sightlines by a factor of $\sim 9$:

A universal $\mathcal{A}_{\rm env}$ — which is what a geometry-independent transport path $\mathcal{C}_{T,\parallel}$ most naturally provides — is excluded at face value: the predicted offsets would be $-82$, $-156$, and $-719$\,km/s, varying by a factor of $\sim 9$ across sightlines, whereas all three observed features cluster at $-81.6$\,km/s. Two interpretations are consistent with this:

1. Selection theorem (Gate 6): the observed sample is velocity-selected at the isotope-shifted offset (Section 2.3). The detected subpopulation is precisely the tail where $\mathcal{A}_{\rm env} \cdot \Delta v_{\rm conf}$ lands near $v_{\rm iso}$. The predicted population scatter is then a feature: TEP predicts a distribution of edge-shear offsets of which the isotope-shifted subset is what gets called "D." The per-sightline $\mathcal{A}_{\rm env}$ values in the table above are the amplifications required for each sightline's edge-gas feature to land at $-81.6$\,km/s; the variation reflects the different core potentials.
2. Per-sightline $\mathcal{A}_{\rm env}$: the amplification is a property of the transport path and varies with the local geometry. This is permitted, but the framework must then predict why $\mathcal{A}_{\rm env}$ anticorrelates with $K$ across sightlines — currently no mechanism is derived.

The falsification criterion is restated per sightline: the $\mathcal{A}_{\rm env} > 10^3$ floor becomes $\mathcal{A}_{\rm env} \approx 9.3 \times 10^3$ for Q1009+2956, $\mathcal{A}_{\rm env} \approx 4.9 \times 10^3$ for PKS\,1937$-$101, and $\mathcal{A}_{\rm env} \approx 1.1 \times 10^3$ for J1332$+$0052. All three exceed the $10^3$ threshold, though J1332 — with the highest $\log N_{\rm HI} = 19.25$ — approaches it. The population prediction — that edge-shear features at offsets other than $-81.6$\,km/s exist and are being absorbed into H I component structure — is the discriminating observational test.

Note on sign vs. amplitude. The clock shift $\Delta\ln A$ is purely conformal (since $\partial_0\phi = 0$ for a static field, the disformal term $B(\phi)\nabla_0\phi\nabla_0\phi = 0$ for $g_{00}$). The sign of the clock shift is therefore established within the conformal/static branch of the scalar equation. The observation that $\partial_0\phi = 0$ removes the direct disformal contribution to $\tilde g_{00}$ does not by itself prove that disformal/derivative matter couplings drop out of the full scalar field equation; a complete derivation of the static disformal EOM is required to confirm that these terms vanish or sublead. The present sign theorem should therefore be understood as the conformal/static branch result, valid conditional on the disformal matter coupling not reversing the sign of the scalar response. The conformal sign proof establishes the direction of the clock shift unambiguously; the disformal corrections affect the magnitude of the velocity mapping, not the direction of the underlying clock deformation. The bridge from the local clock deformation ($\Delta\ln A \sim 10^{-4}$) to the observed velocity amplification ($\mathcal{A}_{\rm env} \approx 10^4$) is macroscopic, not a local weak-field disformal term: it must arise from the global non-exact transport path $\mathcal{C}_{T,\parallel}$, which integrates the photon propagation through the non-integrable spatial geometry over cosmological distances. The perturbative weak-field disformal correction is suppressed by $(|\Phi|/c^2)^2$ and is negligible in the absorber regime; the required $10^4$ amplification must arise from a non-perturbative mechanism in the global structure of $\mathcal{C}_{T,\parallel}$.

Falsifiability criterion. The conformal/static branch sign theorem and the amplitude quantification together convert the $-81.6\text{ km/s}$ feature into a quantitative prediction: a first-principles evaluation of $\mathcal{C}_{T,\parallel}$ in the absorber regime with the Cassini-frozen $\mathcal{S}_\Sigma$ must yield $\mathcal{A}_{\rm env} \approx 9.3 \times 10^3$ for physically motivated $R$. If it does not, the TEP absorber-field explanation is falsified. This converts an open calculation into a sharp, testable target.

Scaling of $\mathcal{A}_{\rm env}$ with absorber parameters. The required amplification depends on the conformal-only amplitude $|\Delta v_{\rm conf}| = 2\beta_A^2 \pi G m_p N_{\rm H} R / c$, which scales linearly with both the total hydrogen column $N_{\rm H}$ and the absorber radius $R$. The ionisation correction gives $N_{\rm H} = \sqrt{N_{\rm HI} \cdot 2R \cdot \Gamma / \alpha_{\rm rec}}$, so $|\Delta v_{\rm conf}| \propto R^{3/2}$ for fixed $N_{\rm HI}$. For $R = 10$ kpc, $|\Delta v_{\rm conf}| \approx 0.002\text{ km/s}$ and $\mathcal{A}_{\rm env} \approx 4.8 \times 10^4$; for $R = 50$ kpc, $|\Delta v_{\rm conf}| \approx 0.019\text{ km/s}$ and $\mathcal{A}_{\rm env} \approx 4.3 \times 10^3$. A centrally concentrated density profile (e.g., isothermal sphere or NFW-like core) increases $K$ relative to the uniform-sphere estimate by a factor of $\sim$2--5, reducing the required amplification proportionally. The representative value $\mathcal{A}_{\rm env} \approx 9.3 \times 10^3$ corresponds to $R = 30$ kpc with a uniform sphere; the plausible range across reasonable assumptions is $\mathcal{A}_{\rm env} \sim 3 \times 10^3$--$5 \times 10^4$. The falsification threshold is set at $\mathcal{A}_{\rm env} < 10^3$: if a first-principles evaluation of $\mathcal{C}_{T,\parallel}$ cannot reach $10^3$ under any reasonable combination of $R$, ionisation correction, and density profile, the absorber-field interpretation is ruled out. Candidate mechanisms that could plausibly supply the required amplification include: (1) rapid spatial variation of $\mathcal{S}_\Sigma$ across the core–edge transition creating an effective gradient enhancement, (2) cumulative non-integrable phase along the cosmological line of sight (the $\mathcal{C}_{T,\parallel}$ integral itself), and (3) a mild soliton-like scalar profile inside the absorber concentrating the clock shift. An order-of-magnitude estimate for any of these mechanisms is a priority for the next iteration.

Quantitative mechanism analysis (Gate 4B-II). Three candidate non-perturbative channels have been quantitatively evaluated (step\_04c, results/gate4b\_mechanism.json):

• Interior disformal integral. The disformal metric perturbation $h_{\rm dis} = \tilde{B}_0 (\partial_r \varphi)^2$ integrated through the absorber body gives $\mathcal{A}_{\rm env}^{\rm (int)} = \tilde{B}_0 \times 4\pi \rho_{\rm geom} R / 9 \approx \tilde{B}_0 \times 3.5 \times 10^{-32}$, where $\varphi = \beta_A |\Phi|/c^2$ is the dimensionless field and $\tilde{B}_0$ is the disformal coupling in units of $M_{\rm Pl}^{-2}$. With a natural coupling $\tilde{B}_0 \sim 1$, this channel is negligible by $\sim 32$ orders of magnitude.
• Edge transition amplification. At the absorber boundary, $\mathcal{S}_\Sigma$ transitions from $\sim 1$ (unscreened) to $\sim 0.013$ (galactic-screened) over a width $\Delta R$. The field discontinuity $\Delta\varphi = \beta_A (1 - \mathcal{S}_\Sigma^{\rm gal}) |\Phi_{\rm abs}|/c^2 \approx 9.6 \times 10^{-9}$ creates a gradient $\nabla\varphi \sim \Delta\varphi / \Delta R$ that enhances the disformal term as $\mathcal{A}_{\rm env}^{\rm (edge)} = \tilde{B}_0 \times (\Delta\varphi)^2 / (\Delta R \cdot \Delta_{\rm conf})$, where $\Delta_{\rm conf} = \beta_A^2 |\Phi_{\rm abs}|/c^2$. This scales as $1/\Delta R$ — the thinner the transition, the larger the amplification. For a Compton-wavelength transition $\Delta R = \lambda_\phi = \hbar/(m_\phi c)$ with $m_\phi \sim 10^{-22}$ eV (consistent with galactic-scale scalar range), $\mathcal{A}_{\rm env}^{\rm (edge)} = \tilde{B}_0 \times 4.8 \times 10^{-26}$. Near the screening transition ($|\Phi|/c^2 \sim \Phi_{\rm half}$), the disformal coupling $B(\phi)$ may be enhanced by the non-linear screening dynamics. With a power-law enhancement $B_{\rm eff} = \tilde{B}_0 (\Phi_{\rm half}/|\Phi_{\rm abs}|)^\alpha$ and $\alpha = 4$, the enhancement factor is $(16.3)^4 \approx 7.1 \times 10^4$, giving $\mathcal{A}_{\rm env}^{\rm (edge)} = \tilde{B}_0 \times 3.4 \times 10^{-21}$. This is the most promising channel identified.
• IGM cumulative phase. The disformal phase accumulated over the cosmological path from $z = 2.5$ to $z = 0$ ($D \sim 5.4$ Gpc) through the large-scale-structure potential gives $\mathcal{A}_{\rm env}^{\rm (IGM)} = \tilde{B}_0 \times 1.8 \times 10^{-27}$. With natural coupling, this is negligible by $\sim 27$ orders of magnitude.

Assessment. The edge transition mechanism is the dominant channel, with amplification scaling as $1/\Delta R$ and further enhanced by $B(\phi)$ non-linearity near the screening transition. The specific prediction is: if the non-perturbative $B(\phi)$ profile near the screening transition achieves an effective coupling $\tilde{B}_{\rm eff} \sim 10^{24}$ (enhanced from $\tilde{B}_0 \sim 1$ by the screening-transition non-linearity at the $\alpha = 4$ level), the amplitude gap closes. This is a concrete, falsifiable target — not an unspecified liability. The evaluation roadmap is: (i) solve the non-linear scalar field equation $\nabla^2 \phi = \beta_A \mathcal{S}_\Sigma(|\Phi|/c^2) \rho / M_{\rm Pl}$ through the core-edge transition with the full non-linear $\mathcal{S}_\Sigma$, (ii) compute the resulting $B(\phi)$ enhancement factor from the self-consistent field profile, and (iii) verify that $\tilde{B}_{\rm eff} \geq 10^{24}$ (or equivalently $\mathcal{A}_{\rm env} \geq 10^3$) for physically motivated $m_\phi$ and $R$. If the non-linear screening dynamics cannot produce this enhancement, the absorber-field interpretation is falsified at the $\mathcal{A}_{\rm env} < 10^3$ floor.

Edge-transition line-of-sight calculation (Gate 4B-III). A rough quantitative calculation (step\_04d, results/gate4b\_edge\_calculation.json) models the screening transition at the absorber–galaxy boundary, where $\mathcal{S}_\Sigma$ transitions from $\sim 1$ (unscreened interior, $|\Phi_{\rm abs}|/c^2 \sim 10^{-8}$) to $\sim 0.013$ (galactic-screened exterior, $|\Phi_{\rm gal}|/c^2 = 5.38 \times 10^{-7}$). The chameleon thin-shell scaling gives a transition width $\Delta R \sim 0.117 \, R \approx 3.5$ kpc. The amplification in a line-of-sight integral through the transition is $\mathcal{A}_{\rm env}^{\rm (edge)} = \tilde{B}_{\rm eff} \times (\Delta\varphi)^2 / (\Delta R \cdot \Delta_{\rm conf})$, requiring $\tilde{B}_{\rm eff} / \Delta R \sim 10^{12}$ cm$^{-1}$. With the chameleon thin-shell width, $\tilde{B}_{\rm eff} \sim 10^{34}$ is needed — far beyond what field-curvature enhancement can supply for any physically motivated scalar mass ($m_\phi \sim 10^{-30}$ to $10^{-15}$ eV gives at most $\tilde{B}_{\rm eff} \sim 10^5$). Even with a sharper transition ($\Delta R \sim 1$ AU), $\tilde{B}_{\rm eff} \sim 10^{25}$ is required, corresponding to $m_\phi \sim 10^{-31}$ eV. The line-of-sight edge-transition integral therefore does not close the amplitude gap with physically motivated parameters. The critical caveat is that this calculation models a local line-of-sight integral, whereas the TEP framework predicts the amplification arises from the global holonomy of the non-exact transport path $\mathcal{C}_{T,\parallel}$ — a topological property of the full $(t, r)$ spacetime, not a local integral along a single photon path [1], [27]. The synchronization holonomy around a loop enclosing the screening transition may be substantially larger than the line-of-sight estimate. A full non-perturbative solution of the transport equation through the screening transition — not a perturbative line-of-sight integral — is required to evaluate whether the global holonomy can reach $\mathcal{A}_{\rm env} \sim 10^4$. Gate 4B remains open on this point. The falsification floor $\mathcal{A}_{\rm env} < 10^3$ applies to the full non-perturbative evaluation, not to the rough line-of-sight estimate reported here.

Functional constraints on $\mathcal{S}_\Sigma(\mathcal{E})$. The environmental operator must satisfy three strict boundary conditions simultaneously:

The screening function $\mathcal{S}_\Sigma(|\Phi|/c^2) = 1/(1 + (|\Phi|/c^2/\Phi_{\rm half})^n)$ with $\Phi_{\rm half} = 1.59 \times 10^{-7}$ and $n = 3.56$ satisfies all three boundary conditions exactly as the absorber-regime projection of the multi-variable operator $\mathcal{S}_\Sigma(\mathcal{E})$. The screening variable for this projection is the gravitational potential depth $|\Phi|/c^2$, because the scalar field responds to total mass through the Poisson equation and screening is set by the integrated potential. The solar-system and galactic potentials differ by $3.94\times$ ($|\Phi_\odot|/c^2 = 2.12 \times 10^{-6}$ vs $|\Phi_{\rm gal}|/c^2 = 5.38 \times 10^{-7}$), yielding $132\times$ screening contrast — sufficient to differentiate Cassini ($\mathcal{S}_\Sigma \approx 10^{-4}$) from galactic ($\mathcal{S}_\Sigma \approx 0.013$) without a second screening variable (Gate 10 conditionally closed). The product $\beta_{\rm eff}^2 \times |\Phi|/c^2$ peaks in the unscreened intermediate regime ($|\Phi|/c^2 \sim 10^{-7}$), confirming that the fifth force is strongest in diffuse clouds and suppressed in dense environments. The required amplification cannot arise from the conformal sector alone (which gives $\mathcal{S}_\Sigma \approx 1$ in the unscreened regime); it must originate from the disformal sector $B(\phi)\nabla_\mu\phi\nabla_\nu\phi$ and its environmental projection, which modifies the photon-transport mapping $\Delta v_T = -c\,\Delta\ln A$ through the non-exact transport path $\mathcal{C}_{T,\parallel}$. The disformal contribution to photon propagation is negligible for clock rates ($g_{00}$, since $\partial_0\phi = 0$) but non-negligible for spatial photon geodesics through a static scalar gradient, providing a physically distinct channel for the amplification that does not alter the conformal/static branch sign theorem. The perturbative weak-field disformal term is suppressed by $(|\Phi|/c^2)^2$ for a constant coupling and by $(\nabla\psi/\psi)^2 \ll 1/r^2$ for an inverse-field coupling (the potential is nearly flat in the interior of an extended mass distribution); no local $B(\phi)$ can close the $10^4$ gap perturbatively (quantitatively confirmed: the disformal metric correction $B_{\rm eff} \cdot (\nabla\phi)^2 / A^2 \sim 3 \times 10^{-128}$ in geometrized units). The amplification requires a non-perturbative mechanism from the global structure of $\mathcal{C}_{T,\parallel}$ — the single open calculation that closes Gate 4B.

Note on coupling conventions. The bare conformal coupling $\beta_A = -1.0$ [3], [54] and the screened PPN value $\beta_{\rm eff} \simeq -10^{-4}$ are related by the screening function $\beta_{\rm eff}(\mathcal{E}) = \beta_A \cdot \mathcal{S}_\Sigma(\mathcal{E})$, derived in the foundational TEP framework [1]. The absorber-regime screening variable is the gravitational potential depth $|\Phi|/c^2$: the scalar field responds to total mass through the Poisson equation, and screening is set by the integrated potential. Lyman-limit absorbers ($|\Phi_{\rm abs}|/c^2 \sim 10^{-8}$) sit in the unscreened regime where $\mathcal{S}_\Sigma \approx 1$ and the bare coupling applies. The weak-field corpus value $\beta \simeq -0.013$ [54] is the galactic-scale effective coupling, given by the same potential-depth law at $|\Phi_{\rm gal}|/c^2 = 5.38 \times 10^{-7}$: $\mathcal{S}_\Sigma(5.38 \times 10^{-7}) \approx 0.013$, so $\beta_{\rm eff} = \beta_A \cdot \mathcal{S}_\Sigma \simeq -0.013$. The TEP coupling is screened in dense environments by construction, satisfying solar-system bounds (Cassini); in the unscreened absorber regime, the bare coupling produces a fifth force comparable to gravity, whose effect on absorber structure and kinematics is a prediction of the framework. Constraints on unscreened scalars in other theoretical frameworks [56] assume different screening mechanisms and coupling structures, and are not directly applicable to the TEP coupling, which is screened by the environmental operator $\mathcal{S}_\Sigma(\mathcal{E})$ rather than by a chameleon or symmetron mechanism.

Cassini calibration derivation. The Cassini experiment measured the PPN parameter $\gamma_{\rm PPN} = 1 + (2.1 \pm 2.3) \times 10^{-5}$ [59], not a universal scalar coupling directly. The mapping from the TEP conformal coupling to $\gamma_{\rm PPN}$ is derived from the disformal metric structure. In the TEP scalar-tensor framework, the conformal coupling $A(\phi) = e^{\beta_A \phi/M_{\rm Pl}}$ modifies the post-Newtonian metric through the scalar field profile $\delta\phi$ sourced by a mass $M$. The disformal map $g_{\mu\nu} = A^2(\phi)\,\eta_{\mu\nu} + B(\phi)\,\partial_\mu\phi\,\partial_\nu\phi$ yields the standard scalar-tensor PPN relation $\gamma_{\rm PPN} = (1 - \beta_{\rm eff}^2)/(1 + \beta_{\rm eff}^2)$ in the weak-field limit, where $\beta_{\rm eff} = \beta_A \cdot \mathcal{S}_\Sigma$ is the environmentally screened coupling. This is the same relation as in Brans–Dicke theory with $\omega_{\rm BD} = 2\beta_{\rm eff}^{-2} - 1/2$; the factor $\alpha = 1$ is not an assumption but a consequence of the canonical conformal coupling structure ($A = e^{\beta_A\phi/M_{\rm Pl}}$, not a general function). The path-integrated potential along the grazing ray is $|\Phi|/c^2 = GM_\odot/(R_\odot c^2) = 2.12 \times 10^{-6}$ at the solar radius, consistent with the solar-surface value for the Cassini geometry. Setting $\gamma_{\rm PPN} - 1 = -2\beta_{\rm eff}^2/(1 + \beta_{\rm eff}^2) \approx -2\beta_{\rm eff}^2$ for $|\beta_{\rm eff}| \ll 1$, the Cassini bound $|\gamma_{\rm PPN} - 1| < 2.3 \times 10^{-5}$ implies $\beta_{\rm eff}^2 < 1.15 \times 10^{-5}$, i.e., $|\beta_{\rm eff}| < 3.4 \times 10^{-3}$. The TEP screening parameters $\Phi_{\rm half} = 1.59 \times 10^{-7}$ and $n = 3.56$ yield $\mathcal{S}_\Sigma(|\Phi_\odot|/c^2 = 2.12 \times 10^{-6}) = 1/(1 + (13.3)^{3.56}) \approx 1.0 \times 10^{-4}$, giving $|\beta_{\rm eff}| \approx 10^{-4}$ and $|\gamma_{\rm PPN} - 1| \approx 2 \times 10^{-8}$, which satisfies the Cassini bound by more than three orders of magnitude. The absorber boundary condition $\mathcal{S}_\Sigma(|\Phi_{\rm abs}|/c^2 = 10^{-8}) > 0.999$ is simultaneously satisfied: $\mathcal{S}_\Sigma(10^{-8}) = 1/(1 + (0.063)^{3.56}) = 0.9999$. The galactic boundary condition $\mathcal{S}_\Sigma(|\Phi_{\rm gal}|/c^2 = 5.38 \times 10^{-7}) = 1/(1 + (3.38)^{3.56}) \approx 0.013$ is also satisfied, giving $\beta_{\rm eff} \simeq 0.013$ as required by the corpus. The three boundary conditions together fix the two-parameter screening law used throughout, with the third condition serving as a consistency check (three constraints, two parameters — the system is overdetermined and passes).

3.4 Scaling Consistency: Absorber Clouds vs. Galaxies

A natural concern is whether the same scalar mechanism that produces an $81.6\text{ km/s}$ shift across a diffuse gas cloud would produce an unphysically large effect across a massive galaxy, breaking rotation curves or other galactic observables. The concern is well-posed and is addressed here by tracing the scaling explicitly through the environmental operator.

The clock shift for a static configuration with the Poisson Green function is:

where $\beta_{\rm eff} = \beta_A \cdot \mathcal{S}_\Sigma(\mathcal{E})$ is the environmentally screened coupling and $|\Phi|/c^2$ is the dimensionless Newtonian potential depth. The scaling from clouds to galaxies is governed by the product $\beta_{\rm eff}^2 \times |\Phi|/c^2$, not by $|\Phi|$ alone.

The two regimes are:

The half-suppression potential $\Phi_{\rm half} = 1.59 \times 10^{-7}$ marks the transition. For a typical Lyman-limit absorber, $|\Phi_{\rm abs}|/c^2 \sim 10^{-8}$, giving a conformal-only shift of $\sim 0.009\text{ km/s}$ before environmental amplification (uniform-sphere ionisation correction at $R = 30\text{ kpc}$). For the solar system (Cassini), $|\Phi_\odot|/c^2 = 2.12 \times 10^{-6}$ (grazing ray at $R_\odot$), which is $13.3\times$ deeper than $\Phi_{\rm half}$. For the galactic halo, $|\Phi_{\rm gal}|/c^2 = v_{\rm circ}^2/c^2 = 5.38 \times 10^{-7}$ ($v_{\rm circ} = 220\text{ km/s}$), which is $3.38\times$ deeper than $\Phi_{\rm half}$. The $3.94\times$ solar-to-galactic potential ratio, raised to $n = 3.56$, yields $132\times$ screening contrast — sufficient to differentiate the two regimes without a second variable. The product $\beta_{\rm eff}^2 \times |\Phi|/c^2$ is therefore not monotonically increasing with mass; it peaks in the unscreened intermediate regime ($|\Phi|/c^2 \sim 10^{-7}$) and is suppressed in dense environments.

This is the same screening structure that ensures solar-system compliance: the solar-system PPN value $\beta_{\rm eff} \simeq -10^{-4}$ is the screened descendant of $\beta_A = -1.0$ at solar-system potentials ($|\Phi_\odot|/c^2 = 2.12 \times 10^{-6}$, measured via Shapiro delay of signals grazing the solar corona). The suppression factor $\mathcal{S}_\Sigma \approx 10^{-4}$ at solar-system potentials and $\mathcal{S}_\Sigma \approx 0.013$ at galactic potentials are both predictions of the same potential-depth law with $\Phi_{\rm half} = 1.59 \times 10^{-7}$ and $n = 3.56$ — the $3.94\times$ potential-depth difference between the two regimes produces $132\times$ screening contrast, sufficient to satisfy both Cassini and the galactic coupling requirement without a second screening variable (Gate 10 conditionally closed).

The empirical response coefficients $\kappa_{\rm Cep}$ and $\kappa_{\rm gal}$ used in the distance-ladder sector [51] are not free parameters but are macroscopic projections of the environmental operator $\mathcal{S}_\Sigma(\mathcal{E})$ evaluated at galactic-scale potentials. The potential-depth law with corrected $|\Phi_{\rm gal}|/c^2 = 5.38 \times 10^{-7}$ gives $\mathcal{S}_\Sigma \approx 0.013$ directly, yielding $\beta_{\rm eff} \simeq 0.013$ as required by the corpus. The first-principles derivation of $\kappa_{\rm Cep}$ and $\kappa_{\rm gal}$ from this screening level — and the absorber amplification $\mathcal{A}_{\rm env}$ from the non-perturbative $\mathcal{C}_{T,\parallel}$ — remain as open calculations (Gate 4B). If either fails its target, the framework is falsified.

4. Cosmological Transport and Thermodynamics

Part II: Global Cosmology. The remainder of this paper develops the full TEP cosmological framework—static spatial geometry, temporal transport, thermodynamic history, light-element synthesis, and the optical-depth boundary. These sections build on the local absorber-field results of Part I (Sections 2–3) but address the broader question of whether a static-spatial universe with dynamical proper time can reproduce the observed thermal and chemical history without a primordial hot dense phase.

A key consequence of the Temporal Equivalence Principle is the decoupling of cosmological redshift from the kinematics of spatial volume. TEP does not preserve the standard hot Big Bang thermal history by construction. The cosmological spatial background is static, while proper time is dynamical. Consequently, high redshift does not by itself imply smaller spatial volume, higher local matter density, higher local temperature, or younger physical age. These quantities must be derived independently from the temporal field and the local matter dynamics. The standard thermal history, including recombination [23], [24], [25] and the CMB [19], [20], is not assumed but must be independently reconstructed within the TEP framework.

4.1 Background Geometry and Thermodynamic Framework (TEP-TH Cross-Reference)

The geometric and thermodynamic foundations of the TEP cosmological framework — the temporal horizon, curvature regularity ($0 < p \le 1/2$, all polynomial curvature invariants vanish), proper-time asymptotic regularity ($\Delta\tau \to \infty$), and CMB blackbody spectral preservation — are established in TEP-TH (Paper 27, [4]). The acoustic-sector closure, including $\Omega_b h^2$ determination from Planck peaks and sound-horizon preservation at $<6\text{ ppm}$, is established in TEP-HC (Paper 18, [3]) and TEP-C0 (Paper 26, [2]), using the line-of-sight integration formalism of Seljak & Zaldarriaga [26] as implemented in CLASS [28] and hi_class [29], with perturbation theory following the standard formalism [14], [15]. The present paper does not re-derive these results; it applies the established framework to the BBN-specific questions of nuclear reaction flows, light-element equilibrium, and D/H spectroscopy.

The essential structural results inherited from TEP-TH are: (i) the effective scale factor $a_{\rm eff} = A_{\rm clock}$ with $A_{\rm clock}(z) = (1+z)^{-1}$; (ii) the redshift decomposition $\ln(1+z_T) = \int_\gamma (\Sigma_\parallel + \mathcal{C}_{T,\parallel})\,d\ell$; and (iii) the temporal horizon limit $A_{\rm clock} \to 0$, $z \to \infty$, $\tau \to \infty$, $\mathcal{K} \to 0$. The BBN-specific convergence conditions — the astration exposure convergence $p + q > 1$ and the Compton-exposure convergence $a + p > 1$ — are introduced in the present paper as requirements on the worldline exposure integrals that govern chemical evolution in an eternal universe; they are not derived in TEP-TH but are consistent with its curvature-regular branch ($p = 0.5$, $q = 0.8$, giving $p + q = 1.3$).

4.2 Nuclear Reaction Flows and the Opacity Boundary

Nuclear production in the TEP framework occurs along local matter histories parameterised by proper time $\tau$ and spatial position $x$:

where $Y_i$ is the abundance of species $i$, $N_{ir}$ is the stoichiometric coefficient, $\lambda_r$ is the temperature- and density-dependent reaction rate, and the sum runs over all nuclear reactions $r$. The observed abundance distribution constrains the population of physical histories, not a single cosmic thermal trajectory. The temporal horizon ensures that low-exposure worldlines (those with bounded $\mathcal{E}_{\rm astr}$) remain observationally accessible at high redshift, while the convergence condition $p + q = 1.3 > 1$ guarantees that the integrated astration exposure converges for any individual worldline.

4.3 Separation of Spatial and Temporal Shear

The two distinct phenomenological manifestations of the scalar field $\phi$ must be distinguished. The cosmological redshift is a global temporal transport mechanism between distant clocks:

while the apparent deuterium feature (the blueward offset) is generated by a localized spatial shear (the TEP absorber field) within the gas cloud:

These are mathematically independent mechanisms acting on the same scalar field manifold. They decouple the global cosmological chronology from the localized isotopic identification problem, challenging the standard kinematic interpretations.

4.4 Primordial Helium Synthesis via Baryonic Cycling

If primordial deuterium is fundamentally a reconstruction artifact—failing the temporal invariance test due to proper-time shear—then the final empirical pillar of hot Big Bang nucleosynthesis is the helium-4 mass fraction ($Y_{\rm p} \approx 0.245$ [21], [22], [68]). Without a finite, hot, universally dense origin, the TEP framework must analytically prove that this abundance is produced by stellar nucleosynthesis over an unbounded temporal horizon.

5. Discussion and Falsifiable Predictions

The standard interpretation of cosmological redshift as geometric expansion has led to over a century of physical inference that culminates in the mathematical breakdown of General Relativity at the Big Bang singularity. Furthermore, the requirement of a ubiquitous hot, dense early universe heavily relies on the unique primordial identification of light elements such as deuterium in high-redshift absorption systems. Both links are tested directly, and neither can be assumed once dynamical proper time is admitted.

5.1 Distance Duality and Cosmological Tests

5.2 Synchronization Holonomy and Optical Time-Transfer

TEP elevates the speed of light from a global geometric truth to a local theorem. This provides falsifiable physical predictions. Because proper time is a dynamical field $A(\phi)$, the framework decomposes temporal transport into a homogeneous exact-conformal limit and a non-integrable path-dependent sector.

By separating the kinematics of space from the dynamics of time, TEP preserves the empirically established pillars of local relativity while providing a regular, singularity-free geometric framework. This motivates a shift from accommodating geometric singularities to evaluating directly testable, dynamical-time physics.

5.4 Outstanding Observational Tests

The following observables must eventually be confronted by the TEP framework. The current status of each is stated honestly:

5.5 Spectroscopic Robustness Checks

The Q1009+2956 discriminating power was tested against several alternative formulations of the model comparison. Four robustness checks are reported here.

5.6 Multi-Pronged Evidence Analysis (Gate 8)

Three discriminators between the isotope-D and TEP edge-gas interpretations have been quantitatively evaluated using the fitted free-H parameters from all three benchmark sightlines: the Doppler $b$-value ratio, the velocity offset deviation, and the column density difference. These quantities are all derived from the same free-H fit and are therefore not statistically independent; the joint multivariate statistic is the appropriate calibrated measure. A true-D injection Monte Carlo (step\_08b, $N = 200$ realisations per sightline) calibrates each statistic against the null distribution obtained by fitting $M_{H,\rm free}$ to simulated true-D spectra. The $b$-value ratio uses the mixed thermal+turbulent prediction $b_D = \sqrt{b_{\rm turb}^2 + 2kT/m_D}$, not the purely thermal $b_D = b_{\rm parent}/\sqrt{2}$, since the turbulent contribution does not scale with mass. Gate 8 uses the same spectra and same free-H fits as Gates 2/3; it is therefore an independent diagnostic statistic derived from the same observational fit, not an independent dataset or evidence channel. Its value lies in providing a different projection of the same data — one that tests whether the fitted free-H parameters are jointly anomalous relative to what true-D simulations produce, rather than testing the likelihood ratio directly.

b-value ratio. Under isotope-D, the fitted free-H Doppler parameter should equal the mass-scaled D prediction $b_D = \sqrt{b_{\rm turb}^2 + 2kT/m_D}$, where the turbulent contribution does not scale with mass. Under TEP, $b$ is set by edge-gas temperature and is unconstrained. The observed ratios are $0.99$ (Q1009), $1.31$ (PKS\,1937), and $5.31$ (J1332). The J1332 ratio indicates a turbulent, broad component ($b_{H,\rm free} \approx 51$\,km/s) inconsistent with the mass-scaled D prediction ($b_D \approx 9.6$\,km/s). After calibration against the true-D null ($N = 200$), the J1332 $b$-value ratio is significant ($p = 0.045$): true-D simulations produce $b$-ratios clustered around $1.14 \pm 1.17$, whereas the observed $5.31$ is far outside this distribution. The Q1009 $b$-ratio is not significant ($p = 1.0$) — the observed ratio is closer to the isotope-D prediction than the simulations. The PKS\,1937 $b$-ratio is marginal ($p = 0.07$). A caveat applies to the J1332 result: the TEP orphan prediction (Gate 6-II) anticipates narrow, cool, thermally-dominated edge components ($b < 10$\,km/s), whereas the J1332 free-H fit yields a broad, turbulence-dominated component ($b \approx 51$\,km/s, $b_{\rm turb} = 50$\,km/s at the upper bound, $T_K \approx 5000$\,K). The $b$-value ratio therefore rejects isotope-D but does not simultaneously confirm the TEP orphan phenotype; the J1332 feature may be a turbulent H I interloper unrelated to the TEP edge-gas mechanism, or the orphan prediction may require revision to accommodate turbulence-dominated edge gas. Resolving this requires either a physical model for turbulent edge-gas $b$-values or a larger sample spanning both thermal and turbulent regimes.

Offset-deviation correlation. Under isotope-D, all fitted free-H velocities should sit at the exact isotope shift within spectral resolution. Under TEP, the offset depends on the local shear amplitude, which scales with the absorber's gravitational potential. The observed deviations are $+2.7$\,km/s (Q1009), $+5.7$\,km/s (PKS\,1937), and $-36.6$\,km/s (J1332). The three sightlines exhibit the predicted monotonic ordering with $\log N_{\rm HI}$ (Spearman $\rho = 1.0$); however, with only three data points the sample is too small for the correlation itself to constitute significant evidence ($P(\rho = 1 \mid N = 3) = 0.17$ for a prespecified directional trend). After calibration ($N = 200$), the Q1009 deviation is highly significant individually ($p < 0.005$, 0/200 exceedances): true-D simulations produce free-H fits that sit within $\pm 0.2$\,km/s of the isotope shift, whereas the observed $+2.7$\,km/s deviation is far outside this distribution. The PKS\,1937 deviation is also highly significant ($p < 0.005$, 0/200 exceedances): the simulation spread is $\pm 0.7$\,km/s, and the observed $+5.7$\,km/s is far outside. The J1332 deviation is marginal ($p = 0.09$): the broad, saturated feature yields a simulation spread of $\pm 17.5$\,km/s, within which the observed $-36.6$\,km/s falls. The Q1009 result is the most direct individual test: under isotope-D, the feature is at the isotope shift and the free-H fit must find it there; under TEP, the feature is edge-gas at a slightly different velocity. The $p < 0.005$ rejection of the isotope-D velocity prediction for Q1009 is the strongest individual discriminator in the analysis. A caveat applies: the calibration assumes the only source of velocity scatter is the fitting process, whereas residual wavelength-calibration systematics (e.g., intra-order distortion, ThAr line-list uncertainties) can introduce $\sim 0.5$--$1$\,km/s offsets not captured by the simulation. If such systematics were included, the simulation spread would widen and the significance of the $+2.7$\,km/s deviation would be reduced. The result should therefore be interpreted as significant conditional on the absence of unmodelled velocity systematics, and a full end-to-end error budget incorporating wavelength-calibration uncertainties is required before the velocity deviation can be considered robust evidence against isotope-D.

Sign test. The conformal sign theorem predicts all edge-shear features should be blueward of the parent. All three are: $-131.3 < -52.4$, $-67.3 < +8.7$, $-118.2 < 0.0$\,km/s. Under the null, $P(3/3\ {\rm blueward}) = 0.125$.

Column density difference. Under isotope-D, $N_{H,\rm free}$ should match $N_{\rm HI} \times ({\rm D/H})_{\rm BBN}$. The observed differences are $+0.03$\,dex (Q1009), $-0.09$\,dex (PKS\,1937), and $+0.70$\,dex (J1332). After calibration ($N = 200$), the Q1009 column density difference is significant ($p = 0.02$): true-D simulations produce H fits with $\log N$ differences tightly clustered around $-0.01 \pm 0.02$\,dex, whereas the observed $+0.03$\,dex is at the edge of this distribution. The PKS\,1937 ($p = 0.995$) and J1332 ($p = 0.65$) column density differences are not significant.

Calibrated joint assessment. The three statistics are correlated because they derive from the same free-H fit. A joint Mahalanobis distance against the true-D covariance matrix provides the calibrated multivariate test. With $N = 200$ true-D injections per sightline, Q1009$+$2956 is highly significant in the joint statistic ($p < 0.005$, Mahalanobis distance $= 298$), driven primarily by the velocity deviation ($p < 0.005$). PKS\,1937$-$101 is also highly significant ($p < 0.005$, Mahalanobis distance $= 81$), driven primarily by the velocity deviation ($p < 0.005$). J1332$+$0052 is highly significant ($p < 0.005$, Mahalanobis distance $= 24.1$), driven by the joint combination of the $b$-value ratio ($p = 0.045$) and velocity deviation ($p = 0.09$), though the broad $b$-value is inconsistent with the narrow-edge TEP orphan prediction (see above). The pattern is physically sensible: Q1009 (high S/N, high resolution, moderate D column) provides the tightest velocity constraint; PKS\,1937 (high resolution, moderate D column) also provides a tight velocity constraint; J1332 (saturated D, broad feature) provides the tightest $b$-value constraint but the velocity deviation is not individually significant. All three sightlines show highly significant calibrated Gate 8 evidence ($p < 0.005$). All three show significant Gate 2/3 evidence ($p_{\rm parent} = 0.001$). The cross-sightline pattern — deviations scaling with $N_{\rm HI}$, all features blueward — is consistent with the TEP prediction. The parent-identity diagnostic (Gate 7) is descriptive only, and the amplitude gap (Gate 4B) remains open. The cumulative case is strengthened by the Q1009 and PKS\,1937 velocity deviation results; the population forecast (Gate 6-II) and orphan search protocol (Gate 6-III) provide the next testable predictions.

5.7 Orphan Search Protocol (Gate 6-III)

The population forecast (Gate 6-II) predicts $\sim 19$ orphan narrow components at non-isotope offsets across $\sim 80$ high-S/N LLS sightlines. A pilot search on $10$--$15$ sightlines from existing archival surveys would turn this prediction into a first observational test.

Archival samples. Three existing surveys provide suitable data without requiring new observations:

• SQUAD (UVES Spectral Quasar Absorption Database, Murphy et al. 2019 [43]): $\sim 200$ UVES sightlines at $R \sim 40\,000$--$50\,000$, covering Ly$\alpha$--Ly$\gamma$ at $z > 2.5$. Approximately $30$--$40$ sightlines have LLS-class absorbers with sufficient S/N for narrow-component detection.
• UVES SLS (UVES Large Programme for D/H, Carswell et al. 2024): $\sim 25$ targeted D/H sightlines at $R \sim 45\,000$--$50\,000$ with S/N $> 30$ per pixel. All have published VPFIT component models that can be searched directly.
• ESPRESSO D/H survey (Cooke et al. 2024): $\sim 10$ sightlines at $R \sim 70\,000$--$140\,000$ with the highest spectral resolution. Ideal for resolving narrow ($b < 10$\,km/s) components.

Search protocol. For each sightline: (i) identify all narrow ($b < 10$\,km/s) H I components with $\log N \sim 12$--$14$ within $\pm 200$\,km/s of the D/H parent complex, (ii) record the velocity offset $\Delta v$ from the nearest H I parent, (iii) classify each as isotope-consistent ($|\Delta v - v_{\rm iso}| < 5$\,km/s) or orphan ($|\Delta v - v_{\rm iso}| > 5$\,km/s), (iv) record $b$, $\log N$, and S/N. The predicted orphan fraction is $\sim 79\%$ (orphan features outnumber isotope-consistent ones by $\sim 3.8:1$). The null hypothesis is not zero orphans — ordinary H I absorbers naturally contain low-column components at many velocities — but rather the conventional H I forest background. A proper null must be constructed from velocity-shuffled windows, matched control absorbers, neighbouring $\Delta v$ intervals, or synthetic conventional H I forests. The TEP prediction is an excess of components with the joint phenotype ($b < 10$\,km/s, $\log N \sim 12$--$14$, non-isotopic offset, systematically related to absorber geometry) beyond this conventional background. A pilot on $10$--$15$ sightlines should detect $\sim 2$--$3$ orphans in excess of the conventional background if the TEP prediction holds.

Falsification criterion. If no excess of orphan components (beyond the conventional H I background) is found across $\geq 20$ high-S/N sightlines (predicting $\sim 4$ excess orphans under TEP), the velocity-selection theorem is falsified. Poisson $P(0 | \lambda = 4) = 0.018$, giving a $>2\sigma$ exclusion.

5.8 Corpus Cross-References: The No-Hot-Big-Bang Framework

The spectroscopic and local-shear results in this paper do not by themselves remove the need for a hot dense early phase. The broader TEP corpus addresses the complementary questions — CMB acoustic peaks, growth of structure, cosmological distances, and high-redshift galaxy assembly — and is cross-referenced here for completeness.

• CMB acoustic peaks and sound horizon (TEP-HC, Paper 18). The TEP conformal scalar field has been implemented natively in hi_class [29] and run through MCMC against Planck 2018, BAO, and Pantheon+. The acoustic sound-horizon ratio $r_s^{\rm TEP}/r_s^{\Lambda{\rm CDM}} = 0.999994$ ($< 6$\,ppm deviation), and the TT/TE/EE peak morphology is preserved without early spatial expansion. The Big Bang is reinterpreted as $A_{\rm clock} \to 0$: a temporal horizon, not a geometric singularity.
• Temporal horizon and absence of singularity (TEP-TH, Paper 27). The FLRW singularity is shown to be a reconstruction artifact. The temporal-horizon conformal boundary $A_{\rm clock}(\eta) = C\eta^{-p}$ with $0 < p \leq 1/2$ yields vanishing curvature invariants, divergent timelike proper time, and divergent null affine parameter — a regular, complete past boundary. The Strong Energy Condition is violated, satisfying the Hawking–Penrose prerequisite for no singularity. The tensor-to-scalar ratio is $r = 9 \times 10^{-6}$.
• Cosmological distances without expansion (TEP-C0, Paper 26). A pure conformal reconstruction exactly reproduces the $\Lambda$CDM distance modulus across $1\,701$ Pantheon+ SNe. A no-$\Lambda$ temporal-shear branch improves $\Delta\chi^2 \approx -3.4$ with Bayes factor $\approx 4.6$--$61.8$ depending on $z_{\rm los}$.
• JWST high-redshift anomalies (TEP-JWST, Paper 12). The canonical TEP cosmology (asymptotic temporal past with unbounded local proper-time history) removes the finite-age assembly bottleneck. Nested model comparison gives $\ln {\rm BF} = +64.1$ (four fewer parameters) vs conventional models, with a dust–$R_{\rm ML}$ correlation $\rho = +0.62$ at $z > 8$.
• Theoretical foundation (TEP, Paper 0). The bi-metric action $\tilde{g}_{\mu\nu} = A^2 g_{\mu\nu} + B\nabla_\mu\phi\nabla_\nu\phi$, field equations, PPN mapping, and the synchronization holonomy as the invariant signature of dynamical time are established in the foundational paper.

The helium-4 equilibrium ($Y_{\rm eq} = 0.247$) and the D/H–$\Omega_b h^2$ concordance are addressed within this paper (Sections 4 and 5.3), while the CMB anchoring of $\Omega_b h^2 \approx 0.022$ is established in TEP-HC. The amplitude gap (Gate 4B) remains the principal open calculation; the spectroscopic evidence (Gates 2, 3, 4E, 7, 8) and the population forecast (Gate 6-II) are independent of the amplitude calculation and stand on their own.

6. Conclusion

Through a systematic, algorithmically controlled analysis of three benchmark D/H absorption systems—Q1009+2956, PKS\,1937$-$101, and J1332$+$0052—the spectroscopic distinguishability of deuterium from ordinary hydrogen is examined under nested model comparison with Monte-Carlo calibrated significance. Isotope identifiability is found to be controlled by both Doppler regime and D column density: at the actual Q1009 D column ($N_{\rm D} = 5.7\times10^{12}\,\mathrm{cm}^{-2}$), the maximum single-pixel discrepancy is $0.52\sigma$ for thermally-dominated gas at Keck/HIRES resolution, falling to $0.10\sigma$ for the fitted mixed-regime parameters and $0.0002\sigma$ for turbulence-dominated gas. At unsaturated LLS-class columns, the intrinsic H/D profile-shape difference is sub-noise on a per-pixel basis, though cumulative discrimination across many pixels and transitions can still be substantial. Under the parent-reassignment statistic with the conservative $\pm 1000$\,km/s parent window—which gives D maximum freedom to select the most favorable parent—the free-H alternative is significantly preferred for all three sightlines: Q1009+2956 ($T_{\rm parent}=161.03$, $p_{\rm parent}=0.001$), PKS\,1937$-$101 ($T_{\rm parent}=59.00$, $p_{\rm parent}=0.001$), and J1332$+$0052 ($T_{\rm parent}=135.76$, $p_{\rm parent}=0.001$). For Q1009+2956, a physically motivated window ($\pm 3b_{\rm max} = \pm 170$\,km/s) was also run, giving results consistent with the conservative-window significance. Physical-window Monte Carlos for PKS\,1937$-$101 and J1332$+$0052 are deferred. After Bonferroni correction for the six-test family (threshold $\alpha/6 = 0.0083$), all six $p$-values survive: $p_{\rm std}$ and $p_{\rm parent}$ are $0.001$ and $0.001$ for Q1009+2956, $0.001$ and $0.001$ for PKS\,1937$-$101, and $0.001$ and $0.001$ for J1332$+$0052. The offset-exactness test (Gate 4E) shows fitted free-H velocities deviating from the isotope-shifted position by $+2.7$\,km/s (Q1009), $+5.7$\,km/s (PKS\,1937), and $-36.6$\,km/s (J1332), providing a per-sightline discriminator. The free-component model is the TEP edge-gas signature: diffuse, unscreened edge gas displaced by temporal shear. Across all three benchmark sightlines, the isotope-tied model is excluded by the free-component alternative under both the canonical-parent and parent-reassignment statistics. A multi-pronged evidence analysis (Gate 8) evaluates three discriminators---b-value ratio, velocity offset deviation, and column density difference---calibrated against a true-D injection Monte Carlo ($N = 200$ per sightline). For Q1009$+$2956, the velocity offset deviation is highly significant individually ($p < 0.005$): true-D simulations produce free-H fits within $\pm 0.2$\,km/s of the isotope shift, whereas the observed $+2.7$\,km/s is far outside. For PKS\,1937$-$101, the velocity offset deviation is also highly significant ($p < 0.005$). For J1332$+$0052, the $b$-value ratio is significant ($p = 0.045$). The joint Mahalanobis statistic is highly significant for all three sightlines ($p < 0.005$): Q1009$+$2956 ($p < 0.005$), PKS\,1937$-$101 ($p < 0.005$), and J1332$+$0052 ($p < 0.005$). For J1332$+$0052 the broad $b$-value is inconsistent with the narrow-edge TEP orphan prediction. The three sightlines exhibit the predicted monotonic ordering of offset deviations with $N_{\rm HI}$ (Spearman $\rho = 1.0$), though with $N = 3$ the sample is too small for the correlation itself to constitute significant evidence. All three features are blueward of the parent as the conformal sign theorem predicts. These results—spanning three instruments (Keck/HIRES, ESPRESSO, UVES+HIRES) and a redshift range $2.5 < z < 3.6$—are consistent with the TEP prediction that temporal shear depends on localized core-edge geometry.

With the expanding-volume requirement removed, the Temporal Equivalence Principle (TEP) is formalized as an alternative geometric foundation. The apparent Big Bang singularity is replaced by an asymptotic temporal horizon ($\mathscr{T}^-$), where ancient clocks tick infinitely slowly without any geometric collapse. It is shown that local thermal processing and chemical evolution can remain bounded by finite exposure measures ($\mathcal{E}_{\rm astr} < \infty$) when the derived temporal-exposure convergence condition is satisfied, ensuring finite integrated astration exposure, and that the helium-4 mass fraction ($Y_{\rm eq} = 0.247$) emerges as the equilibrium of baryonic-cycling reaction flows under temporal-horizon metal sequestration for the adopted stellar histories. The TEP absorber field predicts the blueward sign of the deuterium-like velocity shift from the conformal/static branch sign theorem; the conformal screening operator $\mathcal{S}_\Sigma(|\Phi|/c^2)$ is calibrated by three boundary conditions (absorber, Cassini, galactic halo) with $\Phi_{\rm half} = 1.59 \times 10^{-7}$ and $n = 3.56$, yielding $132\times$ screening contrast between solar-system and galactic regimes; the conformal-only amplitude is quantified ($|\Delta v_{\rm conf}| \approx 0.009\text{ km/s}$ for $R = 30$ kpc with uniform-sphere ionisation correction), and the required amplification $\mathcal{A}_{\rm env} \approx 9.3 \times 10^3$ is identified as a falsifiable target for the non-perturbative transport path $\mathcal{C}_{T,\parallel}$. A quantitative mechanism analysis (Gate 4B-II) identifies the edge transition with $B(\phi)$ enhancement near the screening transition as the most promising non-perturbative channel; a rough line-of-sight calculation (Gate 4B-III) shows the local integral does not close the gap, but the full non-perturbative holonomy of $\mathcal{C}_{T,\parallel}$ — a global topological property, not a local integral — remains the correct evaluation framework. A quantitative population forecast (Gate 6-II) predicts $\sim 20$ orphan narrow components at non-isotope offsets across $\sim 80$ high-S/N LLS sightlines, testable with existing archival data (SQUAD, UVES SLS, ESPRESSO); a pilot search on $10$--$15$ sightlines would provide a first observational test (Gate 6-III). Finally, in the static spatial geometry, the line-of-sight optical depth is shown to diverge at high redshift under an asymptotic electron-density condition, creating an observable boundary without a physical plasma wall. The broader TEP corpus establishes the complementary results: CMB acoustic peak preservation at $< 6$\,ppm (TEP-HC), temporal-horizon regularity and absence of singularity (TEP-TH), cosmological distance reproduction without expansion (TEP-C0), and resolution of JWST high-redshift assembly anomalies (TEP-JWST). These results reduce the classical astration objection to a quantitative convergence condition plus a testable sequestration mechanism, decoupling physical chemical evolution from an eternal coordinate manifold without invoking an explosive spatial origin.

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