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 Monte-Carlo-calibrated model comparison. 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 Model Comparison and Significance Testing

A complete structural analysis of the Q1009+2956 spectrum is performed under a controlled constrained-versus-unrestricted model hierarchy embedded in the joint D+H model. 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 142 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 identically, the resulting $p$-value is calibrated conditional on the adopted published architecture. Q1009 scaffold freedom is tested explicitly in §5.5.

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 eligible H component within the conservative $\pm 1000\,{\rm km\,s^{-1}}$ parent window (14 parent candidates). 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 conservative $\pm 1000\text{ km/s}$ window is the cross-sightline headline statistic; the $\pm 3b_{\rm max}$ physical window has also been calibrated for all three sightlines (Section 5.5), with results consistent with the conservative window. 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:

The signed residuals in this table are measured relative to the isotope position, not relative to the parent. Thus the $+2.7$ and $+5.7$\,km/s residuals mean that Q1009+2956 and PKS\,1937$-$101 are slightly less blueward than exact D; they do not represent redward core-edge shifts. All three fitted components remain blueward of their respective parents.

Q1009+2956 sits $2.7$\,km/s from the isotope shift — lying approximately one instrumental Gaussian width ($\sigma_{\rm inst}\simeq2.6\,{\rm km\,s^{-1}}$) from the isotope position, but highly inconsistent when calibrated against the true-D simulation distribution ($\pm 0.2$\,km/s, $p ≤ 0.001$, Section 5.6). PKS\,1937$-$101 deviates by $5.7$\,km/s — also highly significant after calibration ($p ≤ 0.001$). J1332$+$0052 deviates by $-36.6$\,km/s — not individually significant after calibration ($p = 0.094$). 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. 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 clustering near $-81.6\,{\rm km\,s^{-1}}$ is therefore strongly conditioned by construction of the D-selected sample and cannot be treated as an independent chance-probability test of isotope identity. 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.

Illustrative Population Forecast (Gate 6-II). An illustrative selection model, calibrated to the three benchmark fitted offsets, gives $\sigma_{\rm shear} \approx 19$\,km/s and an orphan fraction of $\sim 79\%$. Because the benchmark sample is D-selected and the J1332 fitted velocity is width-prior-sensitive, these numbers are exploratory rather than a frozen TEP population prediction. The observed cross-sightline deviations ($+2.7$, $+5.7$, $-36.6$\,km/s for Q1009, PKS\,1937, J1332 respectively) give: 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. This is not the expected detected fraction: the archival feasibility analysis in §5.7 shows that saturation and forest contamination reduce an 80-sightline parent sample to roughly 10 usable systems, corresponding to approximately two detectable excess candidates. Because the shear width is inferred from only three D-selected systems, this fraction is provisional; the robust TEP prediction is an excess of non-isotope displaced components with the joint edge-gas phenotype relative to a matched conventional H I background. 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 model-comparison framework, providing the first cross-sightline application of the TEP spectroscopic methodology. The same spectroscopic framework is applied to the two additional benchmark sightlines in §2.4; their cross-sightline amplitude comparison is presented in §3.3.

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 common free-H preference across all three sightlines provides the displaced-H phenomenology to which the TEP interpretation is subsequently applied. A cross-sightline amplitude ranking from the density-weighted Green function $K$ is computed in Section 3.3 (Gate 5).

Multiplicity correction. The primary confirmatory family comprises the six Gate 2/3 statistics and is Bonferroni-corrected at $\alpha/6=0.0083$; all six survive. Gate 8 consists of three secondary correlated multivariate diagnostics derived from the same fitted spectra and is reported separately, not added to the primary family as independent confirmation. Correcting the three Gate-8 joint statistics among themselves gives $\alpha/3=0.0167$, under which the calibrated joint results for Q1009+2956 ($p ≤ 0.001$) and PKS\,1937$-$101 ($p ≤ 0.001$) remain significant; J1332$+$0052 ($p = 0.023$) is significant at the uncorrected $\alpha = 0.05$ level but does not survive the corrected threshold. Parent maximization is already inside $T_{\rm parent}$ and its Monte Carlo calibration; it is not a further Bonferroni factor.

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 free-H preference motivates a separate physical question: whether TEP can provide a non-kinematic contribution to the displaced-H phenomenology. TEP does not predict the isotope constant; it predicts geometry-dependent temporal shifts, while the D-search protocol preferentially selects features lying near the $-81.6\,{\rm km\,s^{-1}}$ isotope window.

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 $g_{\mu 0}$ (taking the local perturbation $\partial_0\delta\phi \simeq 0$; the non-zero cosmological background derivative $\partial_0\bar\phi$ is treated separately in Gate 4B-IV), 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, the conformal absorber solution fixes the blueward sign of the core-edge clock deformation. The sign is a consistency prediction rather than a unique TEP discriminator; the quantitative test is whether the full matter metric can reproduce the observed amplitude and its variation between sightlines.

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 $|\Delta v_{\rm conf}| \approx 0.009\text{ km/s}$. This conformal-only scale is far below the wavelength-calibration/systematic floor of the present spectra and is not separately detectable in these data. It should therefore be interpreted as the calculated signed baseline of the TEP absorber field, not as an observed spectral shift.

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 ratio $\mathcal{A}_{\rm env}\equiv|\Delta v_H|/|\Delta v_{\rm conf}|$ is a conditional diagnostic. If the full fitted free-H displacement is attributed to TEP rather than ordinary gas kinematics, it measures the additional non-conformal contribution required relative to the conformal baseline. It is not itself an observed amplification factor or an independent measurement of the TEP field. The ratio varies across sightlines by a factor of $\sim 9$:

The $\mathcal{A}_{\rm env}$ values in this table use the Poisson Green function with the ionisation-corrected neutral-hydrogen column as the source density (the “gas-column basis”). The halo-mass calculation (Gate 4B-0-III below) uses the total host halo mass as the source density and gives $|\Delta v_{\rm conf}| \approx 0.005$\,km/s for all three sightlines (same $10^{10}\,M_\odot$ host), corresponding to $\mathcal{A}_{\rm env} \approx 1.6$--$2.5 \times 10^4$. The halo-mass basis provides a more complete total-gravitating-mass benchmark within this stress test, while remaining a conventional proxy rather than a TEP-native source solution. Both bases are retained because the gas-column calculation provides the per-sightline ranking while the halo-mass calculation provides the total-mass benchmark amplitude.

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: applying Q1009's $\mathcal{A}_{\rm env} \approx 9.0\times10^3$ to all three sightlines would predict offsets of $-79$, $-151$, and $-694$\,km/s, whereas the fitted free-H displacements are $-78.9$, $-76.0$, and $-118.2$\,km/s. Under the limiting pure-TEP attribution, TEP must account for the unrestricted-H displacement rather than the isotope constant; the per-sightline $\mathcal{A}_{\rm env}$ values quantify the corresponding maximal non-conformal requirement. 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 the geometry-dependent TEP offset happens to lie 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 reflect the different core potentials and halo environments.
2. Per-sightline $\mathcal{A}_{\rm env}$: the ratio is a property of the temporal-disformal transport 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 decisive empirical test is the orphan population (Gate 6): the velocity-selection theorem predicts that edge-shear features at offsets other than $-81.6$\,km/s exist and are being absorbed into H I component structure. The theoretical test is the three-sightline predictive closure (Gate 4B-IV / Gate 10): a single frozen microscopic $B(\phi)$, combined with independently specified absorber environments, must predict the temporal contribution without using the observed velocities, with the fitted free-H displacements serving as maximal pure-TEP amplitude targets. The full covariant calculation (Gate 10) has been performed and confirms that no single power-law $B(\phi)$ achieves this closure; the $A^2/\dot{\bar\phi}^{\,2}$ variation across redshift is the binding constraint.

One-line-of-sight transport calculation (Gate 4B-0-III). A benchmark one-line-of-sight, two-parcel calculation (step\_04g, results/gate4b\_0\_honest\_transport.json) confirms the static-sector predictions of the framework. The scalar field is solved using the Jakarta action architecture with the frozen conformal coupling $\beta_A = -1$ and benchmark weak-field choices $V(\phi) = \frac12 m_\phi^2\phi^2$ and $B(\phi) = B_0\varphi^2$, where $\varphi \equiv \phi/M_{\rm Pl}$, with $m_\phi$ and $B_0$ taken from external constraints rather than from the observed velocity. The source density is a conventional $10^{10}\,M_\odot$ dynamical host halo mass, employed here as an external benchmark proxy for the total gravitating mass; the TEP-consistent source content (which does not invoke particle dark matter) will be treated in a companion paper. For this benchmark proxy at $R = 30$ kpc, the conformal piece gives $|\Delta v_{\rm conf}| \approx 0.005$\,km/s with the correct blueward sign. Paper 29 screening suppresses massive halos ($M > 10^{11}\,M_\odot$) and leaves intermediate halos ($10^{10}$--$10^{11}\,M_\odot$) weakly screened — exactly the LLS host halo mass range — confirming that the fifth force is strongest where TEP requires it. The screening separation is the central TEP prediction. Using the frozen Paper-0 PPN mapping $\gamma_{\rm PPN} - 1 = -2\alpha_{\rm eff}^2/(1 + \alpha_{\rm eff}^2)$ with $\alpha_{\rm eff} = \mathcal{S}_\Sigma^{(\odot)}\,\alpha_0$, the bare unscreened coupling ($\alpha_{\rm eff} = \alpha_0 = -1$) gives $\gamma_{\rm PPN} - 1 = -1$ in the Solar System, violating the Cassini bound $|\gamma - 1| < 2.3 \times 10^{-5}$ by over four orders of magnitude; with Paper 29 screening at the solar grazing potential $|\Phi_\odot|/c^2 \approx 1.3 \times 10^{-6}$, the source-charge screening factor $\mathcal{S}_\Sigma^{(\odot)} \approx 5.3 \times 10^{-4}$ suppresses the effective coupling to $\alpha_{\rm eff} \approx 5.3 \times 10^{-4}$, giving $\gamma_{\rm PPN} - 1 \approx -5.5 \times 10^{-7}$, so $|\gamma_{\rm PPN} - 1| < 2.3 \times 10^{-5}$ (Cassini PASS). Under the absorber-regime amplitude ansatz, LLS-scale potentials place the absorber in the effectively unscreened limit $S_A^{\rm abs}\approx 1$ at LLS host potentials $|\Phi_{\rm abs}|/c^2 \sim 10^{-8}$, so the bare coupling operates in the absorber regime. Note that the Cassini PPN bound constrains the source-charge projection $\mathcal{S}_\Sigma^{(\odot)}$ (which enters $\gamma_{\rm PPN}$), while the absorber clock shift is governed by the amplitude projection $S_A^{\rm abs}$ (which enters $\Delta\ln A$); the same potential-depth environmental variable characterizes both projections, but their response functions are distinct and are not assumed numerically identical (Jakarta §2.2). This environmental separation — screened in the Solar System, unscreened in diffuse LLS clouds — is not a fine-tuning but a prediction of the TEP screening mechanism. The Cassini grazing ray at $b = 1.6\,R_\odot$ lies inside the solar transition radius $r_{\rm trans} = 13.3\,R_\odot$, but the disformal frequency shift is zero for a static field ($\tilde g_{00} = A^2 g_{00}$ when $\partial_0\phi = 0$), and $D|\nabla\phi|^2 \ll 10^{-15}$ (GW170817 PASS). The static conformal piece provides a signed lower bound; the disformal contribution $\mathcal{A}_{\rm env} \sim 10^4$ would have to arise from the non-static disformal transport path $\mathcal{C}_{T,\parallel}$, the identified Gate 4B channel. The same law applied to all three benchmark sightlines (Q1009+2956, PKS\,1937$-$101, J1332$+$0052) with no retuning gives the same conformal lower bound, confirming that the static sector is consistent across the sample.

Disformal clock mechanism: cosmological $\partial_0\bar\phi$ channel (Gate 4B-IV). The static-field calculation above correctly identifies that the disformal $g_{00}$ contribution vanishes when $\partial_0\phi = 0$. However, the v0.11 cosmological architecture $1+z = A_0/A_{\rm em} = \exp(|\beta_A|\,\phi/M_{\rm Pl})$ requires a non-zero cosmological time derivative of the scalar field. Defining the TEP temporal-rate function $H_T(z) \equiv -d\ln A/dt$, the scalar background satisfies $\dot{\bar\phi} = M_{\rm Pl}\,H_T(z)$, so for $A = e^{-\phi/M_{\rm Pl}}$ (step\_04h, results/gate4b\_disformal\_clock.json). The cosmological time derivative activates a disformal $g_{00}$ term that is the candidate amplitude mechanism to the TEP velocity shift. TEP predicts a geometry-dependent $\Delta v_T$, not a universal $-81.6$\,km/s shift; the $-81.6$\,km/s clustering is explained by the velocity-selection theorem (Gate 6-I) as a selection effect of the D-search protocol. The maximal amplitude targets under a pure-TEP attribution are the fitted free-H displacements: $-78.9$, $-76.0$, $-118.0$\,km/s for Q1009+2956, PKS\,1937$-$101, and J1332$+$0052 respectively.

For the reconstructed TEP background used in this calculation, $H_T(z)$ is identified with the effective observational Hubble-rate function corresponding to the fitted distance-redshift relation. At the absorber redshift $z = 2.5042$, the cosmological field is $\bar\phi/M_{\rm Pl} = \ln(1+z)/|\beta_A| = 1.254$, and the cosmological time derivative is $\partial_0\bar\phi = M_{\rm Pl}\,H_T(z) = 1.34 \times 10^{-5}\text{ eV}^2$. The total scalar field at the absorber is $\phi = \bar\phi(t) + \delta\phi(r)$, where the local perturbation $\delta\phi/M_{\rm Pl} \sim 10^{-8}$ is the Poisson-sourced contribution from the host halo. The cosmological time derivative activates the disformal $g_{00}$ term:

producing a differential clock shift between core and edge. The exact matter-metric lapse is $N^2(\phi) = A^2 - B(\phi)\,(\partial_0\bar\phi)^2$, and the exact core-edge clock difference is:

With $B(\phi) = B_0(\phi/M_{\rm Pl})^n$ and the cross-term expansion $B(\phi_{\rm core}) - B(\phi_{\rm edge}) \approx n\,B_0\,(\bar\phi/M_{\rm Pl})^{n-1}\,\Delta(\delta\phi/M_{\rm Pl})$, the first-order expansion gives the linearized velocity shift:

valid only when $B(\phi)\,(\partial_0\bar\phi)^2/A^2 \ll 1$. The exact matter-metric lapse is $N^2(\phi) = A^2 - B(\phi)\,(\partial_0\bar\phi)^2$, and the exact core-edge clock difference is $\Delta_{\rm ec}^{\rm clock} = \frac12 \ln\frac{N_{\rm edge}^2}{N_{\rm core}^2}$. The exact solution, obtained by solving $-c\,\Delta_{\rm ec}^{\rm clock} = \Delta v$ for $B_0$ subject to the Lorentzian condition $N^2 > 0$, gives $B_0 = B_0^{\rm lin}/(1 + \beta\,B_0^{\rm lin})$ where $\beta = (\bar\phi/M_{\rm Pl})^n\,(M_{\rm Pl}\,H_T)^2/A^2$ and $B_0^{\rm lin}$ is the linearized value.

Pure-TEP amplitude stress test. The exact-lapse equation has a divergence: as $B_0 \to 1/\beta$, the invariant $\mathcal{I} = 1 - B(\bar\phi)\,(\partial_0\bar\phi)^2/A^2 \to 0$ and the predicted shift $\Delta v \to \infty$. The original analysis targeted the full $81.6$\,km/s isotope shift, but TEP does not predict a universal $-81.6$\,km/s shift; the velocity-selection theorem (Gate 6-I) explains the clustering as a selection effect. Under the limiting pure-TEP attribution, the fitted free-H displacement defines the maximal amplitude target: $78.9$\,km/s for Q1009+2956. The exact-lapse solution under the first-order core-edge field expansion for this target gives $B_0 \sim 10^8$\,eV$^{-4}$ with $\mathcal{I} \sim 10^{-4}$, sitting near the Lorentzian boundary. The near-critical regime is a property of the exact-lapse equation: any target above the linear-regime maximum ($\sim 0.01$\,km/s) drives $B_0$ toward $1/\beta$. Because $B_0$ is inferred here by solving backward from the observed displacement, the near-critical value of $\mathcal{I}$ is a diagnostic property of this ansatz rather than an independent prediction or evidence for TEP. Predictive significance requires $B(\phi)$ to be frozen independently of these velocities.

Constraint analysis. The same $B_0$ enters both the temporal disformal term (clock shift, the candidate amplitude contribution) and the spatial disformal term (Shapiro delay, constrained by Cassini). The spatial term at the Sun is $\gamma_{\rm dis} = B(\phi_\odot)\,(\partial_r\phi_\odot)^2/(2U_\odot)$, where $U_\odot = |\Phi_\odot|/c^2 = 2.12 \times 10^{-6}$. For this reduced Solar-System consistency calculation, the nonlinear environmental response is represented phenomenologically by an effective source-charge projection $S_\Sigma^{(\odot)}$. In the microscopic Jakarta formulation, $S_\Sigma$ is not an independent factor inserted into the fundamental scalar source equation; it is an observable projection of the environment-dependent scalar solution. The reduced solution gives $\phi_\odot/M_{\rm Pl} = 2.26 \times 10^{-10}$, with $\partial_r\phi_\odot = 156\text{ eV}^2$. For the quadratic continuation $n = 2$, the exact-lapse $B_0 = 2.88 \times 10^{8}\text{ eV}^{-4}$ gives $\gamma_{\rm dis} \approx 8.4 \times 10^{-2}$, exceeding the Cassini bound $|\gamma - 1| < 2.3 \times 10^{-5}$ by a factor $\sim 10^3$. The quadratic $B \propto \phi^2$ realization at the normalization required to reproduce the Q1009 amplitude remains incompatible with Cassini even with the exact lapse. A steeper field dependence can satisfy the local Cassini, GW170817 and clock constraints. For $n = 5$, the Q1009-normalised solution gives $\gamma_{\rm dis}\approx4.9\times10^{-31}$, $\delta c/c\approx8.6\times10^{-20}$, and a negligible local clock shift. However, as shown by the cross-sightline transfer test below, the same $n = 5$ power law loses Lorentzian signature at the two higher-redshift absorbers. Thus $n = 2$ is excluded by local constraints and $n = 5$ by cross-redshift transfer; neither tested power law provides a viable universal microscopic closure. The v0.11 theory states that the microscopic form of $B(\phi)$ is not uniquely frozen; the field-space envelope $B(\phi) = B_0\,\varphi^2/(1+\varphi^2)\exp(-\varphi^4/2\sigma_B^4)$ with $\varphi \equiv \phi/M_{\rm Pl}$ is a named benchmark, not a unique prediction. The Cassini and three-sightline constraints derived here bound the still-open $B(\phi)$ closure: the quadratic realization at the required normalization is excluded, and the combined constraints sharply restrict the still-open microscopic $B(\phi)$ closure.

For this Gate-IV consistency calculation an illustrative Solar-System source-charge profile is adopted with $\mathcal{S}_\Sigma^{(\odot)} = 5.3 \times 10^{-4}$, safely below the Cassini ceiling $3.4 \times 10^{-3}$. This value is not inferred from the absorber $S_A^{\rm abs}$ ansatz.

Three-sightline test: diagnostic, transfer, and predictive closure. The test is performed in three stages. Diagnostic mode: infer $B_0$ per sightline from the fitted free-H displacement. Each sightline admits an individually fitted Lorentzian solution ($\mathcal{I} > 0$), establishing algebraic feasibility of the temporal-disformal channel under this ansatz. The inferred $B_0$ values are $1.46 \times 10^8$, $1.52 \times 10^7$, and $2.00 \times 10^7$\,eV$^{-4}$ for Q1009+2956, PKS\,1937$-$101, and J1332$+$0052 respectively. Transfer mode: freeze a single $B_0$ from Q1009+2956 and transfer it unchanged to the other two sightlines. The invariant $\mathcal{I} = 1 - \beta(z)\,B_0$ depends only on the cosmological background ($\beta(z) = (\bar\phi/M_{\rm Pl})^n\,(M_{\rm Pl}\,H_T)^2/A^2$), not on the halo environment. With $B_0$ from Q1009+2956 ($z = 2.5042$), the invariant becomes $\mathcal{I} < 0$ at PKS\,1937$-$101 ($z = 3.572$) and J1332$+$0052 ($z = 3.421$), crossing the Lorentzian boundary. This signature change is a constraint on the microscopic form of $B(\phi)$: the simple power-law $B(\phi) \propto \phi^5$ calibrated at $z = 2.5$ does not transfer to $z > 3.4$. This is a failure of that particular transfer ansatz, not a falsification of the temporal-disformal channel. Predictive closure (Gate 10, step\_10\_covariant\_transport, results/gate10\_covariant\_transport.json): the full covariant calculation has been performed with a TEP-native background $\bar\phi(z)$, $\partial_0\bar\phi = M_{\rm Pl}\,H_T(z)$, a three-dimensional screened Poisson solution $\delta\phi(r)$ for each absorber using the per-sightline column densities ($\log N_{\rm HI} = 17.36$, $17.92$, $19.25$ for Q1009+2956, PKS\,1937$-$101, J1332$+$0052 respectively), the complete disformal matter metric $\tilde g_{\mu\nu} = A^2 g_{\mu\nu} + B(\phi)\nabla_\mu\phi\nabla_\nu\phi$, Hamiltonian null-geodesic propagation, and the matter-frame frequency $\omega = -u^\mu k_\mu = -k_0/N$ (cone-tilt terms cancelling exactly for a static observer). The three-dimensional screened field gives $\Delta(\delta\phi/M_{\rm Pl}) = 1.26\times10^{-8}$, $2.41\times10^{-8}$, $1.02\times10^{-7}$ for the three sightlines respectively, versus the uniform $1.60\times10^{-8}$ of the reduced two-point approximation; the screening reduction factor at Q1009+2956 is $0.79$. With the corrected sign convention $\Delta v = c\,(N_{\rm core}/N_{\rm edge} - 1)$, positive $B$ gives positive $\Delta v$ (blueward edge-minus-core, consistent with observation). The diagnostic Q1009+2956 shift at the exact-lapse $B_0 = 1.46\times10^8$\,eV$^{-4}$ is $+62.5$\,km/s, reduced from the two-point $+78.9$\,km/s by the screened three-dimensional field. The predictive closure with a single frozen $B(\phi) = B_0(\phi/M_{\rm Pl})^5$ gives $\chi^2 = 2.97$ at $B_0 = 1.50\times10^7$\,eV$^{-4}$ (the simultaneous Lorentzian bound, set by PKS\,1937$-$101 at $B_0 < 1.52\times10^7$\,eV$^{-4}$), with predicted shifts of $0.00$, $1.18$, $0.15$\,km/s versus observed $78.9$, $76.0$, $118.0$\,km/s. The near-criticality and cross-redshift signature loss persist under the full covariant treatment. The root cause is that $A^2/\dot{\bar\phi}^{\,2}$ varies by a factor $3.68$ across the three sightlines; no monotonically increasing $B(\phi)$ can compensate this variation while preserving the correct sign at all three redshifts. The simultaneous Lorentzian bound keeps Q1009+2956 far from critical ($\mathcal{I} = 0.896$), limiting its predictive shift to $\sim10^{-3}$\,km/s. The matter-frame maximum is $|\Delta v| = 0.051$\,km/s (at J1332$+$0052, benefiting from the $8\times$ larger field perturbation at $\log N_{\rm HI} = 19.25$), and the negative-$B$ branch gives the wrong sign ($\Delta v < 0$) with magnitude $\sim0.05$\,km/s. The covariant calculation therefore confirms the reduced Gate-4B finding: the temporal-disformal channel is algebraically feasible per sightline (diagnostic mode), but no single power-law $B(\phi)$ achieves predictive closure across all three sightlines.

Limit of the reduced Gate-4B closure. The failed power-law transfer constrains the specific reduced construction tested here. The present calculation represents the cosmological scalar evolution through an effective background-rate function and reduces the absorber observable to a core-edge lapse difference. In the static-space TEP formulation, predictive closure requires instead a TEP-native background solution $\bar\phi(t)$, an inhomogeneous absorber solution $\delta\phi(x)$, and covariant propagation of the photon through the complete matter metric, with the measured frequency evaluated as $\omega=-u^\mu k_\mu$. This calculation has now been performed (Gate 10, step\_10\_covariant\_transport). The near-criticality and cross-redshift signature loss persist under the full covariant treatment: the $A^2/\dot{\bar\phi}^{\,2}$ variation across redshift is intrinsic to the TEP cosmological background and cannot be removed by replacing the two-point lapse difference with the full three-dimensional screened field and covariant frequency projection. The constraint is therefore not an artefact of the reduced core-edge approximation but a property of the temporal-disformal channel under any single power-law $B(\phi)$.

Environmental discrimination. Solar-System and absorber observables probe distinct projections of the same environment-dependent scalar configuration. Cassini constrains the source-charge projection $S_\Sigma^{(\odot)}$, whereas conformal clock amplitudes depend on $S_A$. Their numerical relationship is not assumed here and must emerge from the common nonlinear scalar solution. The reduced Gate-4B calculation shows that a temporal-disformal contribution can be large in the absorber parametrisation while remaining constrained locally. The full covariant transport (Gate 10) has been completed and confirms the reduced calculation's finding that no single power-law $B(\phi)$ achieves predictive closure; the viable microscopic $B(\phi)$ remains open.

Gate 4B status. The conformal field fixes the sign but gives only $0.005$--$0.01\,{\rm km\,s^{-1}}$. Static perturbative disformal terms are also insufficient. The cosmological temporal-disformal term $B(\phi)\dot{\bar\phi}^{\,2}$ can reach much larger amplitudes in the Einstein-frame parametrisation explored here, although the solutions are near-critical. Individually fitted sightlines demonstrate algebraic feasibility, while the simple $B(\phi)\propto\phi^5$ form fails the cross-redshift transfer test. The full covariant calculation (Gate 10) with $\omega=-u^\mu k_\mu$ confirms this finding: the $A^2/\dot{\bar\phi}^{\,2}$ variation across redshift prevents any single increasing $B(\phi)$ from achieving simultaneous near-criticality at all three sightlines. The viable microscopic $B(\phi)$ must therefore be non-power-law or environment-dependent; its identification remains open.

Functional constraints on $S_A^{\rm abs}$ and $S_\Sigma^{(\odot)}$. The amplitude and source-charge projections must satisfy their respective boundary conditions:

The absorber amplitude ansatz $S_A^{\rm abs}(|\Phi|/c^2) = 1/(1 + (|\Phi|/c^2/\Phi_{\rm half})^n)$ with $\Phi_{\rm half} = 1.59 \times 10^{-7}$ and $n = 3.56$ is the absorber-regime amplitude transfer employed in this paper. The screening variable for this ansatz 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. Cassini independently constrains the source-charge response $S_\Sigma^{(\odot)}$. The relation between $S_A^{\rm abs}$ and $S_\Sigma^{(\odot)}$ is not established by this ansatz. The product $\alpha_{\rm eff}^2 \times |\Phi|/c^2$ 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 disformal contribution cannot arise from the conformal sector alone (which gives $S_A^{\rm abs} \approx 1$ in the unscreened regime); it must originate from the disformal sector $B(\phi)\nabla_\mu\phi\nabla_\nu\phi$ and its environmental projection. The conformal/static contribution maps through $\Delta v_{\rm conf}\simeq-c\,\Delta\ln A$. The temporal-disformal contribution instead modifies the matter-frame lapse $N^2=A^2-B(\phi)\dot{\bar\phi}^{\,2}$; its observable spectral shift must be derived from the covariant frequency $\omega=-u^\mu k_\mu$. No identification of the disformal contribution with $-c\Delta\ln A$ is assumed. For the quasistatic local perturbation, $\partial_0\delta\phi \simeq 0$, so the static disformal term does not modify $g_{00}$. The cosmological background has $\partial_0\bar\phi \neq 0$, activating the distinct temporal-disformal clock channel tested in Gate 4B-IV. 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 perturbative $B(\phi)$ can provide the required disformal contribution (quantitatively confirmed: the disformal metric correction $B_0 \cdot (|\Phi|/c^2)^2/r^2 \sim 10^{-62}$ in geometrized units for $B_0 = 1$). The static spatial perturbative disformal channels are excluded (Gate 4B-II). The temporal disformal clock channel $B(\phi)(\partial_0\bar\phi)^2$ activated by the cosmological scalar time derivative is the candidate amplitude mechanism (Gate 4B-IV). The full covariant core-edge observable $\omega = -u^\mu k_\mu$ has been evaluated (Gate 10, step\_10\_covariant\_transport), confirming the reduced calculation's finding.

Note on coupling conventions. Following Paper 0, the frozen microscopic coupling is $\alpha_0 = \beta_A = -1.0$ ^{[3], [54]}. The environmentally screened effective scalar charge for source-charge/fifth-force observables is $\alpha_{\rm eff} = S_\Sigma(\mathcal{E})\,\alpha_0$ ^[1]; the environmentally screened effective amplitude for clock/Cepheid observables is $\alpha_{A,\rm eff} = S_A(\mathcal{E})\,\alpha_0$. These are distinct projections. 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 $S_A^{\rm abs} \approx 1$ and the bare coupling applies. No numerical galactic $S_A$ anchor is assumed in the present calculation; the relation between weak-field conventions across the TEP corpus remains part of the common microscopic closure. The potential-depth law is a phenomenological amplitude-transfer ansatz for the absorber calculation. Cassini independently constrains the source-charge response $S_\Sigma^{(\odot)} \lesssim 3.4 \times 10^{-3}$. The relation between $S_A$ and $S_\Sigma$ must be derived from the common microscopic scalar solution and is not established by this two-parameter fit. 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 rather than by a chameleon or symmetron mechanism.

Cassini constraint derivation. The Cassini experiment measured the PPN parameter $\gamma_{\rm PPN} = 1 + (2.1 \pm 2.3) \times 10^{-5}$ ^[59]. Following Paper 0's frozen convention, $\gamma_{\rm PPN} - 1 = -2\alpha_{\rm eff}^2/(1 + \alpha_{\rm eff}^2)$, where $\alpha_{\rm eff} = S_\Sigma\,\alpha_0$ is the environmentally screened effective scalar charge and $\alpha_0 = \beta_A = -1$ is the frozen microscopic coupling. 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. For $|\alpha_{\rm eff}| \ll 1$, the Cassini bound $|\gamma_{\rm PPN} - 1| < 2.3 \times 10^{-5}$ requires $S_\Sigma^{(\odot)} \lesssim 3.4 \times 10^{-3}$ in the Solar-System environment. This is an independent constraint on the source-charge response $S_\Sigma^{(\odot)}$; it is not used to calibrate the absorber amplitude ansatz $S_A^{\rm abs}$. The relation between $S_A^{\rm abs}$ and $S_\Sigma^{(\odot)}$ must be derived from the common microscopic scalar solution.

3.4 Scaling Consistency: Absorber Clouds vs. Galaxies

A natural concern is whether a scalar mechanism capable of accounting for a $\sim 80\text{ km/s}$ shift across a diffuse gas cloud would produce an unphysically large effect across a massive galaxy. The clock shift for a static configuration with the Poisson Green function is:

The conformal amplitude does not scale with potential depth alone because the effective amplitude projection is environmentally screened. Diffuse absorbers remain effectively unscreened ($S_A^{\rm abs} \approx 1$ at $|\Phi_{\rm abs}|/c^2 \sim 10^{-8}$). Solar-System fifth-force constraints probe the distinct source-charge projection $S_\Sigma$ ($S_\Sigma^{(\odot)} \lesssim 3.4 \times 10^{-3}$). No numerical galactic $S_A$ anchor is assumed here. A common microscopic scalar solution must ultimately derive both projections; this cross-domain closure is not required for the spectroscopic result tested here.

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 is interpreted in the TEP hypothesis as arising from localized absorber shear. For the local absorber, the conformal/static contribution is $\Delta v_{\rm conf}\simeq-c\,\Delta\ln A$. When the temporal-disformal channel is active, however, the matter-frame lapse becomes $N^2=A^2-B(\phi)\dot{\bar\phi}^{\,2}$, and the complete observable must be obtained from the covariant measured frequency $\omega=-u^\mu k_\mu$. The full absorber displacement is therefore not identified with the conformal expression alone.

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 the spectroscopic free-H alternative is borne out and D/H ceases to provide a unique primordial abundance anchor, the remaining light-element constraints—particularly helium-4—must be reproduced within any non-hot-BBN framework. The helium-4 mass fraction ($Y_{\rm p} \approx 0.245$ ^{[21], [22], [68]}) is the principal such constraint. Without a finite, hot, universally dense origin, the TEP framework must quantitatively demonstrate 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 Line-of-Sight Transport 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, $N = 1000$ 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 ($N = 1000$), the J1332 $b$-value ratio is marginal ($p = 0.059$): 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 ($p = 0.998$) — the observed ratio is closer to the isotope-D prediction than the simulations. The PKS\,1937 $b$-ratio is marginal ($p = 0.05$). 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). A boundary check with $b_{\rm turb,max}=100$ and $150\,{\rm km\,s^{-1}}$ shows that the fitted turbulent width does not stabilise, rising to $\sim 112\,{\rm km\,s^{-1}}$, while $v_H$ shifts from $-118.0$ to $-115.3\,{\rm km\,s^{-1}}$ and $T_{\rm obs}$ increases from $137.0$ to $140.6$. The likelihood preference therefore does not weaken, but no expanded-bound Monte-Carlo significance is assigned here. J1332-specific fitted width and velocity diagnostics are consequently treated as bound-sensitive. The $b$-value ratio is marginally inconsistent with 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). Using the primary bounded fits, the three sightlines show a monotonic ordering with $\log N_{\rm HI}$; however, the J1332 velocity is sensitive to the turbulent-width prior, so this $N=3$ ordering is descriptive rather than a robust correlation test ($P(\rho = 1 \mid N = 3) = 0.17$ for a prespecified directional trend). After calibration ($N = 1000$), the Q1009 deviation is highly significant individually ($p ≤ 0.001$, 0/1000 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 ($p ≤ 0.001$, 0/1000 exceedances): the simulation spread is $\pm 0.5$\,km/s, and the observed $+5.7$\,km/s is far outside. The J1332 deviation is not individually significant ($p = 0.094$): 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 $p ≤ 0.001$ rejection of the isotope-D velocity prediction for Q1009 is the strongest individual discriminator in the analysis. For scale: among the six H I components of Q1009 within the $\pm 3b_{\rm max} = \pm 170$\,km/s physical window, the mean nearest-neighbour velocity spacing is $56$\,km/s (median $45$\,km/s); the fitted free-H displacement of $-78.9$\,km/s is therefore $\sim 1.4$ times the mean spacing, comparable to — but somewhat larger than — ordinary H I component structure. The displacement is not anomalously large relative to the absorber's own kinematic texture. 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. Folding a Gaussian systematic with $\sigma_{\rm sys} = 1$\,km/s into the simulation spread (step\_08c, results/gate8\_wavelength\_systematics.json) reduces the Q1009 deviation from $16.9\sigma$ ($p ≤ 0.001$) to $2.7\sigma$ ($p \approx 0.008$), while PKS\,1937 remains at $5.0\sigma$ ($p \approx 4.7 \times 10^{-7}$), robustly significant. At $\sigma_{\rm sys} = 0.5$\,km/s both sightlines remain highly significant ($5.2\sigma$ and $7.9\sigma$ respectively). The Q1009 result should therefore be interpreted as significant conditional on wavelength-calibration systematics remaining below $\sim 1$\,km/s, 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. However, the D-selected benchmark sample does not constitute an independent sign test, because its selection protocol already targets blueward isotope-shifted absorption. The sign becomes genuinely predictive in an independently selected population of non-isotope displaced components (Gate 6-III).

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 ($N = 1000$), the Q1009 column density difference is significant ($p = 0.017$): 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 ($p = 1.0$) and J1332 ($p = 0.688$) 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 $N = 1000$ true-D injections per sightline, Q1009$+$2956 is highly significant in the joint statistic ($p ≤ 0.001$, 0/1000 exceedances, Mahalanobis distance $= 288$), driven primarily by the velocity deviation ($p ≤ 0.001$). PKS\,1937$-$101 is also highly significant ($p ≤ 0.001$, 0/1000 exceedances, Mahalanobis distance $= 130$), driven primarily by the velocity deviation ($p ≤ 0.001$). J1332$+$0052 is significant ($p = 0.023$, 23/1000 exceedances, Mahalanobis distance $= 14.7$), driven by the joint combination of the $b$-value ratio ($p = 0.059$) and velocity deviation ($p = 0.094$), 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. Q1009$+$2956 and PKS\,1937$-$101 show highly significant calibrated Gate 8 evidence ($p ≤ 0.001$); J1332$+$0052 is significant at the uncorrected level ($p = 0.023$) but does not survive the $\alpha/3 = 0.0167$ correction among the three Gate-8 statistics. 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) is the principal remaining calculation. 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.

Wavelength-calibration systematics. The velocity-deviation statistic measures a relative offset between the fitted free-H component and the isotope-shifted position within the same spectrum. The relevant wavelength systematic is therefore the intra-spectrum relative velocity distortion (intra-order or long-range), not the absolute wavelength offset or inter-night drift, which cancel in the relative measurement. Instrument-specific residual distortion amplitudes are taken from supercalibration studies: $\sigma_{\rm sys} = 0.25$\,km/s for Keck/HIRES with ThAr calibration, based on iodine-cell measurements of intra-order distortions $\sim 250$\,m/s ^[71]; $\sigma_{\rm sys} = 0.05$\,km/s for VLT/ESPRESSO with ThAr+Fabry-Perot calibration, based on laser-frequency-comb discrepancies $\lesssim 24$\,m/s ^[73]; and $\sigma_{\rm sys} = 0.20$\,km/s for VLT/UVES+Keck/HIRES, based on solar-twin supercalibration long-range distortions $\sim 200$\,m/s per $1000$\,\AA ^[72]. Folding these into the calibrated spread via $\sigma_{\rm total} = \sqrt{\sigma_{\rm fit}^2 + \sigma_{\rm sys}^2}$ (step\_08c, $N = 100\,000$ Monte Carlo draws per case) gives Gaussian-equivalent significances of $9.1\sigma$ (Q1009$+$2956), $11.0\sigma$ (PKS\,1937$-$101), and $2.1\sigma$ (J1332$+$0052), corresponding under a normal-tail approximation to $p\sim10^{-19}$, $10^{-27}$, and $0.032$, respectively. The Monte-Carlo sampling itself does not directly resolve tail probabilities below its finite simulation floor. The Q1009 and PKS\,1937 velocity-deviation rejections of isotope-D therefore survive at high significance even under the literature-calibrated systematic floor; the J1332 result remains not individually significant after the $\alpha/3$ correction, consistent with the unsystematic calibration. A sensitivity scan over $\sigma_{\rm sys} \in [0, 1]$\,km/s is reported in the supplementary results (results/gate8\_wavelength\_systematics.json); the $9.1\sigma$ Q1009 result degrades to $2.7\sigma$ only at $\sigma_{\rm sys} = 1.0$\,km/s, a value exceeding all measured intra-spectrum distortions for the relevant instruments.

5.7 Orphan Search: Protocol and Pilot Results (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 the three benchmark sightlines has been carried out using both the published VPFIT component decompositions and direct spectral inspection of the Ly$\alpha$ region (step\_06\_orphan\_search, results/gate6\_orphan\_search.json).

Pilot search results. The VPFIT models and HIRES/ESPRESSO spectra were searched for narrow ($b < 15$\,km/s), low-column ($\log N < 15$) H I components at non-isotope offsets ($|\Delta v - v_{\rm iso}| > 10$\,km/s) within $\pm 300$\,km/s of the parent. One orphan candidate is identified: a narrow H I component in Q1009$+$2956 at $\Delta v = -155.5$\,km/s from the parent ($|\Delta v - v_{\rm iso}| = 73.9$\,km/s), with $b = 10.3$\,km/s and $\log N = 11.5$, detected at $5\sigma$ on a single coadd in the HIRES spectrum. No global significance is assigned because the search over velocity, width and column density has not been Monte-Carlo calibrated. A second absorption feature at $\Delta v = -170$\,km/s in PKS\,1937$-$101 is detected at $5\sigma$ on a single coadd in the ESPRESSO spectrum, but the VPFIT decomposition fits it as a broad component ($b = 31.5$\,km/s); a narrow orphan may be hidden within the broad fit. J1332$+$0052 is not searchable: the Ly$\alpha$ is completely saturated ($\log N_{\rm HI} = 19.25$) and Ly$\beta$ falls outside the UVES wavelength coverage.

The Q1009 orphan candidate is already present in the published VPFIT decomposition (one of the 43 H I components in the Zavarygin et al. 2018 model). The pilot count is not an independent test of the revised forecast, because the usable-fraction correction is estimated from this same archival exercise. It is therefore reported as a feasibility result rather than confirmatory evidence. Three factors suppress the yield: (i) the parent H I is saturated from $\Delta v \approx -50$ to $+100$\,km/s at all three sightlines, hiding any orphans in that window; (ii) the VPFIT models only contain components above the statistical significance threshold, so sub-threshold orphans are invisible; (iii) $\sigma_{\rm shear}$ is estimated from only 3 data points (the D-identified features) and is biased small, understating the predicted velocity spread. The full population test requires the $\sim 80$-sightline archival search described below.

Archival search. An archival search has been carried out using publicly available data from two surveys (step\_06\_orphan\_search, results/gate6\_archival\_search.json):

• SQUAD DR1 (UVES Spectral Quasar Absorption Database, Murphy et al. 2018 ^[43]): 467 UVES quasar spectra at $R \sim 40\,000$--$50\,000$, accessed via the ESO Phase 3 archive. Three benchmark sightline spectra were downloaded and inspected: Q1009$+$2956 (UVES, 3\,000\,s), Q0913$+$072 (UVES, 77\,525\,s), and J212912$-$153841 (UVES, 101\,016\,s). The SQUAD UVES spectrum of Q1009$+$2956 shows a feature at $\Delta v \approx -155$\,km/s consistent with the HIRES orphan candidate, though at lower S/N ($\sim 33$ vs $\sim 87$ for HIRES); this is a consistency check on the same sightline, not an independent detection. Q0913$+$072 is a DLA ($\log N_{\rm HI} = 20.15$) with all Lyman series lines through Ly$\varepsilon$ saturated. J212912$-$153841 has an LLS at $z = 2.968$ ($\log N_{\rm HI} = 17.32$), but the Ly$\alpha$ forest at $z \sim 3$ produces near-zero flux throughout the $\pm 300$\,km/s search window.
• KODIAQ DR2 (Keck Observatory Database of Ionized Absorption toward Quasars, O'Meara et al. 2017): 593 continuum-normalised, coadded HIRES spectra of 300 quasars, downloaded from the Keck KOA. The KODIAQ Z survey (Lehner et al. 2016) provides a catalog of 26 pLLSs/LLSs at $2.3 < z < 3.3$ with $\log N_{\rm HI} = 16.16$--$18.65$, of which 9 are LLS-class ($17.0 \leq \log N_{\rm HI} \leq 19.0$). Spectra for 7 of these 9 LLS-class sightlines were extracted and searched.

Archival search results. All 7 KODIAQ LLS-class sightlines have broad LLS core absorption that saturates the isotope-shift region, preventing orphan detection at the predicted velocity. The saturation arises from two compounding effects: (i) the LLS itself produces Ly$\alpha$ absorption covering $\sim 100$\,km/s, and (ii) the Ly$\alpha$ forest at $z > 2.4$ contributes additional absorption that drives the flux to near-zero at most velocities. The only sightline where the isotope-shift region is recovered is Q1009$+$2956, where the LLS has a relatively narrow core (saturating only from $\Delta v \approx -50$ to $+100$\,km/s) and the forest at $z = 2.504$ is sufficiently sparse. This identifies a critical selection criterion for the population test: only sightlines with (a) narrow LLS cores ($< 50$\,km/s saturated width), (b) S/N $> 30$ per pixel, and (c) $z < 2.6$ are suitable for orphan detection. The usable fraction of the KODIAQ Z LLS sample is $\sim 12\%$ (1 of 8 searched sightlines), substantially lower than the $\sim 30\%$ detectable-geometry fraction assumed in the Gate 6-II forecast.

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. An illustrative selection model calibrated to the three benchmark offsets gives an orphan fraction of $\sim 79\%$ (orphan features outnumber isotope-consistent ones by $\sim 3.8:1$); because the shear width is inferred from only three D-selected systems, this fraction is provisional. 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 robust 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.

Pilot feasibility forecast. The archival search demonstrates that the effective survey size is reduced by the saturation selection effect. If the usable fraction is $\sim 12\%$ rather than the assumed $\sim 30\%$, the effective survey size for a 80-sightline parent sample is $\sim 10$ sightlines, predicting $\sim 2$ excess orphans under TEP. A pilot on $10$--$15$ usable sightlines (requiring a parent sample of $\sim 80$--$120$ LLS sightlines to account for the saturation selection) should detect $\sim 2$--$3$ orphans in excess of the conventional background. A pilot on $\geq 20$ usable sightlines would provide initial evidence; the Poisson threshold $P(0 | \lambda = 4) = 0.018$ is a planning benchmark rather than a hard theory-falsification threshold, given the exploratory nature of the $N=3$ shear calibration. The KODIAQ Z survey (26 pLLSs/LLSs) and the HD-LLS survey (Prochaska et al. 2015; 46 additional LLSs) together provide $\sim 72$ LLS sightlines, yielding $\sim 9$ usable sightlines after the saturation selection — sufficient for a pilot but not for a definitive falsification. A definitive test requires either a larger parent sample or a search strategy that recovers orphans in saturated regions via higher-order Lyman lines or metal-line proxies.

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). Pantheon+ supports the temporal-shear distance reconstruction in SN-only fits, while the joint Planck+Pantheon analysis strongly constrains the additional homogeneous shear amplitude toward zero. The conformal reconstruction remains compatible with the observed distance and acoustic background.
• JWST high-redshift anomalies (TEP-JWST, Paper 12). The TEP predictor carries substantial information beyond a standard mass-plus-redshift baseline and removes the finite-age assembly constraint in the static-temporal interpretation. Complete-model Bayesian comparisons are mixed, however, with flexible bursty-star-formation and AGN-feedback models remaining competitive or preferred in some joint tests. The JWST results are therefore supportive of the temporal interpretation but are not uniquely discriminating.
• 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 line-of-sight transport integral as the invariant signature of dynamical time are established in the foundational paper.

The helium-4 equilibrium is calculated within this paper. The D/H–$\Omega_b h^2$ relation is reformulated as Gate 9, while its catalogue-level population test remains to be performed; the independent CMB anchoring of $\Omega_b h^2\approx0.022$ is inherited from TEP-HC. The amplitude gap (Gate 4B) is the principal remaining calculation; the primary Gates 2/3 spectroscopic result is independent of the unresolved Gate-4B amplitude closure. Gates 4E, 7 and 8 provide secondary diagnostics, while Gate 6 defines the exploratory population test.

6. Conclusion

Across three benchmark D/H absorption systems, unrestricted H I is significantly preferred over isotope-tied D I within the tested Voigt architecture, including under adversarial parent reassignment and the primary multiplicity correction. Q1009+2956 and PKS\,1937$-$101 also show significant departures from exact isotope locking in the calibrated Gate-8 analysis, while J1332$+$0052 is less discriminating. The result therefore demonstrates that the deuterium interpretation is not spectroscopically unique within the tested model class; the isotope-tied model is strongly disfavoured relative to unrestricted H within the tested architecture.

The fixed $-81.6\,{\rm km\,s^{-1}}$ clustering is not an independent discriminator in a sample selected through the deuterium search window. The corresponding falsifiable population test is an excess of suitable displaced H I components outside that window. The archival pilot establishes the practical selection limits of this test and identifies one candidate feature in Q1009+2956, but a substantially larger usable sample is required.

TEP provides a candidate physical interpretation of this displaced-H phenomenology. The conformal absorber field gives the blueward sign but only a $0.005$--$0.01\,{\rm km\,s^{-1}}$ shift. A temporal-disformal channel can reach the observed scale in the explored parametrisation, but its microscopic $B(\phi)$ and full covariant frequency mapping remain open. Independently, the baryonic-cycling calculation shows that a helium-4 equilibrium near the observed abundance is possible under the adopted VMO-dominated, metal-sequestering branch. The spectroscopic, population and light-element predictions therefore provide separate empirical routes for testing the proposed non-hot-BBN interpretation.

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