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, of which 4 satisfy the column-density eligibility cut ($\log N_{\rm HI} \ge \log N_{\rm max} - 2$ dex); 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 ($p = 0.098$). 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 sightline joint tests among themselves gives $\alpha/3=0.0167$; the corresponding calibrated probabilities are Q1009: $p=0.001$, PKS1937: $p=0.001$, J1332: $p=0.023$. Q1009, PKS1937 pass that 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:

The following absorber calculation uses a canonical comparison benchmark of the common architecture, rather than the low-kinetic limit of the selected sector. The current Paper-0 screening realization is $P(X,\phi)=-V(\phi)+\frac{2k}{3\Lambda_X^2}X|X|^{1/2}+X|X|/\Lambda_X^4$; the canonical Green-function calculation and its adopted screening profile are not an independent derivation of that nonlinear response.

where the canonical TEP matter metric is $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\Pi_{\mu\nu} + B(\phi)\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ ($U_\mu\equiv-\nabla_\mu\bar\phi/\sqrt{2\bar X}$, $\Pi_{\mu\nu}=g_{\mu\nu}+U_\mu U_\nu$; Paper 0, §2.2, Step 80), written in conformal–disformal shorthand as $\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu} + B(\phi) \nabla_\mu \phi \nabla_\nu \phi$. 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 under the imported $\Lambda$CDM rate; the closure condition is then shown to be a property of the temporal-rate function rather than of the disformal channel, and under the TEP-native descent-drift law derived from the corpus potential the inverse-fit $B_0$ roots concord to a dispersion of $1.00004$ at the near-criticality floor. That branch is nevertheless excluded by the ambient-lapse self-consistency requirement: the tracked-criticality profile holds the ambient boundary factor at $u(\bar\varphi) \approx 1-10^{-5}$ at every epoch, under which the matter-metric redshift map cannot host the observed $z = 2.5$–$3.6$ absorber population without a negative ambient field. The unscreened temporal-disformal channel is thereby bounded sub-critical at these epochs, and an amplitude of the observed scale is admissible only through an environmentally modulated coupling $B_{\rm eff} = B(\phi)\,G(X_{\rm local})$ — a gate closed in the ambient (restoring the conformal map) that opens inside nonlinear wells. Gate 10-b quantifies the condition: the three sightlines require gate contrasts ${\rm d}G/{\rm d}\ln\delta\varphi = 4.2$–$6.8\times10^{-4}$ per e-fold of well depth, agreeing across the sample within a factor $1.6$ — consistent with a single saturating gate $G\sim10^{-3}$ inside wells (which also bounds the induced displacement in deeper structures below cluster-velocity systematics) — so the channel survives only under a beyond-$B(\phi)$ coupling of $D_2(\phi,X)$ class, reported as a measurable functional requirement rather than an achieved closure. Gate 10-c (step_10c, results/gate10c_ambient_lapse_margin.json) closes the loop from the admissibility side: evaluated on the realized absorber-epoch ambients ($\bar\varphi=\ln(1+z)$, $\dot u=H_T(z)\approx4$–$5\,H_0$ under the imported $\Lambda$CDM rate, $A\approx0.22$–$0.29$), the ambient lapse condition $u(\bar\varphi)=B(\bar\varphi)\dot{\bar\phi}^{\,2}/A^2<1$ caps the ungated canonical envelope at $B_0\lesssim0.03$ (cap minimum $z\simeq3$), and the negative branch is bounded at $|B_0|\lesssim10^{-2}$ by the ambient fluid wall $Z_t>0$ at absorber-epoch matter densities. Under the TEP-native drift ($H_T^2/H_0^2 \approx 0.22$–$0.44$ across the descent and selected-tracking branches) the cap loosens to $B_0\lesssim1.0$–$4.5$ and the required ambient suppression relaxes to $G_{\rm ambient}\lesssim0.01$–$0.06$ at $B_0=52$–$78$: phenomenological amplitudes remain supercritical on the bare ambient by a factor $\sim 30$–$50$ under the native rate ($\sim10^{3}$ under the imported rate), so ambient closure is lapse-mandated on either drift rather than merely phenomenological. The gate the amplitude analysis selects is therefore also the gate the signature ledger requires: the two requirements converge on the same $D_2(\phi,X)$-class form.

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 unscreened scalar charge — the Einstein-frame anomalous acceleration — is strongest where TEP requires it. The screening separation is the central TEP prediction. Using the Paper-0 PPN mapping $\gamma_{\rm PPN} - 1 = -2\beta_A^2\mathcal{S}_\Sigma^{(\odot)}/(1 + 2\beta_A^2\mathcal{S}_\Sigma^{(\odot)})$ under the unified unstretched-space metric (or $-4\beta_A^2\mathcal{S}_\Sigma^{(\odot)}/(1 + 2\beta_A^2\mathcal{S}_\Sigma^{(\odot)})$ on the isotropic-conformal benchmark) — linear in the screened source charge — the bare unscreened source charge ($\mathcal{S}_\Sigma^{(\odot)} = 1$) gives $\gamma_{\rm PPN} - 1 = -2/3$ ($-4/3$ on the isotropic benchmark) in the Solar System, violating the Cassini bound $|\gamma - 1| < 2.3 \times 10^{-5}$ by over four orders of magnitude. The solar screening value is an adopted single-source benchmark, not derived from the absorber-regime ansatz: an adopted conservative profile following the Paper 0 §7 source-charge convention takes $\mathcal{S}_\Sigma^{(\odot)} \approx 1.9 \times 10^{-6}$ (the derived kinetic operator returns $\mathcal{S}_\Sigma^{(\odot)} \simeq 1.6 \times 10^{-15}$ at 1 AU — nine orders stronger suppression; step_21), giving $\gamma_{\rm PPN} - 1 \approx -7.7 \times 10^{-6}$, 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 v0.16 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, $H_T(z)$ enters through $\partial_0\bar\phi = M_{\rm Pl}H_T(z)$ and is a property of the temporal-field background, not of the fitted distance–redshift map: the earlier identification $H_T = H_{\Lambda{\rm CDM}}$ substitutes the FLRW lookback–redshift relation for the field dynamics TEP replaces it with, and is retained below only as a foreign-rate baseline against which the TEP-native candidates are evaluated (Gate 10, section 7d). At the absorber redshift $z = 2.5042$, the cosmological field is $\bar\phi/M_{\rm Pl} = \ln(1+z)/|\beta_A| = 1.254$, and under the $\Lambda$CDM baseline 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 $\delta\phi/M_{\rm Pl} \sim 10^{-9}$–$10^{-8}$ is the Poisson-sourced contribution from the host halo (evaluated with the axiom-convention source $\rho_* A_{,\phi}$, i.e. carrying the ambient coupling $A(\bar\phi) \simeq 0.2$–$0.3$ at these epochs). 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 $\gamma_{\rm dis} \approx 8.4 \times 10^{-2}$ without environmental screening of $B$, exceeding the Cassini bound $|\gamma - 1| < 2.3 \times 10^{-5}$ by a factor $\sim 10^3$. However, if $B(\phi)$ is environmentally screened through the same $\mathcal{S}_\Sigma$ operator that screens the conformal coupling, the Solar-System projection is suppressed by $\mathcal{S}_\Sigma^{(\odot)} \lesssim 1.15 \times 10^{-5}$ under the unified unstretched-space metric ($5.8 \times 10^{-6}$ isotropic), bringing $\gamma_{\rm dis}$ within the Cassini bound. That screening is no longer a conditional option: under the derived Paper-0 §7 operator the solved solar gradient already carries $\mathcal{S}_\Sigma^{(\odot)} \simeq 1.6\times10^{-15}$ at 1 AU ($7.7\times10^{-25}$ at the solar surface), so the disformal spatial term at $B_0 = 2.88\times10^8\,{\rm eV}^{-4}$ evaluates to $\gamma_{\rm dis} \simeq 4\times10^{-42}$ — the Cassini bound is non-binding for every tested power-law normalisation (step_21). The earlier $n=2$ 'Cassini exclusion' was an artifact of the adopted conservative profile, not a derived bound; the binding constraints are the ambient-lapse self-consistency and cross-sightline transfer evaluated below. The quadratic $B \propto \phi^2$ realization at the normalization required to reproduce the Q1009 amplitude therefore remains viable, consistent with the TEP framework. A steeper field dependence can satisfy the local Cassini, GW170817 and clock constraints without screening. 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 without environmental screening of $B$, while $n = 5$ is excluded by cross-redshift transfer; neither tested power law provides a viable universal microscopic closure without additional structure. The v0.16 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 $\mathcal{S}_\Sigma^{(\odot)} = 1.9 \times 10^{-6}$, consistent with the Paper 0 §7 source-charge convention and below the Cassini ceiling $1.15 \times 10^{-5}$ ($5.8 \times 10^{-6}$ on the isotropic-conformal benchmark). This value is not inferred from the absorber $S_A^{\rm abs}$ ansatz; it is also conservative relative to the derived operator, which returns $\mathcal{S}_\Sigma^{(\odot)} \simeq 1.6\times10^{-15}$ at 1 AU (step_21, cross-checked against the Paper-0 step_54 tabulation).

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 $0.79$. The covariant evaluation uses the signed scalar source $\beta_A = -1$, which places the absorber on the physical branch $\delta\phi_{\rm core} > \delta\phi_{\rm edge} > 0$ required by the sign convention $\phi > 0$ in mass concentrations: the core perturbations are $+5.18\times10^{-9}$, $+7.56\times10^{-9}$, and $+3.36\times10^{-8}$ in units of $M_{\rm Pl}$ under the axiom source $\rho_* A_{,\phi}$ (the ambient factor $A(\bar\varphi)\simeq0.22$–$0.29$ suppresses the per-unit-density coupling relative to the earlier bare-$\rho$ convention). With the exact quasi-static form $\Delta v_T = c\,\ln(N_{\rm core}/N_{\rm edge})$, the physical branch gives $N_{\rm core} < N_{\rm edge}$ and hence $\Delta v_T < 0$ — a blueward edge feature relative to the deep-core anchor, matching the sign of the fitted displacements $-78.9$, $-76.0$, $-118.0$ km/s. The full three-dimensional field gives core–edge contrasts $\Delta(\delta\varphi) = 3.61\times10^{-9}$, $5.27\times10^{-9}$, and $2.30\times10^{-8}$ — factors of $0.23$, $0.33$, and $1.44$ relative to the reduced two-point value ($1.60\times10^{-8}$ for all three). The predictive closure with a single frozen $B(\phi) = B_0(\phi/M_{\rm Pl})^5$ gives $\chi^2 = 2.99$ 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 signed shifts of $-0.001$, $-0.41$, and $-0.06$ km/s versus the fitted $-78.9$, $-76.0$, $-118.0$ km/s. Per-sightline inverse fits exist on the corrected branch — each sightline admits a $B_0$ at $\sim 0.999$ of its own Lorentzian boundary ($1.46\times10^8$, $1.52\times10^7$, $2.00\times10^7$ eV$^{-4}$) — but the three values differ by a factor $\sim 10$, so no single $B_0$ closes the sample. 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 keeping all three absorbers simultaneously near-critical. The matter-frame maximum is $|\Delta v| \sim 0.005$ km/s, and the negative-$B$ branch is lapse-safe on this timelike-gradient channel with no amplification boundary, bounding $|\Delta v|$ at the conformal scale $\sim 0.05$ km/s; it remains subject to the ambient fluid-hyperbolicity wall $|B_0|\lesssim10^{-2}$ at these epochs (Gate 10-c). The Paper-0/28 disformal envelope $B(\phi) = B_0\,\varphi^2/(1+\varphi^2)\,\exp(-\varphi^4/2\sigma_B^4)$ was then scanned over $\sigma_B \in [1.0, 3.0]$ in the same covariant pipeline. The scan isolates the sign condition cleanly: the blueward displacement requires the boundary factor $u(\phi) = B(\phi)\dot{\bar\phi}^{\,2}/A^2(\phi)$ to rise into the well, i.e. a local logarithmic slope $n_{\rm eff} \equiv d\ln B/d\ln\varphi > -2\bar\varphi$ ($\approx -2.5$ to $-3.0$ for these absorbers), while simultaneous closure across the sample requires $B(\bar\varphi) \propto A^2/\dot{\bar\phi}^{\,2}$, i.e. $n_{\rm eff} = -2 - d\ln H_T^2/d\bar\varphi \approx -4.9$ under the imported $\dot{\bar\phi} = M_{\rm Pl}H_{\rm LCDM}$ rate. For $\sigma_B \lesssim 1.25$ the envelope declines too steeply ($n_{\rm eff} < -2\bar\varphi$) and the correct-sign branch closes for at most one sightline (at $\sigma_B = 1.25$ only Q1009+2956 admits an inverse fit); for $\sigma_B \gtrsim 1.5$ the correct-sign branch exists for all three sightlines and each admits a signed inverse-fit $B_0$ (e.g. at $\sigma_B = 2.0$: $8.0\times10^8$, $2.1\times10^8$, $2.5\times10^8$ eV$^{-4}$ for Q1009+2956, PKS 1937$-$101, J1332$+$0052), but the three values still differ by a factor $\sim 3$–$4$. The two requirements are disjoint under the imported rate, so the envelope improves on the pure power law — the correct sign is accessible and the inferred $B_0$ dispersion shrinks — without yet achieving three-sightline closure. The failure is thereby converted into a quantitative compatibility condition on the TEP temporal-rate function: with $s(z) \equiv d\ln H_T^2/d\ln(1+z) = d\ln\dot{\bar\phi}^{\,2}/d\bar\varphi$, simultaneous closure requires a local coupling slope $d\ln B/d\bar\varphi = -2 - s$ while the blueward sign requires $d\ln B/d\bar\varphi > -2$, so for a coupling of monotone slope the two requirements are compatible if and only if $s < 0$ — a descent-type ambient drift, non-increasing with field depth. This is not an ad hoc freedom: $H_T \equiv -d\ln A/dt$ is fixed by the cosmological temporal-field solution.

TEP-native rate evaluation (Gate 10, section 7d). The disjointness diagnosed above is a property of the imported rate, not of the disformal channel: $H_{\Lambda{\rm CDM}}$ is the metric-expansion rate of the cosmology TEP rejects, whereas $\partial_0\bar\phi$ is the drift of the temporal field on the static TEP background. Two corpus-derived candidates were therefore evaluated in the same covariant pipeline (step_10_covariant_transport, results/gate10_covariant_transport.json). The homogeneous closed-benchmark drift $H_t^2/H_0^2 = 1 + 3\Omega_0 z/(1+z)$ (Paper 0, step_12; $\Omega_0 \simeq 0.2998$) saturates at high redshift and carries $s \approx +0.12$–$0.16$ over the absorber range: under this rate the envelope inverse-fit dispersion collapses from a factor $\sim 3$–$4$ to $1.24$ at $\sigma_B = 1.31$, but the marginal sign gap remains ($s > 0$). The unsourced ambient sector on the reconstructed potential branch $V_{\rm rec}(u) = V_0 - (\bar\rho/2)\,e^{-u}$ obeys free-roll energy conservation on the static manifold, $\dot{\bar\phi}^{\,2} = \bar\Pi_\infty^2 + \bar\rho\,e^{-\bar\varphi}$, i.e. $H_T^2/H_0^2 = (1 + \kappa/(1+z))/(1+\kappa)$ with $\kappa = \bar\rho/\bar\Pi_\infty^2$; this descent law carries $s = -\kappa/(1+z+\kappa) < 0$ and satisfies the compatibility criterion for every $\kappa > 0$. The required coupling profile then reduces to $B(\bar\varphi) \propto e^{-(2+s)\bar\varphi}$, which in the descent-dominated limit $\kappa \to \infty$ is $B(\bar\varphi) \propto A(\bar\varphi)$: the disformal coupling tracks the conformal factor across the absorber field range.

Evaluated at the pure-descent limit, the descent-tracking realisation $B(\varphi) = B_0\,\varphi^4/(\varphi_c^4+\varphi^4)\,e^{-\lambda\varphi}$ ($\lambda = 1$, $\varphi_c = 0.1$) yields inverse-fit normalisations $B_0 = 7.5585$, $7.5583$, and $7.5582\times10^{10}$ eV$^{-4}$ for Q1009+2956, PKS 1937$-$101, and J1332$+$0052 respectively — a dispersion of $1.00004$, at the near-criticality floor set by the differing observed amplitudes. Each inverse root reproduces its own sightline by construction. Forward evaluation at their geometric mean, the single common $B_0$ inside that cluster ($7.5583\times10^{10}$ eV$^{-4}$), predicts $-18.41$, $-102.94$, and $-366.55$ km/s against the observed $-78.9$, $-76.0$, and $-118.0$ km/s (fractional $\chi^2 = 5.15$). The sign is blueward on every sightline. The amplitudes are not a common-$B_0$ prediction: the near-critical map is steep enough that a factor-$1.00004$ agreement among the inverse roots does not produce a common velocity. An earlier evaluation stepped below the cluster and reported $-12.7$, $-28.3$, and $-118.0$ km/s; that point is not the common root. The same coupling satisfies the local bounds with orders of magnitude to spare: $\gamma_{\rm dis} \approx 1.1\times10^{-14}$ at the Solar-System field depth (Cassini bound $|\gamma-1| < 2.3\times10^{-5}$) and a vanishing $\delta c/c$ at the ambient field (GW170817 bound $3\times10^{-15}$). The evaluation also closes a reporting gap in the envelope scan: the inverse-fit envelope solutions carry $\gamma_{\rm dis} \sim 5\times10^{-2}$–$17$ across the rate models, exceeding the Cassini bound by $\sim 10^3$–$10^6$ because the $\varphi^2$ low-field suppression is insufficient at the required normalisations; Cassini-compatible closure requires suppression at least as steep as $\varphi^3$, which the quartic-suppressed descent-tracking form satisfies by construction. The concordance is nevertheless tautological on this branch, and the branch fails ambient-lapse self-consistency. On the descent-tracking form $u(\varphi) = B(\varphi)\dot{\bar\phi}^{\,2}/A^2(\varphi) \propto e^{\delta\varphi}$ is constant along the entire descent: evaluated at the joint $B_0$, the ambient boundary factor is $u(\bar\varphi) = 0.99997$–$0.99999$ at all three sightline epochs (Gate 10, section 7d; results/gate10_covariant_transport.json, ambient_self_consistency). The factor-$1.00004$ agreement of the inverse roots is therefore a structural property of the flat-$u$ branch — every sightline's inverse fit returns the same ambient boundary factor by construction — not an empirical coincidence. The ambient sector on this branch sits permanently at near-criticality: the matter-metric lapse is $\tilde N = A\sqrt{1-u} \approx 3$–$5\times10^{-3}\,A$, so the placement map $1+z = \tilde N(0)/\tilde N(\bar\varphi)$ used to locate the absorbers is no longer the conformal map $\bar\varphi = \ln(1+z)$. An absorber at the conformal placement would present at $1+z \approx 646$–$1651$, and reproducing the observed $z = 2.5$–$3.6$ requires $\bar\varphi \approx -4.0$ to $-4.4$ — a negative ambient field in conflict with the corpus sign convention $\bar\varphi \ge 0$. The tracked-criticality branch is accordingly excluded by the ambient-lapse self-consistency requirement, and the argument generalises: because $u$ varies with field depth only through $e^{\delta\varphi}$ ($\delta\varphi \lesssim 10^{-7}$), any realisation that is near-critical inside a well is near-critical in the ambient field on the same branch. At sub-critical ambient $u$ the disformal correction to $\tilde N$ across the absorber cannot reach the observed $\sim 80$ km/s scale. The unscreened temporal-disformal amplitude channel is thereby excluded at these epochs; a contribution of the observed scale is possible only where $B(\phi)$ is itself environmentally screened, so that $u$ is sub-critical in the ambient while rising inside the well. The residual open element is thereby sharpened rather than removed: the absorber data bound the ambient drift realisation to sub-critical $u$ and select environmental screening of the disformal coupling, not the descent-tracking form, as the live candidate channel (Paper 0 §8).

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 under the imported expansion rate is not 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; the Gate-10 rate evaluation identifies it instead as a property of the imported $\Lambda$CDM drift, under which no single power-law $B(\phi)$ closes the sample. Under the TEP-native descent-drift law the same machinery admits a Cassini/GW170817-compliant descent-tracking coupling whose inverse roots agree to a factor $1.00004$; the common-$B_0$ forward velocities do not reproduce the three amplitudes (section 7d of the Gate-10 evaluation, reported above); and the ambient self-consistency check excludes the branch outright — the flat-$u$ realisation places the ambient boundary factor at $\approx 1-10^{-5}$ at every epoch, under which the matter-metric redshift map cannot produce the observed absorber population without a sign-violating negative ambient field. Environmental modulation of the disformal coupling, $B_{\rm eff}=B(\phi)\,G(X_{\rm local})$, is therefore not an additional option but the only surviving channel for a temporal-disformal amplitude of the observed scale, with the gate's required form measured by Gate 10-b.

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 emerges from the common nonlinear scalar solution (Paper 0 §7): the single kinetic operator $\mathcal{S}_\Sigma = 1/P_{,X}$ returns $\simeq 1.6\times10^{-15}$ at the solar field; at the LLS ambient the value is branch-dependent — $\sim 1.5$–$21$ on the static acceleration proxy, $\sim 0.13$–$0.18$ on the solved drift-dominated field at $z \sim 2.5$–$3.6$ — a discrimination factor of order $10^{16}$ (drift branch) to $10^{17}$ against the solar value, produced by the same stiffness function, not assumed (step_21, results/step_21_canonical_projection.json). 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 under the imported expansion rate; under the TEP-native descent-drift law the descent-tracking coupling $B(\varphi) \propto \varphi^4/(\varphi_c^4+\varphi^4)\,e^{-\lambda\varphi}$ concords the inverse roots at the criticality floor while satisfying Cassini and GW170817, but is excluded by ambient-lapse self-consistency (section 7d), leaving an environment-modulated disformal coupling — active inside wells, closed in the ambient, with the required gate slope measured in Gate 10-b — as the only admissible amplitude channel.

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 under the imported $\Lambda$CDM drift: 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 corpus disformal envelope $B(\phi) = B_0\,\varphi^{2}/(1+\varphi^{2})\,\exp(-\varphi^{4}/2\sigma_B^{4})$ (Paper 0 §2; employed as the strong-field realization in Paper 28), whose essential-singularity damping is designed for the near-critical regime, has now been evaluated in the same covariant pipeline over $\sigma_B \in [1.0, 3.0]$ (Gate 10, results/gate10_covariant_transport.json): under the imported rate the correct blueward sign is accessible for $\sigma_B \gtrsim 1.5$ and per-sightline inverse-fit solutions exist, but the inferred $B_0$ values retain a factor $\sim 3$–$4$ dispersion. The solar comparison distinguishes assigned-profile diagnostics from a nonlinear source solution. The adopted-gradient and acceleration-proxy columns in Gate 10 use independently assigned solar field values; their numerical PASS/FAIL flags do not establish a complete Cassini prediction. The selected Paper-0 sector already supplies a potential-plus-source solar profile (steps 53/57), with a surface gradient response of order $10^{-8}$ relative to the bare source. The disformal calculation must evaluate $B$ on the total field and use the gradient from that same solution, with the environmental boundary and propagation observable specified. The quartic small-field taper supplies strong suppression in the consistent-profile benchmark; a universal non-binding bound for every scanned coupling is not inferred from the proxy columns. The residual requirement reduces to a quantitative bound on the cosmological rate function: joint sign-and-closure compatibility requires $s \equiv d\ln H_T^{2}/d\ln(1+z) < 0$ over $z \simeq 2.5$–$3.6$, which the imported $\Lambda$CDM rate ($\approx +2.9$) fails, the corpus's homogeneous closed-benchmark drift ($\approx +0.12$–$0.16$) approaches but marginally misses, and the unsourced ambient descent law on the reconstructed potential — free-roll energy conservation $\dot{\bar\phi}^{\,2} = \bar\Pi_\infty^2 + \bar\rho\,e^{-\bar\varphi}$ on the static manifold — satisfies ($s = -\kappa/(1+z+\kappa) < 0$). Under the pure-descent limit the descent-tracking coupling $B(\varphi) = B_0\,\varphi^4/(\varphi_c^4+\varphi^4)\,e^{-\varphi}$ concords the inverse-fit $B_0$ values to a dispersion of $1.00004$, the criticality floor set by the observed amplitude scatter, with quartic suppression at small total field. A zero-field propagation diagnostic does not replace evaluation at the nonzero field and drift of a realized environment. The ambient self-consistency check then excludes the branch: $u(\bar\varphi) \approx 1-10^{-5}$ uniformly along the descent places the ambient sector at near-criticality, under which the matter-metric lapse map $1+z = \tilde N(0)/\tilde N(\bar\varphi)$ requires $\bar\varphi \approx -4.0$ to $-4.4$ to reproduce the observed absorber redshifts — a sign-violating field. The unscreened temporal-disformal channel is thereby bounded sub-critical at these epochs and cannot reach the observed amplitudes; the absorber data select environmental screening of $B(\phi)$, rather than the descent-tracking form, as the admissible realisation.

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; it is supplied by the derived kinetic operator of Paper 0 §7, evaluated at the absorber environment in step_21. That evaluation is branch-dependent: on the static acceleration proxy it returns $\mathcal{S}_\Sigma \sim 1.5$–$21$, while on the solved drift-dominated field at the absorber epoch (where the spatial/temporal gradient ratio is $\sim 10^{-4}$) it returns $\mathcal{S}_\Sigma \sim 0.13$–$0.18$, i.e. the conformal-gradient response is screened by a factor $\sim 6$. The two projections remain distinct: $\mathcal{S}_\Sigma$ acts on the source-charge/gradient response, while the ansatz's unscreened assignment concerns the amplitude channel $S_A^{\rm abs}$, which the corpus's amplitude/shear split preserves independently. 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 Einstein-frame anomalous acceleration 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 $\beta_A = -1.0$, i.e. $\alpha_0 = \sqrt{2}\,\beta_A = -\sqrt{2}$ in DEF normalization ^{[3], [54]}. The environmentally screened effective scalar charge for source-charge observables (the sector that fifth-force tests constrain) 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 $S_\Sigma^{(\odot)} \lesssim 1.15 \times 10^{-5}$ ($5.8 \times 10^{-6}$ on the isotropic benchmark). The relation between $S_A$ and $S_\Sigma$ is supplied by the derived kinetic operator of Paper 0 §7 — evaluated at the absorber ambient it is branch-dependent: $\mathcal{S}_\Sigma \sim 1.5$–$21$ on the static-branch acceleration proxy and $\mathcal{S}_\Sigma \sim 0.13$–$0.18$ on the solved drift-dominated field at $z \sim 2.5$–$3.6$ (step_21, drift-branch evaluation) — rather than by this two-parameter fit. The TEP coupling is screened where the ambient landscape is steep, satisfying solar-system bounds (Cassini); in the unscreened absorber regime, the bare coupling produces an Einstein-frame anomalous acceleration 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, $\gamma_{\rm PPN} - 1 = -2\beta_A^2 S_\Sigma/(1 + 2\beta_A^2 S_\Sigma)$ under the unified unstretched-space metric (or $-4\beta_A^2 S_\Sigma/(1 + 2\beta_A^2 S_\Sigma)$ on the isotropic-conformal benchmark; equivalently $-2\alpha_0\alpha_{\rm eff}/(1+\alpha_0\alpha_{\rm eff})$ with $\alpha_{\rm eff}=S_\Sigma\alpha_0$ and $\alpha_0=\sqrt{2}\,\beta_A$ in DEF normalization). 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 $S_\Sigma \ll 1$, the Cassini bound $|\gamma_{\rm PPN} - 1| < 2.3 \times 10^{-5}$ requires $S_\Sigma^{(\odot)} \lesssim 1.15 \times 10^{-5}$ ($5.8 \times 10^{-6}$ on the isotropic benchmark) in the Solar-System environment, a suppression supplied by the kinetic-sector stiffness mechanism — the slope of the clock-rate landscape pinned where the ambient shear is already steep, rather than the field amplitude flattened (Paper 0 §2.2, v0.16). 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 $S_\Sigma^{(\odot)}$ is supplied by the common kinetic operator evaluated in step_21, which reproduces the Cassini suppression at solar gradients ($\mathcal{S}_\Sigma \sim 10^{-15}$) while returning the absorber on the unscreened-to-anti-screening branch ($\mathcal{S}_\Sigma \sim 1.5$–$21$).

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 tests of the source-charge sector (conventionally reported as fifth-force constraints) probe the distinct source-charge projection $S_\Sigma$ ($S_\Sigma^{(\odot)} \lesssim 1.15 \times 10^{-5}$ unified, $5.8 \times 10^{-6}$ isotropic). No numerical galactic $S_A$ anchor is assumed here. The common microscopic scalar solution derives both projections (Paper 0 §7; evaluated for the BBN environments in step_21); that cross-domain closure is now in place and is not load-bearing 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.

Pressure sensitivity. HC Step 26 evaluates the prescribed sound speed through a supported near-dust sequence, with a zero-pressure CDM control and successive-limit checks. Over multipoles 30–2000, the limiting finite-pressure comparison changes TT and EE by approximately 5.44% and 6.08% RMS relative to its zero-pressure fluid control, and changes the lensing-potential spectrum substantially (99.57% RMS fractional difference). The zero-pressure control differs from CDM by at most 0.25% RMS across the tested spectra. These are phenomenological sensitivity results without a likelihood refit; the pressureless full-solver fit therefore cannot be identified with the finite-pressure effective model merely from agreement of a few peak locations.

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. The reference width uses the mixed thermal+turbulent prediction $b_D=\sqrt{b_{\rm turb}^2+2kT/m_D}$. The true-D injections use the fitted $M_{D,\rm free}$ temperature and turbulent width, which can differ from this parent reference; the ratio need not have null mean 1. The predictive consistency test therefore measures departures from the empirical injection mean, with each null draw excluded from its own reference estimate. The results are Q1009: observed ratio $0.99$, null mean $0.919$, calibrated $p=0.003$; PKS1937: observed ratio $1.31$, null mean $1.187$, calibrated $p=0.092$; J1332: observed ratio $5.31$, null mean $1.162$, calibrated $p=0.065$. J1332 remains a broad, prior-sensitive feature ($b_{H,\rm free}\approx51\,{\rm km\,s^{-1}}$), distinct from the narrow, cool edge-gas orphan phenotype. Its unrestricted-H likelihood preference does not by itself identify that physical mechanism. Expanded turbulent-width fits retain a likelihood preference but are not assigned an expanded-bound Monte-Carlo significance here.

Velocity deviation. The observable is the free-H velocity relative to the isotope-shifted parent position. The injection-calibrated results are Q1009: $+2.72\,{\rm km\,s^{-1}}$, centered calibrated $p=0.001$; PKS1937: $+5.66\,{\rm km\,s^{-1}}$, centered calibrated $p=0.001$; J1332: $-36.60\,{\rm km\,s^{-1}}$, centered calibrated $p=0.098$. The two high-resolution unsaturated systems test isotope locking most tightly; the saturated J1332 system has a much broader, non-Gaussian injection distribution. The calibration describes fitting scatter under the specified true-D model. Wavelength and parent-centroid uncertainties are evaluated separately below, and the $N=3$ column-density ordering is descriptive rather than an independently calibrated correlation test.

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. The reference is $N_{\rm HI}({\rm D/H})_{\rm BBN}$. Because the fitted true-D generating model has a free column density, its injected free-H column differences need not center on zero. The predictive test uses the empirical null mean consistently with the width and velocity diagnostics. The results are Q1009: observed $+0.034$ dex, null mean $-0.010$ dex, calibrated $p=0.001$; PKS1937: observed $-0.094$ dex, null mean $-0.204$ dex, calibrated $p=0.010$; J1332: observed $+0.698$ dex, null mean $-0.848$ dex, calibrated $p=0.015$. These correlated diagnostics are projections of the same spectra and are not multiplied as independent evidence.

Calibrated joint assessment. The three diagnostic coordinates are standardized by their null standard deviations. The covariance metric uses the correlation matrix with a dimensionless $10^{-6}$ ridge, preserving the result under changes of units. Each null draw is scored against the mean and covariance estimated without that draw; the observation is scored against all $N=1000$ injections. The table reports finite-sample $(k+1)/(N+1)$ probabilities and squared Mahalanobis distances. Zero exceedances give $p=0.000999$ at this resolution and a one-sided 95% binomial upper bound of approximately 0.003, rather than a measured zero tail probability.

Q1009, PKS1937 survive the separate Gate-8 family threshold $\alpha/3=0.0167$. Gate 8 reuses the Gate 2/3 spectra and fits and supplies diagnostic interpretation, not independent confirmation. J1332-specific width and velocity conclusions remain conditional on the declared turbulent-width bound.

Wavelength-calibration systematics. The relevant uncertainty is relative wavelength distortion within the same spectrum. The documented instrument-specific assumptions are $\sigma_{\rm sys}=0.25$, $0.05$ and $0.20\,{\rm km\,s^{-1}}$ for Q1009, PKS1937 and J1332, respectively. The Gaussian sensitivity approximation uses the observed displacement from the empirical injection mean and $\sigma_{\rm total}^2=\sigma_{\rm fit}^2+\sigma_{\rm sys}^2$. It gives Q1009: $9.16\sigma$, Gaussian-model $p=5.06e-20$; PKS1937: $10.99\sigma$, Gaussian-model $p=4.16e-28$; J1332: $1.97\sigma$, Gaussian-model $p=4.94e-02$. These small Gaussian tails are model extrapolations, distinct from the empirical injection p-values. The $100\,000$ sensitivity draws report exceedance counts and finite-sampling bounds; they do not resolve arbitrarily small tails. The common $[0,1]\,{\rm km\,s^{-1}}$ scan is retained in results/gate8_wavelength_systematics.json.

Parent-centroid blend bias. A blend can bias the fitted parent centroid and thereby shift the isotope anchor. The combined Gaussian sensitivity calculation uses the empirical-null-centered deviation and $\sigma_{\rm total}^2=\sigma_{\rm fit}^2+\sigma_{\rm sys}^2+\sigma_{\rm blend}^2$, with $100\,000$ Monte-Carlo draws per case. At $\sigma_{\rm blend}=1\,{\rm km\,s^{-1}}$, the Gaussian-model results are Q1009: $p=9.02\times10^{-3}$; PKS1937: $p=5.20\times10^{-7}$. For the distinct physical scenario in which centroid drag supplies the entire raw observed offset, a two-sided blend-only probability of 0.05 requires $\sigma_{\rm blend}=1.39$ and $2.89\,{\rm km\,s^{-1}}$ for Q1009 and PKS1937, respectively. The corresponding offset-to-parent-linewidth ratios are $0.15$ and $0.35$. The illustrative unresolved-companion estimate $f\gtrsim2|\Delta v|/b_{\rm parent}$ gives $0.29$ and $0.69$; this is a blend-geometry approximation rather than a fitted exclusion of every companion profile. The sign-matched blend-only product assumes independent, sign-random Gaussian biases across the two sightlines; it is a conditional sensitivity result and is not multiplied into the main spectral evidence. The full scans, assumptions and finite-sampling bounds are reported in results/gate8_parent_centroid_systematics.json.

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. Under the corrected bounded gate (Gate 10-b; Section 3) the disformal transport shift saturates near $|\Delta v| \approx 140$ km/s for any gate that reproduces the observed amplitudes, so features at $|\Delta v| > 140$ km/s cannot be TEP-shifted edge gas under the same activation law: the two reported pilot features at $-155.5$ and $-170$ km/s lie beyond this ceiling and are re-classified as ordinary forest H I, not viable orphan candidates. The pilot window is accordingly ceiling-constrained to $|\Delta v| \leq 140$ km/s in the corrected search.

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.

Fixed-window mirror pilot. Seven LLS systems in six KODIAQ quasar spectra were compared over the illustrative interval $|\Delta v|\in[50,140]$ km/s. The upper edge is an analysis-window choice, not a universal TEP transport ceiling: the existing activated scalar benchmark permits larger exterior clock contrasts. Candidates outside this window are outside this pilot selection, rather than physically excluded. The redward mirror provides a useful comparison for saturation and forest structure, but genuine deuterium also absorbs blueward near the isotope slot, so the true-D null is not mirror-symmetric. Across the seven sightlines the raw blue-wing depth excess is $+0.154$ (mean over sightlines); the largest single case is J162557$+$264448 ($+0.475$ depth, $+13$ narrow features, random-centre forest $p=0.10$). Random centres in the same spectra give a mean forest asymmetry of $+0.02$ and per-sightline probabilities of $0.10$–$0.57$; two of the seven systems share the same quasar spectrum, so an independence-based Fisher combination is not quoted as a calibrated probability. These quantities describe the forest comparison, not an isotope-D-versus-TEP likelihood or a TEP exclusion. A population discriminant requires matched true-D and shifted-H injections with the same line selection, saturation, forest and instrumental treatment. The primary Gate 2/3 spectral comparisons remain the calibrated model tests.

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 native tensor benchmark gives $r(k_{\rm pivot})=1.9\times10^{-9}$; the fiducial band maximum is $5.2\times10^{-7}$ and the scanned curvature-regular envelope is $1.4\times10^{-5}$ (Paper 27). These are different statistics and are not interchangeable.
• 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 canonical preferred-foliation matter metric $\tilde{g}_{\mu\nu} = -A_{\rm loc}^2(u_{\rm grav}(\mathbf x))\,U_\mu U_\nu + A_{\rm sp}^2(u_{\rm drift}(t,\mathbf x))\,\Pi_{\mu\nu} + B(\phi)\,\Pi_\mu^\alpha\Pi_\nu^\beta\nabla_\alpha\phi\nabla_\beta\phi$ (written in isotropic conformal–disformal shorthand as $\tilde{g}_{\mu\nu} = A^2 g_{\mu\nu} + B\nabla_\mu\phi\nabla_\nu\phi$), the two-branch scalar action, 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 scope of the deuterium argument is stated explicitly: the present analysis establishes that the benchmark absorber identifications are non-unique, not that a deuterium production channel exists. In an eternal recycling cosmology this distinction is binding — astration is a pure deuterium sink (deuterium burns at $T \gtrsim 5\times10^5$ K, below every stellar-core temperature), so the standing Galactic interstellar value ${\rm D/H} \approx 1.5$–$2\times10^{-5}$, measured independently of the quasar-absorber architecture criticised here, is a production requirement rather than a relic. Galactic-cosmic-ray spallation ($\sim10^{-7}$), pre-main-sequence envelope recycling, and pristine inflow each fail to supply it; a post-sequestration channel — horizon-adjacent spallation from the divergent external-time exposure of sequestered-core surfaces — remains contingent on the horizon-transport solution, but the channels quantified in Step 15 show that closure does not require it. The supply-versus-demand ledger is now quantified (Step 13, results/gate13_deuterium_production_ledger.json): holding the ISM at the observed abundance requires a yield of $2.0\times10^{-5}$ D per unit astrated mass, while cosmic-ray spallation and $\nu$-process helium-shell production supply $\sim10^{-7}$ and $\sim10^{-8}$ — shortfalls of $\sim120\times$ and $\sim2000\times$. For the horizon-adjacent channel the binding constraint is throughput rather than rate — external exposure is unbounded, so the emissivity per transit is not the limiter; the requirement is that a fraction $f$ of astrated mass transit a spallation-active surface and return with deuterium enrichment $\varepsilon_D$ satisfying $f\,\varepsilon_D = 2.0\times10^{-5}$ (e.g., $\varepsilon_D = 10^{-2}$ requires $f \approx 0.2\%$; enrichment only to the ISM level requires $f \approx 60\%$). The debt is thereby converted from an open-ended problem into a single throughput condition on the sequestered-core geometry. Step 15 (results/gate15_deuterium_extended_channels.json) then evaluates the channels that operate before or beside temporal-well formation, so that nothing needs to climb out of a formed well (the non-relativistic escape bound $\mathcal{Z}_{\rm max} \sim 6.7\times10^{3}$ of TEP-BH is not invoked). (i) Partial-burning and explosive-freezeout ejecta from very massive objects — the classical Wagoner–Fowler–Hoyle channel — carry documented enrichments $X_D \sim 2\times10^{-5}$–$4\times10^{-4}$, requiring a processed fraction $f \approx 0.05$–$1.0$ of astrated mass; at a representative envelope-transit fraction $f = 0.5$ the supply exceeds demand by a factor $\approx 5$, and the ejection geometry is the same one the helium equilibrium already requires at $f_{\rm VMO} = 0.994$. The Carr–Bond–Arnett metal co-production objection that historically removed this channel is answered by the architecture itself: the remnant is a temporal well that sequesters core metals while the processed envelope returns, not a standard event-horizon black hole. (ii) The collapse neutrino process (Fuller–Woosley–Weaver; Fuller & Shi 1997) reaches the same ejecta enrichment through $\bar\nu_e + p \to n + e^+$ irradiation of the envelope; its transport leg is fixed by the matter-metric axiom — MeV neutrinos are near-null on $\tilde g$ and irradiate the envelope during the collapse transient while the lapse is still of order unity. The irradiation budget is now parameterized (Step 17, results/step_17_inversion_and_channel_budgets.json): inverse-square antineutrino fluence through envelope shells gives capture probabilities $y_n$ exceeding the $10^{-3}$ level required for ejecta D/H $\sim 10^{-4}$ only inside a radius $r_{\rm crit} \simeq 0.002$–$0.22\,R_\star$ across the documented $10^{57}$–$10^{59}$ erg burst range — what fraction of the ejecta lies inside that radius is not determined by a static polytrope and awaits the dynamical temporal-well collapse solution; the transport itself is not a blocker, since MeV neutrinos are near-null on $\tilde g$ and the burst precedes deep-well formation. (iii) Bounded-exposure worldlines, which exist by the $p + q = 1.3 > 1$ astration-convergence condition, accumulate deuterium without a destruction term and close the ledger for $N_{\rm exp} \gtrsim 10^{3}$ exposure equivalents at mixing fractions $f_{\rm mix} \sim 1$–$12\%$, carrying the falsifiable signature of metal-poor reservoirs with super-ISM D/H — the same signature Fuller & Shi 1997 assigned to supermassive-star ejecta. The closure region is sharpened in Step 17 in net terms: returning gas at the ambient ISM abundance is not production, so at an envelope-transit fraction $f_{\rm ej} = 0.5$ the ejecta must reach $X_D \ge 7.5\times10^{-5}$ (D/H $\sim 3.7\times10^{-5}$) if the feed arrives at the ambient value — inside the documented $10^{-5}$–$2\times10^{-4}$ window but requiring a VMO structure solution to establish either number. For the absorber side, Step 17 also records an honest limit: published $\tau_{\rm eff}(z)$ cannot be inverted to mean neutral columns because $\tau_{\rm eff} = -\ln\langle F\rangle$ is nonlinear in saturated regions, so the wing-density confrontation is withheld rather than claimed; the assumed LLS population implies a LyC reach of only $\Delta z \sim 0.6$ at $z \sim 3$, confirming that trough ionization cannot draw on external background light. The deuterium requirement is therefore reduced to a VMO ejecta-fraction/enrichment calculation and a reservoir longevity/mixing calculation, with a single shared observational discriminator. The Gate 4B amplitude channel has been bounded by the ambient-lapse self-consistency requirement (Gate 10, section 7d): the descent-tracking realisation that concords the inverse roots places the ambient boundary factor at $u(\bar\varphi) \approx 1-10^{-5}$ on the entire branch, under which the matter-metric redshift map cannot host the observed absorber population without a negative ambient field; the unscreened channel is thereby excluded at these epochs, and a temporal-disformal amplitude of the observed scale is admissible only under an environment-modulated coupling $B_{\rm eff}=B(\phi)\,G(X_{\rm local})$; Gate 10-b (results/gate10b_disformal_gate_test.json) derives the measurable condition — a gate closed in the ambient and opening with a common slope ${\rm d}G/{\rm d}\ln\delta\varphi\sim(4$–$7)\times10^{-4}$ per e-fold across the three wells (agreement within a factor $1.6$), saturating near $G\sim10^{-3}$ so that deeper structures are not over-displaced — converting the amplitude question into a well-posed target for the disformal-sector derivation rather than an open loophole. The primary Gates 2/3 spectroscopic result is in any case independent of the 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. The full covariant evaluation (Gate 10) shows that the earlier closure obstruction is a property of the imported $\Lambda$CDM drift rate rather than of the disformal channel: joint sign-and-closure compatibility requires a descent-type ambient drift ($d\ln H_T^2/d\ln(1+z) < 0$), which the unsourced free-roll on the reconstructed corpus potential supplies, and under that rate a descent-tracking coupling $B(\varphi) = B_0\,\varphi^4/(\varphi_c^4+\varphi^4)\,e^{-\varphi}$ passes Cassini and GW170817, with inverse roots agreeing to a factor 1.00004. Forward evaluation at that common $B_0$ does not reproduce the three amplitudes, and ambient-lapse self-consistency excludes the branch outright: the flat-$u$ realisation holds the ambient sector at near-criticality, where the matter-metric redshift map would require a sign-violating negative ambient field. The unscreened temporal-disformal channel is accordingly bounded sub-critical at these epochs. 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. The scope of the deuterium argument is bounded accordingly: the analysis establishes that the benchmark absorber identifications are non-unique, not that a deuterium production channel exists. In an eternal recycling cosmology the distinction is binding — astration is a pure deuterium sink, so the standing Galactic interstellar value ${\rm D/H} \approx 1.5$–$2\times10^{-5}$, measured independently of the quasar-absorber architecture criticised here, is a production requirement rather than a relic; galactic-cosmic-ray spallation, pre-main-sequence envelope recycling, and pristine inflow each fail to supply it. The extended ledger (Step 15, results/gate15_deuterium_extended_channels.json) quantifies three channels that operate before or beside temporal-well formation — VMO partial-burning/freezeout ejecta, the collapse neutrino burst irradiating the envelope through near-null neutrinos on the matter metric, and protected-reservoir accumulation on bounded-exposure worldlines — of which the first and third can close the required net yield within documented ranges — at a transit fraction $f_{\rm ej} = 0.5$ the ejecta must reach $X_D \ge 7.5\times10^{-5}$ if the feed carries ambient ISM deuterium — reducing the open item to a VMO ejecta-enrichment calculation and a reservoir mixing rate rather than a missing mechanism. Independently, the opacity three-point requirement is shown to admit an explicit species-resolved profile (Step 14, results/step_14_opacity_three_point.json): the wing datum fixes $n_e \approx 0.51\,n_{b,0}$, the intervening trough is derived to be neutral ($\bar n_e < 3.7\times10^{-4}\,n_{b,0}$), and the ionization front at the Gate-11B thermalization depth places the $\tau = 1$ surface at $z \approx 1080$. The boundary's self-consistency items are then closed at toy-model level (Step 18, results/step_18_boundary_selfconsistency.json): the $\sim 4\times10^{10}$ wing-to-trough ionization-rate contrast is supplied by self-shielding over $\sim 24$ LyC e-folds; the front gas temperature is derived ($T_{\rm gas} = T_{\rm bath} = 2946$ K via Compton equilibrium on a $\sim$yr coupling time), making the Saha ionization a consequence of the Gate-11B bath rather than an assumption; the visibility tail is shown to be a closure artifact removed when the edge is absorptive; and the deuterium channels' residual budgets are bounded (transit neutrino irradiation under a stated homologous-collapse model, reservoir longevity reduced to a $\sim 10^{3}$–$10^{4}$ Gyr sequestration requirement). The remaining open item in the opacity sector is the perturbation mode spectrum on the boundary — the eternal-native Boltzmann treatment — together with the absorber-statistics inversion on a full LLS/Lyman-forest catalogue, which is the identified pipeline extension. The cosmological stability evidence must be read by sector. The homogeneous closed-background benchmark has an Eddington-type secular mode. The explicit Euclidean zero-mean scalar projection in Paper 0 (step_79) gives a positive lowest restricted eigenvalue, $+38.8145$ at $N=24$ and $+38.9129$ at $N=32$; its negative direction is almost spatially constant, with relative variation $2.2\times10^{-5}$. The discrete periodic Poincaré bound independently certifies all zero-mean modes above $38.80$ and $38.90$ on these grids. This is substantive stability evidence for the computed reduced sector. Identifying the constant direction as a gauge mode and establishing the full coupled constraint-reduced spectrum require the corresponding action-level analysis. The chemical attractors evaluated here therefore coexist with a tested stable nonconstant scalar sector; a global stability theorem is not inferred from either chemical convergence or that reduced spectrum.

TEP-BBN/
├── data/
│   ├── raw/                       # Spectroscopic exposures (Keck/HIRES, UVES+HIRES, ESPRESSO)
│   │   ├── atomic/                # Physical H I, D I line registries (NIST ASD)
│   │   └── reduced_products/      # Pre-reduced and co-added normalized spectra
│   ├── literature_components/     # Published VPFIT model files and component tables
│   └── processed/                 # Pipeline-ready union manifests
├── scripts/
│   ├── steps/                     # pipeline steps (01-17 + lettered substeps, secondary, finalize)
│   ├── lib/                       # Physical RT engine, Voigt fitters, model parsers
│   ├── utils/                     # Logging, isotopic shift, shear model utilities
│   └── run_pipeline.py            # Master orchestration script (4-phase pipeline)
├── configs/
│   ├── sightlines/                # Per-sightline JSON configs (bounds, MC settings, noise models)
│   └── TEP_BBN_FOUNDATION_FREEZE.json  # Frozen TEP prior parameters
├── results/                       # Generated parameter ledgers and significance matrices
├── logs/                          # Per-step execution logs
├── site/
│   └── components/                # Manuscript source components
├── requirements-lock.txt          # Locked Python dependencies
└── README.md                      # Documentation

# 1. Clone repository
git clone https://github.com/matthewsmawfield/TEP-BBN.git
cd TEP-BBN

# 2. Install dependencies
pip install -r requirements-lock.txt

# 3. Run full pipeline (default: Q1009+2956 sightline, 1000 MC sims)
python scripts/run_pipeline.py

# 3b. Quick mode (50 MC sims, for faster testing)
python scripts/run_pipeline.py --quick

# 3c. Run a specific sightline
python scripts/run_pipeline.py --sightline PKS1937-101

# 3d. Run all sightlines with ingested data
python scripts/run_pipeline.py --all-sightlines

# 3e. Skip secondary robustness analyses
python scripts/run_pipeline.py --skip-secondary

# 3f. Custom MC simulation count
python scripts/run_pipeline.py --n-sims 500

# 4. Results will be stored in results/ and logs/
#    Pipeline summary: results/pipeline_summary.json
#    SHA-256 manifest: results/checksums_sha256.json

Ionization balance. The opacity calculations use the stable positive root of $x_e^2=q(1-x_e)$. At the declared baryon density, photoionization equilibrium is $\Gamma(1-x_e)=\alpha_B n_b x_e^2$. The step-16 LTE-equivalent rates therefore agree at fixed temperature across its density ladder; these are conditional equilibrium requirements, not a measured radiation field.

Likelihood and tail evaluation. All finite data residuals contribute to the likelihood. Numerical failures are rejected. Gaussian sensitivity tails use the survival function; Monte-Carlo outputs include exceedance counts, sampling resolution and one-sided 95% binomial upper bounds.

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