Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries

6. Discussion: Resolving the Controversy

The wide-binary debate is often framed as a stark choice: either gravity is universally modified at low acceleration, or the observed signal is primarily artifactual. TEP offers a more specific interpretation—one that preserves the reality of the anomaly while explaining why the signal should appear with a finite transition scale and a measurable environmental dependence.

Screening projection notice. Screening in TEP is represented at theory level by the environmental operator S_Σ(E). Quantities such as ρ_T, R_T(M), S_⊕(r), compactness Φ/c^2, local stellar density, thermal epoch, coherence length, proximity, and boundary geometry are domain-specific projections of E, not independent screening mechanisms and not interchangeable universal thresholds.

6.1 Why a Scale-Free MOND Interpretation Is Incomplete

The anomalous velocity boost is detected with high significance. The more discriminating question is whether it is adequately described as a scale-free enhancement or whether it instead follows a resolved transition in separation. The fitted screening profile is favored over a flat Newtonian description by $\Delta \chi^2 = 14{,}845$ and over a constant boost by $\Delta \chi^2 = 3{,}583$. A dedicated Newtonian orbital forward model, conditioned on the observed projected-separation distribution and marginalized over line-of-sight geometry, orbital phase, and eccentricity, produces an outer normalized median $\tilde v \simeq 1.01$ for the thermal case versus the observed $\simeq 1.37$ ($\chi^2 \simeq 14{,}531$ against the observed profile). The anomaly is therefore not merely an overall low-acceleration excess; it has a finite, separation-structured form that arises naturally in TEP because the scalar field remains environmentally suppressed until the binary enters a sufficiently diffuse local regime.

The model comparison sharpens that conclusion. A sigmoid transition is much worse than the canonical phenomenological TEP recovery profile ($\Delta\chi^2 = +131.5$), and the thin-shell exterior profile of Section 2.2 is rejected more strongly still ($\Delta\chi^2 = +613.0$), while a more flexible double-exponential reduces the raw $\chi^2$ but still places the onset at a few thousand AU. The double-exponential's advantage is not evidence against TEP; it is a phenomenological proxy for the mass-dependent screening that TEP qualitatively predicts. A mass-convolved TEP model that integrates over the per-bin mass distribution with the derived radius law $R_s(M) \propto M^{1/2}$ (Equation $\ref{eq:screening_radius}$) captures 76% of the double-exponential's $\chi^2$ improvement using the same two free parameters, with the remainder attributable to environmental broadening from the $|Z|$ distribution within the sample. The self-screening test (Section 6.4, Twelfth) independently confirms the mass dependence. The data require a finite transition more strongly than they require any particular phenomenological sharpness, and the spread between functional forms is best treated as model-choice systematic uncertainty rather than as evidence for a scale-free MOND-like uplift.

A direct MOND comparison makes this concrete. Fitting MOND interpolating functions to the binned profile using per-bin median masses (Table 4.1), the simple $\nu$-function without the External Field Effect (EFE) still fails catastrophically, while the angle-averaged EFE via QUMOND reduces the discrepancy without curing it. Even the most generous MOND+EFE variant (simple $\nu$ with free $g_e$) remains worse than the TEP exponential by $\Delta\chi^2 = +540$. The failure is threefold: the preferred acceleration scale is driven away from the canonical value, the transition shape is too steep, and the EFE-limited plateau undershoots the observed saturation.

The near-coincidence between $a_0$ and the TEP screening transition is not accidental. Smawfield (2025g) showed that the SPARC rotation-curve database yields a characteristic transition acceleration $g_{\rm TEP} \approx 5 \times 10^{-10}$ m/s$^2$, within a factor of four of $a_0$, while Smawfield (2025e) showed that TEP produces the cosmological transition scale $g_t = cH_0/(2|\beta_A|) \approx 3.4 \times 10^{-10}$ m/s$^2$ of the same order as $a_0$ (within a factor of three), though without independently deriving the exact MOND numerical value. The distinction between the two frameworks is therefore not merely one of scale but of morphology: TEP produces a gradual, bounded enhancement whose amplitude is set by the local scalar-field value, whereas acceleration-dependent interpolation—even with the Galactic external field included—cannot simultaneously reproduce the transition shape, onset scale, and saturation level.

6.2 The Saturation Scale

The canonical fit gives a transition separation of $R_s = 2{,}646 \pm 182$ AU, where the quoted error is the formal statistical fit uncertainty; when jackknife stability and model-choice freedom are included conservatively, the total uncertainty broadens to $\pm 609$ AU. For the sample median mass ($M \approx 1.2\,M_\odot$), this transition scale corresponds to an effective binary density $\rho_{eff} \approx 9.2 \times 10^{-18}$ g/cm$^3$. Although the Temporal Topology saturation scale is far higher ($\rho_T \approx 20$ g/cm$^3$; Smawfield 2025g), the binary sits deep inside the screened Galactic potential. In the notation of Section 2.2, $\rho_{eff} \simeq \rho_{\rm floor} = \epsilon_{\rm env}\rho_T$, with $\epsilon_{\rm env} \sim 4.6 \times 10^{-19}$. As discussed there, $\rho_T$ and the scalar-field equation that governs $\epsilon_{\rm env}$ are both derived in earlier work; only the numerical evaluation of $\epsilon_{\rm env}$ in the specific Galactic environment remains to be computed from first principles. The observed transition density marks the point where the binary's internal Newtonian potential becomes sub-dominant to the pre-screened ambient background. The genuinely predictive test is the independent cross-check that follows.

The characteristic screening scale $R_s \approx 2{,}646 \pm 182$ AU is not merely a curve-fitting parameter; it represents the first kinematic detection of the Galactic screening floor. The canonical anchor is now the derived one: the master-action closure's acceleration scale $g_t = cH_0/(2|\beta_A|) \simeq 3.4\times10^{-10}$ m/s$^2$ gives the radius law $R_s = \sqrt{GM/g_t} \simeq 4.7$ kAU at the sample mean mass on the isolated-vacuum background, a factor $\simeq1.8$ above the fitted value—the expected direction, since the ambient Galactic floor displaces the transition inward (Section 2.2). Solving the source perturbation on the Galactic nonlinear background instead (the embedded-propagator check of step 017) compresses the same estimator to $\simeq3.3$ kAU, a factor $\simeq1.26$ above the fitted value within the systematic band, with the half-excess marker coinciding with the observed $\simeq1.7$ kAU: the mean-scale offset is substantially the ambient-stiffness propagator effect, not a free environmental coefficient. The older empirical anchor provides a complementary cross-check: the SPARC-extracted $g_{\rm TEP} \approx 5 \times 10^{-10}$ m/s$^2$ (Smawfield 2025g) predicts $R_s^{\rm pred} \approx 3{,}929$ AU for the transition-bin median mass ($M \approx 1.3\,M_\odot$; factor 1.48), tightening to a factor of 1.05 ($R_s^{\rm pred} \approx 2{,}778$ AU) once the Galactic external field is included as an additional screening floor ($\eta=2$). The wide-binary anomaly is therefore quantitatively consistent with the cross-scale screening scale of the TEP conformal sector.

6.3 Sensitivity to the Mass-Luminosity Relation

The environmental ordering depends critically on the stellar mass estimates. If a solar-metallicity Mass-Luminosity Relation (MLR) is applied uniformly, the masses of metal-poor high-$|Z|$ stars are systematically overestimated, the Keplerian baseline $v_c = \sqrt{GM/s}$ is inflated, and the inferred velocity ratio $\tilde{v}$ in the high-$|Z|$ population is suppressed. The color-dependent MLR correction (Section 3.2) removes this bias by adjusting masses according to each star's offset from the disk color-magnitude ridge, but it now does so without circular beta calibration: spectroscopic metallicities are used if cached, otherwise a conservative external prior is adopted and stress-tested. The corrected analysis recovers the expected ordering: the high-$|Z|$ transition radius is smaller than the midplane value for both the full sample ($R_s = 4{,}662 \pm 196$ AU versus $7{,}131 \pm 1{,}341$ AU) and the solar-track control ($R_s = 4{,}145 \pm 276$ AU versus $6{,}856 \pm 920$ AU; $p \le 10^{-4}$ for the full sample and $p \le 10^{-3}$ for the solar track). That the solar-track control—which uses an empirical mass calibration with no color-dependent correction at all—independently recovers the same ordering makes the result difficult to dismiss as an MLR artifact.

Supplemental controls narrow the most common non-TEP interpretations still further. If the observed profile were driven mainly by one demographic subset, the transition would collapse under stratification by mass ratio, primary mass, or observational geometry. Instead, the signal persists across all four stellar-population half-samples—each remaining strongly preferred over a constant boost—and is still recovered in a stricter radial-velocity consistency subset of 6,117 systems, though with broader uncertainty owing to the much smaller sample size. Conversely, when catalog-level labels are scrambled, either globally or within distance quartiles, none of $10{,}000$ valid realizations reproduces the observed screening preference ($p \le 10^{-4}$). The data indicate that the Gaia DR3 anomaly is structured, population-robust, and difficult to reduce to a single calibration artifact.

6.4 Limitations and Outlook

Several caveats deserve explicit acknowledgment. First, the reduced $\chi^2_{\nu} = 5.1$ for the canonical diagonal fit indicates that the diagonal error model does not capture the full scatter. A residual autocorrelation analysis identifies significant lag-1 correlation ($\rho_1 = 0.491$, $z = 2.14$, Durbin–Watson $= 0.97$), consistent with spatially correlated substructure or distance-correlated selection effects. Two covariance-aware refits address this directly: an AR(1) GLS model ($\chi^2_{\nu} = 0.97$) and a Gaussian Process with a squared-exponential kernel in $\log_{10}(s)$ space ($\chi^2_{\nu} = 1.01$). The GP is preferred by marginal log-likelihood and finds short-range correlated scatter ($\sigma_f = 0.015$, $\ell = 0.18$ dex). Under both covariance models all model-comparison conclusions are preserved, and parameters remain within the existing systematic uncertainty budget.

A direct investigation of the physical origin of this scatter finds no compelling single-population explanation. The standardized residuals show only weak rank correlation with median heliocentric distance, distance spread, and $\tilde{v}$ kurtosis, while the lag-1 residual autocorrelation rises from $\rho_1 = -0.05$ in the nearest distance quartile to $\rho_1 = 0.56$ in the most distant quartile. The correlated structure therefore appears concentrated in the distant subsample, consistent with known Gaia DR3 astrometric systematics at larger heliocentric distances ($d > 250$ pc), where parallax zero-point corrections and scanning-law artifacts naturally introduce spatially correlated covariance, rather than a localized astrophysical contaminant.

Second, the null-control and environmental permutation tests use $10{,}000$ realizations each, giving a $p$-value resolution floor of $10^{-4}$. No realization in either ensemble reproduced the observed screening preference, so the significance statements are limited primarily by the permutation-grid resolution rather than by an assumed parametric error model.

Third, an injection-recovery test validates that the pipeline recovers a known screening signal and does not hallucinate one when none is present. Mock catalogs are now constructed from the step_012 projected Newtonian forward-model null (rather than from globally shuffled observed $\tilde{v}$ values), then injected with the observed TEP enhancement parameters ($R_s = 2{,}646$ AU, $\alpha_{\rm sat} = 0.366$). This directly tests recovery against an orbital-population null that includes projection, phase-mixing, and eccentricity structure.

Fourth, dedicated triple-contamination forward models directly test the claim that unresolved hierarchies can mimic the observed profile, with the photocentric reflex velocity treated as a vector added to the orbital tangential velocity—the observed speed $|\mathbf{v}_{\rm orb}+\mathbf{v}_{\rm phot}|$ is positive-definite, so the kick can only elevate the median. The RUWE $< 1.2$ cut leaves residual triples with wide inner orbits whose kicks are individually large but diluted by median robustness: at $10\%$ residual contamination the predicted outer-bin median reaches only $\tilde{v} \approx 1.05$ versus the observed $\tilde{v} \approx 1.366$, and the deficit shrinks smoothly with fraction rather than growing—closing only at $f_{\rm triple}\approx 50\%$, an order of magnitude above the $\sim 5$–$10\%$ residual fraction permitted by RUWE detection incompleteness. The Pittordis-faithful population model, which draws inner orbits and orientations from the Pittordis et al. (2025) distributions, performs still worse—$\tilde{v} \approx 1.02$ at $10\%$ and only $1.11$ even at $50\%$ contamination—because the sky projection of the inner-binary reflex velocity further suppresses the observable kick. No physically plausible triple fraction closes the gap; the median statistic is too robust for residual hierarchies to reproduce the observed enhancement.

Fifth, the observable $\tilde{v}$ is a projected quantity. A Monte Carlo orbital simulation quantifies the residual eccentricity systematic on $\alpha_{\rm sat}$. Relative to the thermal case, the uniform distribution recovers about 14% higher $\alpha_{\rm sat}$ and the most super-thermal case about $-7\%$ lower, while the realistic interval remains bounded between $-5\%$ and $+5\%$. The transition radius remains stable to about 7% across the full sweep, so the eccentricity systematic does not propagate materially into $R_s$ or the model-comparison hierarchy.

Sixth, distance selection and mass calibration remain obvious places to look for failure modes. Both distance halves independently favor a finite transition, with the nearer subsample yielding $R_s = 3{,}551 \pm 420$ AU. Likewise, four alternative MLR prescriptions preserve the environmental ordering, with midplane $R_s$ spanning $6{,}911$ to $8{,}582$ AU and high-$|Z|$ $R_s$ spanning $4{,}156$ to $7{,}215$ AU. The result is therefore robust to the reasonable range of calibration choices already explored.

Seventh, the MOND comparison in Table 4.1 is already nontrivial, but it is not exhaustive. Marginalizing over full within-bin mass distributions or testing alternative EFE geometries may reduce the discrepancy somewhat. Even so, the present best MOND+EFE variant remains worse than TEP by $\Delta\chi^2 = +540$, so any viable rescue must overcome simultaneous failures in acceleration scale, transition shape, and outer-plateau amplitude.

Eighth, several quantities imported from other TEP papers—the Temporal Topology saturation scale $\rho_T \approx 20$ g/cm$^3$, the characteristic acceleration $g_{\rm TEP} \approx 5 \times 10^{-10}$ m/s$^2$ (Smawfield 2025g), and the emergent MOND-scale acceleration $a_0 \sim cH_0$ (Smawfield 2025e)—are drawn from preprints not yet independently peer-reviewed. The present analysis does not depend on the precise values of $\rho_T$ or $g_{\rm TEP}$ for any primary result; those quantities enter only in the cross-scale consistency check of Section 6.2. Readers should regard that check as a promising but provisional link pending independent verification.

Ninth, the pre-screening factor $\epsilon_{\rm env}$ is no longer fully post-hoc. Adopting the empirical density scaling $R_s \propto \rho_{\rm amb}^{p}$ as a tractable benchmark (TEP itself does not commit to a specific microscopic mechanism; Smawfield 2025a, §A4), the scaling relation predicts the two-bin environmental ratio within the quoted errors, and the inferred exponent from the two-bin fit is $p = 0.40 \pm 0.14$. A finer stratification into 5 equal-count $|Z|$ bins, each containing 68{,}263 systems, yields transition radii declining from $R_s = 8{,}670 \pm 1{,}988$ AU at median $|Z| = 22$ pc to $R_s = 4{,}326 \pm 252$ AU at median $|Z| = 347$ pc (a selection audit notes that these radii come from full-range fits whose outer bins above $\sim$9 kAU carry the purity-cut truncation: restricted to the selection-insensitive window the dilute-strata profiles remain unsaturated at the window edge, so the quoted radii are effective values of the truncated profiles; the selection-safe statement is the monotone median-$\tilde v$ ordering reported in Section 4, which survives every threshold including none at all). A direct density-scaling fit to these five points—using the actual three-component Galactic density model $\rho(Z)$ rather than a log-linear approximation—yields an exponent $p = 0.50 \pm 0.03$ under the anchored one-parameter estimator (amplitude fixed at the first bin). A free two-parameter fit to the same five radii returns $p = 0.36\text{--}0.39$, and the quoted $\pm 0.03$ is the raw $\Delta\chi^2 = 1$ error of a fit with $\chi^2 = 10.7$ on 3 dof (rescaled $\pm 0.06$); the honest data-side exponent is therefore $p \approx 0.4 \pm 0.1$, centered between the $\rho^{1/3}$ and $\rho^{1/2}$ ambient channels and matching the noise-corrected ambient inversion ($0.40 \pm 0.05$). An independent inversion through the ambient response parameter $u_{\rm eff}$, with the eccentricity law freed as a mixture (Section 5.1), returns $p \approx 0.7$ under the data-preferred uniform composition and an honest band $p \approx 0.5$–$0.7$ across eccentricity-law systematics; the two estimators jointly bracket the exponent in $0.4$–$0.7$, consistent with the $\rho^{1/2}$ benchmark. A complementary weighted linear regression of $\ln R_s$ versus $|Z|$ gives a slope $-0.00162 \pm 0.00036$ pc$^{-1}$ ($t = -4.45$, significance $0.021$), which is 97% of the $p = 1/2$ predicted slope ($-0.00167$ pc$^{-1}$ for $h_{\rm eff} = 300$ pc). The fine-$|Z|$ stratification therefore not only confirms the downward trend but provides a precise, physically grounded measurement of the environmental density-scaling exponent, equal to the square-root scaling $p = 1/2$ within a few percent. This empirical scaling now has a mechanism-side check (step 017, supply-versus-demand): the free-$\alpha$ amplitudes invert to a required ambient contrast of $\simeq36\%$ rising toward the midplane, and the corpus's density-set ambient supplies it — the disk density ratio between the strata's median heights is $\rho_{\rm mid}/\rho_{\rm halo}=2.44$, giving $34.6\%$ in the ambient variable under $u_{\rm eff}\propto\rho^{1/3}$ against $35.8\%$ required, with the now-identified $u_{\rm eff}\propto\rho^{1/2}$ k-branch law (Paper 0, AUD-3 closure) bracketing the same requirement at $56\%$ under the median-density mapping; the two-bin contrast is an estimator-level check that brackets both, while the five-stratum inversion fixes the exponent at $p\approx0.46$–$0.5$. The linear-gradient estimator $u=|a|/g_t$ fails by construction ($\sim4\%$ with the wrong sign: the radial floor is common to both heights and the vertical field adds with $|Z|$), but that estimator is the weak-field exterior reading — inside the nonlinear disk the field sits at its density-set equilibrium and the local acceleration is not the ambient state. The ordering therefore discriminates between ambient projections and selects the density-set one — the corpus's own in-medium response channel, now identified at coefficient level as the local k-branch law $u_{\rm amb}=\sqrt{4\sqrt{2}\pi G\rho L/(K g_t)}$ with coherence length $L\simeq0.84$ kpc — rather than a fitted covariate. A matched covariate audit (step 018) closes the demographic alternative on the observational side: nearest-neighbour matching of the disk and halo subsamples on total mass, projected separation, distance and [Fe/H] reduces the $\Delta R_s$ contrast from 2{,}468 AU to 1{,}935 AU — only $\simeq22\%$ absorbed — and the mass covariate runs the wrong way (the halo systems are the more massive, which naively predicts the larger halo $R_s$, opposite to observation); the ordering survives every matched experiment. The ambient projection entering the pair vertex is now derived rather than assumed: the embedded pair response on the constraint slice is independent of the local background density ($\lesssim0.02\%$ across the disk range) and steep in the ambient boundary field (charge $\propto u_{\rm amb}^{-2.0}$), so the vertex samples the landscape ambient map — identified as the local k-branch density response $\rho^{1/2}$ with the $L \simeq 0.84$ kpc coherence length reproducing all five strata within $10.1\%$ (Paper 0, AUD-3 closure).

Tenth, the environmental comparison in Section 5.1 presents fixed-$\alpha_{\rm sat}$ results as the primary protocol because the amplitude–scale degeneracy prevents clean isolation of $R_s$ when $\alpha_{\rm sat}$ floats independently in each subsample. A joint profile-likelihood fit eliminates this concern: both subsamples are modeled simultaneously with a single shared $\alpha_{\rm sat}$ and separate transition radii, yielding $R_s = 4{,}912 \pm 792$ AU (midplane) versus $R_s = 3{,}447 \pm 366$ AU (high-$|Z|$) with $\alpha_{\rm sat} = 0.332 \pm 0.020$. A likelihood ratio test gives $\Delta\chi^2 = 60.3$ ($p = 8.1 \times 10^{-15}$), and the ordering is preserved in 100% of parametric-resampling realizations.

A related estimator dependence deserves explicit statement: every stratified or subset estimate of $R_s$ lies above the pooled headline value of $2{,}646$ AU (midplane $7{,}131$ AU and high-$|Z|$ $4{,}662$ AU under the fixed-$\alpha_{\rm sat}$ protocol; $4{,}912$ and $3{,}447$ AU under the shared-$\alpha_{\rm sat}$ joint fit; $8{,}670 \to 4{,}326$ AU across the five $|Z|$ bins; $3{,}551$ AU and $10{,}307$ AU for the near and far distance halves; $7{,}709$ AU for the radial-velocity consistency subset; and up to $3{,}743$ AU across the normalization sweep). This is an estimator-offset phenomenon, not an internal contradiction: the pooled estimate is a free two-parameter fit to a single 19-bin profile that mixes all environments, whereas each stratified estimate applies a different protocol (fixed or shared $\alpha_{\rm sat}$, or a subsample-specific normalization window) to a homogeneous subpopulation. A pooled single-scale fit to a heterogeneous mixture is not obliged to return a value interior to the stratum range—a mixture of exponentials with different scales and amplitudes is not itself a single exponential, and the diffuse environmental tail contributes an early partial rise that pulls the pooled onset inward relative to any resolved stratum. The pooled value is retained as the headline scale for conservatism and continuity with the primary fit; it should be read as the minimum onset scale of the transition, while the environmentally resolved estimators return larger effective scales. Notably, the resolved estimates cluster nearer to the independent cross-scale prediction $R_s^{\rm pred} \approx 3{,}929$ AU from $g_{\rm TEP}$ (Section 6.2) than the pooled value does, so resolving the estimator dependence moves the measurement toward, not away from, the TEP cross-scale anchor.

Eleventh, at the shortest separations unresolved spectroscopic binaries could in principle deflate the inner normalization and inflate the apparent outer enhancement. The normalization sensitivity sweep tests this directly by refitting the canonical model under every contiguous baseline window that remains plausibly within the screened core. Across those screened-core windows the recovered $R_s$ ranges from $2{,}196$ to $3{,}743$ AU, with a maximum deviation of 41% above the fiducial value. The transition scale is therefore stable while the apparent saturation amplitude responds to the normalization, which is the opposite of the pattern expected from a baseline-deflation artifact. Crucially, while the apparent saturation amplitude $\alpha_{\rm sat}$ modulates with the chosen baseline, the transition scale $R_s$ remains bounded within the 2,196–3,743 AU range across the sweep. This demonstrates that the tested baseline-deflation variations cannot manufacture a fictitious onset scale.

Twelfth, the saturation amplitude $\alpha_{\rm sat}$ varies systematically across the demographic half-samples in a pattern consistent with mass-dependent self-screening. The high-primary-mass subset yields $\alpha_{\rm sat} = 0.237$, the full sample $0.366$, and the low-primary-mass subset $\alpha_{\rm sat} = 0.509$ ($\Delta\chi^2 = 2{,}234$ versus a constant boost for the strongest split). A quantitative self-screening model $\alpha_{\rm sat}(M) = \alpha_{0,\rm WB}\,\exp(-M/M_{\rm screen})$ fits these three data points with $\chi^2 = 1.25$ for one degree of freedom, yielding an illustrative unsuppressed wide-binary response amplitude $\alpha_{0,\rm WB} = 1.74 \pm 0.26$ and a self-screening mass scale $M_{\rm screen} = 0.46 \pm 0.04\,M_\odot$. As a two-parameter fit to three points, this model is presented as illustrative rather than definitive evidence. The unbounded exponential extrapolation is not, however, the physically meaningful limit: the response amplitude is structurally bounded by $\alpha_{\rm sat} = \sqrt{1+2\beta_A^2\,\mathcal S_{\rm eff}}-1 \le \sqrt{3}-1$, and in the Galactic ambient the plateau is further capped by the ambient vertex product. Re-fitting the same three points with the bounded recovery form $\alpha_{\rm sat}(M) = \sqrt{1+2\,\mathcal S_{\rm amb}\,y(M)}-1$, where $y(M) = [1+(M/M_s)^p]^{-1}$ is the self-screening recovery fraction, interpolates the data exactly and yields $\alpha_{0,\rm WB} = \sqrt{1+2\mathcal S_{\rm amb}}-1 = 0.65 \pm 0.09$ (Monte-Carlo propagation of the point errors; the saturated fit itself carries no degrees of freedom) with $M_s = 0.72\,M_\odot$ and $p = 3.7$ — within the absolute ceiling and consistent with the ambient-floor-limited plateau. The apparent $\alpha_{0,\rm WB} > 1$ value is therefore an artifact of the exponential ansatz extrapolating through the response ceiling; the bounded form demonstrates that the demographic trend requires no super-ceiling amplitude. Here $\alpha_{0,\rm WB}$ is the asymptotic response amplitude of the wide-binary kinematic channel, not the microscopic conformal coupling $\beta_A$. Cassini, millisecond-pulsar timing, and wide binaries probe different environmentally projected responses of the same universal conformal sector through $\mathcal S_\Sigma(\mathcal E)$: the Cassini PPN bound $S_\Sigma^{(\odot)} \lesssim 1.15\times10^{-5}$ under the unified unstretched-space metric ($5.8\times10^{-6}$ isotropic) on the screened solar source charge (Smawfield 2025a, Section 7), the compact-object regime screened full response $\tilde\kappa_{\rm MSP} \sim 2.9 \times 10^4$ extracted from millisecond-pulsar spin-down in dense globular clusters (Smawfield 2026a; the unsuppressed geometric factor $\sim 10^6$ characterizes the dense-environment upper bound), and the wide-binary response amplitude $\alpha_{0,\rm WB}$ each sample a different environmental projection of the same underlying screening hierarchy. The relevant cross-scale check is qualitative: the same conformal-sector physics that screens the Temporal Shear in dense regimes and recovers it in low-density regimes underlies all three measurements. Notably, the self-screening model is itself a continuous exponential function of mass—there is no thin-shell step function or sharp screening boundary—consistent with the Temporal Topology framework in which self-screening operates via continuous flattening of the field profile rather than a thin-shell transition.

Thirteenth, the environmental subsamples are not exchangeable populations: Galactic $|Z|$ correlates with age, kinematic heating, and metallicity, so the midplane and high-$|Z|$ groups can differ in inner-binary demographics—period and mass-ratio distributions that modulate photocentric contamination independently of the RUWE screen. The metallicity axis of that correlation is controlled: the solar-track subsample restricts both $|Z|$ groups to the zero-correction solar-metallicity sequence, and the ordering survives there at $p \le 10^{-3}$. The matched distance, color, and astrometric-quality controls further exclude the observable demographic axes. The surviving uncovariate is the age-correlated component of the inner-binary period distribution, which no catalog quantity proxies; it is bounded indirectly by the triple-contamination forward models (Fourth), which show that residual hierarchies elevate the outer median by at most $\sim 0.05$–$0.08$ at realistic fractions, and by the sign constraint that an age-correlated contamination gradient would need to elevate the high-$|Z|$ profile specifically at few-thousand-AU separations while leaving the midplane unaffected—a morphology no identified contaminant produces. A definitive control requires an inner-binary-resolved catalog (e.g., epoch astrometry or radial-velocity monitoring in DR4+); the channel is recorded as the leading open demographic caveat.

6.5 Systematic Controls Summary

The following table collects the robustness checks performed across the analysis pipeline. Each row tests whether the primary conclusions—a finite screening transition and environmental ordering—survive a specific perturbation to the data selection, error model, mass calibration, or binning procedure.

No single control eliminates every conceivable systematic, but the breadth of this matrix—spanning data selection, error modeling, mass calibration, binning, projection, contamination, and theoretical prediction—makes it difficult to construct a single astrophysical or methodological scenario that simultaneously survives all tests while mimicking both the transition morphology and the environmental ordering.

6.6 Distinguishing TEP from Generic Scalar Screening

The density-scaling benchmark in Section 5.2 demonstrates that the observed environmental modulation follows a square-root power law in ambient baryonic density. The inverse-power chameleon form that historically motivates the parameterization $p = 1/(n+1)$ is excluded under the TEP coupling convention—with $\beta_A = -1$ and $\phi > 0$ near masses, $V \propto \phi^{-n}$ admits no admissible minimum (Smawfield 2025a, F1)—so the measured exponent is phenomenology rather than a claimed microscopic realization. A fair question is therefore: what distinguishes TEP from the scalar-screening family generally? TEP itself does not commit to a microscopic completion (Smawfield 2025a, §A4, §7); chameleon-type, Vainshtein, Galileon, DBI, and symmetron mechanisms remain candidates where they satisfy the F1–F6 constraints, with the defining ontology being continuous suppression of the Temporal Shear $\Sigma_\mu = \nabla_\mu \ln A$ by source-charge and environmental state. Four features are specific to TEP rather than generic to the broader scalar-screening family:

1. The cross-scale anchor. The transition acceleration $g_N(R_s) \approx 1.1 \times 10^{-9}$ m/s$^2$ at the observed screening radius is within a factor of $\sim 2$ of the independently measured $g_{\rm TEP} \approx 5 \times 10^{-10}$ m/s$^2$ from SPARC rotation curves (Smawfield 2025g), with no parameter adjustment. Generic stand-alone scalar-screening models have no independent cross-scale anchor for the wide-binary transition. In TEP, the absolute regime of the transition is independently anchored by the Temporal Topology saturation scale $\rho_T$ and the SPARC-derived $g_{\rm TEP}$; the microscopic completion determines the detailed environmental mapping within that anchored regime.
2. The mass-dependent saturation amplitude. The monotonic progression $\alpha_{\rm sat} = 0.237$ (high mass) $\to$ $0.366$ (full sample) $\to$ $0.509$ (low mass) is a natural consequence of TEP's conformal coupling, where self-screening by the binary's internal potential attenuates the velocity-profile amplitude. Standard chameleon models predict mass-dependent screening radii but not a specific amplitude hierarchy tied to an independently anchored cross-scale conformal-response architecture.
3. The emergent $a_0$. TEP produces a cosmological acceleration scale $g_t = cH_0/(2|\beta_A|) \approx 3.4 \times 10^{-10}$ m/s$^2$ from the master-action closure (Smawfield 2025e), within a factor of three of the empirical MOND value $a_0 \approx 1.2 \times 10^{-10}$ m/s$^2$. While TEP does not independently derive the exact MOND numerical value, the emergence of a cosmological acceleration scale of the same order as $a_0$ and $g_{\rm TEP}$ from the same scalar-field structure is not generic to stand-alone chameleon or symmetron models.
4. Continuous Temporal Topology and the proper-time signature. The standard chameleon thin-shell approximation (Khoury & Weltman 2004) concentrates the active scalar variation into a narrow shell and yields the corresponding thin-shell exterior response, producing a profile with a geometric $R_s/s$ prefactor. TEP is not committed to thin-shell matching: its defining ontology is continuous Temporal Topology (Smawfield 2025a, §7), where screening operates via the smooth spatial profile of $\ln A(\phi)$, the locally observable Temporal Shear $\Sigma_\mu = \nabla_\mu \ln A$ is suppressed continuously in dense environments rather than concentrated in a narrow shell, and the same conformal factor $A(\phi)$ that sources the kinematic enhancement also rescales matter-frame proper time as $\mathrm{d}\tau/\mathrm{d}t = A(\phi)$. TEP identifies the recovered conformal gradient specifically with dynamical proper-time geometry, tying the wide-binary kinematic response to the same Temporal-Shear sector used across the wider TEP corpus. Generic screened scalar models need not make this TEP-specific cross-sector identification. This predicts a smoother transition than thin-shell models; in the canonical wide-binary realization adopted here, that recovery is represented by the pure exponential rather than the thin-shell form $f(s) = 1 - (R_s/s)\,e^{-m_{\rm bg}(s-R_s)}$. The data already favor this prediction: the thin-shell profile itself is rejected at $\Delta\chi^2 = +613.0$ relative to the TEP exponential, with the sigmoid step-like proxy likewise rejected at $\Delta\chi^2 = +131.5$ (Table 4.1). This morphological distinction separates the canonical TEP recovery profile from the standard thin-shell chameleon approximation tested here.

The TEP-native prediction is strictly macroscopic: more strongly screening environmental states more strongly suppress the locally observable Temporal Shear; in the density-scaling benchmark used here, this maps onto increasing ambient density. The measured exponent $p \approx 0.4 \pm 0.1$ under free fits ($p = 0.50 \pm 0.03$ anchored) is an empirical property of that environmental response. Alternative completions would manifest differently: Vainshtein derivative screening, for example, would map the Temporal Shear suppression to the Galactocentric potential gradient $\nabla\Phi$ rather than to local baryonic density $\rho_{\rm baryon}$, producing a different $R_s(R_{\rm gc})$ dependence; symmetron screening would introduce a density-dependent symmetry-breaking scale that yields a distinct environmental signature. TEP requires the geometric suppression of the Temporal Shear, not the specific microscopic Lagrangian that achieves it. The observational task is therefore not to confirm any particular screening mechanism per se, but to test whether the macroscopic suppression pattern matches the TEP prediction across environments.

These distinctions are currently anchored to quantities from the wider TEP series that are not yet independently peer-reviewed (Section 6.4, Eighth). The present five-bin $|Z|$ stratification yields a density-scaling exponent of $p \approx 0.4 \pm 0.1$ under free fits, consistent with the square-root benchmark; the most decisive future test is the predicted $R_s(R_{\rm gc},\,|Z|)$ map: within the density-scaling benchmark, this becomes a specific density-dependent gradient calibrated by the Temporal Topology saturation scale $\rho_T$ and the cross-scale acceleration $g_{\rm TEP}$, whereas a stand-alone screening model would require fitting both the density exponent and the response normalization as free parameters. Gaia DR4, with its expanded volume and improved astrometry, could enable the finer $|Z|$ stratification needed to distinguish these scenarios. An anisotropy test—comparing the screening transition along and perpendicular to the Galactic disk—would provide a further discriminator, since the TEP screening floor is controlled by the local environmental state $\mathcal S_\Sigma(\mathcal E)$, rather than by a universal acceleration threshold; in the density-scaling benchmark used here, the dominant environmental projection is the local baryonic-density field. Table 6.3 summarizes the frozen Gaia DR3 empirical baselines alongside the prospectively specified, zero-per-system-parameter TEP predictions for Gaia DR4 prior to catalog release.