SPARC Diagnostics of MOND-like Weak-Field Dictionaries with a Projected Scalar-Sector Interpretation

1. Introduction

Disk-galaxy rotation curves remain one of the sharpest empirical arenas for testing MOND-like weak-field phenomenology [5,7]. The radial acceleration relation [8] and the baryonic Tully–Fisher relation [4,6] set well-established regularities that any proposed weak-field dictionary must confront. The multimessenger bound $|c_T/c-1|\lesssim {10}^{-15}$ [2,3] further motivates infrared postures compatible with tensor-sector luminality [19,20,21].

We focus on the observational identifiability of weak-field response dictionaries and on the additional consistency conditions introduced by a constrained PTQ interpretation:

• Which phenomenological responses are required by SPARC under transparent parameter counting?
• Which model labels correspond to genuinely different observable radial laws?
• What structural and cross-scale diagnostics become available once a constrained projected scalar-sector interpretation is imposed?

Companion structural work on a PT-even Palatini torsion framework argues for tensor-sector coefficient locking at quadratic order [1]. Here that material is used only as structural motivation for a projected scalar-sector dictionary. It is not treated as a complete gravitational theory being validated by SPARC, and the published luminality theorem is not claimed to inherit automatically onto the exact Route-A one-form posture used below.

Contributions. 

This paper provides:

• explicit observational-equivalence identification of the PTQ-$\nu$ and MOND simple-$\nu$ likelihood families under $a_0=\epsilon c{H}_0$, and of the shared ${\nu }_q$ family (code labels ptq-screen/mond-screen);
• transparent full-covariance maximum-likelihood (MLE) accounting with corrected parameter counts including learned ${\sigma }_{\mathrm{sys}}$;
• separation of likelihood equivalence from prior-measure effects under the priors adopted in the Bayesian analysis;
• an external kinematic–structural audit based on an independent epicyclic-frequency-like diagnostic ${\kappa }_{\mathrm{kin}}$;
• a strict cross-scale closure stress test under an adopted diagnostic ${\Omega }_{\Lambda }\left(\epsilon \right)$ map, allowed to fail;
• an open reproducibility record in which both positive associations and failures remain visible.

Section 2 separates structural motivation, selected matching, adopted phenomenology, and the cosmological diagnostic map. Section 3 defines data, covariance, models, and parameter counts. Section 4 reports MLE comparisons, equivalence, geometry, and closure. Section 5 interprets the results; Section 6 states bounded conclusions.

2. Structural Motivation and Epistemic Layers

We organize the PTQ-facing content into four epistemic layers that must not be collapsed into one another.

2.1. Layer A: Structural Motivation

Observable scalar quantities are defined after a PT-even projection onto a real scalar sector [1,25]. The present empirical posture adopts Route A: projective equivalence is implemented by a one-form compensator $A_{\mu }$ treated as a non-dynamical spurion, with invariant residue

$${\mathcal{T}}_{\mu }\equiv T_{\mu }-A_{\mu },$$

(1)

following the projective-Palatini IR residual construction [26]. A scalar representative appears only on admissible domains where ${\mathcal{T}}_{\mu }$ is longitudinal; it is not used as a compensator for full projective invariance. Related conditional scalar-gradient / trace-torsion branch constructions [27] provide additional structural motivation for studying constrained single-response branches.

Scope. 

These public structural sources motivate the constrained interpretation used below; they do not derive the SPARC likelihood dictionary, the simple-$\nu$ response, or the matching relation $a_0=\epsilon c{H}_0$.

2.2. Layer B: Selected Weak-Field Matching Relation

The empirical dictionary anchors the acceleration scale by the matching relation

$$a_0\left(\epsilon \right)=\epsilon c{H}_0,$$

(2)

with $H_0=67.4\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ in the analysis pipeline used in this study [22]. Equation (2) is a selected matching relation: it is dimensionally natural and enables cross-scale bookkeeping, but it is not established here as a unique first-principles prediction.

2.3. Layer C: Adopted Phenomenology

The radial response functions used in the likelihood are adopted phenomenology. We use the fixed simple-$\nu$ interpolating function

$$\nu \left(y\right)={\displaystyle \frac{1}{2}}+\sqrt{{\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{y}}},y\equiv {\displaystyle \frac{g_{\mathrm{bar}}}{a_0}},$$

(3)

and the one-parameter generalization

$${\nu }_q\left(y\right)={\displaystyle \frac{1}{2}}+\sqrt{{\displaystyle \frac{1}{4}}+y^{-q}},$$

(4)

implemented in the analysis pipeline used in this study (repository labels: ptq-screen/mond-screen). These forms are empirical realizations shared with standard MOND benchmarks; they are not derived from the upstream PTQ theorems cited for structural motivation.

2.4. Layer D: Cosmological Diagnostic Map

For the specified cross-scale stress test, we adopt the diagnostic map

$${\Omega }_{\Lambda }\left(\epsilon \right)={\displaystyle \frac{{\epsilon }^2}{1+{\epsilon }^2}},$$

(5)

with the exact inversion on $0\le {\Omega }_{\Lambda }<1$,

$${\epsilon }_{\cos }=\sqrt{{\displaystyle \frac{{\Omega }_{\Lambda }}{1-{\Omega }_{\Lambda }}}}.$$

(6)

Equation (5) is an adopted diagnostic dictionary for that stress test, not a unique cosmological prediction established by the SPARC analysis. Alternate normalizations, when considered, enter only as sensitivity variants and are not promoted to primary equations.

3. Data, Likelihood, Models, and Parameter Accounting

3.1. Sample

We use the SPARC compilation [9]. Baseline cuts require inclination $i>{30}^{\circ }$, fractional distance uncertainty $\delta D/D<0.2$, and quality flag $\mathtt{Qual}\le 2$. Table 1 records the frozen sample flow; intermediate cuts are applied jointly rather than as separately optimized stages.

$$N_{\mathrm{gal}}=91,N_{\mathrm{points}}=1782.$$

For each galaxy we use radii $\left\{r_j\right\}$, observed speeds $\left\{v_{\mathrm{obs}}\left(r_j\right)\right\}$, baryonic templates $\{v_{\mathrm{gas}},v_{\mathrm{d}},v_{\mathrm{b}}\}$, and geometry used in error propagation (D, i, $R_d$).

3.2. Full Within-Galaxy Covariance

For galaxy g with residual vector $r_g=v_{\mathrm{obs}}-v_{\mathrm{mod}}$ of length $n_g$, the primary likelihood contribution is the multivariate Gaussian

$$log{L}_g=-{\displaystyle \frac{1}{2}}\left[r_g^{\mathsf{T}}{C}_g^{-1}{r}_g+log\left|C_g\right|+n_{g}log\left(2\pi \right)\right],$$

(7)

with full sample $logL={\sum }_{g}log{L}_g$. The covariance implemented in the analysis pipeline is

$$C_g=\mathrm{diag}\left({\sigma }_{\mathrm{meas}}^2\right)+{\sigma }_D^2{J}_D{J}_D^{\mathsf{T}}+{\sigma }_i^2{J}_i{J}_i^{\mathsf{T}}+{\sigma }_{\mathrm{sys}}^{2}I,$$

(8)

where the distance and inclination Jacobians follow the linearized propagation used in the code,

$$J_D\approx {\displaystyle \frac{\partial v}{\partial r}}{\displaystyle \frac{r}{D}},J_i\approx vcoti,$$

(9)

with $\partial v/\partial r$ estimated by centered finite differences on the model curve. Distance and inclination therefore enter through linearized covariance propagation; they are not treated as explicitly sampled nuisance parameters that are marginalized in the likelihood. This approximation can affect absolute and descriptive model-comparison values. The analytical PTQ-$\nu$/MOND identity below is unaffected when the identical covariance construction is applied consistently to both parameterizations. The global floor ${\sigma }_{\mathrm{sys}}$ is learned in the fit. Radial points within a galaxy are consequently not treated as independent when evaluating Eq. (7).

3.3. Model Definitions

Let $g_{\mathrm{bar}}=v_{\mathrm{bar}}^2/r$ and $v_{\mathrm{mod}}^2=r{g}_{\mathrm{mod}}$. The implementations used here are:

• Baryon: $v_{\mathrm{mod}}=v_{\mathrm{bar}}$.
• MOND (simple-$\nu$): $g_{\mathrm{mod}}=\nu (g_{\mathrm{bar}}/a_0)g_{\mathrm{bar}}$ with free $a_0$ and Eq. (3).
• PTQ linear: $v_{\mathrm{mod}}^2=v_{\mathrm{bar}}^2+a_0\left(\epsilon \right)r$ with Eq. (2).
• PTQ-$\nu$: same simple-$\nu$ map as MOND, with $a_0=a_0\left(\epsilon \right)$.
• ${\nu }_q$ family: $g_{\mathrm{mod}}={\nu }_q(g_{\mathrm{bar}}/a_0)g_{\mathrm{bar}}$ (Eq. (4)), with either free $a_0$ (MOND-${\nu }_q$) or $a_0\left(\epsilon \right)$ (PTQ-${\nu }_q$; code label ptq-screen), and coded domain $q\in [0.25,8]$.
• NFW-1p: restricted one-halo-parameter NFW benchmark [13] with fixed concentration–mass hyperparameters [14]; not a full $\Lambda$CDM test.

Observational equivalence (identifiability). 

Proposition 1 

(Matched-domain identifiability). Fix $H_0$ and suppose $a_0=\epsilon c{H}_0$ is one-to-one on the compared domain. If all shared nuisance definitions (baryonic ${{\rm Y}}_{★}$, ${\sigma }_{\mathrm{sys}}$, and the covariance construction (8)) are matched, then

$$L_{\mathrm{PTQ}-\nu }(\epsilon ,\theta )=L_{\mathrm{MOND}}\left(a_0=\epsilon c{H}_0,\theta \right)$$

as functions on that domain. The same statement holds for the shared ${\nu }_q$ family under Eq. (4).

This is an observable-model identifiability result, not a new PTQ theorem. The numerical differences $|\Delta maxlogL|\approx 4.7\times {10}^{-8}$ (simple-$\nu$) and $\approx 4.5\times {10}^{-12}$ (${\nu }_q$) are implementation-level verifications of the analytical identity.

Status of the ${\nu }_q$ deformation.

In the deep-MOND regime $y\ll 1$,

$${\nu }_q\left(y\right)\sim y^{-q/2},g\sim a_0^{q/2}{g}_{\mathrm{bar}}^{1-q/2}.$$

Only $q=1$ recovers the usual deep-MOND scaling $g\sim \sqrt{a_0{g}_{\mathrm{bar}}}$. The best recovered value in the present full-covariance MLE is $q\approx 0.984$, close to unity; the likelihood gain relative to simple-$\nu$ is too small to offset the additional parameter under AIC. Thus the present SPARC analysis does not require a material departure from the $q=1$ simple-$\nu$ limit within this phenomenological family. The parameter q is not treated as a derived PTQ quantity.

Independent kinematic diagnostic ${\kappa }_{\mathrm{kin}}$.

The thickness audit uses an independent single-radius kinematic diagnostic constructed from the observed rotation curve. It does not depend on fitted PTQ/MOND parameters, ${{\rm Y}}_{★}$, $\epsilon$, or $a_0$, and it never enters the likelihood. At a representative outer-disk radius $R_{★}$ chosen by the vdisk-peak rule (radius of maximum disk contribution $v_{\mathrm{disk}}$, falling back to maximum $v_{\mathrm{obs}}$), let v denote the observed circular speed and $\beta \equiv \mathrm{d}lnv/\mathrm{d}lnr$ the local logarithmic slope estimated from that observed curve. Then

$${\kappa }_{\mathrm{kin}}\left(R_{★}\right)=\sqrt{2}{\displaystyle \frac{v\left(R_{★}\right)}{R_{★}}}\sqrt{1+\beta \left(R_{★}\right)}.$$

(10)

Dimensionally $\left[{\kappa }_{\mathrm{kin}}\right]={\mathrm{T}}^{-1}$: Eq. (10) is a local epicyclic-frequency-like kinematic diagnostic, not a dimensionless geometric efficiency. A historically discussed dimensionless PTQ-inspired efficiency involving residual acceleration over $a_0$ and a response factor is a distinct observable; no frozen normalization identifying the two is used here. The schematic aperture concept $\langle h/r\rangle$ is likewise dimensionless ($[\langle h/r\rangle ]=1$) and is not equated to ${\kappa }_{\mathrm{kin}}$ (Appendix A).

Characteristic surface-density proxy ${\Sigma }_{\mathrm{tot}}$.

When resolved per-radius surface-density profiles are unavailable, ${\Sigma }_{\mathrm{tot}}\left(R_{★}\right)$ is a characteristic surface-density proxy at $R_{★}$,

$${\Sigma }_{\mathrm{tot}}\left(R_{★}\right)\approx {\displaystyle \frac{{{\rm Y}}_{3.6}{L}_{3.6,\mathrm{disk}}}{2\pi R_{★}^2}}+{\displaystyle \frac{M_{\mathrm{HI}}{f}_{\mathrm{gas}}{f}_{\mathrm{He}}}{2\pi R_{★}^2}},$$

(11)

with the frozen construction constants ${{\rm Y}}_{3.6}=0.5$, $f_{\mathrm{gas}}=1.7$ (an adopted construction constant in the coded proxy), and $f_{\mathrm{He}}=1.33$ (helium mass correction). These factors play distinct roles and are not a duplicated helium correction. The quantity is not a direct catalog surface-density measurement at every radius.

3.4. Parameter Counting

With learned ${\sigma }_{\mathrm{sys}}$ and one stellar mass-to-light ratio ${{\rm Y}}_{★}$ per galaxy, the effective fitted counts used for information criteria are:

Table 2. Effective parameter counts k including learned ${\sigma }_{\mathrm{sys}}$. Distance and inclination enter only through $C_g$ and are not counted. The independent kinematic diagnostic ${\kappa }_{\mathrm{kin}}$ is not counted.

Model family	Global (+${\mathit{\sigma }}_{\mathbf{sys}}$)	k
Baryon	${\sigma }_{\mathrm{sys}}$	92
MOND (simple-$\nu$)	$a_0+{\sigma }_{\mathrm{sys}}$	93
PTQ linear	$\epsilon +{\sigma }_{\mathrm{sys}}$	93
PTQ-$\nu$	$\epsilon +{\sigma }_{\mathrm{sys}}$	93
PTQ-${\nu }_q$ / MOND-${\nu }_q$	$(\epsilon ,q)$ or $(a_0,q)$ $+{\sigma }_{\mathrm{sys}}$	94
NFW-1p	$M_{200}$ per galaxy $+{\sigma }_{\mathrm{sys}}$	183

Table 2. Effective parameter counts k including learned ${\sigma }_{\mathrm{sys}}$. Distance and inclination enter only through $C_g$ and are not counted. The independent kinematic diagnostic ${\kappa }_{\mathrm{kin}}$ is not counted.

Model family	Global (+${\mathit{\sigma }}_{\mathbf{sys}}$)	k
Baryon	${\sigma }_{\mathrm{sys}}$	92
MOND (simple-$\nu$)	$a_0+{\sigma }_{\mathrm{sys}}$	93
PTQ linear	$\epsilon +{\sigma }_{\mathrm{sys}}$	93
PTQ-$\nu$	$\epsilon +{\sigma }_{\mathrm{sys}}$	93
PTQ-${\nu }_q$ / MOND-${\nu }_q$	$(\epsilon ,q)$ or $(a_0,q)$ $+{\sigma }_{\mathrm{sys}}$	94
NFW-1p	$M_{200}$ per galaxy $+{\sigma }_{\mathrm{sys}}$	183

Fitted bounds follow the production implementation: ${{\rm Y}}_{★}\in [0.05,1.5]$, $\epsilon \in (0,4)$ for $\epsilon$-parameterized families, $a_0\in [5\times {10}^{-11},2\times {10}^{-10}]\mathrm{m}{\mathrm{s}}^{-2}$, and $q\in [0.25,8]$. Global scale parameters recovered at the reported maxima lie in the interior of those intervals (e.g. $\epsilon \approx 0.156$ for PTQ-$\nu$). For the simple-$\nu$/PTQ-$\nu$ and ${\nu }_q$ families, 15 per-galaxy ${{\rm Y}}_{★}$ values lie on or at the hard bounds in the reported best starts; NFW-1p shows 42 near-bound parameters in its best recovered start. Bound contact is disclosed here as optimization bookkeeping and is not reinterpreted as physical support for either family. Because some per-galaxy parameters contact hard bounds, AIC/BIC are used as descriptive complexity diagnostics rather than as a unique theory-selection authority.

4. Results

4.1. Full-Covariance MLE Accounting

Table 3 reports maximum log-likelihood values under the full-covariance objective with no priors in the optimization (block-coordinate ascent on $logL$). AIC uses k from Table 2 [15]. ${\mathrm{BIC}}_{\mathrm{NPOINTS}}$ uses $N=1782$ as a descriptive accounting choice [16] and is reported only with that caveat; it is not treated as a unique theory-selection rule.

No single-IC winner narrative.

Among the listed families, NFW-1p attains the highest recovered full-covariance likelihood and lowest AIC in the present multistart search, but ${\mathrm{BIC}}_{\mathrm{NPOINTS}}$ reverses the NFW versus simple-$\nu$ ordering because of the large k. Criterion dependence is therefore material, and AIC/BIC remain descriptive diagnostics under bound contact. Because the NFW cross-start scatter is $\Delta logL\sim 29$, we do not claim a globally certified maximum for that family; no paper conclusion depends on NFW-1p being globally optimal. NFW-1p remains a restricted halo benchmark, not a test of complete $\Lambda$CDM halo modeling. The ${\nu }_q$ family improves $maxlogL$ only marginally relative to simple-$\nu$ while adding one global parameter, so AIC does not prefer it ($q\approx 0.984$). The unscreened linear dictionary is strongly disfavored within the present SPARC design relative to the nonlinear simple-$\nu$ family; this is a dictionary-level result, not a theory-level falsification of PTQ.

4.2. Reparameterization Identity and Prior Measures

Proposition 1 implies exact likelihood-family identity on the matched domain. The full-covariance MLE differences

$$\left|maxlog{L}_{\mathrm{PTQ}-\nu }-maxlog{L}_{\mathrm{MOND}}\right|\approx 4.7\times {10}^{-8},$$

and for the shared ${\nu }_q$ family after mapped refinement,

$$\left|\Delta maxlogL\right|\approx 4.5\times {10}^{-12},$$

are implementation-level numerical verifications of that analytical identity. Historical Bayesian comparisons under the priors adopted in the Bayesian analysis are a different statement: PTQ places a flat prior on $\epsilon \in (0,4)$, which pushes forward to a broad flat prior on $a_0$, whereas MOND places a flat prior on ${log}_{10}{a}_0$ over $[5\times {10}^{-11},2\times {10}^{-10}]\mathrm{m}{\mathrm{s}}^{-2}$, i.e. $p\left(a_0\right)\propto 1/a_0$ on a narrower window. These priors are not pushforward-equivalent, so posterior and related predictive rankings can differ without any difference in the radial likelihood law.

Archived pointwise WAIC/PSIS-LOO scores in the repository use diagonalized per-radius likelihood contributions rather than the full $C_g$ used in fitting. They are retained only as sensitivity/reproducibility diagnostics (Appendix E) and are not used here as canonical full-covariance theory-ranking statistics. NFW predictive diagnostics in that archived construction are not valid for headline ranking.

4.3. External Kinematic–Structural Audit

On the thickness-annotated subsample ($N=19$; external structural catalogs [10,11,12]), we evaluate ${\kappa }_{\mathrm{kin}}\left(R_{★}\right)$ and ${\Sigma }_{\mathrm{tot}}\left(R_{★}\right)$ from Eqs. (10)–(11) and perform a weighted bivariate regression

$$logh=a+b_{{\kappa }_{\mathrm{kin}}}log{\kappa }_{\mathrm{kin}}+c_{\Sigma }log{\Sigma }_{\mathrm{tot}}.$$

(12)

The structural subsample was determined by external thickness availability and baseline quality requirements, not by ${\kappa }_{\mathrm{kin}}$, rotation-curve residuals, or fit performance. Weights follow the frozen WLS convention: inverse-variance weights from the observational uncertainty on h, with ${\sigma }_{{log}_{10}h}=({\sigma }_h/h)/ln10$ and ${\sigma }_h$ taken as the mean of the asymmetric h error bars when both are available (else the available one-sided error). The fit yields

$$\begin{array}{cc}\hfill b_{{\kappa }_{\mathrm{kin}}}& =-2.559\pm 0.334,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill c_{\Sigma }& =+1.118\pm 0.140,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill R^2& =0.818,\hfill \end{array}$$

(15)

with AICc $=-48.35$ [17,18]. Relative to nested alternatives, $\Delta \mathrm{AICc}\approx 22.36$ versus ${\kappa }_{\mathrm{kin}}$-only and $\Delta \mathrm{AICc}\approx 19.42$ versus $\Sigma$-only. Galaxy-level bootstrap resampling ($n=5000$) gives 68% intervals $b_{{\kappa }_{\mathrm{kin}}}\in [-2.949,-2.188]$ and $c_{\Sigma }\in [0.991,1.282]$. Leave-one-galaxy-out coefficient means remain within one standard error of the full-sample values ($b_{{\kappa }_{\mathrm{kin}}}^{\mathrm{LOO}}\approx -2.561$, $c_{\Sigma }^{\mathrm{LOO}}\approx 1.120$). These geometry-regression LOO checks are distinct from the legacy radius-level PSIS-LOO scores discussed in Appendix E. A distance-scaled companion diagnostic weakens to $R^2\approx 0.217$.

Figure 1. Added-variable visualization of the frozen bivariate WLS thickness regression ($N=19$). Axes show partial residuals of $logh$ and $log{\kappa }_{\mathrm{kin}}$ after controlling for $log{\Sigma }_{\mathrm{tot}}$; the displayed slope equals $b_{{\kappa }_{\mathrm{kin}}}=-2.559$. The reported $R^2=0.818$ belongs to the full bivariate model (12). Because $\left[{\kappa }_{\mathrm{kin}}\right]={\mathrm{T}}^{-1}$ while $[\langle h/r\rangle ]=1$, this panel reports a kinematic–structural association; it is not a direct geometric-efficiency confirmation.

Figure 1. Added-variable visualization of the frozen bivariate WLS thickness regression ($N=19$). Axes show partial residuals of $logh$ and $log{\kappa }_{\mathrm{kin}}$ after controlling for $log{\Sigma }_{\mathrm{tot}}$; the displayed slope equals $b_{{\kappa }_{\mathrm{kin}}}=-2.559$. The reported $R^2=0.818$ belongs to the full bivariate model (12). Because $\left[{\kappa }_{\mathrm{kin}}\right]={\mathrm{T}}^{-1}$ while $[\langle h/r\rangle ]=1$, this panel reports a kinematic–structural association; it is not a direct geometric-efficiency confirmation.

Table 4. Thickness regression on ${\kappa }_{\mathrm{kin}}$ ($N=19$, WLS) with frozen bootstrap/LOO robustness. The result is a kinematic–structural association; a map to dimensionless PTQ geometric efficiency remains open.

Coefficient	Value	SE	Bootstrap 68%
Intercept a	$4.679$	$0.648$	$[4.007,5.435]$
$log{\kappa }_{\mathrm{kin}}$ (b)	$-2.559$	$0.334$	$[-2.949,-2.188]$
$log{\Sigma }_{\mathrm{tot}}$ (c)	$1.118$	$0.140$	$[0.991,1.282]$
$R^2=0.818$; AICc$=-48.35$; $\Delta$AICc(${\kappa }_{\mathrm{kin}}$-only)$=22.36$; $\Delta$AICc($\Sigma$-only)$=19.42$
Bootstrap $n=5000$; LOO means within $1$SE of full-sample coefficients

Table 4. Thickness regression on ${\kappa }_{\mathrm{kin}}$ ($N=19$, WLS) with frozen bootstrap/LOO robustness. The result is a kinematic–structural association; a map to dimensionless PTQ geometric efficiency remains open.

Coefficient	Value	SE	Bootstrap 68%
Intercept a	$4.679$	$0.648$	$[4.007,5.435]$
$log{\kappa }_{\mathrm{kin}}$ (b)	$-2.559$	$0.334$	$[-2.949,-2.188]$
$log{\Sigma }_{\mathrm{tot}}$ (c)	$1.118$	$0.140$	$[0.991,1.282]$
$R^2=0.818$; AICc$=-48.35$; $\Delta$AICc(${\kappa }_{\mathrm{kin}}$-only)$=22.36$; $\Delta$AICc($\Sigma$-only)$=19.42$
Bootstrap $n=5000$; LOO means within $1$SE of full-sample coefficients

Interpretation.

We classify the result as a robust kinematic–structural association whose mapping to a projected-sector geometric efficiency remains open. The coefficient $b_{{\kappa }_{\mathrm{kin}}}$ itself is negative and stably so under bootstrap/LOO. Because ${\kappa }_{\mathrm{kin}}$ and the dimensionless aperture $\langle h/r\rangle$ are dimensionally inequivalent, a direct sign test of a geometric-efficiency hypothesis ${\kappa }_{\mathrm{geom}}\sim \langle h/r\rangle$ is not established by this regression. Additional closure-oriented $\kappa$-like diagnostics (data-defined, model-control, and stacked/profile) use distinct constructions and are reported separately in Appendix B; the stacked/profile control remains near-null ($R^2\approx 0.008$).

4.4. Cross-Scale Consistency Test

Using the adopted map (5) and the PTQ-$\nu$ inference from this study,

$${\epsilon }_{\mathrm{RC}}\approx 0.155,{\epsilon }_{\cos }\approx 1.492.$$

Direct strict equality ${\epsilon }_{\mathrm{RC}}={\epsilon }_{\cos }$ fails, including throughout a pre-specified ${\Omega }_{\Lambda }$-band $3\sigma$ sensitivity scan under Eqs. (5)–(6). No point in that scan restores equality. The numerical factor required to reconcile the two inferred scales is

$${\kappa }_{\mathrm{req}}\equiv {\displaystyle \frac{{\epsilon }_{\mathrm{RC}}}{{\epsilon }_{\cos }}}\approx 0.10.$$

(16)

In the absence of a validated dimensionless observable bridge, ${\kappa }_{\mathrm{req}}$ is retained only as a bookkeeping requirement and is not interpreted as evidence for closure. A hypothetical bridge-mediated completion would take the form ${\epsilon }_{\mathrm{RC}}={\kappa }_{\mathrm{geom}}{\epsilon }_{\cos }$, where ${\kappa }_{\mathrm{geom}}$ is a dimensionless physical geometric efficiency distinct from the kinematic diagnostic ${\kappa }_{\mathrm{kin}}$. Such a completion would require a theory-derived or independently motivated definition of ${\kappa }_{\mathrm{geom}}$, a dimensionally valid observable map, an independent empirical estimate of that quantity, and quantitative agreement with ${\kappa }_{\mathrm{req}}$. None of these is established here, so bridge-mediated closure is not yet operationally testable in the present analysis. The failure reported in this section applies to the conjunction of the selected matching relation with the adopted cosmological diagnostic map; it is not a general falsification of complete PTQ.

Figure 2. Cross-scale closure sensitivity under the adopted diagnostic map. With ${\epsilon }_{\mathrm{RC}}\approx 0.155$ and ${\epsilon }_{\cos }\approx 1.492$, strict cross-scale equality fails, and no point in the pre-specified $3\sigma$ sensitivity scan restores that equality.

Figure 2. Cross-scale closure sensitivity under the adopted diagnostic map. With ${\epsilon }_{\mathrm{RC}}\approx 0.155$ and ${\epsilon }_{\cos }\approx 1.492$, strict cross-scale equality fails, and no point in the pre-specified $3\sigma$ sensitivity scan restores that equality.

5. Discussion

Likelihood analysis, independent structural diagnostic, and cross-scale stress test. 

The empirical chain is intentionally staged. Rotation-curve fits determine the likelihood parameters under the full-covariance model. An independent kinematic diagnostic ${\kappa }_{\mathrm{kin}}$ is then compared with external structural information; ${\kappa }_{\mathrm{kin}}$ is not inferred from the PTQ/MOND fit. Cross-scale consistency is tested separately under the selected matching relation and the adopted diagnostic map. Interpretive quantities are therefore not rewarded with additional likelihood freedom.

What SPARC establishes. 

Under the present matched full-covariance likelihood, a MOND-like weak-field response describes the data compactly relative to baryons-alone and to the unscreened linear dictionary. Proposition 1 implies that PTQ-$\nu$ and MOND are observationally identical once $a_0=\epsilon c{H}_0$ and the same simple-$\nu$ are imposed, so model-label preference is not evidence for a distinct PTQ radial law. Within the ${\nu }_q$ family, $q\approx 0.984$ does not require a material departure from simple-$\nu$ under AIC. The near-unity best recovered value, together with the absence of AIC support for the additional deformation parameter, makes the $q=1$ response a natural consistency target for future PTQ-specific derivations intended to reproduce this phenomenological branch. Such a derivation should account for why its observable weak-field response remains close to the $q=1$-like square-root scaling over the acceleration regime probed by SPARC.

What the constrained PTQ interpretation adds. 

It supplies an $\epsilon$-anchored bookkeeping relation, soft structural motivation for a projected scalar-sector posture [1,25,26,27], and additional structural/cross-scale diagnostics that do not feed back into the fit.

What the external audits show. 

The thickness analysis identifies a robust kinematic–structural association based on ${\kappa }_{\mathrm{kin}}$, while its mapping to a dimensionless PTQ geometric efficiency remains open. A standard stellar-disk dynamical heuristic provides one possible non-PTQ interpretation of this association [23,24]. For a stellar disk near a common Toomre stability level, $Q_{★}\propto {\sigma }_R\kappa /\left(G{\Sigma }_{★}\right)$ links radial velocity dispersion, epicyclic frequency, and surface density [23]; vertical equilibrium likewise links scale height to vertical velocity dispersion and surface density through ${\sigma }_z^2\sim CG\Sigma h$ [24]. If $Q_{★}$ and the velocity-ellipsoid ratio ${\sigma }_z/{\sigma }_R$ vary only weakly across the relevant subsample, and if the characteristic surface-density proxy tracks the dynamically relevant surface density, these relations schematically imply $h\propto \Sigma {\kappa }_{\mathrm{kin}}^{-2}$, i.e. an exponent pair near $(-2,+1)$. The fitted pattern $b_{{\kappa }_{\mathrm{kin}}}\approx -2.56$ and $c_{\Sigma }\approx +1.12$ has the same sign structure and a comparable scale to that simplified relation. Because $Q_{★}$, ${\sigma }_z/{\sigma }_R$, and the exact dynamically relevant surface density are not measured as constant across the $N=19$ sample, and because gas, halo gravity, and radial structure may modify the scaling, we offer this only as an astrophysical hypothesis for the association—not as its explanation, not as a quantitative prediction, and not as evidence for a PTQ geometric mechanism. The classification therefore remains a kinematic–structural association only. Strict selected cosmology-to-galaxy equality fails under the adopted diagnostic map. A bridge-mediated closure remains a future theory target: it would require a dimensionless physical efficiency with an independently defined observable map, whereas the present analysis determines only the numerical factor ${\kappa }_{\mathrm{req}}$ required for such a reconciliation. These outcomes define concrete targets for future theory work.

Open-source positioning. 

The companion repository exposes model equations, parameter accounting, covariance assumptions, fits, deterministic equivalence tests, full-covariance MLE tables, prior-measure statements, geometry diagnostics, closure failure, and a reproducibility record.

6. Conclusion

SPARC discriminates among the tested response architectures under transparent full-covariance accounting, but does not distinguish PTQ-$\nu$ from MOND when the same $\nu$ law and one-to-one matching $a_0=\epsilon c{H}_0$ are adopted. The near-unity ${\nu }_q$ recovery makes $q=1$-like square-root scaling a useful consistency target for future theory intended to reproduce this phenomenological branch. The external structural audit identifies a robust association between disk thickness and the kinematic diagnostic ${\kappa }_{\mathrm{kin}}$; a standard disk-dynamical heuristic may qualitatively motivate its sign pattern, while a dimensionally valid map from ${\kappa }_{\mathrm{kin}}$ to a projected-sector geometric efficiency remains open. Strict selected cosmology–galaxy equality fails under the adopted diagnostic map, and a bridge-mediated completion remains observationally and theoretically undefined. Together these results define a constrained observational interface for PTQ-motivated weak-field phenomenology and clear targets for future theory derivation.