A Current-Variational Anisotropic Source for Halo-like Galactic Scaling

1. Introduction

Galaxy rotation curves exhibit a well-known asymptotic behaviour: the circular velocity tends to approach a constant value at large radii, and the asymptotic velocity obeys the baryonic Tully-Fisher relation [1,2] $v_f^4\propto G{M}_b{a}_0$, where $M_b$ is the baryonic mass. These observations are usually interpreted either in terms of dark matter halos, with the NFW profile as the standard cold-dark-matter reference [3,4], or in terms of modified low-acceleration dynamics, starting from MOND [5,6,7]. Later developments and current issues are reviewed in [8], while the broader dark-sector setting is summarized in [9]. The observed radial acceleration relation provides an additional empirical benchmark for rotationally supported galaxies [10].

An alternative possibility is that the observed phenomenology is generated by additional structure within the matter energy-momentum tensor. Non-trivial exchange and effective dark-sector behaviour within stress-energy descriptions have been considered in related contexts [11]. The construction used below keeps only the current and rotational sector of matter. The rotationally extended stress-energy tensor is taken from the variational Alena Tensor source [12]. The reduced Rainich-Codazzi description in [13] is used only as comparison language for the anisotropic source type.

The calculation is carried out for one explicit branch metric. A normalized vorticity flux selects a spatial direction, and the transverse two-plane is warped by one scalar. The corresponding current scalar is varied in curved spacetime, so the source entering the Einstein equation is fixed before the halo collar is evaluated. In the weak-field regime the resulting $r^{-2}$ active component gives flat rotation curves and MOND-like scaling [14]. The interpretation is close in spirit to models in which halo phenomenology is produced by stress-energy components rather than by new particle species [15].

The purpose of this Letter is to isolate the minimal source mechanism. First, the current action and its Hilbert tensor are written explicitly. Second, the Einstein tensor of the one-field halo collar is computed and its large-radius scaling is displayed. Third, the baryonic Tully-Fisher normalization is reduced to a local transition rule in which the asymptotic branch slope is matched to the baryonic areal-radius slope at an acceleration-selected radius. The result is a source-level construction of the $r^{-2}$ halo component and its transition normalization.

2. Current Scalar and Branch Action

The branch metric is denoted by $k_{\mu \nu }$, and the sign convention is $(+---)$. The action is taken in the form

$$S[k,J]=\int d^{4}x\sqrt{-k}\left(AR\left[k\right]-P\varphi [J,k]\right).$$

(1)

Here A and P are constants, and the current scalar is written as

$$\varphi =1-{\chi }_J,{\chi }_J={\zeta }^2+{\mu }_{\zeta }{R}_{\omega }.$$

(2)

The scalar ${\zeta }^2$ is the translational-current coefficient. The quantity $R_{\omega }$ is the normalized rotational response, and ${\mu }_{\zeta }$ is the corresponding response coefficient. In the current notation the translational current may be written as $J_{\mathrm{tr}}^{\mu }=P{\zeta }^2{U}^{\mu }$, with $k_{\mu \nu }{U}^{\mu }{U}^{\nu }=1$. The rotational part is the vorticity-response contribution to the same matter current sector. In the reduced branch calculation no independent dark component is introduced; the current dependence of (2) fixes the additional Hilbert source.

The metric variation of ${\chi }_J$ is defined by

$$\delta {\chi }_J={\mathcal{E}}_{\mu \nu }\left[{\chi }_J\right]\delta k^{\mu \nu }+{\nabla }_{\alpha }{\Theta }^{\alpha }.$$

(3)

The boundary term is not used below. Since $\delta \sqrt{-k}=-(1/2)\sqrt{-k}{k}_{\mu \nu }\delta k^{\mu \nu }$, the variation of (1) gives

$$A{G}_{\mu \nu }+{\displaystyle \frac{P}{2}}\varphi k_{\mu \nu }-P{\mathcal{E}}_{\mu \nu }\left[\varphi \right]=0.$$

(4)

Using (2), this becomes

$$G_{\mu \nu }+\Lambda k_{\mu \nu }=\kappa T_{\mu \nu }^{\left(J\right)},$$

(5)

where

$$\kappa ={\displaystyle \frac{1}{2A}},\Lambda ={\displaystyle \frac{P}{2A}}=\kappa P,$$

(6)

and

$$T_{\mu \nu }^{\left(J\right)}=P{\chi }_J{k}_{\mu \nu }-2P{\mathcal{E}}_{\mu \nu }\left[{\chi }_J\right].$$

(7)

If the constant is parametrized as $P=p_{\Lambda }/2$, then (6) gives $\Lambda =\kappa p_{\Lambda }/2$. The phenomenological density normalization below uses the usual gravitational coupling, $\kappa =8\pi G/c^4$. In the unexcited current sector, ${\chi }_J=0$ and (5) reduces to the vacuum equation with cosmological constant. Since (1) is diffeomorphism invariant, the source (7) is covariantly conserved on the current-sector equations. The reduced collar calculation therefore evaluates a conserved variational source, not an externally prescribed density profile.

The tensor (7) contains the current and rotational response. The first part of (2) gives the translational contribution, while the variation of ${\mu }_{\zeta }{R}_{\omega }$ gives the anisotropic rotational contribution. In the continuum reading of the Alena Tensor source [12], this sector contains the rotational energy flux and the vortex stress. The reduced calculation below uses only the one-field radial branch generated by this variational source.

3. Branch Metric and Halo Collar

The stationary axisymmetric reference metric is written as

$$d{s}^2=N^2{c}^{2}d{t}^2-A^2(d{r}^2+r^{2}d{\theta }^2)-B^2{r}^2{sin}^2\theta {(d\phi -\omega dt)}^2.$$

(8)

Here N, A, B, and $\omega$ are metric functions; the symbol A in this section is independent of the constant A in (1). The associated orthonormal coframe is ${\Theta }^0=Ncdt$, ${\Theta }^1=Adr$, ${\Theta }^2=Ard\theta$, and ${\Theta }^3=Brsin\theta (d\phi -\omega dt)$. In the halo region the normalized vorticity flux selects the branch direction, so $U={\Theta }^0$, $N_R={\Theta }^1$, and the transverse plane is spanned by $W={\Theta }^2$ and $S={\Theta }^3$. The subscript on $N_R$ only separates the branch vector from the metric coefficient N in (8).

The anisotropic branch metric is taken as

$$k_{\mu \nu }=U_{\mu }{U}_{\nu }-N_{R\mu }{N}_{R\nu }-e^{2\psi \left(r\right)}(W_{\mu }{W}_{\nu }+S_{\mu }{S}_{\nu }).$$

(9)

The corresponding line element is

$$d{s}_k^2=N^2{c}^{2}d{t}^2-A^{2}d{r}^2-e^{2\psi \left(r\right)}\left[A^2{r}^{2}d{\theta }^2+B^2{r}^2{sin}^2\theta {(d\phi -\omega dt)}^2\right].$$

(10)

This is the one-field branch used in the asymptotic calculation. The metric (9) is used to compute the active anisotropic source associated with (7). Test-particle and photon observables are read from the weak physical field sourced by this tensor. In the weak-branch reduction the active component computed from (9) enters the Poisson source of that physical field. If the two transverse coefficients in (10) do not share the same large-radius scaling, the remaining part is a shear inside the transverse plane.

The reduced flow closure fixes the anisotropy through the normalized shear-vorticity scalar,

$$tanh\psi ={\mu }_{\zeta }{\displaystyle \frac{2\varpi {\sigma }_{NS}+{\varpi }^2}{2|D_{\omega }|}}.$$

(11)

Here $D_{\omega }$ is the vorticity-flux scale, $\varpi$ is the vorticity amplitude, and ${\sigma }_{NS}$ is the corresponding shear component. The sign is fixed by the oriented shear-vorticity response. Equation (11) is used only as the one-field closure for $\psi$, while the positive-density branch is selected below. The source tensor is the tensor (7).

For an asymptotically flat rotation curve one has $\Omega \sim r^{-1}$. The leading terms in the numerator of (11) and the normalization $|D_{\omega }|$ then have the same radial order in the large-radius disk regime. The asymptotic branch is therefore

$$\psi \left(r\right)\to {\psi }_{\infty }.$$

(12)

The corresponding asymptotic transverse slope and the weak positive-branch amplitude are denoted by

$$\alpha =e^{{\psi }_{\infty }},\epsilon =-{\psi }_{\infty }.$$

(13)

The sign of $\epsilon$ is fixed below by the active source.

4. Einstein Tensor of the Halo Collar

The large-radius calculation is isolated by the halo collar

$$d{s}_k^2=c^{2}d{t}^2-d{r}^2-e^{2\psi \left(r\right)}{r}^{2}d{\Omega }^2.$$

(14)

Put $a\left(r\right)=e^{\psi \left(r\right)}r$. With the curvature-sign convention used in (5), the mixed Einstein tensor of (14) is

$$\begin{array}{cc}\hfill {G^t}_t& ={\displaystyle \frac{1-a^{\prime 2}-2a{a}^{\prime \prime }}{a^2}},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {G^r}_r& ={\displaystyle \frac{1-a^{\prime 2}}{a^2}},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {G^{\theta }}_{\theta }& ={G^{\phi }}_{\phi }=-{\displaystyle \frac{a^{\prime \prime }}{a}}.\hfill \end{array}$$

(17)

Equivalently, for the warp scalar,

$$\begin{array}{cc}\hfill {G^t}_t& ={\displaystyle \frac{e^{-2\psi }-1}{r^2}}-{\displaystyle \frac{6{\psi }^{\prime }}{r}}-3{\left({\psi }^{\prime }\right)}^2-2{\psi }^{\prime \prime },\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {G^r}_r& ={\displaystyle \frac{e^{-2\psi }-1}{r^2}}-{\displaystyle \frac{2{\psi }^{\prime }}{r}}-{\left({\psi }^{\prime }\right)}^2,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {G^{\theta }}_{\theta }& ={G^{\phi }}_{\phi }=-{\psi }^{\prime \prime }-{\left({\psi }^{\prime }\right)}^2-{\displaystyle \frac{2{\psi }^{\prime }}{r}}.\hfill \end{array}$$

(20)

The flat collar is recovered for $a\left(r\right)=r$. If (12) holds, then $a^{\prime \prime }/a$ is subleading and

$${G^t}_t\simeq {G^r}_r\simeq {\displaystyle \frac{{\alpha }^{-2}-1}{r^2}},{G^{\theta }}_{\theta }\simeq {G^{\phi }}_{\phi }\simeq 0.$$

(21)

The trace-free part is

$$\mathrm{tf}{G^{\mu }}_{\nu }\simeq {\displaystyle \frac{{\alpha }^{-2}-1}{2r^2}}\mathrm{diag}(1,1,-1,-1).$$

(22)

The cosmological term in (5) contributes an isotropic mixed tensor $\Lambda {{\delta }^{\mu }}_{\nu }$ and leaves (22) unchanged.

The sign branch is chosen so that ${T^{\left(J\right)t}}_t>0$ in the halo regime after subtraction of the constant vacuum term. Equation (21) then gives $\alpha <1$, or equivalently ${\psi }_{\infty }<0$ and $\epsilon >0$. With the normalization stated above, the active density scale satisfies

$${\rho }_{\psi }\left(r\right)c^2\simeq {\displaystyle \frac{1}{\kappa }}{\displaystyle \frac{{\alpha }^{-2}-1}{r^2}},{\rho }_{\psi }\left(r\right)\simeq {\displaystyle \frac{c^2}{8\pi G}}{\displaystyle \frac{{\alpha }^{-2}-1}{r^2}}.$$

(23)

For $\epsilon \ll 1$, this gives the usual $r^{-2}$ scaling. In the weak-branch reduction, (23) is the active contribution associated with the current source.

5. Weak-Branch Profiles and Rotation Curves

The weak branch is parametrized by $x=r/r_c$ and

$$\psi \left(r\right)=-\epsilon f\left(x\right),f\left(x\right)={\displaystyle \frac{x^n}{1+x^n}},n>2,\epsilon >0.$$

(24)

Using (18) to first order in $\epsilon$, with primes on f denoting derivatives with respect to x, the normalized active source is

$$Q\left(x\right)={\displaystyle \frac{x^2{G^t}_t}{C_{\infty }}}=f\left(x\right)+3x{f}^{\prime }\left(x\right)+x^2{f}^{\prime \prime }\left(x\right),C_{\infty }={\displaystyle \frac{2\epsilon }{r_c^2}}.$$

(25)

The branch contribution to the circular velocity is then

$${\displaystyle \frac{v_{\psi }^2\left(x\right)}{v_{\infty }^2}}={\displaystyle \frac{1}{x}}{\int }_0^{x}Q\left(u\right)du.$$

(26)

Equations (25) and (26) are the quantities shown in Figure 1.

The same calculation gives the large-radius behaviour

$$v_{\psi }\left(r\right)\to v_{\infty }.$$

(27)

The normalization is

$$v_{\infty }^2\simeq {\displaystyle \frac{c^2}{2}}({\alpha }^{-2}-1)\simeq c^2\epsilon .$$

(28)

For $v_{\infty }=200\mathrm{km}{\mathrm{s}}^{-1}$, (28) corresponds to $\epsilon \simeq 4.4\times {10}^{-7}$, so the weak-branch expansion is sufficient for galactic velocities.

For a flat curve, the corresponding spherical-equivalent density is

$${\rho }_{\mathrm{eff}}\left(r\right)={\displaystyle \frac{v_{\infty }^2}{4\pi G{r}^2}}.$$

(29)

For $v_{\infty }\sim 2\times {10}^5\mathrm{m}{\mathrm{s}}^{-1}$ and $r=10\mathrm{kpc}$ this gives

$${\rho }_{\mathrm{eff}}\sim 5\times {10}^{-22}\mathrm{kg}{\mathrm{m}}^{-3}\simeq 7\times {10}^{-3}{M}_{\odot }{\mathrm{pc}}^{-3}.$$

(30)

The active energy density fixed by (30) is macroscopic on galactic scales. For a representative outer-disk baryonic density ${\rho }_b\sim {10}^{-24}\mathrm{kg}{\mathrm{m}}^{-3}$, the ordinary kinetic rotational energy density ${\rho }_b{v}_{\infty }^2/2$ is much smaller. The branch source is therefore not identified with the ordinary post-Newtonian gravitomagnetic correction.

The transition profile in (24) is not arbitrary if the active source is required to remain positive. For the family used in Figure 1,

$$Q\left(x\right)={\displaystyle \frac{x^n\left[x^{2n}+(-n^2+2n+2)x^n+{(n+1)}^2\right]}{{(1+x^n)}^3}}.$$

(31)

The minimum of (31) becomes negative for sufficiently sharp transitions. Positivity for all $x>0$ gives

$$n\le n_c=2+2\sqrt{2}\simeq 4.83.$$

(32)

The scan in Figure 2 displays this threshold.

6. Rotational Transport Interpretation

The branch amplitude has a natural transport interpretation. For stationary axisymmetric systems the rotational Killing vector is ${\xi }_{\left(\phi \right)}^{\mu }={\partial }_{\phi }$, and the corresponding Killing current is

$$J_{\left(L\right)}^{\mu }={T^{\mu }}_{\phi }.$$

(33)

Conservation of the matter tensor with respect to the transport metric gives

$${\nabla }_{\mu }{J}_{\left(L\right)}^{\mu }=0.$$

(34)

In axisymmetric coordinates this reduces to

$${\partial }_r\left(\sqrt{-g}{F}_r\right)+{\partial }_{\theta }\left(\sqrt{-g}{F}_{\theta }\right)=0,$$

(35)

where g denotes the physical transport metric, and $F_r$ and $F_{\theta }$ denote the radial and meridional angular-momentum fluxes carried by the rotational stress sector. A stream potential is introduced by

$$\sqrt{-g}{F}_r={\partial }_{\theta }\Psi ,\sqrt{-g}{F}_{\theta }=-{\partial }_r\Psi .$$

(36)

The angular expansion is

$$\Psi (r,\theta )=\sum _{\ell }{R}_{\ell }\left(r\right)P_{\ell }(cos\theta ).$$

(37)

On the galactic plane only odd multipoles contribute to the radial flux. The lowest non-vanishing mode is

$$\Psi (r,\theta )\simeq R_1\left(r\right)cos\theta .$$

(38)

Assuming the standard large-radius hierarchy, higher modes decay faster than the lowest odd contribution [16]. Similar hierarchies occur in gravitational transport settings [17]. In the weak-field large-radius regime,

$$F_r\sim {\displaystyle \frac{R_1\left(r\right)}{r^2}}.$$

(39)

Thus a finite asymptotic transport amplitude gives the same scaling as the active source in (23).

The coefficient $R_1\left(r\right)$ is related to the radial angular-momentum flux by

$$\dot{J}\left(r\right)=\int J_{\left(L\right)}^{r}dS=2\pi {\int }_0^{\pi }d\theta \sqrt{-g}{F}_r.$$

(40)

Using (36) and (38),

$$R_{\infty }=-{\displaystyle \frac{{\dot{J}}_{\infty }}{4\pi }}.$$

(41)

Approximately constant angular-momentum fluxes are standard in rotating astrophysical systems [18,19]. Large-scale redistribution by torques and correlated stresses also appears in cosmological disk settings [20,21]. If the transport operates on the local orbital timescale ${\tau }_J\sim R_d/v_{\infty }$ [22], then $J_b\sim M_b{R}_d{v}_{\infty }$ gives ${\dot{J}}_{\infty }\sim M_b{v}_{\infty }^2$. This scaling fixes the transport amplitude to which the branch response is matched at the transition.

7. BTFR as Transition Closure

The empirical baryonic Tully-Fisher relation is written as

$$v_{\infty }^4\propto G{M}_b{a}_0.$$

(42)

In the branch picture the first normalization is (28). The remaining input is local. Under a universal current-transition threshold, the asymptotic branch slope is matched to the baryonic areal-radius slope at the transition radius. For the baryonic transition geometry

$$d{s}_b^2=N_b^2{c}^{2}d{t}^2-d{\ell }_b^2-R_b^2\left({\ell }_b\right)d{\Omega }^2,$$

(43)

the transverse scale is matched at the level of the proper-radial areal slope. The matching condition is

$$e^{{\psi }_{\infty }}={\left.{\displaystyle \frac{d{R}_b}{d{\ell }_b}}\right|}_{R_b=r_c}.$$

(44)

For a weak exterior monopole,

$${\left.{\displaystyle \frac{d{R}_b}{d{\ell }_b}}\right|}_{R_b=r_c}={\left(1-{\displaystyle \frac{2G{M}_b}{c^2{r}_c}}\right)}^{1/2}.$$

(45)

Equations (44) and (45) give

$$\epsilon =-{\displaystyle \frac{1}{2}}ln\left(1-{\displaystyle \frac{2G{M}_b}{c^2{r}_c}}\right)\simeq {\displaystyle \frac{G{M}_b}{c^2{r}_c}}.$$

(46)

Thus (28) gives

$$v_{\infty }^2\simeq {\displaystyle \frac{G{M}_b}{r_c}}.$$

(47)

The transition radius is selected by the current threshold

$${\chi }_J\left(r_c\right)={\chi }_{*}.$$

(48)

In the weak monopole-dominated closure used below, the same selection is parametrized by the baryonic acceleration invariant

$${\mathcal{A}}_b=c^2{\left(h_b^{ij}{\partial }_{i}ln{N}_b{\partial }_{j}ln{N}_b\right)}^{1/2}.$$

(49)

The branch transition is imposed at

$${\mathcal{A}}_b\left(r_c\right)=a_{*}.$$

(50)

If the baryonic field is monopole-dominated at $r_c$, then

$$r_c^2\simeq {\displaystyle \frac{G{M}_b}{a_{*}}}.$$

(51)

Equations (47) and (51) imply

$$v_{\infty }^4\simeq G{M}_b{a}_{*}.$$

(52)

Thus the BTFR normalization is reduced to the areal-slope matching (44) and the current-transition selection (48). The acceleration scale $a_{*}$ is the transition scale of the current sector. If it is universal, (52) has the same structure as (42). In the transport language, the same transition constrains the product of the asymptotic transport amplitude and the branch response. If the response coefficient is universal, (41) fixes the mass dependence of the asymptotic transport amplitude. For $a_{*}$ of the galactic low-acceleration order, comparable to $a_0$ in (42), the transition radius from (51) is galactic for ordinary disk masses, while (28) keeps $\epsilon \sim v_{\infty }^2/c^2\ll 1$. The weak branch is therefore consistent with the normalization range used above.

8. Observational Consequences

The following consequences are tied to the source geometry rather than to a global rotation-curve fit. The primary geometric signature is the disk-aligned anisotropic lensing component. The dynamical residuals are expected to enter through the transition radius, baryonic angular momentum, and disk structure. Systems without a coherent rotating branch provide the corresponding suppression tests.

8.1. Disk-Aligned Lensing Anisotropy

The branch metric (9) produces a flattened effective source aligned with the baryonic disk. In $\Lambda$CDM, halo triaxiality can also produce anisotropic shear with sensitivity to baryonic alignment and projection. A systematics-corrected 3.8$\sigma$ detection of galaxy-scale halo flattening aligned with galaxy light was reported in [23]. In MOND-like disk models, lensing anisotropy can arise from the baryonic disk and the phantom-matter distribution; inclination-dependent disk-lensing effects were calculated in [24]. In the present source reading the relevant quantity is the Ricci focusing generated by (7).

For a leading-order null lensing vector ${\ell }^{\mu }$ of the weak optical field, the cosmological term in (5) drops out of the Ricci focusing. The source projection is

$$R_{\mu \nu }{\ell }^{\mu }{\ell }^{\nu }=\kappa T_{\mu \nu }^{\left(J\right)}{\ell }^{\mu }{\ell }^{\nu }.$$

(53)

The prediction is therefore a weak-lensing quadrupole tied to the vorticity-selected transverse plane. At fixed $M_b$, the leading projection gives an anisotropic shear contribution scaling with disk inclination as $\Delta \gamma \propto {sin}^{2}i$, with the orientation following the disk-plane geometry. Inclination-dependent stacking in Euclid or LSST data would test the source geometry encoded in (22).

8.2. Transition Radii and RAR Residuals

The transition radius $r_c$ in (24) is expected to correlate with disk structure. Higher modes in (37) decay faster and may couple to scales such as $R_d$, the bar corotation radius, or the stellar-to-gas transition radius.

SPARC data constrain this picture. The RAR scatter after fitting individual galaxies is about $0.057$ dex [25], and no strong galaxy-to-galaxy variation in the critical acceleration scale is found in [26]. A proportional scaling between $r_c$ and disk structure can preserve the mean RAR while allowing secondary residuals. Non-monotonic features in the $[g_{\mathrm{bar}},g_{\mathrm{obs}}]$ plane [27] would correspond to a locally non-trivial branch profile.

The predicted partial correlation is between RAR residuals and disk geometry at fixed $g_{\mathrm{bar}}$ and $M_b$. In the present closure the residuals may enter through deviations from the monopole limit in (45), through disk-dependent corrections to (49), or through a non-universal transport response. A universal MOND interpolation depending only on $|g_{\mathrm{bar}}|/a_0$ provides the corresponding baseline without this secondary dependence [25,26]. In $\Lambda$CDM the analogous residuals are usually associated with concentration, feedback, and assembly history.

8.3. Baryonic Angular Momentum and UDG Systems

Because (41) relates the asymptotic amplitude to ${\dot{J}}_{\infty }$, the main secondary parameter at fixed $M_b$ is expected to be the baryonic specific angular momentum $j_b=J_b/M_b$. Illustris and EAGLE simulations find weak correlation between stellar disc spin and host halo spin [28,29], with angular-momentum retention discussed in [28]. The disk-scale distinction is therefore observationally relevant.

The BTFR shows no strong correlation between residuals and galaxy structural parameters at fixed $M_b$ [30]. This can be compatible with a tight transport response if the branch follows the observed Fall relation $j_b\propto M_b^{0.60}$ [31]. The Fall relation has larger scatter than the BTFR [32], and its residuals are sensitive to gas fraction [31]. A useful residual test is

$$\delta log{V}_f{|}_{M_b}\propto {\alpha }_s\delta log{j}_b{|}_{M_b},{\alpha }_s>0.$$

(54)

In the usual $\Lambda$CDM interpretation, BTFR residuals are primarily associated with halo concentration variations [33].

Systems with low coherent baryonic angular momentum are expected to have a reduced branch contribution. UDGs with suppressed or absent halo-like scaling, including NGC 1052-DF2 [34,35], NGC 1052-DF4 [36], and FCC 224 [37], are natural candidates for this low-$j_b$ tail. Some UDG morphologies have been associated with halo spin in simulations [38]. Recent EAGLE results indicate that UDG sizes are driven by high spin in the star-forming gas rather than by high halo spin [39]. The kinematic proxy ${\lambda }_R$ measured by integral-field spectroscopy [40] can therefore be compared directly with the halo-like velocity excess. The MUSE separation between rotation-supported and dispersion-dominated UDGs in [37] gives an existing test case.

8.4. Non-Disk Systems and Clusters

The galactic normalization is tied to a coherent disk-transport branch. On BCG and cluster scales the effective acceleration scale is higher than the galactic RAR scale by a factor of order seventeen [41]. Kinematic, lensing, and X-ray thermodynamic mass profiles give the cluster comparison in [42]; cluster mass profiles based on lensing and X-ray data are also discussed in [43,44]. The baryonic component of clusters is dominated by a hot quasi-isotropic intracluster medium, so the disk branch (9) is not expected to set the cluster scale.

NGC 1052-DF2 was confirmed as a galaxy whose stellar velocity dispersion is consistent with its baryonic mass alone [34,35]. Similar claims have been made for NGC 1052-DF9 [45]. Formation channels involving strong interactions or collisions can disrupt disk structure [46]. Such systems should not sustain the transport channel entering (33), even if a formal low-acceleration radius can be assigned to the baryonic field.

8.5. High-Redshift Disk State

The branch amplitude should build up as a quasi-stationary transport channel is established. Dynamically immature systems with low $V/\sigma$, disturbed morphology, or young mean stellar age may therefore show negative BTFR residuals at fixed $M_b$. Dynamically cold or giant disks have already been observed at $z\simeq 3.25-4.2$ [47,48], and no strong BTFR evolution to $z\sim 2.5$ was reported in [49]. A compact parametrization is

$$\delta log{v}_f\propto -f\left({\displaystyle \frac{t_{\mathrm{age}}}{{\tau }_J}}\right),f\left(x\right)\to 0(x\to \infty ).$$

(55)

A partial correlation between $\delta log{v}_f$ and $V/\sigma$ at fixed $M_b$ is a direct test.

8.6. Weak-Lensing Measurements

Weak-lensing measurements already provide a qualitative comparison class. Early galaxy-galaxy lensing analyses reported evidence for halo flattening aligned with galaxy light [50]. Later SDSS work emphasized systematic contaminants and found weaker evidence [51]. The systematics-corrected detection in [23] is consistent with a flattened source aligned with the baryonic component. These measurements do not isolate the vorticity-selected plane in (9), but they motivate the inclination-dependent test described above.

9. Discussion and Limitations

The calculation isolates the asymptotic weak-field branch of a rotational-current source. The scalar (2) gives, after Hilbert variation, the source tensor (7). Its constant part fixes the cosmological term through (6), and its current-dependent part supplies the anisotropic source in (5). The halo collar computation shows that an asymptotically constant transverse warp gives an $r^{-2}$ active source with trace-free type (22). The weak-branch profiles in Figure 1 display the corresponding rotation curves, while Figure 2 shows that positivity restricts the transition sharpness.

The normalization has been written as a local transition closure. The positive branch is an areal-slope deficit, and (44) identifies this deficit with the baryonic transition geometry through the proper-radial areal slope. The compactness condition (46) is the weak-field monopole limit of the slope matching, while (48) selects the radius at which the branch freezes into its asymptotic collar value. With a universal transition scale, (52) follows in the monopole-dominated regime. The present Letter isolates the asymptotic one-field branch. The remaining disk problem concerns the radial completion of $\psi \left(r\right)$, the vorticity flux, and the large-radius transport amplitude.

The astrophysical setting is compatible with known mechanisms of angular-momentum redistribution in disks [52]. Correlated stresses and torques in the baryonic component provide the transport channel, while the branch geometry supplies the active source geometry. Relativistic local spin-vorticity and transport treatments give the corresponding comparison class [53,54].

The next step is the radial branch problem for realistic disks, including the transition implied by (11), the energy conditions of (7), and joint fits to rotation curves and weak lensing. The most direct observational tests are disk-aligned lensing anisotropy, residual dependence on $j_b$ and gas fraction, suppression in systems without coherent rotation, and the scaling of the transition radius across disk structural classes.