Nonlocal Torque Coupling in Disk Galaxies: A Continuum Framework for Interpreting Rotation Curves

1 Nonlocal Torque Coupling in Disk Galaxies: A Continuum Framework for Interpreting Rotation Curves Ridwan Sakidja*, Armitha Dutta Dept. of Physics, Astronomy and Materials Science, Missouri State University, USA Justus Lau Dept. of Physics and Astronomy, Heidelberg University, Germany * Corresponding author ([email protected]) Abstract Observed galactic rotation curves rise, bend, and remain flat at radii where a purely baryonic Newtonian potential predicts declining velocities. The conventional interpretation introduces non-luminous dark matter to supply the missing rotational support. Here we propose an alternative dynamical architecture based on rotational coherence transmission. In this framework the stellar disk is treated not as a set of mechanically isolated Keplerian annuli but as a continuum of energy-coupled shells capable of inheriting, transmitting, and eventually saturating rotational coherence from inner regions. From this premise we derive three scale-free, falsifiable observables, namely the Curl Index, the Fractional Uplift, and the Surplus Index, quantify torque inheritance, rotational energy surplus, and radial coherence export. These indices can be computed directly from observed rotation curves without invoking additional mass. Applying the framework to the full SPARC database, we find that galaxies cluster into three dynamical regimes predicted by the model. The structured scaling of outer rotation curves is not explained by strictly local baryonic dynamics but follows naturally from nonlocal torque transport. This particular work concerns only the dynamical inference drawn from galactic rotation curves. It does not address cosmological evidence for dark matter from lensing, CMB anisotropies, or large scale structure. The framework is therefore offered as a galactic-scale dynamical alternative to the enclosedmass interpretation. 1. Introduction The standard explanation for flat galactic rotation curves is the presence of extended dark matter halos. This framework successfully reproduces observed velocities but does so by introducing a non-luminous mass component whose microscopic identity remains unknown after more than five decades of experimental and observational searches. In practice, constraining dark matter halo profiles at the galactic scale is challenging, as discrepancies between prediction and measurement can often be accommodated by adjusting halo parameters within their plausible ranges. This work explores whether such adjustments might be obviated by a more complete treatment of baryonic disk dynamics. The empirical foundation of the rotation-curve problem itself is not in dispute. Seminal observations by Rubin, Ford, and Thonnard[1] established that disk galaxies exhibit rotation curves that remain flat or 2 continue rising well beyond their optical extents. The subsequent interpretation of these measurements as evidence for large reservoirs of unseen mass, however, relies on an additional dynamical assumption: that the observed mean azimuthal velocity at each radius may be identified with a local circular velocity determined solely by the enclosed mass. This identification implicitly treats each radius as mechanically isolated, with no transfer of angular momentum or rotational energy between neighboring annuli except through the local gravitational field. While historically natural, this assumption is rarely examined and may be overly restrictive. The question addressed in this work is therefore narrow and specific: whether the apparent need for dark matter on galactic scales arises, at least in part, from a mechanical blind spot in the classical interpretation of disk dynamics. Importantly, this issue does not concern the observations themselves, but the dynamical closure imposed when interpreting them. Observations and theory alike indicate that real disks are not mechanically isolated systems. Spiral patterns maintain phase coherence across tens of kiloparsecs. Bars redistribute angular momentum throughout the disk. Density waves, wakes, and resonant structures propagate torque and information non-locally. These phenomena demonstrate that disk galaxies behave as gravitationally coupled continua rather than as collections of independent circular orbits. Motivated by these facts, this paper develops a constrained and falsifiable framework based on the idea that rotational coherence can be transmitted non-locally within the disk, supplying additional rotational support in outer regions without modifying gravity or introducing new mass components. The approach is grounded in three principles: 1. Coherence transmission rather than modification of gravity. The mechanism operates entirely within Newtonian dynamics and established disk physics. 2. Scalar diagnostics rather than adjustable parameters. The model introduces falsifiable indices that quantify rotational uplift and radial structure. 3. Direct comparison with observational data. Predictions are tested against the full SPARC database[2] rather than hypothetical idealized curves. 2. Context and Positioning Within Established Galaxy Dynamics The standard interpretation of disk–galaxy rotation curves assumes a strictly local relation between the circular velocity and the enclosed baryonic mass. In this picture, the disk is treated as a sequence of dynamically isolated test masses, where the equilibrium speed at radius 𝑅is given by 𝑣circ 2(𝑅)=𝑅 ∂Φ(𝑅) ∂𝑅 ,(1) where Φ(𝑅)is determined solely by the baryonic mass interior to 𝑅. Under this assumption, any excess of the observed velocity 𝑣obs(𝑅)over 𝑣circ(𝑅)is attributed to additional unseen mass. This methodology originates from classical disk models [3], is formalized in standard texts such as Galactic Dynamics [4], and underlies the historical inference of dark matter [5], [6]. However, this assumption of local isolation is neither required by Newtonian gravity nor consistent with the kinetic description of stellar disks. A collisionless galaxy is governed not by a set of independent 3 orbits but by the Vlasov–Poisson system [7], which describes the full phase-space evolution of stars under their collective gravitational field: ∂𝑓 ∂𝑡+𝐯⋅∇𝑥𝑓−∇𝑥Φ⋅∇𝑣𝑓=0, (2) ∇2Φ=4𝜋𝐺∫𝑓 𝑑3𝑣. (3) The Vlasov equation specifies how the phase-space density 𝑓(𝐱,𝐯,𝑡)is transported along stellar trajectories, while the Poisson equation enforces a global coupling between all radii through the gravitational potential. Because Φ(𝐱)depends on the full mass distribution, the gravitational field cannot be assigned to any locally isolated annulus. This nonlocal coupling is absent in ring-isolation treatments and is precisely what enables collective dynamical responses. Such collective behavior is well established observationally and theoretically, appearing through spiral density waves transmitting angular momentum [8], [9], swing amplification coupling adjacent annuli [10], bar-driven resonances redistributing angular momentum over kiloparsec scales [11] [12], gravitational wakes and dynamical friction [13], and radial migration without heating [14]. These processes are direct consequences of the Vlasov–Poisson dynamics and cannot occur in models that treat annuli as dynamically independent. Within this kinetic framework, the angular-momentum surface density inside cylindrical radius 𝑅 is obtained directly from the Vlasov distribution (see Supplementary A). With specific angular momentum ℓ=𝑅𝑣𝜙, 𝐿(𝑅,𝑡)=∫𝑅 𝑣𝜙 𝑓 𝑑3𝑣 𝑑𝑧.(4) Since 𝑅is independent of all velocity coordinates and of 𝑧, this may be written as 𝐿(𝑅,𝑡)=𝑅∫𝑣𝜙 𝑓 𝑑3𝑣 𝑑𝑧.(5) Defining the surface density Σ(𝑅,𝑡)=∫𝑓 𝑑3𝑣 𝑑𝑧,(6) and the mean azimuthal velocity ⟨𝑣𝜙⟩(𝑅,𝑡)= 1 Σ(𝑅,𝑡)∫𝑣𝜙 𝑓 𝑑3𝑣 𝑑𝑧,(7) the angular-momentum surface density becomes 𝐿(𝑅,𝑡)=Σ(𝑅,𝑡) 𝑅 ⟨𝑣𝜙⟩. (8) In the axisymmetric limit of nearly circular motion, ⟨𝑣𝜙⟩=𝑅Ω(𝑅,𝑡), yielding 𝐿(𝑅,𝑡)=Σ(𝑅,𝑡) 𝑅2Ω(𝑅,𝑡), (9) which is the familiar axisymmetric expression recovered as a special case of the full Vlasov moment. 4 Multiplying the Vlasov equation (2) by 𝑅𝑣𝜙 and integrating over velocity space gives an exact evolution equation for the angular-momentum surface density (see Supplementary A for the full derivation), ∂𝐿 ∂𝑡=−1 𝑅∂ ∂𝑅[𝑅 𝐹𝐿(𝑅,𝑡)],(10) where the radial flux of angular momentum is 𝐹𝐿(𝑅,𝑡)=∫𝑅 𝑣𝜙 𝑣𝑅 𝑓(𝑅,𝑣𝑅,𝑣𝜙,𝑣𝑧,𝑡) 𝑑3𝑣 𝑑𝑧.(11) Equation (10) is already the complete torque law for a collisionless disk. No source term appears. All changes in 𝐿(𝑅,𝑡) arise solely from the radial divergence of the Vlasov flux 𝐹𝐿. Integrating Eq. (10) over the full disk, 𝑑 𝑑𝑡∫ 𝐿(𝑅,𝑡) 𝑑𝑅=− ∞ 0∫1 𝑅 ∞ 0∂ ∂𝑅[𝑅 𝐹𝐿]𝑑𝑅,(12) and assuming vanishing boundary flux, lim 𝑅→0𝑅𝐹𝐿=lim 𝑅→∞𝑅𝐹𝐿=0, (13) yields 𝑑𝐽tot 𝑑𝑡 =0, 𝐽tot≡∫ 𝐿(𝑅,𝑡) 𝑑𝑅. ∞ 0(14) Thus, total angular momentum is strictly conserved, as required for a Hamiltonian Vlasov–Poisson system. The flux divergence redistributes angular momentum in radius but does not create or destroy it. Although the radial integral of Eq. (10) vanishes, the local divergence −1 𝑅∂ ∂𝑅[𝑅 𝐹𝐿(𝑅,𝑡)](15) need not vanish pointwise. Regions with 𝜕𝑅(𝑅𝐹𝐿)<0 lose angular momentum, while regions with 𝜕𝑅(𝑅𝐹𝐿)>0 gain angular momentum. Because the specific angular momentum scales as ℓ=𝑅𝑣𝜙, a small transfer from the inner disk can generate a large rotational response at large radius, where the energetic cost is low. The rotational kinetic-energy surface density, 𝐸rot(𝑅,𝑡)=1 2Σ(𝑅,𝑡)𝑅2Ω2(𝑅,𝑡), (16) therefore evolves through spatial redistribution rather than creation of energy. Quantities such as the rotational surplus measure where conserved rotational energy resides within the disk, not whether additional energy has been introduced. Equation (10) thus reframes the classical torque picture [15], [16], [17] as a transport law: angular momentum growth and redistribution arise from the divergence of a measurable Vlasov flux rather than from localized source terms. Any approximation enforcing 𝐹𝐿= 0 deletes this transport channel and collapses the dynamics back to the isolated-ring limit. Once non 5 axisymmetric structure is admitted, correlated radial and azimuthal motions activate this channel, enabling outward coherence transmission without invoking additional mass components. Physically, this expresses the fact that the specific angular momentum 𝑅𝑣𝜙 is transported across radius by stars with nonzero radial velocity 𝑣𝑅. Because eq. (10) is obtained as a direct moment of the Vlasov equation, it is exact and applies to any self-gravitating collisionless system—stellar disks, collisionless plasmas, or more general Vlasov–Poisson media. This torque-divergence structure is also an instance of a broader class of transport-driven phenomena discussed in the Curvature–Transport Correspondence (CTC), which interprets such flux divergences as the local drivers of dynamical curvature (see [18]). As a consequence, any approximation that enforces 𝐹𝐿=0 therefore deletes this transport channel and collapses the dynamics back to the isolated-ring limit (see Supplementary A). To further support this interpretation, we examine the SPARC dataset [2] for signatures that would not be expected in the isolated annulus limit, where 𝐹𝐿=0. In that limit, the torque density vanishes identically, and no radial coupling exists between annuli. Consequently, both of our diagnostics—Surplus Index and Curl Index—would be (close to) zero for every galaxy. Instead, SPARC galaxies populate structured, low-dimensional manifolds in uplift space (Surplus Index vs. Curl Index), demonstrating systematic, radius-spanning coherence that cannot arise in any model with 𝐹𝐿=0. These nonzero values are fully consistent with Vlasov–Poisson dynamics, which naturally predicts radial transport and collective modes, but they are absent from enclosed-mass formulations. The present work translates these classical Vlasov principles into a direct explanation of observed rotation-curve behavior. Mathematically, it differs from prior treatments in three essential ways: 1. Nonlocal coupling is retained. No closure is imposed that would force 𝐹𝐿=0; radial transport remains operative at all radii. 2. A transmission factor encodes 𝐹𝐿-driven coherence between annuli. This restores the continuum character of the Vlasov torque law and allows uplift to propagate outward with decreasing amplitude. 3. Two empirical diagnostics—Surplus Index and Curl Index—quantify uplift and radial structure. Both must vanish in a fully local disk, yet SPARC galaxies do not. Their nonzero structure is a direct observational signature of nonlocal angular-momentum flow. Together, these elements generate the observed three-regime response of disk galaxies: a linear inheritance region, a nonlinear bending region, and a saturated outer plateau. These regimes emerge naturally from the structure of 𝐹𝐿-mediated torque transport and require no modification of gravity or mass. For completeness, we emphasize that the standard dark matter halo framework [19] and MOND [20] has been the foundational benchmarks for explaining rotation curves. Both numerically succeed by altering the mass distribution or the force law. In this study, our model remains strictly Newtonian and attributes the observed phenomenology to coherence transmission and angular-momentum transport inherent in the dynamical architecture of rotating disks. 6 2.1 Transport versus Mass Augmentation At this point it is important to clarify how the present framework differs procedurally from dark matter– and MOND-based approaches. Many dark matter models implicitly rely on energy and angularmomentum redistribution as a central dynamical ingredient. In numerical simulations, dark matter halos are not passive mass reservoirs; they exchange angular momentum with the disk, absorb torques from non axisymmetric structure, and provide long-range dynamical coupling that stabilizes rotation over secular timescales. The framework developed here assigns this transport role to the visible collisionless disk itself, through the angular-momentum flux that arises naturally in 7. SPARC Diagnostics, Correlations, and the Failure of Radius as an Organizer We test the transmission framework using all 175 late-type galaxies in the SPARC Rotmod_LTG sample [2] (see Supplementary E, F and G). For each galaxy we reconstruct the baryonic circular velocity and compute three diagnostics derived directly from the model: the Curl Index, measuring torque contrast; the mean uplift, measuring fractional velocity enhancement; and the Surplus Index, measuring fractional excess rotational energy in the outer disk. We also extract characteristic radii, including the outermost measured radius 𝑅max, to test whether uplift is controlled by physical extent. Here, the SPARC velocities are reconstructed using a tilted-ring, axisymmetric model. Mild non-circular motions—such as warps, oval flows, asymmetric drift, or small inclination uncertainties—can alter the local slope of the baryonic curve. These effects tend to reduce the apparent torque contrast and therefore bias the Curl Index downward rather than produce spurious high-curl structure. The fact that the empirical Curl–Surplus distribution displays a strong, coherent sequence despite these conservative biases indicates that the transport signal should be interpreted as a lower bound. (a) Radius plots. When Surplus Index and mean uplift are plotted against 𝑅max (Figures 1a and 2a), the SPARC galaxies show little systematic organization. Galaxies spanning more than an order of magnitude in size occupy overlapping ranges of surplus and uplift, and systems with nearly identical radii can differ strongly in both diagnostics. No monotonic relation between galaxy size and rotational enhancement is observed. This indicates that radius alone is not the controlling variable of outer-disk rotational support. (b) Curl plots. In contrast, when the same diagnostics are plotted against Curl Index (Figures 1b and 2b), the SPARC sample forms a highly structured sequence. At low curl (≲1), both surplus and uplift rise approximately linearly, corresponding to the linear inheritance regime. At intermediate curl (∼2–5), the trends bend: surplus continues to increase while uplift begins to flatten. At high curl (≳6–8), Surplus Index settles into a stable band (∼0.3–0.6) while Curl Index spreads horizontally, indicating saturation of transmission. The central empirical result is that Curl Index, not radius, organizes rotational uplift in disk galaxies. The SPARC sample is highly structured in curl space but essentially unorganized in radius space, exactly as expected if outer rotational support arises from torque-mediated coherence export rather than from purely radial mass augmentation. 7 The SPARC scaling relations identified by Ghari et al. (2019) [21] provide important statistical context. Within the present framework, these relations can be interpreted as observational signatures of torquemediated coherence transmission: Curl Index parallels the baryon-induced acceleration correlation, while Surplus Index mirrors the core-density relation. The three-stage progression of inheritance, bending, and saturation reflects the same structured behavior observed in the SPARC sample. Inner–Outer Energy Balance and the Absence of a Required Inner Deficit In reference to Section 5.1, it is important to emphasize that the transmission framework does not require a clearly resolved inner deficit in the observed rotation curve. In practice, baryonic rotation curves are constructed relative to an inferred inner baseline that rarely corresponds to a true dynamical zero point. The innermost regions of disk galaxies are typically influenced by bulges, bars, beam smearing, finite spatial resolution, and uncertainties in mass to light ratios. Any early withdrawal of rotational coherence occurring at radii smaller than the observational core is therefore naturally absorbed into the reconstructed baryonic profile and does not appear as an explicit inner dip. From an energetic standpoint, this behavior is expected. The rotational energy surface density is given by 𝐸rot(𝑅)=1 2 Σ(𝑅) 𝑅2 Ω2(𝑅), (17) and for late type disks with approximately flat rotation curves, Ω(𝑅)≈𝑣0/𝑅. In this regime, the dominant radial dependence enters through the surface density Σ(𝑅). Because Σ(𝑅)declines exponentially, the cumulative rotational energy stored in the inner disk exceeds that of the outer disk by several orders of magnitude. As a result, only a percent level redistribution of inner rotational coherence is sufficient to generate the observed velocity excess at large radii. In systems with extended radial coverage and high spatial resolution, a shallow inner deficit or transition region is sometimes observed. Its absence in most galaxies does not contradict the transmission picture. Rather, it reflects the fact that the energetic withdrawal required to support the outer disk is dynamically small and often occurs at radii below the observational core, where it is difficult to resolve observationally. For a small subset of SPARC galaxies, the reported rotation velocities are systematically low at all measured radii. Such behavior is inconsistent with any physical disk equilibrium and is most naturally attributed to geometric or observational systematics, including inclination uncertainties, distance errors, asymmetric drift, or non circular motions associated with bars and spiral structure. Because these effects uniformly suppress the inferred rotation curve, these galaxies do not provide reliable constraints on dynamical models and are excluded from further analysis. More generally, once non axisymmetric structure is admitted, angular momentum and energy transport arise naturally within the collisionless disk. The distinction between competing dynamical interpretations is therefore not whether transport is required, but where that transport is permitted to reside. In dark matter based models, the halo functions as the carrier of angular momentum and energy flow. In the present work, the same dynamical role is supplied by the stellar disk itself through nonzero phase space flux, without introducing additional mass or modifying gravity. 8 Accordingly, the present framework should be understood not as an alternative force law, but as a logically prior step. It asks whether the fully closed collisionless Vlasov Poisson dynamics of visible matter are sufficient to account for the observed rotation curves before assigning the required transport to an unseen component. 2.2. Distinction Between Circular Velocity and Mean Azimuthal Velocity A crucial point often left implicit in rotation-curve modeling is that the baryonic circular velocity, 𝑣bar 2(𝑅)=𝑅 ∂𝑅Φbar, (18) is a local equilibrium diagnostic valid only in the limit where each annulus behaves as a dynamically isolated test ring. By contrast, the observed rotation curve measures the mean azimuthal velocity, ⟨𝑣𝜙(𝑅)⟩=Σ−1(𝑅)∫𝑣𝜙𝑓(𝑅,𝐯,𝑡) 𝑑3𝑣, (19) which is a phase-space moment of the full distribution function. These two quantities are equal only when radial angular-momentum flux vanishes. Taking the angular-momentum moment of the Vlasov equation yields ∂𝐿 ∂𝑡=−1 𝑅∂ ∂𝑅(𝑅𝐹𝐿), (20) where 𝐹𝐿=∫𝑅𝑣𝑅𝑣𝜙𝑓 𝑑3𝑣is the radial flux of angular momentum. Dividing by Σ𝑅and integrating in time gives ⟨𝑣𝜙(𝑅)⟩=𝑣bar(𝑅)+ 1 Σ(𝑅)𝑅∫[−1 𝑅∂ ∂𝑅(𝑅𝐹𝐿)] 𝑡𝑑𝑡′, (21) showing explicitly that deviations between the observed velocity and the baryonic circular speed arise from the divergence of the angular-momentum flux. The resulting velocity uplift therefore reflects redistribution of conserved angular momentum rather than the presence of additional mass or modified gravity.This distinction explains how outer disks can exhibit velocities exceeding the baryonic circular expectation while remaining fully consistent with Newtonian dynamics and global energy conservation. 2.3. Why This Connection Has Not Been Emphasized in Rotation-Curve Modeling Although the ingredients of nonlocal disk dynamics have long been established within galactic dynamics and kinetic theory, they have not traditionally been incorporated into the interpretation of galactic rotation curves. This reflects a historical separation between complementary research traditions rather than any deficiency in the underlying theory. Rotation-curve analyses developed primarily within observational astronomy, where enclosed-mass constructions and halo fitting provided a practical phenomenology. Within this framework, discrepancies between baryonic predictions and observed velocities were naturally interpreted as evidence for additional mass, and disk galaxies were treated, to leading order, as collections of dynamically decoupled annuli. 9 In parallel, studies of bars, spiral structure, resonances, and secular evolution focused on collective behavior in collisionless disks, emphasizing wave–particle coupling, gravitational torques, and angularmomentum redistribution. In these analyses, rotation curves typically entered as background constraints rather than as primary dynamical diagnostics. As a result, the continuum transport structure implied by the Vlasov–Poisson system was not routinely applied to the interpretation of the observed circular velocity profile 𝑣(𝑅). Formally, the enclosed-mass relation commonly used in rotation-curve modeling corresponds to a restricted subset of Vlasov–Poisson solutions in which the radial angular-momentum flux 𝐹𝐿vanishes. This limit describes an idealized, perfectly axisymmetric disk lacking nonaxisymmetric structure. While useful, it does not represent the generic behavior of real galactic disks. The present work relaxes this restriction by retaining the angular-momentum flux term in the exact moment equation derived from the Vlasov equation. Writing the equation in cylindrical coordinates and multiplying by the specific angular momentum 𝑙=𝑅𝑣𝜙yields a conservation law in which 𝐹𝐿governs nonlocal angular-momentum transport. In disks hosting persistent or slowly evolving asymmetries, this flux is generically nonzero. A related objection is that visible-matter–only models have already been tested and found insufficient, necessitating the introduction of dark matter. This conclusion follows only if the visible disk is evaluated under the restrictive assumption of vanishing angular-momentum flux, imposed implicitly through axisymmetry and isolated-annulus closures. When the resulting system fails to sustain the observed rotation, dark matter is introduced to supply the missing gravitational support and angular-momentum exchange. This procedure effectively introduces an external reservoir to supply transport that is suppressed by the imposed closure and therefore does not test whether the fully closed collisionless Vlasov–Poisson system of visible matter can self-consistently generate the required torque. This distinction concerns the closure of the Vlasov–Poisson system rather than its use: standard treatments employ the same equations but close them by assigning the required transport to a dark matter component, whereas the present work tests whether that transport can arise self-consistently from the visible disk alone. The broader context includes the extensive literature on secular evolution, reviewed comprehensively by Sellwood [12], in which bars, spiral density waves, and associated resonances redistribute angular momentum over long timescales. Secular evolution refers to the slow internal reconfiguration of a galaxy that is driven by nonaxisymmetric gravitational torques and is most efficient at the resonances first formalized by Lynden-Bell and Kalnajs [8]. These phenomena represent structured solutions of the Vlasov–Poisson system in which coherent asymmetries generate sustained gravitational torques. Classical work correctly demonstrated that resonances can transfer angular momentum and energy, but this was usually presented as a resonance-localized mechanism rather than as one instance of a more general continuum transport process. The transmission framework developed here is fully consistent with this classical picture and extends it by showing that any persistent or slowly varying asymmetry produces a nonlocal flux, even far from formal resonances. Secular evolution is therefore a wellestablished and important subset of the more general continuum coupling described in this work. The present model connects that transport directly to observable rotation-curve structure through the Surplus and Curl Indices, which serve as measurable projections of the underlying angular momentum flux 𝐹𝐿. 16 The transmission law T(S) should be understood as an effective, coarse-grained response rather than a new dynamical interaction. It does not modify the Poisson equation or introduce any additional gravitational source; instead, it constrains the efficiency with which existing non axisymmetric torque flux is transmitted once finite absorptive capacity is reached. From a kinetic-theory perspective, the saturating hyperbolic-tangent form represents the minimal response consistent with collisionless dynamics under energetic constraints, and emerges naturally from a Lagrangian extension of the Vlasov–Poisson system (Supplementary C and D). 6.3 Effective transmission coefficient The effective local transmission rate is the product of geometric coupling and curl-limited response: 𝑘eff(𝑟)=𝑘(𝑟) 𝑇(𝑆(𝑟)). (32) A radial zone therefore transmits rotational energy efficiently only if: 1. it lies in a high-coupling phase of the radial chain, and 2. its local curl-stress remains below the saturation threshold. Low-coupling phases, large radii, and high-curl zones naturally suppress transmission. This alternating, radius-decaying, stress-regulated structure constitutes the core transmission rule of the model. 6.4 A simplified energy-lock surrogate To evaluate how this transmission modifies the rotation curve, we introduce an energy-lock surrogate. Instead of solving full hydrodynamics, we augment the Newtonian rotation curve by adding the fraction of inner rotational energy deposited at radius 𝑟: 𝑣obs 2(𝑟)=𝑣𝑁 2(𝑟)+𝑘eff(𝑟) (Ωinner 𝑟)2.(33) This surrogate is intended as an illustrative mapping of how nonzero angular-momentum flux reshapes the rotation profile. It does not, by itself, enforce full Jeans equilibrium. A steady-state closure would follow from solving the angular-momentum flux equation together with the axisymmetric Jeans equation under a prescribed 𝐹𝐿(𝑅). In this paper, we focus on the surrogate to isolate the observable imprint of torque transmission, while the formal flux derivation is provided in Supplementary AThis surrogate captures: • how much energy a shell inherits from the inner engine, • how rapidly that energy attenuates with radius, • how local stress limits further inheritance, and • how a small leakage from the inner reservoir produces a large outer response. The alternating-layer structure should not be interpreted as literal rigid shells. It approximates the heterogeneous coupling landscape of real disks, where different radial zones respond differently to the same torque field. 6.5 Why this formulation matters 17 This construction delivers three essential outputs directly from the model: • a Newtonian baseline 𝑣𝑁(𝑟), • a transmission-enhanced surrogate curve 𝑣obs(𝑟), and • a complete set of SPARC-consistent diagnostics, including the Surplus Index, mean uplift ⟨𝑈⟩, and Curl Index. Because transmission is local and radial, while diagnostics integrate deposited energy globally, the two need not organize in the same variable. Section 6 defines the transmission rules—radius-dependent, coupling-limited, and stress-saturated—while Section 7 shows how the resulting energy redistribution organizes the observed phenomenology. A full synthetic parameter sweep illustrating the linear, bending, and saturated regimes is provided in Supplementary E. 6.6 Falsifiable Diagnostics The transmission model leads to three observational diagnostics that follow directly from the torque formulation and require no adjustable parameters. The first is the Curl Index, defined by 𝐂𝐮𝐫𝐥 𝐈𝐧𝐝𝐞𝐱= ∫|dτobs dR −dτbar dR |dR ∫|dτbar dR |dR (34) This index measures the extent to which the observed torque gradient differs from that implied purely by the baryonic mass distribution. In any model that enforces 𝐹𝐿=0, the torque is generated locally and the two profiles coincide, forcing the Curl Index to vanish. A nonzero value therefore represents a direct observational signal of torque inheritance across radii: it records the imprint of nonlocal angularmomentum transport predicted by the Vlasov formulation but eliminated in enclosed-mass models. We should note that because the Curl Index is constructed to probe large scale radial structure, smoothing suppresses small-scale noise; the diagnostic is intended to capture global transport signatures rather than fine-grained fluctuations. A second quantity, the Fractional Uplift, 𝐔(𝐑)=vobs 2−vbar 2 vbar 2 (35) captures the local enhancement of the rotational velocity relative to the baryonic expectation. Because uplift responds point-by-point to the arrival of transmitted coherence, its radial structure traces the same progression that appears in the transmission map: a region of linear inheritance at small radii, a bending zone where the disk begins to saturate, and an outer plateau where additional inheritance becomes dynamically ineffective. In a system without nonlocal coupling, uplift cannot show such structured behavior and reduces to noise around zero. The third diagnostic, the Surplus Index, 18 𝐒𝐮𝐫𝐩𝐥𝐮𝐬 𝐈𝐧𝐝𝐞𝐱= ∫[vobs 2−vbar 2]dR Rmax 0.5Rmax ∫vobs 2 Rmax 0dR (36) provides a global measure of the excess rotational energy stored in the outer disk. The choice of integration limits is deliberate. The numerator integrates only over the outer half of the galaxy, from 0.5𝑅max to the edge of the measured rotation curve at 𝑅max. This isolates the radial domain in which the disk cannot generate rotational support through its own local baryonic mass and must therefore rely on energy inherited from interior radii. A nonzero surplus indicates rotational support beyond what the local baryonic mass alone provides. Within the present framework, this is naturally interpreted as exported angular momentum from interior radii, although alternative explanations such as additional mass distributions cannot be ruled out purely from this diagnostic. The denominator integrates the total observed rotational energy over the entire disk; normalizing by this quantity expresses the outer surplus as a fraction of the galaxy's overall rotational budget. Note that all three diagnostics rely only on quantities directly provided by the SPARC database[2]. For each galaxy, SPARC tabulates both the observed circular speed 𝑉obs(𝑅) and the full baryonic contribution 𝑉bar(𝑅) on a common radial grid. From these profiles the Newtonian torque follows simply as 𝜏(𝑅)=𝑅𝑉2(𝑅), and its radial derivative is obtained using standard spline smoothing without introducing additional dynamical assumptions. Likewise, the integrals defining the uplift and surplus follow directly from the tabulated velocities. Thus, each diagnostic is an observational construct, obtained entirely from SPARC without modeling freedom or hidden parameters. In the transmission model, this quantity is expected to be positive because torque is transported outward whenever 𝐹𝐿≠0, leading to a measurable accumulation of rotational support in the outer annuli. By contrast, in any model where the outer disk is dynamically decoupled, whether through the enclosed-mass framework or by implicitly enforcing 𝐹𝐿=0, the outer region cannot acquire additional rotational energy from the interior, and the Surplus Index necessarily collapses to negligible values. It is worth noting that other recent approaches, such as the dual-density formulation of Suleiman (2023) [23], also seek to extract structural information from the rotation curve. These valuable approaches often operate as algebraic mappings from the velocity field to a density field. Our framework is complementary but distinct: it introduces an explicit, physically motivated mechanism for angular momentum transmission, which in turn generates the new diagnostics (Curl Index, Surplus Index) that quantify this process. Our diagnostics are therefore tied to a specific dynamical transport mechanism, rather than to a reinterpretation of the density profile itself. 7. SPARC Diagnostics, Correlations, and the Failure of Radius as an Organizer We test the transmission framework using all 175 late-type galaxies in the SPARC Rotmod_LTG sample [2] (see Supplementary E–G). For each galaxy, we reconstruct the baryonic circular velocity and compute three diagnostics derived directly from the model: the Curl Index, measuring torque contrast; the mean uplift, measuring fractional velocity enhancement; and the Surplus Index, measuring fractional excess rotational energy in the outer disk. We also extract characteristic radii, including the outermost measured radius 𝑅max, to test whether rotational uplift is controlled by physical extent. Interpretation of near-zero Surplus values. Because the Surplus Index is defined as an integrated, signed measure of excess rotational energy in the 19 outer disk, values near zero do not necessarily indicate the absence of angular-momentum transport. Such values may arise either from genuinely weak transport or from partial cancellation between positive and negative radial contributions. For this reason, galaxies with Surplus Index ≈ 0 are retained in the sample rather than reclassified. Their radial uplift profiles, shown in Supplementary G, frequently exhibit coherent uplift of a single sign over most of the disk, interrupted by localized countercontributions, indicating balanced but physically active transport rather than dynamical quiescence. The diagnostic therefore encodes net transport balance, not a binary on/off criterion. The SPARC velocities are reconstructed using a tilted-ring, axisymmetric model. Mild non-circular motions—such as warps, oval flows, asymmetric drift, or small inclination uncertainties—can alter the local slope of the inferred baryonic curve. These effects tend to reduce the apparent torque contrast and therefore bias the Curl Index downward rather than generate spurious high-curl structure. The presence of a strong, coherent Curl–Surplus sequence despite these conservative biases indicates that the inferred transport signal should be interpreted as a lower bound. (a) Radius as an organizer When the Surplus Index and mean uplift are plotted against 𝑅max(Figures 1a and 2a), the SPARC galaxies show little systematic organization. Galaxies spanning more than an order of magnitude in size occupy overlapping ranges of surplus and uplift, and systems with nearly identical radii can differ substantially in both diagnostics. No monotonic relation between galaxy size and rotational enhancement is observed, indicating that radius alone does not control outer-disk rotational support. (b) Curl as an organizer In contrast, when the same diagnostics are plotted against Curl Index (Figures 1b and 2b), the SPARC sample forms a highly structured sequence. At low curl (≲1), both surplus and uplift increase approximately linearly, corresponding to the linear inheritance regime. At intermediate curl (∼2–5), the trends bend: surplus continues to rise while uplift begins to flatten. At high curl (≳6–8), the Surplus Index settles into a stable band (∼0.3–0.6) while Curl Index spreads horizontally, indicating saturation of transmission. The central empirical result is that Curl Index, not radius, organizes rotational uplift in disk galaxies. The SPARC sample is highly structured in curl space but essentially unorganized in radius space, exactly as expected if outer rotational support arises from torque-mediated coherence export rather than from purely radial mass augmentation. The SPARC scaling relations identified by Ghari et al. (2019) [21] provide useful statistical context. Within the present framework, these relations can be interpreted as observational signatures of torquemediated coherence transmission: Curl Index parallels the baryon-induced acceleration correlation, while Surplus Index mirrors the core-density relation. The three-stage progression of inheritance, bending, and saturation reflects the same structured behavior observed across the SPARC sample. Inner–Outer Energy Balance and the Absence of a Required Inner Deficit As discussed in Section 5.1, the transmission framework does not require a clearly resolved inner deficit in the observed rotation curve. In practice, baryonic rotation curves are constructed relative to an 20 inferred inner baseline that rarely corresponds to a true dynamical zero point. The innermost regions of disk galaxies are typically influenced by bulges, bars, beam smearing, finite spatial resolution, and uncertainties in mass-to-light ratios. Any early withdrawal of rotational coherence occurring at radii smaller than the observational core is therefore naturally absorbed into the reconstructed baryonic profile. From an energetic standpoint, this behavior is expected. The rotational-energy surface density is 𝐸rot(𝑅)=1 2 Σ(𝑅) 𝑅2Ω2(𝑅), (eq. 17) and for late-type disks with approximately flat rotation curves, Ω(𝑅)≃𝑣0/𝑅. Because Σ(𝑅)declines exponentially, the cumulative rotational energy of the inner disk exceeds that of the outer disk by several orders of magnitude. Consequently, only a percent-level redistribution of inner rotational coherence is sufficient to generate the observed outer-disk velocity excess. In systems with extended radial coverage and high spatial resolution, a shallow inner deficit or transition region is sometimes observed. Its absence in most galaxies does not contradict the transmission picture but reflects the fact that the required energetic withdrawal is dynamically small and often occurs below the observational core. For a small subset of SPARC galaxies, the reported rotation velocities are systematically low at all measured radii. Such behavior is inconsistent with any physical disk equilibrium and is most naturally attributed to geometric or observational systematics, including inclination uncertainties, distance errors, asymmetric drift, or non-circular motions associated with bars and spiral structure. Because these effects uniformly suppress the inferred rotation curve, these galaxies do not provide reliable constraints on dynamical models and are excluded from further analysis. 8. Surplus–Curl Structure and the tanh Transmission Law The Surplus Index and Curl Index probe complementary aspects of inherited rotational support. Defining Δ𝑉2(𝑅)=𝑉obs 2(𝑅)−𝑉bar 2(𝑅), (37) the Surplus Index (Eq. 23) measures the amplitude of inherited rotational energy in the outer disk, while the Curl Index measures the radial variation of this support through the torque gradient. Using 𝜏(𝑅)=𝑅𝑉2(𝑅), (38) one finds 𝑑𝜏obs 𝑑𝑅 −𝑑𝜏bar 𝑑𝑅 =Δ𝑉2(𝑅)+𝑅𝑑Δ𝑉2 𝑑𝑅 .(39) Thus, Surplus Index responds mainly to Δ𝑉2, whereas Curl Index responds to both Δ𝑉2 and its radial gradient. When inherited support is rising, both indices increase together; when Δ𝑉2 flattens, surplus saturates while curl spreads. This directly reproduces the SPARC pattern. The tanh transmission law, 21 𝑇(𝑆)=𝑇maxtanh (𝑆/𝑆0), (40) implements the minimal physically required behavior: linear inheritance at low stress, bending at intermediate stress, and saturation at high stress (Eqs. 28–33). Its success in reproducing the threeregime SPARC structure supports the interpretation that rotational-coherence transmission, not local mass augmentation, governs disk rotation curves. 9. Relation to the Standard Vlasov–Poisson Dynamics Within the Vlasov–Poisson framework [7], angular momentum evolution follows directly from the collisionless Boltzmann equation. Taking the angular-momentum moment yields ∂𝐿 ∂𝑡=−1 𝑅∂ ∂𝑅(𝑅𝐹𝐿),(41) with 𝐹𝐿(𝑅,𝑡)=⟨𝑅𝑣𝑅𝑣𝜙⟩. (42) Crucially, Eq. (10) contains no source term. Angular momentum is conserved locally and globally; all redistribution arises solely from the divergence of the flux 𝐹𝐿. Nonzero 𝐹𝐿 therefore does not indicate angular-momentum or energy creation, but the presence of correlated radial and azimuthal motions. Whether these correlations remain localized or organize coherently over extended radii is controlled by disk geometry and symmetry, not by conservation laws. Classical results on spiral density waves, swing amplification, and bar-driven resonances [9], [10], [11], [12] establish that collisionless disks behave as gravitationally coupled continua. The transmission model developed here does not modify this theory; it restores the nonlocal coupling implicitly omitted when disks are treated as isolated annuli. Setting 𝐹𝐿=0 recovers the isolated-ring limit and suppresses precisely the transport processes responsible for secular evolution in real disks. 10. Extreme Transport Regimes and Control Cases To delineate the dynamical regimes relevant to the transmission framework, we examine two complementary sets of extreme systems. The first consists of galaxies drawn from the SPARC sample that exhibit unusually large rotational surplus, with Surplus Index values approaching ≃0.6. These objects are selected to highlight disks in which strong non-axisymmetric structure—such as prominent spiral arms, bars, warps, or large-scale curvature asymmetries—enforces sustained angular-momentum and energy transport. In these systems, rotational uplift emerges at large radii in direct association with persistent torque generation. In parallel, we consider a second, independent set of five galaxies widely cited in the literature as darkmatter-suppressed systems. These objects are not selected from SPARC by surplus, but are included as a contrasting extreme. Their morphologies and kinematics place them near the zero-transport or transport-irrelevant limit: they are either pressure-supported or dynamically regular and nearly axisymmetric, and they lack the structural features required to generate significant nonlocal torque flux. Consequently, they exhibit negligible rotational surplus and rotation curves that remain consistent with the local baryonic potential within the regions probed. 22 The purpose of this comparison is not to equate the two populations, but to bracket the boundaries of the transmission framework. High-surplus SPARC disks occupy a transport-dominated regime in which rotational energy is redistributed outward through non-axisymmetric structure, while dark-mattersuppressed galaxies define the limiting case in which transport is ineffective or dynamically unnecessary. Together, these extremes demonstrate that rotational surplus emerges only when angular-momentum transport is structurally enabled, and collapses when such channels are absent. The two regimes and their representative systems are summarized in Table 1 and detailed in Supplementary H. Table 1. Extreme-Case Galaxies Used to Anchor the Transmission Framework Galaxy Regime Morphology / Kinematics Diagnostic Signature Transport Interpretation NGC 3109 High-surplus (Transmissiondominant) Elongated stellar disk; strongly warped H I; tidal interaction with Antlia; filamentary group with coherent velocity gradient High Surplus Index; large Curl Index; positive mean uplift Sustained non-axisymmetry enforces strong torque flux (FL ≠ 0), driving coherent redistribution of rotational energy NGC 3741 High-surplus (Inheritance regime) Extremely extended H I disk (∼42 scale lengths); persistent warp; non-circular motions across most radii High Curl Index with linearly rising surplus Radial flows and warps enforce long-range angularmomentum transport; surplus tracks torque divergence DDO 154 Inheritance / falsifiability anchor Gas-dominated dwarf; extended warped H I disk; asymmetric outer velocity field; quiescent inner reservoir Moderate surplus and curl; surplus collapses where asymmetry vanishes Demonstrates falsifiability: surplus appears only where flux divergence exists and collapses in transport-free zones UGC 01281 Bending / saturation regime Low-surface-brightness disk; asymmetric baryonic distribution; slowly rising rotation curve Intermediate surplus and curl Moderate torque gradients sustain partial transmission; surplus saturates as coherence export weakens ESO 444 G084 Inheritance (compact end) Lopsided dwarf irregular; outer H I distortions; kinematic warp beyond inner disk High surplus relative to size Structural asymmetry alone enforces transmission, independent of scale or stellar mass NGC 1052-DF2 Transportsuppressed Ultra-diffuse; pressuresupported; no disk rotation or asymmetry Near-zero surplus and curl Absence of torque pathways places system near the isolated-annulus (FL ≈ 0) limit NGC 1052-DF4 Transportsuppressed Ultra-diffuse twin of DF2; dispersion-dominated Near-zero surplus and curl Dark-matter deficiency coincides with intrinsically inactive transport geometry FCC 224 Transportsuppressed Quenched ultra-diffuse dwarf; no disk rotation; old stellar populations Near-zero surplus and curl Random stellar motions dominate; nonlocal transport dynamically irrelevant AGC 114905 Transportsuppressed (rotating) Gas-rich UDG; thin, regular H I disk; weak shear; no bars or spirals Low surplus despite rotation Rotation without torque: coherent circular motion but negligible angular-momentum flux NGC 1277 Control (FL ≈ 0) Compact, massive ETG; axisymmetric; equilibrium stellar kinematics No surplus; equilibrium Jeans/orbit models Absence of inferred dark matter reflects lack of transport demand, not suppressed transport 11. Geometry-Dependent Angular-Momentum Transport: Spherical versus Spiral Disks To verify that the transmission mechanism arises directly from collisionless dynamics, we perform a minimal N-body realization of the Vlasov–Poisson system using the REBOUND integrator. In this 23 representation, the phase-space distribution is sampled by discrete particles, and the angularmomentum flux 𝐹𝐿(𝑟,𝑡) = ∑𝑚𝑖𝑖 𝑟𝑖 𝑣𝑟,𝑖 𝑣𝜙,𝑖 (43) provides a direct discrete estimator of the continuum torque flux appearing in the Vlasov–Poisson moment equation. When the disk is constrained to remain axisymmetric, this quantity fluctuates around zero and produces no secular transport. When weak non-axisymmetry is permitted, coherent correlations between radial and azimuthal motions emerge, generating sustained outward angularmomentum flux without mass accumulation at large radii (Figures 3–6). Full numerical details are provided in Supplementary I. To isolate the role of geometry in enabling angular-momentum and energy transport, we compare two controlled N-body simulations evolved under identical gravitational and numerical conditions but differing only in their initial symmetry: • Case A (spherical control): an axisymmetric disk initialized with perturbation amplitude 𝜖=0, preserving spherical symmetry. • Case B (spiral disk): the same disk seeded with a weak non axisymmetric perturbation (𝜖= 0.05), generating a long-lived spiral pattern. Both systems are evolved to 𝑡=60 using the same particle number, softening length, and timestep. All diagnostics are constructed from azimuthally averaged quantities in radial bins (see Supplementary I). Particle Trajectories and Disk Morphology The geometric origin of the contrasting behavior is illustrated in Figures 3 and 4, which show representative particle snapshots for the two cases. In Case A (Figure 3), particle trajectories remain quasi-circular and radially confined. Orbits exhibit only weak diffusion associated with discreteness noise, and no coherent collective motion develops. The disk retains its near-spherical symmetry throughout the evolution. In Case B (Figure 4), particles organize into spiral-like trajectories that mediate correlated radial and azimuthal motion. Individual particles intermittently gain or lose angular momentum as they interact with the spiral pattern, producing sustained radial transport. The spiral acts as a long-lived conduit for angular-momentum exchange rather than as a transient disturbance. Surface-Density Evolution Figure 5 shows the time–radius evolution of the surface density Σ(𝑅,𝑡)for both cases. In Case A, the surface density remains smooth and monotonic throughout the integration. Apart from stochastic fluctuations associated with finite particle number, no systematic radial redistribution of mass is observed. The outer disk remains persistently low-density, and the overall profile retains its initial exponential-like form. 24 In Case B, the spiral perturbation generates coherent overdense and underdense features that propagate radially over time. Although the absolute surface density at large radii remains small, these features persist over multiple dynamical times. The spiral therefore introduces organized structure into the disk without requiring a net outward migration of mass. This comparison demonstrates that the key distinction between the two cases is not the magnitude of the surface density in the outskirts, but the presence or absence of spatially coherent non axisymmetric structure. Angular-Momentum Flux The dynamical consequences of these geometric differences are quantified by the angular-momentum flux 𝐹𝐿(𝑅,𝑡)=⟨𝑅 𝑣𝑅 𝑣𝜙⟩. (44) In Case A, 𝐹𝐿(𝑅,𝑡)fluctuates around zero with no persistent radial structure. While instantaneous nonzero values arise from discreteness noise, the flux divergence averages out over time, and no systematic redistribution of angular momentum occurs. In Case B, a coherent, outward-propagating structure appears in 𝐹𝐿(𝑅,𝑡). This feature persists over many dynamical times and produces a nonzero flux divergence. Although the surface density at large radii remains low, angular momentum is efficiently transported outward because the transported quantity scales with the specific angular momentum 𝑅𝑣𝜙, rather than with mass alone. As a result, the spiral case develops a substantial outer-disk rotational surplus, whereas the axisymmetric case does not. This demonstrates that geometry—not mass content, force law, or numerical resolution—controls the existence of a transport channel. Conservation and Symmetry It is essential to emphasize that the angular-momentum flux equation contains no source term. In the collisionless Vlasov–Poisson system, angular momentum is conserved both locally and globally; any redistribution must therefore arise exclusively from the divergence of the flux itself. Nonzero values of 𝐹𝐿do not indicate angular-momentum creation, but the presence of correlated radial and azimuthal motions. The physical distinction between the two cases is whether these correlations remain spatially localized or instead organize coherently over extended radial ranges. In perfectly symmetric systems, angular momentum is locally exchanged but globally trapped. Once non axisymmetric structure is admitted— even at low amplitude—angular momentum and rotational energy propagate outward naturally through an expanding radial domain. Implication 25 These results show that the absence of angular-momentum transport in spherical systems is not a failure of dynamics, but a consequence of symmetry. When realistic disk geometry is admitted, outward transport arises naturally and robustly, providing a purely dynamical pathway for sustaining extended rotation curves without invoking additional mass components. 12. Limitations and Scope of the Present Analysis The present work is intentionally structural and dynamical in nature, focusing on the consequences of the Vlasov Poisson system viewed from an energy and transport based perspective, rather than as a fully closed phenomenological model of galaxy formation. While we demonstrate that only a small redistribution of inner disk rotational energy is sufficient to generate the observed outer velocity excess, several limitations and qualifications must be stated clearly. First, although the Vlasov Poisson framework rigorously governs collisionless gravitational dynamics, it places stringent requirements on how energy transfer and angular momentum flux are analyzed. Our results highlight that even a percent level leakage of inner rotational coherence can produce a substantial velocity increase at large radii. This sensitivity underscores the need for careful treatment of energy bookkeeping within the Vlasov Poisson system, especially when interpreting small residuals in rotation curves. The present analysis demonstrates this principle but does not yet constitute an exhaustive dynamical accounting of all transport channels. Second, the SPARC database, while the most comprehensive and internally consistent rotation curve compilation currently available, is not free of observational limitations. These include incomplete radial coverage for some galaxies, uncertainties in inclination and distance estimates, nonuniform data quality, and residual systematics associated with mass to light ratio assumptions. Such factors can introduce scatter and local discrepancies that are not fully captured by any single diagnostic. Consequently, our results should be interpreted statistically and structurally, rather than as precise reconstructions for every individual system. Third, the governing Vlasov Poisson equations are fundamentally conservative, whereas real galaxies are subject to complex and time dependent processes such as disk heating, secular evolution, bar formation, spiral winding, and environmental interactions. The present work does not attempt to model these processes explicitly. Instead, it isolates the minimal dynamical requirement for rotational inheritance and torque transmission within a collisionless framework. In this sense, our analysis should be viewed as a baseline dynamical constraint, rather than a complete evolutionary model. Fourth, the transmission law employed in this work remains empirical in form, even though its structure is motivated by saturation arguments and supported by numerical experiments. While the observed correlations strongly suggest that galaxies exhibiting large surplus values are characterized by efficient geometrically mediated transfer mechanisms such as spirals, bars, or other non axisymmetric structures, this study does not yet provide a first principles derivation linking specific morphological features to quantitative transmission efficiencies. That connection remains an important target for future work. Finally, while we include an analysis of dark matter deficient galaxies and show that their behavior is consistent with the same transport based interpretation, the present sample remains limited. A more systematic expansion of such cases, including a broader characterization of their morphology, 32 REFERENECES [1] V. C. Rubin, W. K. Ford Jr., and N. Thonnard, “Extended rotation curves of high-luminosity spiral galaxies. IV. Systematic dynamical properties, Sa -> Sc.,” The Astrophysical Journal, vol. 225, pp. L107–L111, Nov. 1978, doi: 10.1086/182804. 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Comerón et al., “The massive relic galaxy NGC 1277 is dark matter deficient,” A&A, vol. 675, 2023, doi: 10.1051/0004-6361/202346291. 35 SUPPLEMENTARY A: Formal Vlasov–Poisson Moment Equations A.1 Purpose and scope This supplement summarizes the standard moment equations of the collisionless Vlasov–Poisson system and highlights the appearance of the angular-momentum flux term that underlies rotational coherence transmission. No new physics is introduced here. The goal is solely to establish notation and to show that a nonzero angular-momentum flux arises generically in non axisymmetric disks. A complete derivation based on Liouville’s theorem and Hamiltonian phase-space flow is standard in galactic dynamics and may be found in textbooks such as Galactic Dynamics[24]. For completeness, we summarize only the essential ingredients relevant to the present discussion. All saturation and energetic considerations are treated separately in Supplementary H. A.2 Vlasov equation and lowest moments The evolution of a collisionless, self-gravitating system is governed by the Vlasov equation, ∂𝑓 ∂𝑡+𝐯⋅∇𝑥𝑓−∇𝑥Φ⋅∇𝑣𝑓=0, (A-1) where 𝑓(𝐱,𝐯,𝑡)is the phase-space distribution function and Φis the gravitational potential. The surface density and mean velocity are defined as Σ(𝐱,𝑡)=∫𝑓 𝑑3𝑣,Σ𝐮=∫𝐯𝑓 𝑑3𝑣. (A-2) Taking the zeroth velocity moment yields the continuity equation, ∂Σ ∂𝑡+∇⋅(Σ𝐮)=0. (A-3) A.3 Angular-momentum moment and flux The specific angular momentum about the symmetry axis is 𝐿𝑧=𝑟 𝑣𝜙. (A-4) Multiplying the Vlasov equation by 𝐿𝑧and integrating over velocity space yields the angular-momentum balance equation, ∂ ∂𝑡(Σ𝐿𝑧)+1 𝑟∂ ∂𝑟(𝑟𝐹𝐿)=𝜏grav,(A-5) where 36 𝐹𝐿=Σ⟨𝑣𝑟𝑣𝜙⟩(A-6) is the radial flux of angular momentum and 𝜏gravis the gravitational torque density arising from non axisymmetric structure. Equation (A-4) is exact and follows directly from collisionless dynamics. A.4 Generic nonzero flux in real disks In a perfectly axisymmetric, time-independent disk, the velocity correlations vanish and 𝐹𝐿=0. However, any non axisymmetric structure—spiral arms, bars, lopsidedness, or transient density waves— induces correlated radial and azimuthal motions, producing a nonzero angular-momentum flux. Thus, 𝐹𝐿≠0is a generic property of realistic galactic disks and does not rely on dissipation, hydrodynamics, or modified gravity. A.5 Connection to the transmission framework The Rotational Coherence Transmission (RCT) framework introduced in the main text does not modify Eq. (A-4). Instead, it provides an effective description of how the net, time-averaged angularmomentum flux is inherited, attenuated, and redistributed with radius. The saturation of this transmission—arising from finite absorptive capacity and energetic constraints—is not addressed here. Its field-theoretic and energetic origin is derived separately in Supplementary H. 37 SUPPLEMENTARY B: Physical Interpretation of the Angular-Momentum Flux FL B.1 Purpose This supplement provides a physical interpretation of the angular-momentum flux 𝐹𝐿(𝑅,𝑡)that appears in the exact Vlasov–Poisson angular-momentum moment equation, ∂ ∂𝑡(Σ𝐿𝑧)=−1 𝑅∂ ∂𝑅[𝑅 𝐹𝐿(𝑅,𝑡)]+𝜏grav(𝑅,𝑡), (B-1) with emphasis on how familiar non axisymmetric disk structures generate nonzero flux. No new formalism is introduced. The goal is to show, mechanism by mechanism, why realistic disks generically violate the isolated-annulus assumption 𝐹𝐿=0. B.2 Stress decomposition In collisionless galactic dynamics, the radial angular-momentum flux naturally decomposes into kinematic and gravitational stress contributions[8], [24] : 𝐹𝐿=𝐹𝐿kin+𝐹𝐿grav,(B-2) where 𝐹𝐿kin∼𝑅 Σ ⟨𝑣𝑅𝑣𝜙⟩ (B-3) arises from correlated radial and azimuthal motions, and 𝐹𝐿grav∼1 4𝜋𝐺∫( 2𝜋 0∂𝑅Φ1)(∂𝜙Φ1) 𝑑𝜙 (B-4) arises from non axisymmetric potential gradients. Either contribution being nonzero is sufficient to produce angular-momentum transport. B.3 Spiral structure Spiral density waves are intrinsically non axisymmetric and therefore generate angular-momentum flux through gravitational stress. In the Lynden-Bell–Kalnajs formalism, spiral patterns transport angular momentum via phase-shifted coupling between the perturbed potential and the stellar response[8]. As a result, any spiral pattern of finite amplitude generically produces 𝐹𝐿grav≠0, (B-5) driving secular redistribution of angular momentum across the disk[8], [24]. B.4 Swing-amplified transients Swing amplification produces short-lived, sheared non axisymmetric patterns[10] [8]. Such transients typically generate: 38 • correlated streaming motions, yielding ⟨𝑣𝑅𝑣𝜙⟩≠0(kinematic stress), and • time-dependent non axisymmetric potentials (gravitational stress). Although intermittent in time, swing-amplified structures therefore realize 𝐹𝐿≠0through mixed stress channels[4], [10]. B.5 Bars Bars are global, large-amplitude non axisymmetric perturbations extending over several kiloparsecs. Their potentials necessarily possess both radial and azimuthal gradients, producing sustained gravitational torque and therefore 𝐹𝐿grav≠0 (B-6) throughout the bar region[11]. While resonances (ILR/CR/OLR) amplify the stellar response locally, the existence of angular-momentum transport does not rely on isolating specific resonant radii; the non axisymmetric bar potential alone guarantees radial coupling [4], [11]. B.6 Gravitational wakes and dynamical friction A massive perturber (e.g., a giant molecular cloud, a satellite sub-halo, a globular cluster) moving through a stellar disk produces a gravitational wake in the surrounding collisionless medium. This wake modifies both the velocity distribution and the gravitational potential, producing simultaneous contributions to the kinematic and gravitational parts of the angular-momentum flux 𝐹𝐿. Formally, the total flux is 𝐹𝐿(𝑅,𝑡)=∫𝑅 𝑣𝑅𝑣𝜙 𝑓 𝑑3𝑣 𝑑𝑧 ⏟ kinematic stress +1 4𝜋𝐺∫ (∂Φ ∂𝑅) 2𝜋 0(∂Φ ∂𝜙)𝑑𝜙 ⏟ gravitational stress .(B-7) A gravitational wake excites both terms. Massive perturbers embedded in disks (e.g. giant molecular clouds, satellites, substructure) generate gravitational wakes. These wakes: • induce correlated velocity distortions (kinematic stress), and • produce asymmetric potential perturbations (gravitational stress). Consequently, wakes provide a clear example where both stress channels contribute simultaneously to 𝐹𝐿, consistent with classical dynamical-friction analyses in collisionless systems[13], [25]. B.7 Radial migration Radial migration corresponds to systematic changes in guiding-center radii driven by weak, transient non axisymmetric patterns. Although heating is minimal, migration involves a nonzero velocity covariance ⟨𝑣𝑅𝑣𝜙⟩, implying 39 𝐹𝐿kin≠0 (B-8) even when gravitational stress is small. Migration thus demonstrates that significant angularmomentum transport can occur purely through kinematic stress, without strong potential distortions [14][12]. B.8 Failure of the isolated-ring rotation-curve assumption The textbook rotation-curve relation 𝑣circ 2(𝑅)=𝑅 ∂𝑅Φ(𝑅) (B-9) corresponds to the special limit in which the disk is stationary, axisymmetric, and torque-free so that 𝐹𝐿=0. However, real disks commonly host spirals, bars, transients, wakes, and migration, each of which implies 𝐹𝐿≠0 through one or both stress channels. The breakdown of the isolated-ring approximation therefore reflects suppressed angular-momentum coupling, not a failure of Newtonian gravity. B.9 Connection to the transmission framework The Rotational Coherence Transmission (RCT) framework introduced in the main text does not modify the Vlasov–Poisson equations. Instead, it provides an effective closure describing how the net, timeaveraged angular-momentum flux implied by 𝐹𝐿≠0is inherited, attenuated, and redistributed with radius. The saturating response used in the model reflects energetic constraints and is derived from a minimal Lagrangian extension in Supplementary H. 40 SUPPLEMENTARY C: Numerical Implementation of the Three-Stage Transmission Model This section documents the Jupyter Notebook/Python implementation (included in GitHub: https://github.com/sakidja/CTC_related_papers) used to generate all synthetic transmission cases summarized in Supplementary E. The code translates the analytic structure of the three-stage transmission framework into a fully reproducible numerical pipeline that produces rotation curves, curlstress fields, and SPARC-style diagnostics (CurlIndex, SurplusIndex, and ⟨U⟩). Several expressions appearing below—such as the definitions of the Newtonian rotation curve, the curlstress field, torque gradients, and SPARC-style indices—are also presented in the main text. They are restated here intentionally. The goal of this supplement is not to introduce additional derivations, but to make the numerical pipeline explicit, self-contained, and directly executable. Repetition ensures that readers can follow each computational step without navigating between sections. All definitions are therefore fully consistent with, and subordinate to, the formulations provided in the main paper. The implementation proceeds in five steps. C.1 Newtonian baseline and curl-stress field The disk is initialized using an exponential surface-density profile, Σ(𝑟)=Σ0𝑒−𝑟/𝑅𝑑,(C-1) from which the enclosed mass 𝑀(<𝑟), the Newtonian circular velocity 𝑣𝑁(𝑟)=√𝐺 𝑀(<𝑟) 𝑟,(C-2) and the baseline angular frequency Ω𝑁(𝑟)=𝑣𝑁/𝑟are computed. The Newtonian curl-stress field is evaluated as 𝑆raw(𝑟)=∣1 𝑟𝑑 𝑑𝑟[𝑟 𝑣𝑁(𝑟)]∣, (C-3) which supplies the dimensionless shear amplitude entering the transmission map. C.2 Alternating transmission geometry 41 To represent heterogeneous coupling across the disk, the code imposes alternating highand lowcoupling radial shells. These shells do not correspond to material components; they are a numerical discretization of spatially varying transmission efficiency. The geometric coupling functions are defined as • high-coupling shells 𝑘high(𝑟)=exp (𝑟/𝑅𝑘,high), (C-4) • low-coupling shells 𝑘low(𝑟)=exp (𝑟/𝑅𝑘,low), (C-5) assigned to even and odd radial indices respectively. The innermost ten grid points are treated as a rigidly locked core with 𝑘=1, reflecting near-perfect coupling in the central region. An optional three-point smoothing pass removes sharp discontinuities and enforces monotonic decay. This alternating geometry provides a coarse-grained representation of the heterogeneous transmission landscape through which rotational energy propagates. C.3 Three-stage transmission map The curl-stress field enters the three-stage response law defined in the main text. First, the raw shear is normalized: 𝑆(𝑟)= 𝑆raw(𝑟) max𝑆raw,(C-6) and then passed through the saturation map 𝑇(𝑟)=𝑇maxtanh (𝑆(𝑟) 𝑆0). (C-7) For 𝑆≪𝑆0, the response is linear (Stage I). For intermediate shear, the response bends (Stage II). For 𝑆≫𝑆0, the transmission approaches the asymptotic value 𝑇max(Stage III). The effective transmission efficiency is then 𝑘eff(𝑟)=𝑘(𝑟) 𝑇(𝑟), (C-8) which determines how much inner-engine rotational coherence can be inherited at each radius. C.4 Energy-lock surrogate for the observed rotation curve 48 SUPPLEMENTARY E: Classification of Synthetic Transmission Cases E.1. Overview To compare the synthetic transmission experiments with the empirical SPARC sample, each model realization is assigned to one of three dynamical groups using three SPARC-style diagnostics, all summarized in Table E1: • SurplusIndex_SP – the primary discriminator; measures whether outward transmission has activated and how efficiently rotational energy accumulates in the outer disk. • CurlIndex_SP – amplitude of non–axisymmetric shear in the disk. • ⟨U⟩ (mean uplift) – the net enhancement of rotational support relative to the Newtonian baseline. The grouping is entirely empirical and mirrors the same structure seen in the SPARC data: a subcritical low-surplus cloud, an intermediate bending regime, and a high-surplus saturated sequence. The individual synthetic realizations, their indices, and their assigned dynamical group are listed in Table E1, while the global trends of these indices are shown in Figure E1(a,b). For the Jupyter Notebook , please see the GitHub repository: https://github.com/sakidja/CTC_related_papers. E2. Summary of Diagnostic Trends The behavior of the synthetic sample is most clearly seen by plotting the diagnostics against CurlIndex_SP. Figure E1(a) shows the mean uplift ⟨U⟩ as a function of CurlIndex_SP, and Figure E1(b) shows SurplusIndex_SP versus CurlIndex_SP. Together with Table E1, they organize the synthetic cases into the three dynamical regimes. (i) Mean uplift vs. CurlIndex_SP In Figure E1(a), the synthetic cases trace out a tight, monotonic trend that is identical in form to the SPARC pattern: • ⟨U⟩ ≈ 0 when CurlIndex_SP ≲ 1 • ⟨U⟩ grows quasi–linearly for 1 ≲ CurlIndex_SP ≲ 10 • High–curl systems (CurlIndex_SP ≳ 10) reach ⟨U⟩ ≈ 6–8 This confirms that uplift behaves as a kinematic response to the shear amplitude: once curl crosses a threshold, uplift rises systematically with CurlIndex_SP rather than with radius or mass alone. (ii) SurplusIndex_SP vs. CurlIndex_SP In Figure E1(b), the SurplusIndex_SP values from Table E1 reproduce the same three-part structure as the SPARC galaxies: • A low-curl, low-surplus floor (subcritical regime), where SurplusIndex_SP ≈ 0 even though CurlIndex_SP > 0. • A rising bending sequence, where SurplusIndex_SP increases with CurlIndex_SP as transmission activates but remains limited in radial reach. • A saturated plateau, where SurplusIndex_SP approaches ~0.30–0.45 at large curl, indicating nearly maximal inheritance of inner rotational support. The synthetic population therefore not only spans the same numerical range in (CurlIndex_SP, SurplusIndex_SP, ⟨U⟩) as the SPARC sample but also arranges itself into the same three dynamical regimes. The detailed case-by-case assignments in Table E1 simply make this organization explicit. 49 Each model realization is assigned to a dynamical group (Group 1: subcritical, Group 2: bending, Group 3: saturated) based on its SurplusIndex_SP, CurlIndex_SP, and mean uplift ⟨U⟩. SurplusIndex_SP serves as the primary discriminator (defining whether transmission has activated), while CurlIndex_SP and ⟨U⟩ provide secondary dynamical context. The table lists each synthetic case together with its three diagnostics and its resulting group assignment. The full distribution of these points is shown in Figures E1 and E2. Table E1. Classification of all synthetic transmission cases by SPARC-style diagnostics Case CurlIndex_SP SurplusIndex_SP ⟨U⟩ (Mean uplift) Group I.1 0.0217 0.0011 0.0098 Group 1 I.2 0.0739 0.0054 0.0337 Group 1 I.3 0.2927 0.0315 0.1478 Group 1 I.4 0.5906 0.0342 0.2696 Group 1 I.5 2.1693 0.0513 0.9761 Group 1 I.6 0.5734 0.0559 0.2736 Group 1 I.7 0.3960 0.0772 0.2481 Group 1 II.1 2.6532 0.1284 1.4039 Group 2 II.2 1.7926 0.1768 1.0257 Group 2 II.3 5.7500 0.2200 3.1623 Group 2 III.1 11.4933 0.3112 6.8143 Group 3 III.2 3.7303 0.3130 1.5390 Group 3 III.3 6.7665 0.3202 4.2280 Group 3 III.4 8.5692 0.3281 5.4642 Group 3 III.5 5.3338 0.3293 3.1979 Group 3 III.6 10.4446 0.3372 6.3638 Group 3 III.7 11.1909 0.4320 4.5910 Group 3 III.8 14.6228 0.4520 5.9989 Group 3 Figure E1. SurplusIndex_SP versus CurlIndex_SP for the synthetic sample. 0 0.1 0.2 0.3 0.4 0.5 0 5 10 15 SURPLUS INDEX CURL INDEX 50 The synthetic realizations reproduce the same three-regime structure found in the SPARC dataset. Lowcurl models cluster on a subcritical floor with SurplusIndex_SP ≈ 0. Intermediate models occupy a rising “bending” branch where surplus increases with curl as transmission activates. High-curl systems saturate at SurplusIndex_SP ≈ 0.30–0.45, reflecting global, radius-limited coherence transfer. Together with Figure E1, this plot shows that the synthetic transmission model naturally generates the three dynamical regimes observed in real disks. Figure E2. Mean uplift versus CurlIndex_SP for all synthetic transmission cases. This plot shows how the mean uplift ⟨U⟩ responds to increasing non axisymmetric shear. The synthetic models follow the same monotonic sequence observed in the SPARC galaxies: (1) a near-zero uplift floor for CurlIndex_SP ≲ 1, (2) a quasi-linear rise for 1 ≲ CurlIndex_SP ≲ 10, and (3) a high-curl plateau with ⟨U⟩ ≈ 6–8. The trend demonstrates that uplift is a direct kinematic response to shear amplitude, not to radius or mass alone. 0 1 2 3 4 5 6 7 8 0 5 10 15 MEAN UPLIFT CURL INDEX 51 Group 1 — Subcritical / Low-Surplus Regime Definition : A model belongs to Group 1 if: SurplusIndexSP<0.10, Physical Interpretation Transmission does not meaningfully activate. The inner engine injects insufficient coherent torque for the signal to propagate beyond the inner few rings. Mild shear produces nonzero curl (i.e., 𝐾𝐿≠0), but the surplus energy never departs from the Newtonian baseline. These runs reproduce the subcritical regime seen in galaxies such as UGC 02455 and Holmberg II, where the disk is mildly distorted but torque inheritance is effectively absent. . 52 CASE I.1 N = 5000 r = np.linspace(0.1, 30.0, N) # kpc dr = r[1] - r[0] idx = np.arange(N) # Disk / gravity Sigma0, R_d = 1e9, 3.5 # Msun/kpc^2, kpc G = 4.302e-6 # kpc·(km/s)^2 / Msun # Inner “engine” Omega_inner = 50.0 # km/s/kpc i_solid = 10 # rigid-core edge index # Alternating-layer geometric coupling Rk_alt = 5.0 # kpc Rk_star = 1.0 # kpc # Three-stage transmission parameters (tune these) T_MAX = 0.3 # max transmitted fraction S0 = 0.2 # curl-stress scale for bending/saturation ===== Energy / L diagnostics (model) ===== Total ΔE_rot: 1.720877e+13 Outer ΔE_rot: 1.720843e+13 Fractional uplift (outer ΔE/Newtonian): 0.019 Total ΔL: 5.945477e+11 Outer ΔL: 5.945461e+11 ===== Model indices (energy-lock vs Newtonian) ===== Curl_Index_model = 0.0098 Surplus_Index_model= 1.0000 ===== SPARC-style indices (using vN vs v_partial) ===== CurlIndex_SP = 0.0217 SurplusIndex_SP = 0.0011 Mean uplift <U> = 0.0098 R_max = 30.00 kpc 53 54 CASE I.2 N = 5000 r = np.linspace(0.1, 30.0, N) # kpc dr = r[1] - r[0] idx = np.arange(N) # Disk / gravity Sigma0, R_d = 1e9, 3.5 # Msun/kpc^2, kpc G = 4.302e-6 # kpc·(km/s)^2 / Msun # Inner “engine” Omega_inner = 100.0 # km/s/kpc i_solid = 10 # rigid-core edge index # Alternating-layer geometric coupling Rk_alt = 6 # kpc Rk_star = 0.2 # kpc # Three-stage transmission parameters (tune these) T_MAX = 0.6 # max transmitted fraction S0 = 0.6 # curl-stress scale for bending/saturation ===== Energy / L diagnostics (model) ===== Total ΔE_rot: 5.387538e+13 Outer ΔE_rot: 5.387339e+13 Fractional uplift (outer ΔE/Newtonian): 0.059 Total ΔL: 1.996647e+12 Outer ΔL: 1.996638e+12 ===== Model indices (energy-lock vs Newtonian) ===== Curl_Index_model = 0.0279 Surplus_Index_model= 1.0000 ===== SPARC-style indices (using vN vs v_partial) ===== CurlIndex_SP = 0.0739 SurplusIndex_SP = 0.0054 Mean uplift <U> = 0.0337 R_max = 30.00 kpc 55 56 CASE I.3 N = 5000 r = np.linspace(0.1, 30.0, N) # kpc dr = r[1] - r[0] idx = np.arange(N) # Disk / gravity Sigma0, R_d = 1e9, 3.5 # Msun/kpc^2, kpc G = 4.302e-6 # kpc·(km/s)^2 / Msun # Inner “engine” Omega_inner = 50.0 # km/s/kpc i_solid = 10 # rigid-core edge index # Alternating-layer geometric coupling Rk_alt = 9 # kpc Rk_star = 4 # kpc # Three-stage transmission parameters (tune these) T_MAX = 0.8 # max transmitted fraction S0 = 0.1 # curl-stress scale for bending/saturation ===== Energy / L diagnostics (model) ===== Total ΔE_rot: 2.073152e+14 Outer ΔE_rot: 2.073143e+14 Fractional uplift (outer ΔE/Newtonian): 0.228 Total ΔL: 7.974187e+12 Outer ΔL: 7.974183e+12 ===== Model indices (energy-lock vs Newtonian) ===== Curl_Index_model = 0.0953 Surplus_Index_model= 1.0000 ===== SPARC-style indices (using vN vs v_partial) ===== CurlIndex_SP = 0.2927 SurplusIndex_SP = 0.0315 Mean uplift <U> = 0.1478 R_max = 30.00 kpc 57 64 CASE I.7 N = 5000 r = np.linspace(0.1, 30.0, N) # kpc dr = r[1] - r[0] idx = np.arange(N) # Disk / gravity Sigma0, R_d = 1e9, 3.5 # Msun/kpc^2, kpc G = 4.302e-6 # kpc·(km/s)^2 / Msun # Inner “engine” Omega_inner = 100.0 # km/s/kpc i_solid = 10 # rigid-core edge index # Alternating-layer geometric coupling Rk_alt = 15 # kpc Rk_star = .6 # kpc # Three-stage transmission parameters (tune these) T_MAX = 1.1 # max transmitted fraction S0 = 0.5 # curl-stress scale for bending/saturation ===== Energy / L diagnostics (model) ===== Total ΔE_rot: 2.306767e+14 Outer ΔE_rot: 2.306727e+14 Fractional uplift (outer ΔE/Newtonian): 0.254 Total ΔL: 1.020012e+13 Outer ΔL: 1.020010e+13 ===== Model indices (energy-lock vs Newtonian) ===== Curl_Index_model = 0.1292 Surplus_Index_model= 1.0000 ===== SPARC-style indices (using vN vs v_partial) ===== CurlIndex_SP = 0.3960 SurplusIndex_SP = 0.0772 Mean uplift <U> = 0.2481 R_max = 30.00 kpc 65 66 Group 2— Intermediate / Bending Regime Definition: A model belongs to Group 2 if: 0.10≤SurplusIndexSP<0.30, Physical Interpretation Transmission is active but throttled. Increasing either the inner-engine strength Ωinneror the gas-coupling length 𝑅𝑘,gasproduces visible uplift and curvature in the rotation curve, but the propagation chain stalls part-way out. This occurs because either (i) the geometric decay across “star-like” rings remains too steep, or (ii) curl-stress saturation halts growth before the torque can be exported globally. These models correspond to the bending regime of SPARC dwarfs where strong non-axisymmetry is present but full surplus saturation is not achieved. 67 CASE II.1 N = 5000 r = np.linspace(0.1, 30.0, N) # kpc dr = r[1] - r[0] idx = np.arange(N) # Disk / gravity Sigma0, R_d = 1e9, 3.5 # Msun/kpc^2, kpc G = 4.302e-6 # kpc·(km/s)^2 / Msun # Inner “engine” Omega_inner = 300.0 # km/s/kpc i_solid = 10 # rigid-core edge index # Alternating-layer geometric coupling Rk_alt = 9 # kpc Rk_star = 4 # kpc # Three-stage transmission parameters (tune these) T_MAX = 0.8 # max transmitted fraction S0 = 0.4 # curl-stress scale for bending/saturation ===== Energy / L diagnostics (model) ===== Total ΔE_rot: 1.979969e+15 Outer ΔE_rot: 1.979940e+15 Fractional uplift (outer ΔE/Newtonian): 2.178 Total ΔL: 5.694597e+13 Outer ΔL: 5.694586e+13 ===== Model indices (energy-lock vs Newtonian) ===== Curl_Index_model = 0.7378 Surplus_Index_model= 1.0000 ===== SPARC-style indices (using vN vs v_partial) ===== CurlIndex_SP = 2.6532 SurplusIndex_SP = 0.1284 Mean uplift <U> = 1.4039 R_max = 30.00 kpc 68 69 CASE II.2 N = 5000 r = np.linspace(0.1, 30.0, N) # kpc dr = r[1] - r[0] idx = np.arange(N) # Disk / gravity Sigma0, R_d = 1e9, 3.5 # Msun/kpc^2, kpc G = 4.302e-6 # kpc·(km/s)^2 / Msun # Inner “engine” Omega_inner = 100.0 # km/s/kpc i_solid = 10 # rigid-core edge index # Alternating-layer geometric coupling Rk_alt = 12 # kpc Rk_star = .6 # kpc # Three-stage transmission parameters (tune these) T_MAX = 1.2 # max transmitted fraction S0 = 0.1 # curl-stress scale for bending/saturation ===== Energy / L diagnostics (model) ===== Total ΔE_rot: 1.062327e+15 Outer ΔE_rot: 1.062322e+15 Fractional uplift (outer ΔE/Newtonian): 1.169 Total ΔL: 3.944952e+13 Outer ΔL: 3.944950e+13 ===== Model indices (energy-lock vs Newtonian) ===== Curl_Index_model = 0.4581 Surplus_Index_model= 1.0000 ===== SPARC-style indices (using vN vs v_partial) ===== CurlIndex_SP = 1.7926 SurplusIndex_SP = 0.1768 Mean uplift <U> = 1.0257 R_max = 30.00 kpc 70 71 CASE II.3 N = 5000 r = np.linspace(0.1, 30.0, N) # kpc dr = r[1] - r[0] idx = np.arange(N) # Disk / gravity Sigma0, R_d = 1e9, 3.5 # Msun/kpc^2, kpc G = 4.302e-6 # kpc·(km/s)^2 / Msun # Inner “engine” Omega_inner = 200.0 # km/s/kpc i_solid = 10 # rigid-core edge index # Alternating-layer geometric coupling Rk_alt = 10 # kpc Rk_star = 1.0 # kpc # Three-stage transmission parameters (tune these) T_MAX = 1.8 # max transmitted fraction S0 = 0.16 # curl-stress scale for bending/saturation ===== Energy / L diagnostics (model) ===== Total ΔE_rot: 3.729374e+15 Outer ΔE_rot: 3.729341e+15 Fractional uplift (outer ΔE/Newtonian): 4.103 Total ΔL: 9.958281e+13 Outer ΔL: 9.958270e+13 ===== Model indices (energy-lock vs Newtonian) ===== Curl_Index_model = 1.2361 Surplus_Index_model= 1.0000 ===== SPARC-style indices (using vN vs v_partial) ===== CurlIndex_SP = 5.7500 SurplusIndex_SP = 0.2200 Mean uplift <U> = 3.1623 R_max = 30.00 kpc 72 73 Group 3 — High-Surplus / Saturated Transmission Regime Definition: A model belongs to Group 3 if: SurplusIndexSP≥0.30, Physical Interpretation Transmission is global and highly efficient. The inner engine produces sufficient coherent torque, and the geometric chain remains open enough for the signal to propagate across the entire disk. Curl grows strongly with radius, surplus builds rapidly, and the outer rotational energy approaches the injected inner-engine budget. This regime matches the upper-locus systems in the SPARC dataset, analogous to disks with strong, large scale non axisymmetric modes. In these models, surplus growth becomes radius-limited rather than stress-limited, matching the observed outer flattening of surplus in large high-curl galaxies 80 CASE III.4 N = 5000 r = np.linspace(0.1, 30.0, N) # kpc dr = r[1] - r[0] idx = np.arange(N) # Disk / gravity Sigma0, R_d = 1e9, 3.5 # Msun/kpc^2, kpc G = 4.302e-6 # kpc·(km/s)^2 / Msun # Inner “engine” Omega_inner = 200.0 # km/s/kpc i_solid = 10 # rigid-core edge index # Alternating-layer geometric coupling Rk_alt = 16 # kpc Rk_star = 1.6 # kpc # Three-stage transmission parameters (tune these) T_MAX = 1.8 # max transmitted fraction S0 = 0.16 # curl-stress scale for bending/saturation ===== Energy / L diagnostics (model) ===== Total ΔE_rot: 5.053581e+15 Outer ΔE_rot: 5.053548e+15 Fractional uplift (outer ΔE/Newtonian): 5.560 Total ΔL: 1.308683e+14 Outer ΔL: 1.308682e+14 ===== Model indices (energy-lock vs Newtonian) ===== Curl_Index_model = 1.7590 Surplus_Index_model= 1.0000 ===== SPARC-style indices (using vN vs v_partial) ===== CurlIndex_SP = 8.5692 SurplusIndex_SP = 0.3281 Mean uplift <U> = 5.4642 R_max = 30.00 kpc 81 82 CASE III.5 N = 5000 r = np.linspace(0.1, 30.0, N) # kpc dr = r[1] - r[0] idx = np.arange(N) # Disk / gravity Sigma0, R_d = 1e9, 3.5 # Msun/kpc^2, kpc G = 4.302e-6 # kpc·(km/s)^2 / Msun # Inner “engine” Omega_inner = 250.0 # km/s/kpc i_solid = 10 # rigid-core edge index # Alternating-layer geometric coupling Rk_gas = 20.0 # kpc Rk_star = 1.0 # kpc # Three-stage transmission parameters (tune these) T_MAX = 0.7 # max transmitted fraction S0 = 0.2 # curl-stress scale for bending/saturation ===== Energy / L diagnostics (model) ===== Total ΔE_rot: 2.606247e+15 Outer ΔE_rot: 2.606227e+15 Fractional uplift (outer ΔE/Newtonian): 2.867 Total ΔL: 8.420015e+13 Outer ΔL: 8.420007e+13 ===== Model indices (energy-lock vs Newtonian) ===== Curl_Index_model = 1.1767 Surplus_Index_model= 1.0000 ===== SPARC-style indices (using vN vs v_partial) ===== CurlIndex_SP = 5.3338 SurplusIndex_SP = 0.3293 Mean uplift <U> = 3.1979 R_max = 30.00 kpc 83 84 CASE III.6 N = 5000 r = np.linspace(0.1, 30.0, N) # kpc dr = r[1] - r[0] idx = np.arange(N) # Disk / gravity Sigma0, R_d = 1e9, 3.5 # Msun/kpc^2, kpc G = 4.302e-6 # kpc·(km/s)^2 / Msun # Inner “engine” Omega_inner = 220.0 # km/s/kpc i_solid = 10 # rigid-core edge index # Alternating-layer geometric coupling Rk_alt = 15 # kpc Rk_star = .6 # kpc # Three-stage transmission parameters (tune these) T_MAX = 1.2 # max transmitted fraction S0 = 0.1 # curl-stress scale for bending/saturation ===== Energy / L diagnostics (model) ===== Total ΔE_rot: 5.811107e+15 Outer ΔE_rot: 5.811081e+15 Fractional uplift (outer ΔE/Newtonian): 6.393 Total ΔL: 1.470448e+14 Outer ΔL: 1.470447e+14 ===== Model indices (energy-lock vs Newtonian) ===== Curl_Index_model = 1.8549 Surplus_Index_model= 1.0000 ===== SPARC-style indices (using vN vs v_partial) ===== CurlIndex_SP = 10.4446 SurplusIndex_SP = 0.3372 Mean uplift <U> = 6.3638 R_max = 30.00 kpc 85 86 CASE III.7 N = 5000 r = np.linspace(0.1, 30.0, N) # kpc dr = r[1] - r[0] idx = np.arange(N) # Disk / gravity Sigma0, R_d = 1e9, 3.5 # Msun/kpc^2, kpc G = 4.302e-6 # kpc·(km/s)^2 / Msun # Inner “engine” Omega_inner = 250.0 # km/s/kpc i_solid = 10 # rigid-core edge index # Alternating-layer geometric coupling Rk_alt = 50.0 # kpc Rk_star = 1.0 # kpc # Three-stage transmission parameters (tune these) T_MAX = 0.6 # max transmitted fraction S0 = 0.2 # curl-stress scale for bending/saturation ===== Energy / L diagnostics (model) ===== Total ΔE_rot: 2.825390e+15 Outer ΔE_rot: 2.825373e+15 Fractional uplift (outer ΔE/Newtonian): 3.108 Total ΔL: 9.474579e+13 Outer ΔL: 9.474572e+13 ===== Model indices (energy-lock vs Newtonian) ===== Curl_Index_model = 1.5373 Surplus_Index_model= 1.0000 ===== SPARC-style indices (using vN vs v_partial) ===== CurlIndex_SP = 11.1909 SurplusIndex_SP = 0.4320 Mean uplift <U> = 4.5910 R_max = 30.00 kpc 87 88 CASE III.8 N = 5000 r = np.linspace(0.1, 30.0, N) # kpc dr = r[1] - r[0] idx = np.arange(N) # Disk / gravity Sigma0, R_d = 1e9, 3.5 # Msun/kpc^2, kpc G = 4.302e-6 # kpc·(km/s)^2 / Msun # Inner “engine” Omega_inner = 350.0 # km/s/kpc i_solid = 10 # rigid-core edge index # Alternating-layer geometric coupling Rk_alt = 50.0 # kpc Rk_star = 1.0 # kpc # Three-stage transmission parameters (tune these) T_MAX = 0.4 # max transmitted fraction S0 = 0.2 # curl-stress scale for bending/saturation ===== Energy / L diagnostics (model) ===== Total ΔE_rot: 3.691843e+15 Outer ΔE_rot: 3.691821e+15 Fractional uplift (outer ΔE/Newtonian): 4.062 Total ΔL: 1.148007e+14 Outer ΔL: 1.148006e+14 ===== Model indices (energy-lock vs Newtonian) ===== Curl_Index_model = 1.8542 Surplus_Index_model= 1.0000 ===== SPARC-style indices (using vN vs v_partial) ===== CurlIndex_SP = 14.6228 SurplusIndex_SP = 0.4520 Mean uplift <U> = 5.9989 R_max = 30.00 kpc 89 96 97 98 99 100 101 102 103 104 105 112 113 114 115 116 117 118 119 120 121 128 129 130 131 132 133 134 135 136 137 144 145 146 147 148 149 150 151 152 153 256 257 258 259 260 261 262 263 264 265 272 Independent studies of the galaxy’s H I bar show that it rotates slowly, consistent with strong dynamical friction from a dominant dark-matter halo. While these measurements do not directly track torque flux, they support the picture of sustained structural asymmetry and ongoing angular-momentum redistribution. With a compact stellar component and an extremely extended, distorted gas disk, NGC 3741 provides a clear example in which large-scale transport is driven by structure rather than radius or stellar mass. References: Gentile et al., A&A, 472, 925 (2007) [31]; Banerjee et al., MNRAS, 434, 1257 (2013) [32]. 3. DDO 154 DDO 154 is a gas-dominated dwarf irregular galaxy with very little stellar mass and one of the most extended H I disks known. Deep H I observations trace the disk to extremely large radii and show clear warps and asymmetries, especially in the outer regions. These features break axisymmetry and produce measurable departures from simple circular rotation. The transmission diagnostics show a high Surplus Index and a substantial Curl Index, placing DDO 154 in a regime where angular-momentum and energy transport are actively sustained by disk asymmetry. The surplus appears where the H I disk is warped or distorted, indicating that torque-driven transport remains active (𝐹𝐿≠0) even at very large radii. Where the disk becomes smoother and more symmetric, the surplus declines, providing a clear falsifiability condition. Independent studies of the stellar population show that much of the outer H I disk is dynamically quiet, with suppressed star formation, while localized asymmetric regions host ongoing activity. In the transmission picture, these quiet zones correspond to regions with little or no torque flux, while warped regions sustain transport and redistribution. DDO 154 therefore demonstrates, in a clean and isolated system, that extended gas asymmetries alone can drive angular-momentum transport without the need for interactions or massive stellar components. References: Carignan & Purton, AJ, 506, 125 (1998) [33]; Watts et al., MNRAS, 477, 5554 (2018) [34]. 4. UGC01281 UGC 01281 is a low–surface-brightness disk galaxy with a slowly rising rotation curve. Studies of large galaxy samples show that it follows the radial acceleration relation with very small scatter, indicating that its observed dynamics are tightly linked to its baryonic mass distribution rather than requiring finetuned modeling. Independent analysis of UGC 01281 shows that its baryonic mass is unevenly distributed. When this asymmetry is taken into account, models based on dark-matter–only assumptions fail to reproduce the inner velocity profile, consistently overpredicting the observed rotation. This demonstrates that baryons, and especially their asymmetric distribution, play a dominant role in shaping the galaxy’s kinematics. Using transmission diagnostics, UGC 01281 shows a moderate Surplus Index and a high Curl Index, placing it in a regime where angular-momentum transport is active but not fully saturated. In this 273 regime, surplus energy is sustained by ongoing torque gradients produced by disk asymmetry. When asymmetry is reduced, the surplus diminishes, directly linking the observed dynamics to internal torquedriven transport rather than hidden mass. Taken together, independent observational analyses and transmission diagnostics point to asymmetry as the controlling factor in UGC 01281. The galaxy therefore provides a clear example where internal structure, rather than unseen matter, governs the redistribution of angular momentum and energy. References: Li et al., A&A, 615, A3 (2018) [35]; Bar et al., Phys. Rev. D, 99, 103020 (2019) [36]. 5. ESO 444 G084 ESO 444 G084 is a compact dwarf irregular galaxy with a clearly asymmetric disk. H I observations show strong lopsidedness and outer distortions that break axisymmetry, placing the galaxy among the many dwarfs known to be morphologically and kinematically asymmetric. This sustained non-axisymmetry provides a natural source of internal torques. The measured diagnostics show a large Surplus Index and Curl Index, indicating active angular-momentum and energy transport. The surplus appears where the disk is most distorted, demonstrating that transmission is controlled by structural asymmetry rather than galaxy size or radial extent, with 𝐹𝐿≠0. More recent MeerKAT observations reveal a warped H I disk beyond the inner region, together with a steeply rising rotation curve and ongoing localized star formation. These features indicate that transport is spatially selective: regions that are symmetric remain dynamically quiet, while warped and lopsided regions sustain angular-momentum flux. ESO 444 G084 therefore provides a clear example of a low-mass dwarf in which internal transport is driven by structural forcing rather than scale. References: Swaters et al., A&A, 390, 829 (2002) [37]; Namumba et al., A&A, 699, A372 (2025) [38]. B. EXTREME TRANSPORT-SUPPRESSED GALAXIES This section documents a small set of galaxies frequently cited as extreme cases of dark matter deficiency and explains, for each system, why their observed velocity fields are dynamically consistent with suppressed internal transport. These galaxies are not treated as counterexamples to the transmission framework, but rather as limiting cases in which angular-momentum redistribution is minimal or dynamically irrelevant. 1 NGC 1052-DF2 NGC 1052-DF2 is an ultra-diffuse dwarf galaxy whose stellar and globular-cluster motions can be explained by the mass of its stars alone, with little evidence for dark matter. The galaxy shows no rotating disk, spiral arms, bar, or other long-lived asymmetric structures. Its stars are supported mainly by random motions rather than rotation, placing the system close to a configuration where angular-momentum transfer is minimal. In this setting, the lack of inferred dark matter is consistent with a geometry in which internal transport is strongly suppressed. References: van Dokkum et al. (2018), Nature, 555, 629[39]; Danieli et al. (2019), ApJ, 874, L1[39]. 274 2 NGC 1052-DF4 NGC 1052-DF4 is an ultra-diffuse galaxy with structural and kinematic properties very similar to those of DF2. It shows no clear signs of disk rotation, spiral structure, or other non-axisymmetric features. The motions of its stars and globular clusters are dynamically cold and are well explained by the gravitational potential of the stellar mass alone, with little evidence for dark matter. Because the galaxy lacks both significant rotation and structural features that could generate internal torques, angular-momentum and energy transport within DF4 is expected to be weak. The system is therefore consistent with a transport-suppressed configuration rather than one undergoing active internal redistribution. References: van Dokkum et al. (2019), ApJ, 880, 91[39]; Shen et al. (2021), ApJ, 907, L7[40]. 3 FCC 224 FCC 224 is an ultra-diffuse dwarf galaxy in the outer Fornax Cluster. Measurements of stellar and globular-cluster velocities show that its internal motions can be explained by the mass of its stars alone, with little evidence for dark matter within one effective radius. The galaxy contains an extended system of bright globular clusters and has an old, metal-poor, and fully quenched stellar population. FCC 224 shows no clear signs of disk rotation, spiral arms, bars, or other asymmetric structures that could drive angular-momentum transfer. Its stellar motions are dominated by random velocities, with only very weak rotation. Because there are no strong torque-producing features, the galaxy is consistent with a state in which angular-momentum and energy transport are strongly suppressed, corresponding to the near-zero-flux (𝐹𝐿≈0) limit of collisionless dynamics. References: Buzzo., et.al., A&A, Volume 695, March 2025[41] 4 AGC 114905 AGC 114905 is a gas-rich ultra-diffuse dwarf galaxy whose H I rotation curve can be explained entirely by the mass of its observed baryons, with little or no need for dark matter within the measured disk. The galaxy is rotationally supported but has very low surface density and weak rotational shear. It shows no strong bars, spiral arms, or other asymmetric features. The H I gas forms a thin, smoothly rotating disk with low velocity dispersion. Because the disk lacks strong non-axisymmetric structures, internal gravitational torques are weak. As a result, angularmomentum and energy transport within the disk are strongly suppressed, and the observed velocity field is consistent with a transport-inefficient, near–steady-state configuration rather than active redistribution. References: Mancera Piña et al. (2019), ApJ, 883, L33[42]; Mancera Piña et al. (2020), MNRAS, 495, 3636[43]. 275 5 NGC 1277 NGC 1277 is a compact, massive early-type galaxy whose stellar motions can be explained by the mass of its stars and central black hole alone, without requiring dark matter within several effective radii. Detailed dynamical models show that the galaxy is highly regular and nearly perfectly axisymmetric. It shows no bars, spiral features, warps, or other asymmetries. Because the system is dynamically smooth and well ordered, there is little need for angular-momentum or energy transport beyond local stellar motions. In this case, the lack of inferred dark matter reflects the fact that large-scale transport is not required, rather than being actively suppressed. NGC 1277 therefore serves as a control example where the absence of dark matter coincides with a naturally transport-inactive dynamical state. References: Yıldırım et al. (2015), MNRAS, 452, 1792[44]; Comerón et al. (2023), A&A, 675, A143[45] Across these extreme systems, the absence of dark matter consistently coincides with morphologies and velocity fields that suppress internal angular-momentum and energy transport. These galaxies therefore define the transport-free boundary of the dynamical continuum explored in the main text, complementing the transport-dominated disk galaxies that populate the SPARC sample. 276 SUPPLEMENTARY I: Geometry-Controlled Angular-Momentum Transport in a Collisionless N-Body Disk I.1. Purpose and Scope This supplement documents the N-body simulations used in Section 10 of the main text to illustrate how disk geometry controls angular-momentum transport in collisionless stellar systems. The goal is not to simulate a realistic galaxy or to demonstrate spontaneous structure formation, but to provide a controlled dynamical experiment showing that, once coherent non axisymmetric geometry is present, the angular-momentum flux predicted by the Vlasov–Poisson moment equations becomes active and produces outer-disk rotational inheritance. All simulations are performed using the REBOUND N-body integrator with Newtonian gravity. No modification of gravity, no additional mass components, and no non-Hamiltonian dissipation are introduced. I.2. Continuum Transport Equation The collisionless evolution of a stellar disk is governed by the Vlasov–Poisson system, ∂𝑓 ∂𝑡+𝐯⋅∇𝑥𝑓−∇Φ⋅∇𝑣𝑓=0,∇2Φ=4𝜋𝐺𝜌. Defining the specific angular momentum 𝑙=𝑅𝑣𝜙, the surface density of angular momentum is 𝐿(𝑅,𝑡)=∫𝑅𝑣𝜙𝑓 𝑑3𝑣 𝑑𝑧. Multiplying the Vlasov equation by 𝑅𝑣𝜙and integrating over velocity yields the exact transport identity ∂𝐿 ∂𝑡=−1 𝑅∂ ∂𝑅[𝑅𝐹𝐿(𝑅,𝑡)], where the angular-momentum flux is 𝐹𝐿(𝑅,𝑡)=⟨𝑅𝑣𝑅𝑣𝜙⟩. This equation contains no source term: angular momentum is conserved locally and globally. Any redistribution arises solely from the divergence of the flux. I.3. N-Body Representation In the N-body realization, the phase-space distribution is approximated by discrete particles, 𝑓(𝐱,𝐯)≃∑𝑚 𝛿(𝐱− 𝑁 𝑖=1 𝐱𝑖)𝛿(𝐯−𝐯𝑖). Within a radial bin centered at 𝑅, the continuum quantities become 𝐿(𝑅) → ∑𝑚 𝑖∈𝑅 𝑅𝑖𝑣𝜙,𝑖, 𝐹𝐿(𝑅) → ⟨𝑅𝑣𝑅𝑣𝜙⟩𝑅=1 𝑀𝑅∑𝑚 𝑖∈𝑅 𝑅𝑖𝑣𝑅,𝑖𝑣𝜙,𝑖. 277 Thus the particle system directly samples the same transport equation as the continuum theory, with the flux measured from correlated radial and azimuthal velocities. I.4. Initial Disk Model The simulations begin with a cold, self-gravitating exponential disk with surface density Σ(𝑅)∝𝑒−𝑅/𝑅𝑑, sampled using 𝑁=3000 equal-mass particles. Particle positions are drawn from an exponential radial distribution and uniform azimuth. Initial velocities are assigned to produce circular motion consistent with a smooth baryonic rotation curve. A Plummer softening length 𝜖=0.05 is used to suppress close two-body encounters and ensure that the evolution remains in the collisionless regime. I.5. Controlled Geometry: Axisymmetric vs Spiral Disk To isolate the role of geometry, two simulations are evolved under identical numerical and gravitational conditions, differing only in their imposed symmetry: Case A: Axisymmetric Control No non axisymmetric forcing is applied. The disk remains close to spherical/axisymmetric throughout the evolution. Case B: Spiral Geometry A weak, rotating 𝑚=2 tangential acceleration is applied using REBOUND’s additional_forces interface. This forcing maintains a long-lived spiral-like pattern but does not add mass or net angular momentum to the system. Its role is purely geometric: to sustain coherent non axisymmetric structure and prevent cancellation of velocity correlations. This setup is intended to mimic the dynamical role of spiral structure, not its origin. I.6. Measured Quantities and Figure Definitions At each output time, particles are binned radially and the following diagnostics are computed: • Mean rotational velocity 𝑣ˉ𝜙(𝑅)=⟨𝑣𝜙⟩𝑅. • Angular-momentum flux 𝐹𝐿(𝑅,𝑡)=⟨𝑅𝑣𝑅𝑣𝜙⟩𝑅. • Rotational energy proxy 𝐸rot(𝑅)=1 2𝑅2𝑣ˉ𝜙 2(𝑅). Figures 3–6 in the main text correspond to: • particle snapshots (orbital geometry), • surface-density evolution Σ(𝑅,𝑡), 278 • angular-momentum flux 𝐹𝐿(𝑅,𝑡), • rotational uplift and surplus diagnostics. I.7. Results Axisymmetric Case In the axisymmetric control simulation, particle orbits remain quasi-circular and radially confined. The angular-momentum flux fluctuates around zero at all radii. While finite-𝑁 noise produces instantaneous nonzero values, the flux divergence averages out over time, and no secular redistribution of angular momentum or rotational energy is observed. The outer disk remains dynamically cold and rotationally under-supported. Spiral Geometry Case In the spiral-geometry simulation, coherent correlations between 𝑣𝑅and 𝑣𝜙 persist over extended radial ranges. As a result, a structured angular-momentum flux develops: the inner disk exports angular momentum (𝐹𝐿<0), while the outer disk inherits it (𝐹𝐿>0). Although the surface density at large radii remains low, the outer disk acquires enhanced rotational support because the transported quantity scales with specific angular momentum 𝑅𝑣𝜙, not mass alone. This produces a clear outer-disk rotational surplus consistent with the behavior shown in Figures 3–6. I.8. Interpretation The simulations demonstrate that angular-momentum transport in collisionless disks is controlled by symmetry rather than conservation laws or numerical resolution. In perfectly symmetric systems, angular momentum remains trapped despite local fluctuations. Once coherent non axisymmetric geometry is present, correlated radial–azimuthal motion organizes into a sustained flux that redistributes rotational support outward. Crucially, this redistribution does not create angular momentum. It reflects the activation of the transport channel already present in the Vlasov–Poisson hierarchy. I.9. Conclusion This N-body experiment provides a controlled dynamical illustration of the transmission mechanism discussed in the main text. When coherent non axisymmetric geometry is maintained, angularmomentum flux emerges naturally from collisionless dynamics and produces outer-disk rotational inheritance without invoking additional mass components or modified gravity. The experiment therefore supports the central claim of the paper: extended rotation curves can arise from geometry-enabled angular-momentum transport within standard Newtonian dynamics.