The Curvature–Transport Correspondence (CTC): From Quantum Effective Mass to Cosmological Dark Matter

1 The Curvature–Transport Correspondence (CTC): From Quantum Effective Mass to Cosmological Dark Matter Ridwan Sakidja Dept. of Physics, Astronomy and Materials Science, Missouri State University, Springfield, MO, USA. Abstract We introduce the Curvature–Transport Correspondence (CTC), a simple but unifying framework that identifies a shared structural role for mass across physical theories. In quantum mechanics, galactic dynamics, and cosmology, the curvature of a fundamental field, whether wavefunction, gravitational potential, or angular momentum, is governed by the divergence of an associated transport flux. Under CTC, quantities conventionally interpreted as inertial mass, effective density, or stress energy are reframed as curvature response coefficients. They measure the energetic cost of field curvature generated by transport processes. This principle recasts dark matter not as unseen substance but as the curvature response parameter required by the observed gravitational field. Without altering the equations of quantum mechanics or general relativity, CTC provides a falsifiable geometric interpretation of mass, shifting the conceptual foundation from material content to transport driven curvature response. 1. Introduction What is mass? Across physics, this foundational quantity appears in very different roles. In quantum mechanics, it is the coefficient that sets the curvature of the wavefunction. In cosmology, it enters as the density that sources gravitational potential. In galactic dynamics, it is the inferred “missing mass’’ required to sustain observed rotation curves. Despite the diversity of these contexts, we propose that a single geometric structure underlies them all: mass, in any domain, functions as a curvature-response coefficient. Most physical field equations relate the curvature of a field to a source term. The Schrödinger equation ties wavefunction curvature to inertial mass. The Poisson equation links gravitational curvature to mass density. The angular-momentum equation in collisionless disks connects the curvature of the momentum profile to the divergence of a torque flux. These equations share a common form: curvature is generated by the divergence of an associated transport process. Traditionally, the source terms in these equations—mass, density, stress—are interpreted as intrinsic properties or as independent substances. The Curvature–Transport Correspondence (CTC) advances a different view. We show that these quantities are more fundamentally understood as curvatureresponse parameters that arise from an underlying transport flux. The unifying structure is Curvature of Field 𝑿=𝛁⋅(Transport Flux 𝑭𝑿) (1) Under the CTC, the inertial mass in quantum mechanics, the torque-flux divergence in galactic dynamics, and the effective density inferred in cosmology are not separate constructs. They are different manifestations of one physical principle: each quantifies how strongly a system resists curvature generated by the divergence of a transport flux. 2 This paper develops that correspondence in detail. In Section 2, we examine the curvature structure of the Schrödinger equation, where mass explicitly sets the cost of wavefunction curvature. Section 3 derives how torque-flux divergence governs angular-momentum curvature in collisionless stellar systems. Section 4 shows the structural equivalence to the Poisson equation and to tidal-torque theory in cosmology. Section 5 states the general CTC principle and presents a cross-domain correspondence table. Sections 6 and 7 discuss the implications of this reframing and outline falsifiable predictions that distinguish the CTC from traditional, substance-based interpretations of mass. The CTC does not modify the mathematics of quantum mechanics, general relativity, or galactic dynamics. It provides a new interpretive lens: one that links curvature directly to measurable transport processes and offers a unified, falsifiable account of mass from the quantum scale to the cosmic scale. Perspective from Materials Physics. The viewpoint developed in this work arises naturally from materials physics, where quantities such as effective mass, mobility, and conductivity are routinely understood as geometric or transport-derived response coefficients rather than intrinsic substances. In solid-state theory the effective mass of an electron, for example, reflects the curvature of the band structure and does not represent a distinct particle or material component. By adopting this established operational perspective, the CTC interprets the source terms appearing in quantum, galactic, and cosmological curvature equations as response coefficients generated by transport-flux divergence, restoring interpretational consistency without altering the underlying physics. 2. Curvature-Source Structure in Quantum Mechanics The role of mass as a curvature-response coefficient finds its most explicit and operational definition in quantum mechanics. The stationary Schrödinger equation for a particle provides the canonical starting point[1]: −ℏ2 2𝑚∇2𝜓+𝑉𝜓=𝐸𝜓. (2) Here, the Laplacian ∇2𝜓 mathematically represents the spatial curvature of the wavefunction 𝜓. The coefficient preceding it, ℏ2 2𝑚, is the crucial parameter that converts this geometric property—curvature— into a dynamical quantity: kinetic energy. Within this structure, the inertial mass m is the curvatureresponse coefficient. It quantifies the wavefunction's resistance to spatial bending: a large mass strongly suppresses curvature, resulting in a slowly varying 𝜓, while a small mass permits rapid spatial variations. This interpretation extends seamlessly into solid-state physics via the concept of effective mass 𝑚∗, defined by the curvature of the electronic energy bands[2]: 1 𝑚∗=1 ℏ2𝑑2𝐸 𝑑𝑘2. (3) In this formulation, 𝑚∗ is literally the parameter that governs how a Bloch electron's state responds to curvature in momentum space, directly determining its acceleration under an applied force. Thus, quantum mechanics provides a precise, non-metaphysical definition of mass: Inertial or effective mass is the curvature-response coefficient of the quantum state. 3 It sets the energetic cost incurred by a curved wavefunction. This perspective reveals the quantum mechanical "mass" not as a primary substance, but as a parameter mediating between geometry (curvature) and dynamics (energy). This is the foundational analogy for the Curvature–Transport Correspondence, establishing a template for identifying analogous curvature-response coefficients in galactic and cosmological systems. 2.1 The Case of the Massless Photon The interpretation of mass as a curvature-response coefficient extends naturally to the case of fundamentally massless particles within a medium. A photon in a vacuum, obeying the wave equation ∇2A− 1 𝑐2∂2A ∂𝑡2=0, (4) has zero inertial mass. This corresponds to an infinite susceptibility to spacetime curvature—the photon's wavefunction can vary arbitrarily without energetic cost. However, inside a plasma or dielectric, the photon acquires an effective mass 𝑚𝛾. In a plasma with frequency 𝜔𝑝, the dispersion relation becomes 𝜔2=𝜔𝑝 2+𝑐2𝑘2, (5) yielding an effective mass of 𝑚𝛾=ℏ𝜔𝑝 𝑐2. This effective mass emerges directly from the material's interaction with the electromagnetic field, and it governs the photon's response to spatial curvature in momentum space, defined by 1 𝑚𝛾 ∗=1 ℏ2𝑑2𝐸 𝑑𝑘2. (6) Here, just as for the electron in a solid, the photon's effective mass is not an intrinsic substance but a curvature-response coefficient. It quantifies how the photon-quasiparticle resists variations in its 𝑘space profile due to interactions with the medium. This example reinforces the central CTC principle: what we call "mass" is a measure of a system's response to curvature generated by transport processes, even for a particle whose fundamental mass is zero. 2.2 The Case of the Emergent Phonon The interpretation of mass as a curvature-response coefficient finds another well-known illustration in the behavior of collective excitations, most notably the phonon. In a crystal lattice, atomic vibrations are quantized into phonons, which can be viewed as emergent quasiparticles that are not fundamental substances but quantized modes of lattice displacement. The phonon dispersion relation 𝜔(𝑘) provides the fundamental link. The curvature of this relation in momentum space directly defines the phonon's effective mass: 1 𝑚phonon ∗=1 ℏ2𝑑2𝐸 𝑑𝑘2, where 𝐸 =ℏ𝜔(𝑘). (7) This is formally identical to the effective mass definitions for electrons (Eq. 3) and photons in a plasma (Eq. 6). 4 The effective mass of a phonon arises directly from the curvature of its dispersion relation. Physically, this effective mass governs how a phonon wave packet accelerates under external forces and how it transports energy and momentum through the lattice. Classic treatments of electron–phonon interaction, for example in the works of Ashcroft and Mermin[3] and in the many body theory of Mahan[4], show that lattice vibrations not only possess their own curvature response coefficients but also reshape those of other excitations. A foundational precursor to this idea appears in Einstein’s explanation of the photoelectric effect[5]. When a photon transfers discrete amounts of energy and momentum to an electron, the electron’s intrinsic mass does not change, but the event illustrates a deeper principle central to the CTC: transport flux can modify a particle’s dynamical response even when it does not alter its rest mass. Fröhlich identified the continuous version of this mechanism. In an ionic crystal, an electron couples to longitudinal optical phonons through the long range interaction he first formulated[6]. The electron becomes dressed by a cloud of virtual phonons and its observed mass increases. Modern analyses, such as the review by Devreese and Alexandrov[7], show that this mass enhancement follows directly from the curvature of the combined electron phonon energy surface. In the curvature transport interpretation, this renormalization does not arise because one form of matter adds mass to another. Instead, one transport channel, the lattice vibrational flux, modifies the curvature structure of another transport channel, the electronic band. Einstein’s discrete momentum transfer and Fröhlich’s continuous dressing are two limits of the same principle: the curvature of a dynamical field is shaped by the transport flux acting upon it. The observed mass is the curvature response coefficient of the composite system. The polaron therefore provides a clear and experimentally verified case where mass arises entirely from transport induced curvature, reinforcing the universality of the CTC principle. 2.3. The Case of the Boson The curvature response interpretation of mass applies directly to bosonic fields. For a scalar field 𝜙, the Klein Gordon equation gives the standard relativistic description of a spin zero boson [8]: (□+𝑚2𝑐2 ℏ2)𝜙=0, (8) where □ is the d Alembert operator that contains the spacetime curvature of the field. The term 𝑚2𝑐2/ℏ2 sets the energetic cost of bending the scalar field in spacetime. In this sense the mass of the boson acts as a curvature response coefficient, measuring the resistance of the field to spacetime curvature. This viewpoint extends naturally to gauge bosons. In the electroweak theory the W and Z bosons acquire mass through the Higgs mechanism [9]. The Higgs field introduces an additional curvature term in the gauge sector, and the resulting boson masses follow from the vacuum expectation value of the Higgs condensate: 𝑚𝑊,𝑍 =1 2𝑔𝑣.(9) 5 Although this is the standard result of the electroweak theory, the CTC interpretation treats these masses as response coefficients that reflect how the gauge field reacts to the transport structure defined by the Higgs condensate. Bosonic excitations in condensed matter systems express the same structure. Quasiparticles such as phonons, magnons, and excitons obey dispersion relations of the form 𝐸(𝑘)=ℏ𝜔(𝑘), and their effective mass is determined by the curvature of the dispersion relation[4], [10]: 1 𝑚boson ∗=1 ℏ2𝑑2𝐸 𝑑𝑘2.(10) Here the effective mass is again a measure of how the excitation energy responds to curvature in the underlying transport field. These examples illustrate that bosonic systems, whether fundamental fields or emergent quasiparticles, fit naturally within the curvature transport correspondence. Their masses arise as curvature response coefficients determined by the behaviour of the relevant transport flux. This places bosons alongside electrons, photons, nucleons, phonons, and strings in the unified set of systems that display curvature driven energetic response[11], [12]. 2.4. The Case of the Nucleon and Quark Confinement The curvature-response interpretation of mass extends naturally into nuclear physics. Protons and neutrons, though often treated as fundamental carriers of mass, acquire their effective mass from the dynamics of the quantum chromodynamic (QCD) field. Within QCD, quarks are bound by gluon exchange, and the confinement potential defines the curvature of the quark wavefunction. The nucleon mass emerges as the energetic cost of resisting this curvature, rather than as an intrinsic substance[13]. Formally, the dispersion relation for quarks inside a nucleon can be expressed as 𝐸(𝑘), where 𝑘is the quark momentum. Expanding near equilibrium yields: 𝐸(𝑘) ≈𝐸0+ℏ2𝑘2 2𝑚𝑁 ∗, (11) with the effective nucleon mass defined by 1 𝑚𝑁 ∗=1 ℏ2𝑑2𝐸 𝑑𝑘2. (12) Here, 𝑚𝑁 ∗quantifies the resistance of the bound quark state to curvature in the QCD potential. Just as the electron effective mass in solids (Eq. 3) reflects band curvature, the nucleon mass reflects the curvature of the QCD energy spectrum. This perspective reframes the nucleon not as a particle with intrinsic mass, but as a transport-derived excitation. The gluon flux acts as the transport process, and its divergence defines the curvature of the quark field. The nucleon mass is therefore the curvature-response coefficient of the QCD system. To illustrate, consider the MIT bag model[14], where quarks are confined in a finite region of space. The energy of the system can be written as 6 𝐸(𝑅) = 𝑍 𝑅+4 3𝜋𝑅3𝐵, (13) where 𝑅 is the bag radius, 𝑍 encodes quark kinetic energy, and 𝐵 is the bag constant representing gluon pressure. Differentiating twice with respect to 𝑅 gives the curvature of the confinement energy, and the effective nucleon mass emerges as the coefficient that balances this curvature against transport flux. Thus, even at the nuclear scale, “mass” is revealed as a curvature-response coefficient: it measures the geometric rigidity of quark states under gluon transport. This further aligns with the broader concept of nucleon effective mass in nuclear matter[15]. This reinforces the universality of the Curvature–Transport Correspondence, showing that the same principle applies from condensed-matter electrons to nucleons in QCD. 2.5. The Case of String Excitations The curvature-response interpretation of mass extends naturally to the framework of string theory[16]. The aim here is not to reinterpret the fundamental theory but simply to note the structural parallel between the dispersion curvature of string vibrational modes and the curvature-based effective-mass viewpoint advanced in this work. In standard string theory, particles arise as quantized vibrational modes of an extended one-dimensional object. Each mode has an energy spectrum 𝐸𝑛(𝑘) =ℏ 𝜔𝑛(𝑘), (14) with 𝑘 the momentum along the string. Expanding near a minimum of the dispersion relation yields 𝐸𝑛(𝑘) ≈𝐸𝑛,0 +ℏ2𝑘2 2𝑚𝑛 ∗,(15) where the effective mass follows from the curvature of the spectrum, 1 𝑚𝑛 ∗=1 ℏ2𝑑2𝐸𝑛 𝑑𝑘2.(16) This relation, presented in its standard form by Green, Schwarz, and Witten [16], shows that the mass level of a string excitation is determined by the curvature of its vibrational energy profile in the compactified directions. Polchinski’s conformal field theory formulation [17] formalizes the same structure by deriving the mode spectrum from the Virasoro constraints and the conformal symmetry of the world sheet. Within the CTC interpretation, these results simply exemplify the central idea: the observed mass level functions as a curvature-response coefficient. It quantifies the energetic cost of deforming a vibrational mode under the divergence of its associated transport flux. This places string excitations in continuity with the earlier examples—photons in plasmas, phonons, bosons, and nucleons—each of which acquires mass from the curvature induced by an underlying transport process. Physically, transport along the string is carried by oscillatory energy flow residing on the world-sheet. The divergence of this vibrational flux determines the curvature of the mode profile, and the 7 corresponding mass parameter measures the resistance of that profile to deformation in the compactified directions. This mirrors the earlier cases: photons in a plasma acquire effective mass from electromagnetic transport (2.1), phonons from lattice vibrational curvature (2.2), bosons from curvature of their scalar or gauge fields (2.3), and nucleons from curvature imposed by gluon confinement (2.4). String modes simply extend this same geometric principle into higher-dimensional settings, where vibrational flux divergence defines the curvature-response coefficient. In this respect, the stringtheoretic mass spectrum fits naturally within the CTC catalogue of systems whose effective mass arises from curvature generated by transport. 2.6. Relation to String Theory and Independence from Supersymmetry The CTC incorporates string excitations through a structural feature of string theory that does not depend on supersymmetry or on any specific high energy particle content. The mass of a string mode is fixed by the curvature of its vibrational dispersion, a relation derived directly from the world sheet conformal field theory of the string spectrum[16], [17]. This curvature reflects the divergence of the transport flux that propagates along the one dimensional string, a point emphasized in standard treatments of string vibrational modes[18]. It is therefore a kinematic property of the vibrational system rather than a supersymmetry dependent constraint. In the CTC framework, a string mode is interpreted as the one-dimensional member of a broader class of curvature transport systems. Quasiparticles in solids, phonons, gluon confined nucleons, and orbit averaged angular momentum transport all share the same geometric structure: an extended or collective transport channel whose flux divergence determines the curvature of the excitation spectrum. A fundamental string fits naturally into this family because its vibrational flux plays the same geometric role as the transport fluxes that set curvature in the other systems. This structural viewpoint also clarifies the relevance of low dimensional condensed matter systems. Quantum Hall edges support chiral one dimensional modes that carry a well-defined transport flux[19]. Spin chains and other one-dimensional quantum fluids exhibit collective excitations whose dispersion curvature follows directly from the internal transport structure[20], [21]. Topological phases contain extended degrees of freedom and stringlike operators whose excitations propagate through a quantized transport channel[22], [23]. In all these systems, the curvature of the vibrational spectrum is determined by the divergence of the underlying flux in the same geometric sense as in a fundamental string mode. They therefore provide (potentially) experimentally accessible realizations of the curvature transport relation that governs the string spectrum, without requiring any attempt to reproduce the full microscopic framework of string theory. From this perspective, supersymmetry remains a consistent and powerful mechanism within string theory, but its role is understood differently. Supersymmetry manages the curvature structure of the spectrum by removing or regulating specific contributions. It is not required for the curvature transport relation itself, which arises directly from the world sheet dynamics of the string. The CTC therefore treats the string spectrum as fully compatible with its geometric framework while remaining independent of supersymmetry and other high energy assumptions. 2.7. Unified Interpretation of Mass Across Physical Scales 8 Taken together, these examples reveal a common geometric structure underlying the appearance of mass in diverse physical settings. In each case, the quantity called “mass” acts as the curvature-response coefficient associated with an underlying transport process: • Electrons in solids: the effective mass arises from the curvature of electronic bands and the associated crystal-momentum transport. • Photons in plasmas: an effective mass appears through dielectric transport, encoded in the curvature of the plasma dispersion relation. • Phonons in lattices: vibrational modes acquire effective mass from the curvature of the lattice dispersion spectrum. • Nucleons in QCD: the nucleon mass reflects the curvature imposed by gluon-confinement flux in the strongly coupled regime. • Bosons in scalar and gauge sectors: the Higgs field and related curvature terms generate mass as a response to field-transport structure. • Strings in higher dimensions: vibrational excitations acquire mass from the curvature of their mode spectra, determined by the divergence of oscillatory flux along the string. Across these domains, mass is not a primitive intrinsic quantity but the response coefficient that quantifies how strongly a system resists curvature generated by a transport flux. This recurring structure is the unifying theme of the Curvature–Transport Correspondence. 3. Transport as Curvature Sourcing in Collisionless Galactic Dynamics Collisionless stellar systems also provide a realization of the curvature transport structure. A complete development of this framework, including the role of non-local torque coupling and its use in reconstructing rotation curves, will appear in a forthcoming companion study. For completeness, the essential derivation is summarized in the Supplementary Materials. The present section focuses only on the geometric result. The dynamics begin with the collisionless Boltzmann (Vlasov) equation: ∂𝑓 ∂𝑡 +𝐯⋅∇𝑓−∇Φ⋅∇𝐯𝑓 =0. (17) Taking its moment with respect to the specific angular momentum ℓ=𝑅𝑣𝜙 yields the exact evolution equation for the angular-momentum surface density 𝐿(𝑅,𝑡): ∂𝐿 ∂𝑡 =−1 𝑅∂ ∂𝑅(𝑅𝐹𝐿), (18) where the radial flux of angular momentum is 𝐹𝐿(𝑅,𝑡)=∫𝑣𝑅 ℓ 𝑓 𝑑3𝑣. (19) This result has a profound interpretation: the temporal curvature (evolution) of the angular momentum field, ∂𝐿/∂𝑡, is sourced explicitly by the negative divergence of the angular momentum torque flux, 𝐹𝐿. This is the direct dynamical analogue of the gravitational Poisson equation, ∇2Φ=4𝜋𝐺𝜌eff,(20) 9 where the spatial curvature of the potential is sourced by an effective density. Both equations share the universal CTC form: curvature = divergence of a flux. This geometric equivalence reframes the physical interpretation: just as 𝐹𝐿 is the explicit transport flux sourcing angular momentum curvature, the source of gravitational curvature (∇2Φ) must be an effective density, 𝜌eff, that functions as a curvature-response coefficient emerging from an underlying gravitational transport flux, 𝐹𝐺. This identity positions 𝜌eff not as an independent substance, but as a measure of the field's energetic resistance to transport-induced curvature. The cosmological role of 𝜌eff is developed in Section 4, and the unified CTC principle is formalized in Section 5. 4. Cosmological Curvature from Effective Density and Tidal Flux 4.1. The Effective Density as a Curvature-Response Coefficient The geometric foundation of cosmological gravity is the Poisson equation, where curvature in the gravitational potential, ∇2Φ, is sourced by an effective density in eq. 20. In the standard ΛCDM model, this density is a sum of components: 𝜌eff =𝜌dm +𝜌b+𝜌rad +𝜌vel/stress, (21) encompassing dark matter, baryons, radiation, and velocity-stress corrections. The CTC reframes the interpretation of this fundamental equation. Observables such as CMB anisotropies, gravitational lensing, and large-scale structure are direct measurements of gravitational curvature ∇2Φ. The inferred 𝜌eff is the parameter required to satisfy the Poisson equation given this observed geometry. Therefore, under the CTC, the effective density 𝜌eff is the curvature-response coefficient for the gravitational potential. It quantifies the collective response of the cosmic medium to the observed curvature, rather than necessarily representing a sum of distinct particulate substances. This perspective aligns with the structure identified in Section 3: just as the curvature of the angular momentum field is sourced by torque-flux divergence, the curvature of the potential is sourced by an effective density that can be understood as emerging from an underlying gravitational transport flux, 𝐹𝐺. This perspective remains fully consistent with the ΛCDM model and all its observational successes. The distinction is one of interpretation and focus: whereas the standard view concentrates on the concentrated entity of density (e.g., a dark matter particle density), the CTC framework shifts the focus to the divergence process from which the effective density emerges. The value of 𝜌eff is the same, but its physical meaning is reframed from a measure of substance to a measure of curvature-response generated by transport-flux divergence. 4.2. Tidal Torque Theory as a Flux-Divergence Law The dynamical growth of cosmic structure provides a complementary view through tidal torque theory (TTT). The standard formulation describes the generation of halo spin: 𝑑𝐿𝑖 𝑑𝑡 =𝜖𝑖𝑗𝑘𝐼𝑗𝑙𝑇𝑘𝑙,(22) where the inertia tensor 𝐼𝑗𝑙 couples to the tidal tensor 𝑇𝑘𝑙. 16 (f) The Low-Energy Pathway to Unification via Curvature-Transport Algebra • CTC Prediction: Emergent, string-like excitations in low-dimensional condensed-matter systems (e.g., quantum Hall edges, spin chains, topological phases) will obey the same curvaturetransport algebra 𝐶[𝑋]=∇⋅𝐹𝑋 that defines the effective mass spectrum of fundamental strings. The key test is not whether these quasiparticles replicate high-energy microphysics, but whether their dispersion relations reveal an effective mass determined by the divergence of a measurable transport flux (e.g., energy current) in the material. Confirmation of this shared geometric skeleton in laboratory systems would provide a practical, low-energy path to validating the unified principles underlying quantum gravity. Together, these predictions provide a direct empirical testbed for the CTC framework. They identify signatures that arise specifically from the divergence of transport fluxes and cannot be naturally reproduced by modifying or tuning unseen mass densities. Verification or refutation of these predictions will therefore decisively distinguish the CTC from all substance-based interpretations. 7.3. A Program of Curvature Diagnostics The CTC reframes cosmological and galactic research as a program of curvature diagnostics. Just as condensed-matter physics progresses by mapping the curvature of electronic band structures to define effective mass, gravitational physics should prioritize the precise mapping of curvature itself—rotation curves, tidal fields, void expansion profiles. Within this framework, the so-called dark components are interpreted as response coefficients that encode how transport drives curvature. This perspective suggests a systematic methodology: treat gravitational measurements analogously to band-structure mapping in solids. Deviations in halo spin alignments, rotation-curve slopes, or void dynamics then become empirical tests of transport-based curvature generation, moving beyond the inference of hidden substances. 7.4 On the Quantum Nature of Transport The transport fluxes that appear in the CTC framework are collective and emergent quantities. Their origins lie in microscopic quantum dynamics, yet the fluxes themselves act as classical fields. Examples include the probability flux derived from the wavefunction, the torque flux generated by ensembles of stellar orbits, the mass flux associated with the classical Higgs condensate, and the effective potentials that govern electron and phonon motion in solids. In each case the flux represents a coarse-grained description of many underlying degrees of freedom. For this reason, there is no requirement and no clear physical meaning in attempting to quantize the flux itself. The situation is closely analogous to hydrodynamics, where fluid flow arises from molecular motion yet the macroscopic fields in the Navier Stokes equations are not quantized. The CTC predictions concern this emergent and classical level of description, where curvature is generated by the divergence of a transport field. The deeper quantum question therefore shifts from the quantization of the flux to the origin of the structure it expresses. The central issue becomes the following: which classes of quantum dynamics naturally produce the curvature transport relation that is seen in systems ranging from solids to plasmas to galactic disks and the large-scale structure of the universe? The answer to this question defines the direction for a future quantum theory that recovers the CTC structure in the appropriate classical limit. 17 8. Summary Quantum mechanics, galactic dynamics, and cosmological gravity share a common transport curvature structure: in every domain the curvature of a physical field is generated by the divergence of an associated transport flux. Quantities usually interpreted as mass, density, or dark matter therefore take the role of curvature response coefficients rather than fundamental substances. The Curvature Transport Correspondence (CTC) formalizes this idea by placing inertial mass, torque flux divergence in stellar disks, effective density in cosmology, and the stress energy tensor in general relativity within a single geometric framework. By treating these source terms as energetic responses to underlying transport, the CTC provides a unified language that links quantum effective mass, angular momentum transport in galaxies, and tidal curvature in cosmology. This approach preserves all established observational results while broadening the geometric interpretation of the quantities inferred from them. In this way the CTC complements standard theories and offers a coherent and empirically grounded perspective on curvature across scales, from the microscopic to the cosmic. Acknowledgement The author gratefully acknowledges support from the Matthew and Patricia Harthcock Fellowship in the College of Natural and Applied Science at Missouri State University. The author also thanks his son Ardian Putra Yudawan, for insightful and energizing discussions during the Thanksgiving break that helped clarify several conceptual elements of this work. 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Misner, “Republication of: The dynamics of general relativity,” General Relativity and Gravitation, vol. 40, pp. 1997–2027, Sept. 2008, doi: 10.1007/s10714-0080661-1. [26] L. B. Szabados, “Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article,” Living Reviews in Relativity, vol. 7, no. 1, p. 4, Dec. 2004, doi: 10.12942/lrr-2004-4. Supplementary Materials: Angular-Momentum Transport in the Vlasov–Poisson Framework This note summarizes the minimal derivation required to obtain the angular-momentum transport law used in the main text. The full version appears in the companion paper (arXiv ) ; here we present a compressed form, retaining only the steps needed to show that torque transport arises directly from the Vlasov–Poisson system. A1. Vlasov–Poisson in Axisymmetric Cylindrical Coordinates 19 A collisionless stellar disk is governed by the Vlasov equation ∂𝑓 ∂𝑡 +𝐯⋅∇𝑓−(∇Φ)⋅∇𝐯𝑓 =0, (S1) with potential determined via ∇2Φ= 4𝜋𝐺 𝜌,𝜌 =∫𝑓 𝑑3𝑣. (S2) In cylindrical coordinates (𝑅,𝜙,𝑧)with velocities (𝑣𝑅,𝑣𝜙,𝑣𝑧), the advective form becomes ∂𝑓 ∂𝑡 +𝑣𝑅∂𝑓 ∂𝑅 +𝑣𝑧∂𝑓 ∂𝑧 +𝐴𝑅∂𝑓 ∂𝑣𝑅+𝐴𝜙∂𝑓 ∂𝑣𝜙+𝐴𝑧∂𝑓 ∂𝑣𝑧=0, (S3) where 𝐴𝑖=−∂𝑖Φ. Axisymmetry gives ∂𝜙𝑓 =0 and 𝐴𝜙=0. To expose transport explicitly, this equation must be written in conservative form, where each term becomes a divergence. Using 𝑣𝑅∂𝑅𝑓 = 1 𝑅∂𝑅(𝑅𝑣𝑅𝑓),𝐴𝑖∂𝑣𝑖𝑓 =∂𝑣𝑖(𝐴𝑖𝑓), (S4) (the velocity derivatives of 𝐴𝑖vanish), the conservative Vlasov equation reduces to ∂𝑓 ∂𝑡 +1 𝑅∂ ∂𝑅(𝑅𝑣𝑅𝑓)+∂𝑣𝑅(𝐴𝑅𝑓)+∂𝑣𝜙(𝐴𝜙𝑓)+∂𝑣𝑧(𝐴𝑧𝑓) =0. (S5) This is the only form needed for the moment calculation. A2. Angular-Momentum Surface Density For axisymmetric disks the specific angular momentum is 𝑙 =𝑅𝑣𝜙. The angular-momentum surface density is the azimuthal moment of 𝑓: 𝐿(𝑅,𝑡)=∫𝑅𝑣𝜙𝑓 𝑑3𝑣 𝑑𝑧.(S6) Using the standard expression for mean rotation Ω=⟨𝑣𝜙⟩/𝑅, 𝐿(𝑅,𝑡) =Σ(𝑅,𝑡) 𝑅2Ω(𝑅,𝑡), (S7) so 𝐿 is the rotational support of the annulus at radius 𝑅. A3. Multiplying the Vlasov Equation by 𝒍= 𝑹𝒗𝝓 Multiply Eq. (A1-1) by 𝑅𝑣𝜙and integrate over all velocities and 𝑧: ∫𝑅𝑣𝜙 ∂𝑡𝑓 𝑑3𝑣 𝑑𝑧+∫𝑅𝑣𝜙 1 𝑅∂𝑅(𝑅𝑣𝑅𝑓) 𝑑3𝑣 𝑑𝑧+∑∫𝑅𝑣𝜙 ∂𝑣𝑖(𝐴𝑖𝑓) 𝑑3𝑣 𝑑𝑧 𝑖=0. (S8) 20 We evaluate the three terms: (1) Time derivative ∫𝑅𝑣𝜙 ∂𝑡𝑓 𝑑3𝑣𝑑𝑧= ∂𝐿 ∂𝑡.(S9) (2) Radial transport ∫𝑅𝑣𝜙 1 𝑅∂𝑅(𝑅𝑣𝑅𝑓) 𝑑3𝑣 𝑑𝑧 =1 𝑅∂ ∂𝑅[𝑅 𝐹𝐿(𝑅,𝑡)], (S10) where the angular-momentum flux is defined as 𝐹𝐿(𝑅,𝑡)=∫𝑅𝑣𝜙𝑣𝑅𝑓 𝑑3𝑣 𝑑𝑧.(S11) This is the flux of angular momentum carried across radius 𝑅by stars with nonzero 𝑣𝑅. (3) Velocity-space terms Integrating by parts gives velocity-space boundary terms that vanish because 𝑓 →0as ∣𝐯∣→∞. Derivative terms vanish because: • ∂𝑣𝑅(𝑅𝑣𝜙)=0 • ∂𝑣𝑧(𝑅𝑣𝜙)=0 • ∂𝑣𝜙(𝑅𝑣𝜙)=𝑅, but multiplies 𝐴𝜙=0under axisymmetry. Thus, ∑∫𝑅𝑣𝜙 ∂𝑣𝑖(𝐴𝑖𝑓) 𝑑3𝑣 𝑑𝑧 𝑖=0. (S12) A4. Exact Angular-Momentum Transport Equation Collecting (A3-2), (A3-3), and (A3-5) gives the exact conservation law: ∂𝐿 ∂𝑡 =−1 𝑅∂ ∂𝑅[𝑅 𝐹𝐿(𝑅,𝑡)] (S13) This result follows solely from the collisionless Vlasov equation , meaning no fluid closure, no approximations, and no special orbit assumptions. It is the precise torque law used in the main text: the curvature of angular-momentum support is the divergence of a radial torque flux. A5. Significance • If 𝐹𝐿=0, each annulus evolves independently → the enclosed-mass rotation-curve formula is valid. • If 𝐹𝐿≠0, inner regions export angular momentum → outer regions maintain elevated rotation → flat or rising rotation curves arise without additional mass. Thus, the enclosed-mass method corresponds to a highly restricted, torque-free subset of Vlasov solutions. The general case, seen in all real disks, requires the full transport law (A4-1).