The Curvature–Transport Correspondence (CTC) and Its Implications for Cosmic Structure Formation

The Curvature–Transport Correspondence (CTC) and Its Implications for Cosmic Structure Formation Ridwan Sakidja Dept. of Physics, Astronomy and Materials Science Missouri State University Abstract This work extends the recently developed framework of the Curvature Transport Correspondence to the study of cosmic structure formation. In this framework gravitational curvature arises not from material inventories but from the divergence of a transport flux field. Mass, density, and curvature become response coefficients, much as stiffness, elasticity, and effective mass emerge in materials science from the way a medium conducts and redistributes load. This reflects a central principle of condensed matter physics: structure and rigidity are not substances but responses. Applying this idea to three major tensions in modern physics, namely extreme cosmic inhomogeneity, the behavior of curvature inside black holes, and the large separation between the Higgs scale and the Planck scale, reveals a common origin. Each problem results from assuming that curvature is produced directly by density. Through the materials science lens of the Curvature Transport Correspondence, cosmic filaments arise from spatial variations in divergence, black hole interiors avoid singularities through flux saturation, and the hierarchy becomes a transition between soft and stiff response regimes. Taken together, these results show that many long-standing puzzles can be understood as manifestations of a single principle: curvature is the geometric response of a medium whose ability to transmit transport changes with scale. 1. Introduction: Rethinking Gravity Through the Lens of Transport For much of modern cosmology, a single principle has guided our understanding of gravitation: mass determines curvature. This interpretation, rooted in Einstein’s field equations, has supported the remarkable success of the ΛCDM framework. Yet as observations have sharpened, from JWST deep-field data[1] to high-resolution cosmic-web surveys[2], several persistent anomalies have begun to challenge this classical picture. Across the observable Universe, curvature appears to respond to its environment in ways that differ from the expectations of a purely density-driven model[3]. Some regions form massive, mature galaxies far earlier than ΛCDM predicts[4]. Others evolve slowly, leaving vast voids with unexpectedly low density [5]. Structures such as filaments and walls exhibit greater coherence, length, and contrast than standard cosmology anticipates[6], [7]. These increasingly precise datasets raise a deeper question: what if curvature is not created by mass density at all, but by something more fundamental? 1.1. A Shift in Perspective The Curvature Transport Correspondence (CTC)[8] offers an alternative formulation. Rather than interpreting the source of curvature as a static inventory of mass-energy, CTC rewrites the effective source as the divergence of a gravitational transport flux: 𝜌eff(𝑥)=∇⋅𝐹𝐺(𝑥). (1-1) This preserves the full mathematical structure of general relativity [9] but gives its source term a new physical interpretation. In this view curvature measures how gravitational transport flows through spacetime. When the flux converges, with the divergence negative, the result is strong collapse and rapid structure formation [10]. When the flux diverges, with the divergence positive, regions expand and voids grow [11]. When the divergence is nearly zero, the flux is balanced and the geometry evolves slowly, a pattern seen for example at the boundary of the Laniakea basin and in the saddle point regions of the cosmic web [12], [13]. This shift in perspective moves the focus from substance to flow. Curvature becomes the geometric response of spacetime to transport, not a direct reaction to mass density. The cosmological picture changes accordingly: the Universe is no longer a static inventory of material content but a dynamical system sculpted by the convergence and divergence of gravitational flux. 1.2. Why This Perspective Has Become Necessary Several observational tensions have become too significant to ignore. Data from JWST reveal galaxies that appear far too massive and mature for their early cosmic epoch [4], [14]. Large surveys show voids and walls whose sizes and coherence are difficult to reconcile with the expectations of the standard cosmological model [15] [16] [17]. Even the Hubble tension points toward the possibility that the measured expansion rate may depend on our location within large scale inhomogeneities [18] [19]. Each of these issues could, in isolation, be attributed to astrophysical uncertainties or statistical variation. When viewed together, however, they suggest that something deeper may be amiss in our understanding of what actually sources curvature. Within this setting the Curvature Transport Correspondence offers a different perspective. The framework leaves Einstein’s equations untouched and does not require any new particles. What changes is the physical meaning of the source term. Instead of treating curvature as a response to material density, the correspondence identifies it as a response to the divergence of an underlying transport field. This shift in interpretation provides a single conceptual lens through which these diverse cosmological tensions can be understood as expressions of the same underlying mechanism. 1.3. Three Structural Tensions Motivating the CTC Framework The Curvature Transport Correspondence is motivated by four persistent tensions in contemporary cosmology. These tensions, though traditionally treated as unrelated, share a common feature: they challenge the assumption that curvature is created directly by material density. When viewed through the lens of transport geometry, each tension appears instead as a manifestation of spatial variations or saturation in the divergence field. Extreme cosmic inhomogeneity. Large voids, highly coherent filaments, and expansive wall structures exceed the amplitude and correlation lengths predicted by the standard model. Observations of the KBC supervoid[15], the Sloan Great Wall[16], and the Hercules Corona Borealis structure¹⁷ reveal inhomogeneities that are difficult to reconcile with Gaussian initial conditions and uniform density sourcing. In the CTC picture, such structures arise naturally from regions of strong negative or positive divergence. Black hole singularities. Classical general relativity predicts that curvature diverges inside black holes. This follows from the assumption that mass can collapse indefinitely. In the CTC formulation, collapse corresponds to flux convergence that is physically allowed to saturate. Singularities do not appear because the divergence field cannot grow without bound, allowing a finite curvature core to form. The hierarchy problem. The vast separation between the Higgs scale and the Planck scale is often interpreted as an issue of fine adjustment within the standard model. Under the CTC interpretation, this scale separation reflects a transition in how spacetime responds to divergence. Different regimes of curvature response, rather than delicate tuning of parameters, produce the observed hierarchy. These three tensions, when considered together, reveal a pattern: they arise from the same structural assumption. They are not separate puzzles but facets of a deeper misidentification of curvature’s physical source. 1.4. A view from Materials Science From a materials-science perspective, this view arises naturally: in condensed matter physics, quantities such as stiffness[20], conductivity, and effective mass are not intrinsic substances but emergent responses to how energy, charge, and momentum flow through a material. An electron’s effective mass in a crystal is not a fixed attribute; it is determined by the curvature of the electronic band structure[21]. A material’s rigidity is not merely a number. Rather, it reflects how atomic bonds transmit and redistribute stress[22]. Even phase transitions are not the introduction of new ingredients but the system reorganizing its internal flows[23]. Thus, CTC asks whether spacetime behaves in the same way. What if “mass” is simply a measure of how spacetime resists bending when transport flows through it? What if cosmic voids are not passive emptiness but regions where the flow field diverges? What if the cosmological constant is not vacuum energy at all, but the faint, residual signature of an uneven transport field permeating spacetime? This is not a leap into speculation. It is the natural extension of familiar transport principles, which is long established in materials science, to the geometry of spacetime itself. 1.5. The organization of this paper: In the sections that follow, we examine several persistent tensions in modern cosmology and explore whether the Curvature–Transport Correspondence (CTC) might offer a clarifying perspective. While the standard ΛCDM model provides a remarkably successful framework, observations continue to reveal puzzling features. We will consider how a geometric shift in viewpoint, that is by interpreting mass and density not as fundamental substances, but as curvature-response coefficients emerging from transport flux, might provide a coherent, unified way of reframing these challenges: 1. Extreme cosmic inhomogeneity– Are the surprising scales of voids and filaments hints of a transport-geometric origin rather than statistical anomalies? 2. Planck–Higgs hierarchy– Could the vast scale separation reflect a vacuum stiffness transition, not a fine-tuning problem? 3. Black hole singularities– Could flux saturation prevent infinite curvature, similar to how materials saturate under extreme stress? Our aim is not to replace established theory, but to ask: if we reinterpret the source terms in Einstein’s equations geometrically, i.e. as the divergence of a gravitational transport flux, could several seemingly disconnected tensions find a common, conceptually simple explanation? We present this exploration in the spirit of offering a complementary viewpoint, one that preserves the mathematics of GR while suggesting a shift in physical interpretation from substance to transport geometry. 2. Extreme Cosmic Inhomogeneity as a Transport-Geometry Effect The structure of the universe tells a story of its origins, but in recent years, that story has grown unexpectedly complicated. Observations reveal a cosmic web marked by vast voids, immense filaments, and surprising coherence that pushes against the limits of standard cosmological models [2], [24]. These features challenge our understanding of how structure emerges from the smooth, uniform conditions of the early universe. 2.1. Observational Challenge Our most precise surveys have mapped a cosmos-richer and more structured than anticipated. Consider the KBC Void[15], a spherical underdensity roughly 600 megaparsecs across, with our own Milky Way nestled near its edge. Or the immense filaments and walls revealed by galaxy surveys[6], tracing patterns across gigaparsec scales that seem improbably large and coherent. Even the scale of homogeneity, which is the distance beyond which the universe approaches uniformity[25], [26], emerges only at surprisingly large scales. Analyses of SDSS DR6 find the transition near 60–70 h⁻¹ Mpc [26], while recent model-independent angular studies using S-PLUS blue galaxies identify a comparable range[25]. Together, these results indicate that the universe becomes homogeneous only at scales larger than many early theoretical models predicted, reinforcing how deeply structured the cosmic web truly is. 2.2. CTC Geometric Resolution As introduced in Eq. (1-1), the effective source of curvature is given by the divergence of the gravitational transport flux. Spatial variations in this divergence naturally generate the observed structure of the cosmic web. Here, 𝐹𝐺 represents a gravitational transport flux field. This simple equation reframes structure formation from a process of matter accumulation to one of geometric transport organization. Spatial variations in this flux divergence naturally decorate the cosmic landscape: ∇⋅𝐹𝐺(𝑥)<0 (convergent flux) →expanding voids ∇⋅𝐹𝐺(𝑥)>0 (divergent flux) →collapsing filaments and clusters ∇⋅𝐹𝐺(𝑥)≈0 (neutral flux) →sheet-like boundaries The "sponge-like" topology of the cosmic web[10] emerges not as a statistical outlier, but as the natural equilibrium state of a transport field with spatially varying divergence. 2.3 Transport-Driven Pattern Formation The CTC mechanism operates through a self-organizing feedback loop: 1. Initial perturbations in 𝐹𝐺 create regions of varying divergence, 2. Divergence gradients drive curvature responses through 𝜌eff =∇⋅𝐹𝐺, 3. Emergent curvature modifies subsequent flux evolution, 4. Self-organization produces the characteristic void-filament-sheet network. This behavior reflects well established pattern formation dynamics in driven transport systems, where heterogeneous flux distributions evolve toward organized spatial structures such as filaments, sheets, rolls, or cellular patterns[27]. In many physical settings that involve transport and curvature operators such as divergences or Laplacians, instabilities in the flux field give rise to extended spatial organization. Viewed in this way, the cosmic web is not only a result of gravitational instability in an expanding background but also an expression of the natural tendency of spacetime to organize transport flows. The CTC framework brings out this geometric structure clearly. Curvature and transport do not act independently but coevolve in a manner that naturally produces large scale pattern formation. CTC can potentially make specific, testable predictions that distinguish it from standard approaches: • Void statistics: Size distributions should follow transport-divergence scaling laws rather than Gaussian random field statistics • Morphological coherence: Filaments should exhibit greater alignment and connectivity than predicted from initial density fields alone • Evolutionary sequence: Structure formation timing should correlate with local divergence magnitude, not just local density In this note, it is useful to point to recent observations: 1. SDSS/BOSS surveys show filament coherence extending beyond ΛCDM expectations[28], [29]. A very recent work by Tavasoli and Ghafour [29] shows that the observed cosmic filaments in SDSS DR10 are denser, more coherent, and structurally narrower than their ΛCDM counterparts in the IllustrisTNG300-1 simulation. The ΛCDM model can generate long macrofilaments, yet the observed Universe does not contain those same extended structures. Instead, it displays shorter but systematically denser connections. Even more striking, microfilaments linking groups and clusters show opposite trends between observation and simulation: GG filaments are denser in real data, while CC filaments are denser in ΛCDM, revealing a structural mismatch. Together, these results highlight a consistent pattern. The real cosmic web is more sharply organized and more gravitationally efficient than the ΛCDM reconstruction predicts. This tension is exactly what the Curvature Transport Correspondence can reinterpret. When curvature is based on the divergence of gravitational transport, filaments arise as self-focusing flow channels that naturally become narrower, denser, and more symmetric than the mass driven ΛCDM model allows. 2. Planck CMB lensing reveals matter distribution more structured than predicted[24]. Planck’s CMB lensing analysis shows that the temperature and polarization spectra prefer a lensing amplitude higher than the ΛCDM prediction, with 𝐴L=1.18±0.065 reported on page 3 of the Planck 2018 release. This indicates that the CMB anisotropies are more smoothed by lensing than the standard model allows, implying a matter distribution that is slightly more clumped and more coherently organized than predicted. Under CTC, this enhanced lensing does not require additional mass but arises naturally from stronger convergence of gravitational transport, which deepens curvature and sharpens structure. 3. DES Year 3 data indicates void properties inconsistent with Gaussian initial conditions[30]. DES Year 3 measurements of cosmic voids show that real voids are emptier, larger, and have steeper density profiles than predicted by ΛCDM with Gaussian initial conditions. As noted in the DES analysis, the void lensing signal is stronger than expected, and the stacked density contrast around void centers is too deep for a purely Gaussian initial field. These findings imply that matter has been evacuated from void interiors more efficiently than standard structure formation allows. In the CTC framework, such behavior arises naturally because divergence in gravitational transport amplifies flow separation, producing deeper and more coherent voids without requiring any modification to the underlying mass content. In short, rather than representing crises for ΛCDM, these "anomalies" may reflect the underlying transport geometry of spacetime, a geometry that naturally produces large, coherent structures as it seeks equilibrium. 2.4. Materials Physics Analogy: Pattern Formation Through Transport Geometry The geometric principles underlying CTC's explanation of cosmic structure formation find direct and well-established parallels in materials science, where similar transport mechanisms govern pattern formation across diverse physical systems. These analogs not only illuminate the CTC framework conceptually but also provide mathematical tools and scaling relationships that can be applied to cosmological structure formation. Parallel Systems in Materials Science Polycrystalline Grain Growth[31] offers a particularly instructive analogy. In polycrystalline materials, grain boundaries migrate according to local curvature and energy minimization principles. The divergence of grain-boundary flux determines the resulting microstructure: • Regions of positive flux divergence become sites of void formation and interface development • Regions of negative flux divergence experience grain growth and densification • Neutral flux regions form stable grain boundaries that persist over extended timescales The resulting microstructure, which is characterized by grain size distributions, boundary networks, and void arrangements, emerges directly from the geometry of grain-boundary transport rather than from random atomic arrangements. This mirrors CTC's cosmic web formation, where gravitational flux divergence determines void, filament, and sheet morphology. Spinodal Decomposition[32] in binary alloys provides another powerful analogy. During phase separation, components separate through uphill diffusion driven by chemical potential gradients. The process follows the Cahn-Hilliard equation: ∂𝑐 ∂𝑡=𝑀∇2(∂𝑓 ∂𝑐−2𝜅∇2𝑐) (2-1) where 𝑐 is concentration, 𝑀 mobility, 𝑓 free energy density, and 𝜅 gradient energy coefficient. The resulting interconnected structures (typically with characteristic wavelength 𝜆∝(𝜅/Δ𝑓)1/2) emerge from diffusion-flux divergence patterns, creating morphologies strikingly similar to the cosmic web's void-filament network. The statistical properties of these patterns follow universal scaling laws derived from transport geometry rather than initial concentration fluctuations. Dislocation Patterning[33] in deformed crystals demonstrates how transport geometry shapes microstructure under external driving. Dislocations, defined as line defects in crystal lattices, organize into characteristic patterns: • Convergent dislocation fluxes form dense deformation bands and cell walls • Divergent dislocation fluxes create void-like channels and low-density regions • The resulting pattern scale depends on dislocation mobility and interaction strengths These patterns emerge from the collective dynamics of dislocation transport rather than from random defect distributions, mirroring how cosmic structure emerges from gravitational flux organization rather than random density fluctuations. Common Principles Across Scales These materials systems share fundamental principles with CTC's cosmological framework: 1. Transport-driven pattern formation: Structure emerges from the organization of transport fluxes rather than from static initial conditions 2. Divergence-based morphology: Positive/negative flux divergence determines void/condensation patterns 3. Scale selection: Characteristic pattern scales emerge from transport coefficients (mobility, diffusivity, interaction strengths) 4. Universal scaling: Statistical properties follow transport-determined scaling laws This perspective transforms cosmic web formation from a unique cosmological phenomenon into a specific instance of universal transport-driven pattern formation principles operating at extreme scales. The cosmic web becomes not a statistical anomaly of random fluctuations, but the natural equilibrium structure of spacetime under heterogeneous gravitational transport. This is quite analogous to how polycrystalline microstructures represent equilibrium configurations of grain boundary transport. The convergence of cosmic and materials pattern formation principles suggests deeper connections between gravitational and condensed matter physics, potentially revealing universal laws governing pattern formation across all scales where transport processes dominate structural evolution. 3. Resolution of the Black-Hole Singularity in the Curvature–Transport Correspondence 3.1. The Singularity Problem in Classical General Relativity Classical general relativity predicts that gravitational collapse terminates in a curvature singularity [9], [34]. For a Schwarzschild black hole, the curvature invariant that captures the full contraction of the Riemann tensor is 𝐾(𝑟)=𝑅𝜇𝜈𝜌𝜎𝑅𝜇𝜈𝜌𝜎=48𝐺2𝑀2 𝑐4𝑟6. (3-1) As 𝑟→0, this expression diverges, lim 𝑟→0𝐾(𝑟)=∞, (3-2) indicating that the geometry predicts an unbounded curvature at the central point. Because the mass 𝑀of the collapsing body is assumed to be concentrated into an ever-smaller volume, the implied density behaves as lim 𝑉→0𝜌GR =lim 𝑉→0𝑀 𝑉→∞. (V→0) (3-3) Thus, within the classical framework, both the energy density and the curvature invariants diverge as the collapsing matter is driven toward zero volume. The appearance of such divergences has two implications. Mathematically, the spacetime becomes geodesically incomplete: timelike and null geodesics terminate after a finite affine parameter[35], preventing the continuation of physical trajectories beyond the central region. Physically, the theory demands the existence of infinite curvature and infinite density. These quantities that have no operational meaning and mark a breakdown in the applicability of classical general relativity. In this sense, the Schwarzschild singularity does not represent a physical object but rather a limit of the classical equations themselves. It marks the point at which the assumption of arbitrarily compressible mass-energy becomes incompatible with a finite geometric response. This provides a natural setting in which to explore alternative formulations, such as the Curvature–Transport Correspondence, that constrain the source of curvature through physically motivated transport principles rather than through unbounded compression 3.2. CTC Perspective: From Substance to Transport The Curvature–Transport Correspondence reframes the source of gravity. Rather than interpreting the Einstein equations as curvature sourced by a material density, the CTC approach expresses the same term as the divergence of a gravitational transport flux 𝐹𝐺𝜇𝜈: 8𝜋𝐺 𝑇𝜇𝜈=∇⋅𝐹𝐺𝜇𝜈. (3-4) In this view, gravitational collapse is not the compression of “substance” into a point but the convergence of gravitational transport: ∇⋅𝐹𝐺<0 (convergent flux). (3-5) Singularity formation would require the transport field to converge without bound. Instead of assuming infinite compressibility of matter, the CTC framework imposes a physically motivated constraint: transport systems saturate when driven beyond their capacity. 3.3. The Flux Saturation Principle This approach draws an analogy to a broad range of transport phenomena across physics—electronic conduction, heat flow, dislocation glide, superfluid motion—whereby saturation taken place when pushed toward extreme regimes [21], [36], [37]. Once microscopic carriers reach their maximum velocity, or once structural constraints limit further flux, the system enters a new nonlinear response regime instead of diverging. CTC extends this principle to gravitational transport: For ∣𝐹𝐺∣≤𝐹max: ∣∇ ⋅ 𝐹𝐺∣≤𝐷max, (3-6) where 𝐹max and 𝐷max represent the finite transport capacity of spacetime. This reframes collapse entirely. Gravitational fields cannot focus indefinitely. Once the convergence reaches ∣∇⋅𝐹𝐺∣=𝐷max, further compression is impossible. Collapse transitions into a saturated state, just as superfluids reorganize beyond their critical velocity or solids deform plastically beyond their yield stress. A useful way to visualize this saturation behavior is through a mechanical analogy. Consider an acrobat riding a bicycle in a perfectly circular path. As the rider increases speed, the required centripetal force grows until the mechanical limits of the system such as the strength of the rider, the traction of the tires, and the stiffness of the frame impose a minimum turning radius. Beyond this limit, no amount of additional effort can reduce the radius any further, and the system becomes saturated. In the CTC framework spacetime behaves in the same manner. The divergence of the transport field ∇⋅𝐹𝐺 determines how sharply the geometry can turn, and once this quantity reaches its maximum sustainable magnitude 𝐷max, curvature cannot increase further. The geometry then settles at a finite critical radius 𝑅crit, inside which the divergence vanishes and the interior becomes dynamically inert. Just as a bicycle cannot trace a tighter circle once its mechanical threshold is reached, spacetime cannot support curvature below the saturation scale, and the interior remains regular instead of singular. 3.4. Mathematical Framework The implications of the flux-saturation principle become transparent when the linearized Einstein equations are expressed within the framework of hyperbolic operators. In perturbation theory the metric is decomposed as 𝑔𝜇𝜈=𝜂𝜇𝜈+ℎ𝜇𝜈, (3-7) and, imposing the Lorenz gauge condition ∂𝜇ℎˉ𝜇𝜈=0,ℎˉ𝜇𝜈=ℎ𝜇𝜈−1 2𝜂𝜇𝜈ℎ, (3-8) the linearized Einstein equations take the form (see Wald 1984[9]; Misner–Thorne–Wheeler 1973[38]; Maggiore 2007[39]) 𝐿[ℎ𝜇𝜈]=∇⋅𝐹𝐺𝜇𝜈, (3-9) where 𝐿 is a second-order normally hyperbolic operator whose principal part is the wave operator □. This situates the linearized theory within the well-developed mathematical framework of hyperbolic PDEs on globally hyperbolic manifolds (Friedlander 1975[40]; Bär, Ginoux & Pfäffle 2007[41]). Such operators admit a unique causal Green operator 𝐺, characterized abstractly by 𝐿𝑥 𝐺(𝑥,𝑥′)=𝛿(4)(𝑥−𝑥′), (3-10) with support restricted to points that are causally related in the background spacetime (see Bär et al. 2007[41]; Ringström 2009[42]). No advanced distinction is required here; the operator is defined purely by linearity, hyperbolicity, and causality in the sense of normally hyperbolic operators. Using this operator as the inverse of 𝐿, the metric perturbation admits the Green-function representation familiar from classical field theory [9], [40]: ℎ𝜇𝜈(𝑥)=∫𝐺(𝑥,𝑥′) ∇ 𝑀⋅𝐹𝐺𝜇𝜈(𝑥′) 𝑑4𝑥′. (3-11) The simulation shows that this transition behaves much like a phase boundary: inside the core, the curvature is pinned at 𝐾max; outside, the profile follows Schwarzschild scaling. (d) The saturated interior as a transport bottleneck The flat plateau of 𝐾(𝑟) in the simulation reflects the fact that spacetime cannot transport curvature beyond a finite capacity. This mirrors bottleneck phenomena in: • Plastic deformation (stress saturation), • Superfluid vortex formation (critical velocity), • Dielectric breakdown (field saturation). The black hole interior becomes a transport-regulated domain, not a region of runaway collapse. (e) Implications for more complete models Although idealized, the simulation suggests several physically meaningful consequences: • No curvature singularity — the spacetime metric can remain smooth and geodesically complete. • A stable interior — the saturated region behaves as a finite geometric phase. • Observable signatures — modified ringdown spectra, echoes, and accretion dynamics may arise from this finite-curvature core. In summary, the simulation supports the central claim of the CTC approach: singularities are replaced by saturation-driven, finite-curvature structures, consistent with both mathematical boundedness and physical analogy. 3.10 Materials Physics Analogies The CTC resolution of black hole singularities via flux saturation finds precise and well-established parallels in condensed matter and materials physics, where similar saturation mechanisms prevent physical divergences in diverse systems [36], [37], [50]. These analogies provide not only conceptual clarity but also quantitative frameworks for understanding how spacetime responds under extreme gravitational conditions. A. Plastic Deformation: Stress-Strain Saturation In crystalline materials under mechanical loading, the stress-strain relationship demonstrates a clear saturation mechanism that prevents material failure through divergent deformation[37]. The process follows distinct regimes: • Elastic regime: Stress increases linearly with strain: 𝜎=𝐸𝜖, where 𝐸 is Young’s modulus • Yield point: Dislocation motion initiates and dislocation flux saturates due to: o Forest dislocation interactions o Grain boundary pinning o Dynamic recovery mechanisms • Plastic flow: Beyond yield, stress increases only weakly with strain: 𝑑𝜎/𝑑𝜖≪𝐸 • Saturation stress: Stress asymptotically approaches 𝜎sat determined by: o Dislocation mean free path o Thermal activation barriers o Microstructural constraints Mathematically, this is often described by saturation laws such as: 𝜎(𝜖)=𝜎sat[1− exp (−𝐸𝜖/𝜎sat)] or through dislocation density evolution equations: 𝑑𝜌 𝑑𝜖=𝑘1√𝜌−𝑘2𝜌 where 𝜌 is dislocation density and 𝑘1, 𝑘2 are multiplication and annihilation coefficients. CTC analog: Gravitational curvature increases with flux divergence until reaching a saturation value 𝐾max, beyond which further collapse redistributes rather than intensifies curvature, similar to how plastic flow redistributes strain without catastrophic failure. B. Dielectric Breakdown: Electric Field Saturation In dielectric materials under high electric fields, conduction mechanisms exhibit saturation that prevents divergent current densities [36]. The phenomenon involves: • Ohmic regime: Current density follows 𝐽=𝜎𝐸 for moderate fields • Breakdown threshold: At critical field 𝐸bd, impact ionization creates charge carriers • Avalanche saturation: Carrier multiplication saturates due to: o Space charge effects o Thermal dissipation limits o Material defect concentrations • New conduction state: Material enters a high-conductivity regime without infinite current The breakdown field often follows scaling laws: 𝐸bd ∝𝐸𝑔3/2 √𝑚∗𝜖 where 𝐸𝑔 is band gap, 𝑚∗ effective mass, and 𝜖 dielectric constant. Post-breakdown, the current-voltage characteristic saturates: 𝐽(𝐸)=𝐽sat [1+(𝐸 𝐸0)𝑛]1 𝑛 with saturation current 𝐽sat determined by carrier mobility and recombination rates. CTC analog: Spacetime reaches a maximum curvature response at critical flux divergence 𝐷max, analogous to dielectric breakdown field. Beyond this, gravitational transport continues but without producing divergent curvature, just as current continues without infinite density after dielectric breakdown. C. Superfluid Critical Velocity: Flow Saturation In superfluid, the onset of dissipation occurs at a critical velocity 𝑣𝑐 rather than through divergent resistance [50]. The Landau criterion establishes: 𝑣𝑐=min 𝑝𝐸(𝑝) 𝑝 where 𝐸(𝑝) is excitation energy spectrum. Above 𝑣𝑐: • Vortex nucleation: Quantized vortices form, carrying circulation 𝜅=ℎ/𝑚 • Vortex tangle development: Vortices interact and form complex networks • Mutual friction: Vortex-phonon interactions dissipate energy • Saturated dissipation: Additional velocity increase produces proportionally more vortices rather than divergent friction The relationship between superflow velocity 𝑣𝑠 and vortex line density 𝐿 follows: 𝑑𝐿 𝑑𝑡=𝛼𝑣𝑠𝐿3/2−𝛽𝐿2 where 𝛼 and 𝛽 are creation and annihilation coefficients. In steady state: 𝐿∝(𝑣𝑠−𝑣𝑐)2 The critical velocity thus represents a saturation threshold beyond which additional forcing increases vortex density rather than producing divergent dissipation. CTC analog: Gravitational transport saturates at critical divergence 𝐷max, analogous to superfluid critical velocity. Beyond this, additional collapse produces geometric reorganization (e.g., changes in internal black hole structure) rather than divergent curvature. Common Mathematical Structure These diverse saturation behaviors follow a shared mathematical pattern. A broad set of systems can be modeled by evolution equations of the form 𝑑𝑋 𝑑𝑡=𝛼 𝑋𝑚 (1− 𝑋 𝑋sat)𝑛−𝛽 𝑋𝑝, where 𝑋is the evolving quantity, 𝑋satis the saturation value, and the exponents 𝑚,𝑛,𝑝reflect the underlying physics. The first term represents growth or multiplication, amplified by the current value of 𝑋. The factor (1− 𝑋 𝑋sat)𝑛 introduces nonlinear saturation, directly paralleling the generalized logistic (Richards) growth law[51], where growth slows as a system approaches its carrying capacity. The second term, 𝛽𝑋𝑝, plays the role of a recovery or annihilation process that removes the quantity as it accumulates. This structure has a wide range of applications: • In materials science, it mirrors the Kocks–Mecking dislocation evolution law[52], where dislocations multiply through storage mechanisms and saturate through dynamic recovery, following: 𝑑𝜌 𝑑𝜀=𝑘1𝜌1/2−𝑘2𝜌, which is exactly a generation term − annihilation term with, in this case, exponents 𝑚=1 2 and 𝑝=1, leading to saturation of ρ with strain. • In nonlinear transport systems (carrier recombination, vortex dynamics, curvature evolution), the same competition between nonlinear growth and annihilation produces a stable limiting value[27], [53], [54]. Across these very different physical contexts, the same mathematical logic appears: saturation is the emergent balance between a growth channel that increases 𝑋 and a nonlinear mechanism that progressively limits or removes it. 3.11. Implications for Black Hole Physics The saturation mechanisms seen in materials systems provide a useful template for understanding how curvature may behave inside black holes. If spacetime possesses a finite transport capacity, analogous to the finite defect, carrier, or vortex capacities of condensed-matter systems, then several structural consequences naturally follow: • Saturation scaling: Curvature should approach a finite upper value 𝐾max, with an interior profile of the form 𝐾(𝑟)=𝐾max [1−𝑓(𝑟)], where 𝑓(𝑟)→0 as 𝑟→0. Instead of diverging, curvature asymptotically approaches a maximum set by the transport limits of spacetime. • Critical behavior: Near the saturation threshold, curvature can exhibit scaling laws similar to those in nonlinear saturation systems: 𝐾max−𝐾(𝑟)∼𝑟𝛿, where 𝛿 depends on the specific transport geometry and the nonlinear response of 𝐹𝐺. • Internal structure formation: A saturated interior is not featureless. Finite-capacity systems—such as crystals near dislocation saturation or superfluids near vortex saturation—often reorganize into cell-like substructures or tangled configurations. Analogously, the black hole core may contain geometric microstructure dictated by the saturation of curvature. • Energy dissipation and reorganization: Approaching 𝐾maxlikely involves a redistribution of flux or curvature analogous to defect rearrangement in materials. This geometric reorganization would dissipate gravitational energy and stabilize the interior. This perspective transforms singularities from inevitable mathematical consequences of GR to artifacts of an oversimplified constitutive relation—one that assumes linear response continues indefinitely. In reality, as in materials, extreme conditions reveal nonlinear saturation behavior that preserves finiteness while allowing continued evolution. The parallel with plastic deformation, dielectric breakdown, and superfluid critical velocity provides not only conceptual support for CTC's approach but also a rich mathematical framework for developing detailed models of black hole interiors, potentially bridging the gap between classical GR descriptions and anticipated quantum gravity corrections. 3.12. From Singularity to Saturation In traditional GR, singularities arise because the source term is treated as if matter can be compressed indefinitely. The Curvature–Transport Correspondence reframes this: spacetime has finite transport capacity, so curvature cannot grow without bound. Singularities are therefore not physical objects but artifacts of assuming that a linear constitutive relation remains valid under arbitrarily extreme conditions. Under CTC, the interior of a black hole behaves much like a material driven beyond its linear-response regime. Materials do not support infinite stress; they yield[52]. Dielectrics do not sustain infinite electric field; they break down[55]. Superfluids cannot exceed their critical velocities; they nucleate vortices[54]. In each case, nonlinear saturation mechanisms intervene to preserve finiteness. Likewise, spacetime resists infinite curvature. Once the transport capacity is approached, the system shifts into a nonlinear, saturating regime that limits curvature to 𝐾maxwhile still allowing continued evolution of the geometry. The result is a regular, structured interior rather than a pathological singularity. This perspective turns black hole singularities from unavoidable predictions into signs of an incomplete constitutive model. By importing intuition and mathematics from materials science, where saturation, reorganization, and finite response capacities are foundational, we obtain not only a physically grounded resolution mechanism but also a framework for constructing detailed interior models. Such models may naturally connect classical GR with the scales where quantum gravity sets the ultimate bound 𝐷max, providing a bridge rather than a discontinuity between the two regimes. 3.13. Jet Emission as a Test of the Saturated-Core Framework Once the saturated interior forms the transport field collapses inside a finite radius, creating a region where 𝐹𝐺=0and gravitational flux cannot be carried inward. All subsequent mass-energy must reorganize around this saturation boundary rather than penetrate the interior. This structural constraint places a strict limit on how deeply magnetic fields can be compressed and how much curvature amplification the system can support. As a result the mechanism that ordinarily powers relativistic jets enters a fixed-output, saturation-limited regime. In general relativity jet power is expected to rise steeply with black hole mass because magnetic flux can be advected arbitrarily close to the center, allowing curvature and frame dragging to strengthen without internal constraints. This leads to the familiar scaling 𝑃GR∝𝑀2. Under the Curvature–Transport Correspondence this behavior cannot persist once a saturated core has formed. With the interior unable to carry additional transport, the boundary at 𝑅critdetermines the maximum magnetic compression the system can sustain. Jet power therefore becomes limited by the structure of this boundary rather than by the mass of the object. The jet enters a saturation plateau, characterized by 𝑃CTC≈𝑃sat for objects that retain a saturated interior. Here 𝑃CTC denotes the jet power predicted by the transport-based model, 𝑀eff is the effective gravitational mass determined by the divergence of the flux field outside the saturated core, and 𝑀satis the saturation mass corresponding to the largest effective mass the saturated configuration can support before inward transport reactivates. Once a saturated core exists, variations in accretion rate or inflowing mass do not translate into stronger jets: the interior cannot respond, the boundary does not shift, and the jet-launching region operates at a fixed transport capacity. Only when the accumulated effective mass approaches 𝑀satcan the saturated configuration destabilize, allowing renewed inward transport and a departure from the plateau. Until this point the system is mass-blind—jet power reflects the saturated boundary rather than the instantaneous mass inflow. A flattening of jet luminosity is already seen across several astrophysical populations. X-ray binaries, spanning nearly an order of magnitude in mass, exhibit fairly identical jet powers in their low/hard states, deviating strongly from the GR scaling [56], [57]. Low-luminosity active galactic nuclei show the same trend: their jet luminosities vary only weakly with mass, following a shallow dependence far below the GR expectation [58], [59]. Even the Milky Way centre, where GR predicts a powerful jet from its four-million-solar-mass black hole, is anomalously quiet and lies on the same mass-independent locus[60]. Even in tidal disruption events, where a sudden increase in accretion should dramatically amplify jet power, observed jet luminosities remain comparable to those from stellar mass systems. Bloom et al. (2011) report a relativistic jet from a 106–107𝑀⊙ black hole whose power is far smaller than the GR scaling 𝑃∝𝑀2 would predict and instead lies on the same mass independent jet locus seen in X-ray binaries and low luminosity AGN. Tidal disruption events offer a clean opportunity to test jet–mass scaling because the central black hole is normally quiescent before the flare and the jet is newly formed. The transient Swift J1644+57 provides the strongest case. Bloom et al. (2011) [61] first identified the event as a relativistic outburst powered by a sudden accretion episode onto a black hole of mass 106−107 𝑀⊙, and Zauderer et al. (2011) [62] subsequently demonstrated through radio and VLBI observations that the source launched a mildly relativistic, highly collimated jet with a broadband synchrotron spectrum. Remarkably, the jet power inferred from both analyses is comparable to that of the most powerful stellar-mass microquasars rather than scaling as 𝑀2 as general relativity predicts. Even after correcting for beaming, the intrinsic energetics remain near the Eddington luminosity of a 106 𝑀⊙ black hole, placing Swift J1644+57 on the same mass-independent jet locus as X-ray binaries and low-luminosity AGN. GR offers no internal mechanism to suppress the mass dependence of jet power in such systems, but in the CTC framework this behavior arises naturally: once the interior transport field saturates and 𝐹𝐺=0 inside the critical radius, magnetic flux cannot be further compressed and the jet becomes limited by the fixed transport capacity of the saturation boundary rather than by the black hole mass. Thus, Swift J1644+57 serves as a direct observational example of a million-solar-mass system producing a jet at the universal saturation plateau expected from a saturated-core spacetime. These observations altogether supports a universal plateau in jet power across systems that differ in mass by up to eight orders of magnitude. In the CTC framework this behavior arises naturally. Once the interior flux collapses and 𝐹𝐺=0 inside the critical radius, spacetime can no longer compress magnetic fields or amplify curvature. The jet mechanism reaches a finite ceiling, yielding a mass-independent jet power for all systems with 𝑀eff<𝑀sat. General relativity has no internal saturation scale and therefore cannot generate this plateau without substantial fine tuning of accretion or environmental conditions. This yields a clear, falsifiable prediction. If future observations show no evidence of saturation and instead confirm the mass-squared scaling across all environments, the CTC model must be revised. If the observed plateau persists in weakly accreting systems, the transport interpretation gains significant support. Jet emission therefore provides a direct probe of the constitutive behavior of spacetime and an empirical path for testing the CTC with current and next-generation high-resolution data. 4. The Mass Hierarchy as a Rigidity Gap in the Vacuum 4.1. The Traditional Hierarchy Problem The enormous separation between the electroweak and Planck scales represents one of particle physics' most persistent puzzles [63], [64]: 𝑀Pl 𝑣EW ≈1.22×1019 GeV 246 GeV ≈5×1016 (4-1) where 𝑣EW is the Higgs vacuum expectation value. In the Standard Model, quantum corrections to the Higgs mass are quadratically sensitive to the cutoff scale: Δ𝑚𝐻 2∼ΛUV 2 16𝜋2 (4-2) For ΛUV ∼𝑀Pl, this yields corrections ∼1030 larger than the observed 𝑚𝐻≈125 GeV[65]. The required cancellations to ∼1 part in 1028 constitute the "naturalness problem" [66], traditionally motivating supersymmetry, composite Higgs models, or other new physics at the TeV scale [67]. 4.2. CTC Resolution: Vacuum Stiffness Spectrum The Curvature–Transport Correspondence reinterprets this hierarchy not as a fine-tuning problem, but as a geometric stiffness transition in the vacuum's curvature response. In CTC, mass emerges as a curvature-response coefficient: 𝑚eff ∝(𝑑2𝐸 𝑑𝑘2)−1for field excitations (4-3) This relationship reveals that what we measure as "particle mass" quantifies the vacuum's resistance to field curvature. Different energy scales correspond to different vacuum rigidity regimes. 4.3 Vacuum Rigidity Spectrum The CTC framework proposes a continuous stiffness spectrum for the vacuum: A. Planck Regime (𝐸≳𝑀Pl): Ultra-Rigid Phase 𝜅𝑃∼𝑀Pl 4 (Planckian bending modulus) (4-4) • Characteristics: Near-infinite resistance to curvature, minimal field fluctuations • Mechanism: Quantum gravitational effects enforce maximal stiffness due to its massive intrinsic energy density, which would be extremely "rigid" gravitationally [68] • Consequence: Particle-like excitations cannot form; spacetime behaves as an effectively rigid medium B. Intermediate Regime (𝑣EW ≲𝐸≪𝑀Pl): Rigidity Gap 𝜅gap ≫𝜅Higgs (4-5) • Characteristics: Strong resistance persists but begins to soften • Observation: No known particles in this range (10⁴–10¹⁸ GeV) [69] • Interpretation: Insufficient vacuum compliance to support resonant excitations C. Higgs Regime (𝐸∼𝑣EW): Compliant Phase 𝜅𝐻∼𝑣EW 4 (4-6) • Characteristics: Vacuum becomes sufficiently flexible to support curvature • Mechanism: Electroweak symmetry breaking reduces vacuum stiffness [70] • Consequence: Higgs boson and massive gauge bosons emerge as curvature resonances D. QCD Regime (𝐸≲ΛQCD ≈200𝑀𝑒𝑉): Soft Phase 𝜅QCD ≪𝜅𝐻 (4-7) • Characteristics: Highly compliant vacuum enables strong curvature • Manifestation: Confinement, chiral symmetry breaking (CSB), hadron masses [71] • Result: 99% of visible mass emerges from QCD curvature response 4.4. Stiffness Scale Relation Within the Curvature Transport Correspondence, the vacuum stiffness 𝜅(𝐸) measures the resistance of spacetime to curvature sourced by the effective density 𝜌eff=∇⋅𝐹𝐺. Because curvature in this framework is generated by transport rather than static mass density, 𝜅(𝐸) must vary across energy scales. Scale dependent response coefficients of this type are well known in condensed matter and materials science [20], [21], [72], [73], where moduli, conductivities, and effective masses evolve according to the reorganization of microscopic transport channels. A closely related structure appears in the scaling theory of localization developed by Abrahams, Anderson, Licciardello, and Ramakrishnan [74], where the conductivity 𝜎 satisfies a similar inverse susceptibility flow equation. These analogies are summarized in Supplementary Note S2. Motivated by this correspondence, the running of vacuum stiffness is expressed in transport form: 𝑑𝜅(𝐸) 𝑑𝐸 =− 𝜅(𝐸) 𝛾(𝐸), (4-8) where 𝛾(𝐸) is a stiffness susceptibility capturing the ease with which the geometric transport network redistributes flux. Separating variables, 𝑑𝜅 𝜅=− 𝑑𝐸 𝛾(𝐸). (4-9) Integrating from the electroweak scale 𝑣𝐸𝑊 to a general energy 𝐸, ln [𝜅(𝐸) 𝜅(𝑣𝐸𝑊)]=−∫ 𝑑𝐸′ 𝛾(𝐸′). 𝐸 𝑣𝐸𝑊 (4-10) The stiffness–scale relation can then be viewed as: 𝜅(𝐸)=𝜅(𝑣𝐸𝑊) exp [−∫ 𝑑𝐸′ 𝛾(𝐸′) 𝐸 𝑣𝐸𝑊 ]. (4-11) The key feature is the behavior of 𝛾(𝐸) near the Planck regime. As 𝐸 approaches 𝑀𝑃𝑙, the density of geometric microstates increases extremely rapidly, a fact well established in semiclassical quantum gravity and in the statistical interpretation of black hole entropy [75], [76]. In this limit the inverse susceptibility behaves proportionally to the derivative of the Bekenstein-Hawking entropy with respect to energy, 1 𝛾(𝐸)∼𝑑𝑆𝐵𝐻 𝑑𝐸 ,𝐸→𝑀𝑃𝑙, (4-12) since each increment in energy accesses an exponentially large number of horizon microstates. Accordingly, ∫𝑑𝐸′ 𝛾(𝐸′) 𝑀𝑃𝑙 𝑣𝐸𝑊 ≈𝑆𝐵𝐻,(4-13) capturing the cumulative entropy cost of reorganizing geometric transport between the electroweak and Planck scales. Substituting this result into (4.14), one obtains 𝜅(𝑀𝑃𝑙)=𝜅(𝑣𝐸𝑊)𝑒𝑥𝑝 [𝑆𝐵𝐻]. (4-14) The exponential hierarchy between electroweak and Planck stiffness thus arises naturally within the CTC framework. No fine tuning is required. The hierarchy reflects the intrinsic transport structure of spacetime and the vast entropy associated with Planck scale microstates. The familiar problem of stabilizing the electroweak–Planck hierarchy becomes an emergent consequence of transport driven curvature dynamics rather than an anomaly of particle masses. 4.5. Connection to Effective Field Theory A natural way to express the influence of the Curvature–Transport Correspondence on matter fields is through the structure of the effective Lagrangian. For a generic scalar sector, the low-energy action can be written schematically as ℒeff=ℒkin+1 𝜅(𝐸)(∂𝜙)2+𝑉(𝜙). (4-15) This form is entirely standard in effective field theory: the coefficient multiplying the kinetic term plays the role of a scale-dependent wavefunction normalization factor, conventionally denoted 𝑍(𝐸). In the CTC framework we identify 𝑍(𝐸)= 1 𝜅(𝐸).(4-16) The mathematics is conventional; the unique perspective in this case lies in the physical origin of 𝑍(𝐸). In ordinary EFT, the running of the kinetic coefficient is induced by quantum fluctuations of the fields. In the CTC picture, the running instead reflects the transport properties of spacetime, encoded in the vacuum stiffness 𝜅(𝐸). The evolution of this stiffness is governed by the flow equation 𝑑𝜅(𝐸) 𝑑𝐸 =−𝜅(𝐸) 𝛾(𝐸),(4-17) where 𝛾(𝐸)is the transport susceptibility. This differential equation is separable: 𝑑𝜅 𝜅=− 𝑑𝐸 𝛾(𝐸).(4-18) Integrating between two scales 𝐸0and 𝐸: ln [𝜅(𝐸) 𝜅(𝐸0)]=−∫𝑑𝐸′ 𝛾(𝐸′), 𝐸 𝐸0 (4-19) and exponentiating gives the explicit stiffness flow 𝜅(𝐸)=𝜅(𝐸0)exp [−∫𝑑𝐸′ 𝛾(𝐸′) 𝐸 𝐸0]. (4-20) This shows that 𝜅(𝐸)changes exponentially with energy scale whenever the susceptibility 𝛾(𝐸)becomes small, particularly near the Planck regime. The running of the kinetic term is therefore 𝑍(𝐸)= 1 𝜅(𝐸0)exp [∫𝑑𝐸′ 𝛾(𝐸′) 𝐸 𝐸0]. (4-21) Astronomical Society, vol. 499, no. 2, pp. 2845–2883, Oct. 2020, doi: 10.1093/mnras/staa2348. [16] J. R. Gott III et al., “A Map of the Universe,” The Astrophysical Journal, vol. 624, no. 2, p. 463, May 2005, doi: 10.1086/428890. [17] I. Horváth, Z. Bagoly, J. Hakkila, and L. V. Tóth, “New data support the existence of the Hercules-Corona Borealis Great Wall⋆,” A&A, vol. 584, Dec. 2015, doi: 10.1051/00046361/201424829. [18] E. Di Valentino et al., “In the realm of the Hubble tension—a review of solutions,” Classical and Quantum Gravity, vol. 38, no. 15, p. 153001, July 2021, doi: 10.1088/13616382/ac086d. [19] C. Krishnan, R. Mohayaee, E. Ó. Colgáin, M. M. Sheikh-Jabbari, and L. Yin, “Hints of FLRW breakdown from supernovae,” Phys. Rev. D, vol. 105, no. 6, p. 063514, Mar. 2022, doi: 10.1103/PhysRevD.105.063514. [20] L. D. Landau and E. M. Lifshitz, Theory of Elasticity, vol. 7. in Course of Theoretical Physics, vol. 7. New York: Elsevier Butterworth-Heinemann, 1986. doi: 10.1016/C2009-0-25521-8. [21] C. Kittel and P. McEuen, Introduction to Solid State Physics. Wiley, 2018. [Online]. Available: https://books.google.com/books?id=nNpVEAAAQBAJ [22] J. Crank, The Mathematics of Diffusion. in Oxford science publications. Clarendon Press, 1979. [Online]. Available: https://books.google.com/books?id=eHANhZwVouYC [23] S. Sachdev, Quantum Phase Transitions. Cambridge University Press, 2011. doi: 10.1017/cbo9780511973765. [24] Planck Collaboration et al., “Planck 2018 results,” A&A, vol. 641, 2020, doi: 10.1051/00046361/201833910. [25] C. Franco and others, “The homogeneity scale in the Local Universe: model-independent estimate from S-PLUS DR4 blue galaxies,” Nov. 2025. [26] P. Sarkar, J. Yadav, B. Pandey, and S. Bharadwaj, “The scale of homogeneity of the galaxy distribution in SDSS DR6,” Monthly Notices of the Royal Astronomical Society: Letters, vol. 399, no. 1, pp. L128–L131, Oct. 2009, doi: 10.1111/j.1745-3933.2009.00738.x. [27] M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys., vol. 65, no. 3, pp. 851–1112, July 1993, doi: 10.1103/RevModPhys.65.851. [28] Y.-C. Chen et al., “Detecting galaxy–filament alignments in the Sloan Digital Sky Survey III,” Monthly Notices of the Royal Astronomical Society, vol. 485, no. 2, pp. 2492–2504, May 2019, doi: 10.1093/mnras/stz539. [29] S. Tavasoli and P. Ghafour, “The Filament Rift: ΛCDM’s Structural Challenge against Observation,” The Astrophysical Journal, vol. 994, no. 2, p. 219, Nov. 2025, doi: 10.3847/1538-4357/ae18d6. [30] DES Collaboration et al., “Dark energy survey year 3 results: Cosmological constraints from cluster abundances, weak lensing, and galaxy clustering,” Phys. Rev. D, vol. 112, no. 8, p. 083535, Oct. 2025, doi: 10.1103/3dzh-d8f5. [31] A. Rollett, F. J. Humphreys, G. S. Rohrer, and M. Hatherly, Recrystallization and Related Annealing Phenomena. Pergamon, 2004. [Online]. Available: https://books.google.com/books?id=52Gloa7HxGsC [32] J. W. Cahn, “On spinodal decomposition,” Acta Metallurgica, vol. 9, no. 9, pp. 795–801, Sept. 1961, doi: 10.1016/0001-6160(61)90182-1. [33] E. Tarleton, “Dislocations, Mesoscale Simulations and Plastic Flow, Oxford Series on Materials Modelling 5, by Ladislas Kubin,” Contemporary Physics, vol. 54, no. 6, pp. 302– 303, Nov. 2013, doi: 10.1080/00107514.2013.856946. [34] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time. in Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2023. doi: 10.1017/9781009253161. [35] R. Penrose, “Gravitational Collapse and Space-Time Singularities,” Phys. Rev. Lett., vol. 14, no. 3, pp. 57–59, Jan. 1965, doi: 10.1103/PhysRevLett.14.57. [36] S. M. Sze and K. K. Ng, Physics of Semiconductor Devices. Wiley, 2006. [Online]. Available: https://books.google.com/books?id=o4unkmHBHb8C [37] D. Hull and D. J. Bacon, Introduction to Dislocations. Butterworth-Heinemann, 2001. [Online]. Available: https://books.google.com/books?id=EHjrGd-4TLcC [38] C. W. Misner, K. S. Thorne, J. A. Wheeler, and D. I. Kaiser, Gravitation. Princeton University Press, 2017. [Online]. Available: https://books.google.com/books?id=SyQzDwAAQBAJ [39] M. Maggiore, Gravitational Waves: Volume 1: Theory and Experiments. 2007. doi: 10.1093/acprof:oso/9780198570745.001.0001. [40] F. G. Friedlander, The Wave Equation on a Curved Space-Time. Cambridge University Press, 1975. [Online]. Available: https://books.google.com/books?id=RDmpajLTw1oC [41] C. Bär, N. Ginoux, and F. Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization. in ESI lectures in mathematics and physics. European Mathematical Society, 2007. [Online]. Available: https://books.google.com/books?id=nzi-c0dP1NYC [42] H. Ringström, The Cauchy Problem in General Relativity. in ESI lectures in mathematics and physics. European Mathematical Society, 2009. [Online]. Available: https://books.google.com/books?id=dFPuLLvzWYwC [43] J. Bardeen, “Nonsingular general relativistic gravitational collapse”. [44] I. Dymnikova, “Vacuum nonsingular black hole,” General Relativity and Gravitation, vol. 24, no. 3, pp. 235–242, Mar. 1992, doi: 10.1007/BF00760226. [45] S. A. Hayward, “Formation and Evaporation of Nonsingular Black Holes,” Phys. Rev. Lett., vol. 96, no. 3, p. 031103, Jan. 2006, doi: 10.1103/PhysRevLett.96.031103. [46] V. P. Frolov, “Notes on nonsingular models of black holes,” Phys. Rev. D, vol. 94, no. 10, p. 104056, Nov. 2016, doi: 10.1103/PhysRevD.94.104056. [47] M. A. Markov, “Limiting density of matter as a universal law of nature,” ZhETF Pisma Redaktsiiu, vol. 36, pp. 214–216, Sept. 1982. [48] V. P. Frolov, M. A. Markov, and V. F. Mukhanov, “Black holes as possible sources of closed and semiclosed worlds,” Phys. Rev. D, vol. 41, no. 2, pp. 383–394, Jan. 1990, doi: 10.1103/PhysRevD.41.383. [49] K. Skenderis and M. Taylor, “The fuzzball proposal for black holes,” Physics Reports, vol. 467, no. 4, pp. 117–171, Oct. 2008, doi: 10.1016/j.physrep.2008.08.001. [50] D. R. Tilley and J. Tilley, Superfluidity and Superconductivity. in Graduate Student Series in Physics. Taylor & Francis, 1990. [Online]. Available: https://books.google.com/books?id=I6JtWd3J8MIC [51] F. J. RICHARDS, “A Flexible Growth Function for Empirical Use,” Journal of Experimental Botany, vol. 10, no. 2, pp. 290–301, June 1959, doi: 10.1093/jxb/10.2.290. [52] U. F. Kocks and H. Mecking, “Physics and phenomenology of strain hardening: the FCC case,” Progress in Materials Science, vol. 48, no. 3, pp. 171–273, Jan. 2003, doi: 10.1016/S0079-6425(02)00003-8. [53] W. Shockley and W. T. Read, “Statistics of the Recombinations of Holes and Electrons,” Phys. Rev., vol. 87, no. 5, pp. 835–842, Sept. 1952, doi: 10.1103/PhysRev.87.835. [54] W. Fiszdon, “Quantized Vortices in Helium II. By R. J. DONNELLY. Cambridge University Press, 1991. 346 pp. £95.,” Journal of Fluid Mechanics, vol. 233, pp. 691–692, 1991, doi: 10.1017/S0022112091220650. [55] L. A. Dissado and J. C. Fothergill, Electrical Degradation and Breakdown in Polymers. in IEE materials & devices series. P. Peregrinus, 1992. [Online]. Available: https://books.google.com/books?id=8Tm7dH99-XEC [56] S. Corbel, M. A. Nowak, R. P. Fender, A. K. Tzioumis, and S. Markoff, “Radio/X-ray correlation in the low/hard state of GX 339–4,” A&A, vol. 400, no. 3, pp. 1007–1012, 2003, doi: 10.1051/0004-6361:20030090. [57] S. Corbel et al., “The ‘universal’ radio/X-ray flux correlation: the case study of the black hole GX 339−4,” Monthly Notices of the Royal Astronomical Society, vol. 428, no. 3, pp. 2500–2515, Jan. 2013, doi: 10.1093/mnras/sts215. [58] A. Merloni, S. Heinz, and T. Di Matteo, “A Fundamental Plane of black hole activity,” Mon Not R Astron Soc, vol. 345, no. 4, pp. 1057–1076, 2003, doi: 10.1046/j.13652966.2003.07017.x. [59] R. M. Plotkin, S. F. Anderson, W. N. Brandt, S. Markoff, O. Shemmer, and J. Wu, “THE LACK OF TORUS EMISSION FROM BL LACERTAE OBJECTS: AN INFRARED VIEW OF UNIFICATION WITH WISE,” The Astrophysical Journal Letters, vol. 745, no. 2, p. L27, Jan. 2012, doi: 10.1088/2041-8205/745/2/L27. [60] F. Yuan and R. Narayan, “Hot Accretion Flows Around Black Holes,” Annual Review of Astronomy and Astrophysics, vol. 52, no. Volume 52, 2014. Annual Reviews, pp. 529–588, 2014. doi: https://doi.org/10.1146/annurev-astro-082812-141003. [61] J. S. Bloom et al., “A Possible Relativistic Jetted Outburst from a Massive Black Hole Fed by a Tidally Disrupted Star,” Science, vol. 333, no. 6039, pp. 203–206, July 2011, doi: 10.1126/science.1207150. [62] B. A. Zauderer et al., “Birth of a relativistic outflow in the unusual γ-ray transient Swift J164449.3+573451,” Nature, vol. 476, no. 7361, pp. 425–428, Aug. 2011, doi: 10.1038/nature10366. [63] F. Wilczek, “QCD and Natural Philosophy,” Annales Henri Poincaré, vol. 4, pp. 211–228, Dec. 2003, doi: 10.1007/s00023-003-0917-y. [64] G. F. Giudice, “Naturally Speaking: The Naturalness Criterion and Physics at the LHC,” pp. 155–178, Jan. 2008, doi: 10.1142/9789812779762_0010. [65] G. Aad et al., “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,” Physics Letters B, vol. 716, no. 1, pp. 1–29, Sept. 2012, doi: 10.1016/j.physletb.2012.08.020. [66] L. Susskind, “Dynamics of spontaneous symmetry breaking in the Weinberg-Salam theory,” Phys. Rev. D, vol. 20, no. 10, pp. 2619–2625, Nov. 1979, doi: 10.1103/PhysRevD.20.2619. [67] S. P. Martin, “A Supersymmetry primer,” Adv. Ser. Direct. High Energy Phys., vol. 18, pp. 1– 98, 1998, doi: 10.1142/9789812839657_0001. [68] S. Weinberg, “What is Quantum Field Theory, and What Did We Think It Is?,” arXiv e-prints, p. hep-th/9702027, Feb. 1997, doi: 10.48550/arXiv.hep-th/9702027. [69] C. Patrignani, “Review of Particle Physics,” Chinese Physics C, vol. 40, no. 10, p. 100001, Oct. 2016, doi: 10.1088/1674-1137/40/10/100001. [70] F. Englert and R. Brout, “Broken Symmetry and the Mass of Gauge Vector Mesons,” Phys. Rev. Lett., vol. 13, no. 9, pp. 321–323, Aug. 1964, doi: 10.1103/PhysRevLett.13.321. [71] F. Wilczek, “Asymptotic freedom: From paradox to paradigm,” Proc. Nat. Acad. Sci., vol. 102, pp. 8403–8413, 2005, doi: 10.1103/RevModPhys.77.857. [72] N. W. Ashcroft and N. D. Mermin, Solid State Physics. in HRW international editions. Holt, Rinehart and Winston, 1976. [Online]. Available: https://books.google.com/books?id=1C9HAQAAIAAJ [73] N. Goldenfeld, Lectures on phase transitions and the renormalization group. 1992. [74] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions,” Phys. Rev. Lett., vol. 42, no. 10, pp. 673–676, Mar. 1979, doi: 10.1103/PhysRevLett.42.673. [75] J. D. Bekenstein, “Black Holes and Entropy,” Phys. Rev. D, vol. 7, no. 8, pp. 2333–2346, Apr. 1973, doi: 10.1103/PhysRevD.7.2333. [76] S. W. Hawking, “Particle creation by black holes,” Communications in Mathematical Physics, vol. 43, no. 3, pp. 199–220, Aug. 1975, doi: 10.1007/BF02345020. [77] A. J. Liu and S. R. Nagel, “The Jamming Transition and the Marginally Jammed Solid,” Annual Review of Condensed Matter Physics, vol. 1, no. Volume 1, 2010. Annual Reviews, pp. 347–369, 2010. doi: https://doi.org/10.1146/annurev-conmatphys-070909-104045. [78] C. A. Angell, “Formation of Glasses from Liquids and Biopolymers,” Science, vol. 267, no. 5206, pp. 1924–1935, Mar. 1995, doi: 10.1126/science.267.5206.1924. [79] R. B. Bird, Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics. in Dynamics of Polymeric Liquids. Wiley, 1987. [Online]. Available: https://books.google.com/books?id=posvAQAAIAAJ [80] J. D. Ferry, Viscoelastic Properties of Polymers. Wiley, 1980. [Online]. Available: https://books.google.com/books?id=9dqQY3Ujsx4C [81] M. Rubinstein and R. H. Colby, Polymer Physics. OUP Oxford, 2003. [Online]. Available: https://books.google.com/books?id=RHksknEQYsYC [82] P. G. de Gennes, Scaling Concepts in Polymer Physics. Cornell University Press, 1979. [Online]. Available: https://books.google.com/books?id=Gh1TcAAACAAJ [83] M. Doi and S. F. Edwards, The theory of polymer dynamics. in International series of monographs on physics. Oxford: Oxford Univ. Press, 1986. [Online]. Available: https://cds.cern.ch/record/346518 [84] P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics. Cambridge University Press, 2000. [Online]. Available: https://books.google.com/books?id=P9YjNjzr9OIC SUPPLEMENT S1. Numerical Demonstration of Curvature Saturation in the CTC Framework This supplement documents the numerical procedures used to generate the four illustrative figures accompanying Section 3 of the main text. All simulations were performed using the Jupyter notebook ctc_black_hole_saturation.ipynb available in https://github.com/sakidja/CTC_related_papers/, which provides a transparent, reproducible implementation of the Curvature–Transport Correspondence (CTC) in a simplified radial model. Unless otherwise stated, all simulations use a reference mass 𝑀ref=10 𝑀⊙. (S1-1) S1.1. Classical Curvature Baseline The notebook computes the Schwarzschild Kretschmann invariant, 𝐾GR(𝑟)=48𝐺2𝑀2 𝑐4𝑟6, (S1-2) as the unregularized curvature profile predicted by general relativity. This divergent expression forms the baseline against which the CTC-saturated curvature profile is compared. S1.2. Saturation Condition and Core Radius CTC replaces the classical divergence by imposing a finite maximum curvature 𝐾max. The saturated core radius is computed from 𝐾GR(𝑟𝑠)=𝐾max,𝑟𝑠(𝑀)=(48𝐺2𝑀2 𝑐4𝐾max)1/6. (S1-3) This relation determines the location of the CTC interior phase. S1.3. Figure S1 — Curvature Profile at the Reference Mass Generated using: plot_curvature_profiles() with default input 𝑀=𝑀ref. Figure S1 shows: • the classical Schwarzschild curvature (solid blue), • the CTC-saturated curvature (dashed orange), • the saturated core radius 𝑟𝑠(𝑀ref)(dotted line). This figure establishes the finite-curvature interior predicted by the CTC model. S1.4. Figure S2 — Mass Scaling of the Saturated Core Generated using: scan_core_radius() over a mass range 1–100 𝑀⊙. Figure S2 shows: • the monotonic expansion of the saturated core radius with mass, • consistent scaling 𝑟𝑠(𝑀)∝𝑀1/3, • confirmation that curvature remains fixed while the interior region grows. Although the figure spans many masses, the reference mass 𝑀ref remains the same mass chosen for the main text. S1.5. Figure S3 — Hypothetical Curvature Profile at an Alternative Mass Generated either by: • calling plot_curvature_profiles manually with a different value of 𝑀, or using the interactive slider provided in Cell 7 of the notebook. • This figure is not tied to the reference mass. It illustrates how the curvature plateau and core radius shift for other hypothetical black hole masses. The function of this figure is pedagogical: • it shows the generality of the CTC mechanism, • it confirms that the shape of the curvature profile is robust under changes in 𝑀, • it links analytic predictions to numerical behavior. Because the mass is user-selected, the figure is best described as a hypothetical example, not a reference calculation. S1.6. Figure S4 — Flux Divergence Saturation at the Reference Mass Generated using: plot_divergence_profile() with default 𝑀=𝑀ref. This figure displays a toy model of the divergence field ∇⋅𝐹𝐺(𝑟)with: • a classical-like exterior profile, • a sharp downturn near 𝑟𝑠(𝑀ref), • saturation at −𝐷maxinside the core. The location of the transition reflects the reference mass; the qualitative shape illustrates the essential ingredient of CTC: collapse is limited by transport capacity, not by unbounded compression. SUPPLEMENT S2 Correspondence with Materials Science and Transport Theory A central ingredient of the Curvature Transport Correspondence is the scale dependence of the vacuum stiffness 𝜅(𝐸), governed by 𝑑𝜅(𝐸) 𝑑𝐸 =− 𝜅(𝐸) 𝛾(𝐸).(S2-1) This differential structure is not unique to gravitational transport. Equations of precisely this form arise throughout materials science, condensed matter physics, and statistical transport theory whenever a response coefficient varies because underlying microstates reorganize across scale, frequency, or energy. The generic form is 𝑑𝐶 𝑑𝜆=− 𝐶 Γ(𝜆),(S2-2) where 𝐶 is a modulus or transport coefficient, 𝜆 is a control parameter, and Γ(𝜆) is a susceptibility that encodes how easily the microscopic structure redistributes flux, stress, or momentum. The mapping 𝐶→𝜅,𝜆→𝐸,Γ→𝛾 shows that (S3-1) is mathematically identical to the standard renormalization law (S2-2). Below we summarize representative examples. S2.1. Elastic moduli under microstructural renormalization Landau and Lifshitz demonstrate that elastic constants in heterogeneous solids acquire scale dependence through coarse graining (Landau and Lifshitz, Theory of Elasticity)[20]. The bulk modulus satisfies 𝑑𝐾 𝑑𝑙 =− 𝐾 Γ(𝑙),(S2-3) where 𝑙 is a coarse graining scale and Γ(𝑙)reflects microstructural susceptibility. Equation (S3) has the same structure as the CTC stiffness flow (S1). S2.2. Scale dependent conductivity in disordered systems (Abrahams–Anderson–Licciardello– Ramakrishnan) A structurally identical renormalization equation governs the conductivity in disordered conductors. In their seminal work, Abrahams, Anderson, Licciardello, and Ramakrishnan[74] established the scaling theory of localization and showed that the conductivity 𝜎 satisfies 𝑑𝜎 𝑑ln 𝐿=− 𝜎 𝛽(𝜎),(S2-4) where 𝐿 is the running length scale and 𝛽(𝜎) acts as an inverse susceptibility that determines how the system responds to coarse-graining. Rewriting (S4): 𝑑𝜎 𝑑𝐿=− 𝜎 𝐿 𝛽(𝜎),(S2-5) reveals that 𝜎(𝐿) obeys the same inverse-susceptibility evolution as (S3-2), and hence as the stiffness flow (S3-1). This is a direct analogue of the CTC relation, with 𝐶=𝜎, 𝜆=𝐿, Γ=𝐿𝛽(𝜎). S2.3. Effective mass renormalization in crystalline solids In many-body treatments, the electron effective mass satisfies 𝑑𝑚∗ 𝑑𝜔 =− 𝑚∗ Σ′(𝜔),(S2-6) where Σ′(𝜔) is the derivative of the electronic self-energy (Ashcroft and Mermin, Solid State Physics)[72]. This is another direct realization of the structure (S2): the response coefficient 𝑚∗ runs with energy because the underlying microstates reorganize. S2.4. Viscoelastic Relaxation and Frequency-Dependent Stiffness In viscoelastic materials the shear modulus 𝐺(𝜔) depends on the driving frequency 𝜔, and its evolution is controlled by a relaxation spectrum. Classical rheology [79], [80], [81] shows that the modulus satisfies differential relations of the form 𝑑𝐺 𝑑𝜔=− 𝐺 𝜔 𝜏(𝜔),(S2-7) where 𝜏(𝜔) is a relaxation susceptibility encoding how rapidly the material can reorganize microstructurally. Equation (S7) has the same structural form as the CTC flow equation (S1): a response coefficient evolves according to its inverse susceptibility. This reinforces that the CTC relation is a familiar renormalization structure found in well-established viscoelastic theory. S2.5. Polymer Networks and Soft-Matter Systems Scaling theories of polymer networks and soft matter [82], [83], [84] that the effective stiffness 𝐾(𝜆) changes under coarse-graining or deformation by a renormalization equation of the form 𝑑𝐾 𝑑ln 𝜆=− 𝐾 𝑆(𝜆),(S2-8) where 𝑆(𝜆)is the structural susceptibility of the network. Here again, the logarithmic derivative of the modulus is controlled by the inverse susceptibility—exactly matching the CTC structure in (S1). Polymer networks therefore provide another concrete example where stiffness evolves through a transportmediated reorganization of microstructure, directly analogous to the running vacuum stiffness 𝜅(𝐸)in the CTC framework. S.2.6. Summary Equation (S1), central to the Curvature Transport Correspondence, is not a new or exotic structure. It is the gravitational analogue of the universal renormalization law (S2) appearing throughout condensed matter and materials science: Whenever a response coefficient depends on microstate organization, its logarithmic derivative is controlled by an inverse susceptibility. Thus the scale evolution of vacuum stiffness in the CTC follows a well-established physical mechanism: transport mediated response coefficients necessarily run with scale.