The Big Start: Cosmogenesis from a Finite Planck-Phase Boundary

The Big Start: Cosmogenesis from a Finite Planck-Phase Boundary Ridwan Sakidja Department of Physics, Astronomy and Materials Science Missouri State University Abstract The standard Big Bang picture assumes that spacetime already exists at the moment of origin and is driven to a singular state of infinite curvature and infinite density. The Curvature Transport Correspondence (CTC) offers a different beginning in which the vacuum is a physical medium with stiffness that controls whether curvature or fields can exist at all. This leads to the Big Start, a finite radius Planck phase vacuum state where gravitational transport is saturated and no geometric degrees of freedom are present. This state is a pre geometric and zero entropy vacuum, directly realizing the insight of Sir Roger Penrose that the Universe must begin in an exceptionally ordered condition. As expansion relaxes vacuum stiffness, the layers of physics appear in sequence: curvature mobility at rG, transverse and quantum modes at rT, Higgs condensation and the appearance of mass at rH, and finally a classical FRW spacetime at rS. Cosmogenesis and gravitational collapse follow the same divergence law in opposite directions, and the Big Start replaces the classical singularity with a natural vacuum phase transition in which spacetime, gravity, quantum behavior, and mass appear only when the vacuum becomes soft enough to support them 1. Introduction Classical general relativity allows the Universe to begin from zero volume and infinite curvature, but such a continuation contradicts the physical limits of the vacuum. In a transport-based formulation of curvature, introduced in the first paper on CTC [1], the curvature of any field 𝑋is generated by the divergence of an associated transport flux 𝐹𝑋𝜇: 𝐶[𝑋]=∇𝜇𝐹𝑋𝜇.(1) When applied to gravity, this general relation yields the effective curvature source 𝜌eff=∇𝜇𝐹𝐺𝜇,(2) identifying gravitational curvature as the vacuum’s response to transport imbalance rather than the product of intrinsic material density. In the second paper of CTC[2], we showed that the vacuum possesses a finite, phase-dependent stiffness that limits how much curvature it can support. This leads to the saturation principle ∣∇𝜇𝐹𝐺𝜇∣≤𝐷Pl,(3) which imposes a maximum admissible flux divergence. This bound prevents curvature from diverging and therefore forbids singularities. Instead of collapsing to a point, the vacuum reaches a saturation radius, a finite region in which curvature attains its upper limit and cannot increase further. The same saturation mechanism also produces dark-matter–like phenomena, resolves the hierarchy problem, and eliminates singularities inside black holes. In this third paper of CTC, we apply this framework to the origin of the Universe. We demonstrate that the Big Bang singularity is replaced by a finite-radius Planck-phase bubble, a saturated region in which curvature reaches its allowable maximum. Cosmogenesis therefore begins not from a point of infinite curvature, but from the boundary of this finite Planck-phase domain. We will also examine the ontology of black-hole interiors and show how the same saturation principle links the structure of a black-hole core to the origin of the Universe. For clarity, we note that the CTC cosmogenesis mechanism differs fundamentally from the standard Big Bang with inflation [3], [4], [5], [6], [7] as well as from other nonsingular proposals[8], [9], [10], [11], [12], [13], [14] [15], [16]. Among the many ideas proposed to address the origin of the Universe, two frameworks are especially relevant for comparison because they touch on similar questions of initial simplicity and the replacement of the singularity. The first is Penrose’s Conformal Cyclic Cosmology[17], [18], which seeks to achieve a low entropy beginning through conformal identification between successive aeons. CTC shares the motivation for an initially simple state but obtains it through a physical mechanism: the early vacuum is a perfectly rigid phase with no degrees of freedom (DOF), and cosmogenesis begins when its stiffness relaxes, with no role for conformal matching. The second is the class of finite radius black hole core cosmogenesis models[19], [20], in which the singularity is replaced by a finite domain that births a new cosmological region. Although both approaches replace the singularity with a finite radius structure, the underlying physics is entirely different. In CTC, the finite radius arises from saturation of curvature transport in a degree of freedom free vacuum, not from black hole interior dynamics or quantum gravitational effects. 2. The Divergence Law and Vacuum Saturation CTC reinterprets gravitational curvature not as a direct response to “mass,” but as the vacuum’s transport reaction to flux loading. In this formulation, curvature is present only to the extent that the effective density 𝜌eff=∇𝜇𝐹𝐺𝜇 is nonzero. However, the vacuum cannot transmit arbitrarily large curvature. Just as real materials cannot sustain unlimited stress or strain, the vacuum possesses a finite curvature-carrying capacity characterized by a universal bound: ∣∇𝜇𝐹𝐺𝜇∣≤𝐷Pl. When the divergence of the gravitational flux approaches this limit, the vacuum enters a saturated regime in which its normal dynamical degrees of freedom (DOF) cannot operate. In this Planck phase: • curvature cannot increase beyond the saturated value, • gravitational propagation shuts down, • transverse gauge modes cannot exist, • the Higgs condensate cannot form, • mass and gauge interactions lose meaning, and • quantum fluctuations are suppressed, since oscillatory modes require finite vacuum stiffness. From this perspective, there are potentially two extreme gravitational situations which drive the vacuum into this saturated state: 1. Black-hole collapse, where the inward flux loading yields ∇𝜇𝐹𝐺𝜇→−𝐷Pl, 2. Cosmogenesis, where the primordial vacuum begins fully outward-loaded at ∇𝜇𝐹𝐺𝜇=+𝐷Pl. The same divergence law governs both phenomena, differing only in the direction of flux loading. Let us examine these two cases in the following section. 3. Saturation in Black Holes and Cosmogenesis Although black holes and the birth of the Universe appear to represent opposite gravitational extremes, CTC reveals that they are controlled by a single underlying mechanism: vacuum saturation under flux divergence, with opposite signs of loading determining the physical outcome. Black holes reach saturation through inward compression; cosmogenesis begins from a fully outwardloaded saturated state. 3.1. Back Holes: Inward Loading to Saturation During gravitational collapse, the inward gravitational transport flux intensifies and the divergence ∇𝜇𝐹𝐺𝜇becomes increasingly negative. In the CTC framework, curvature cannot grow without bound. As shown in the second CTC paper[2], the transport field reaches a universal saturation limit 𝐾max, defined by 𝐾(𝑟𝑠)=𝐾max, with the interior curvature profile 𝐾(𝑟)=48𝐺2 𝑚(𝑟)2 𝑐4𝑟6. Solving (3–23) yields 𝑟𝑠3∼√48𝐺2𝑚(𝑟𝑠)2 𝑐4𝐾max ⟹𝑟𝑠∝𝑀1/3. This is the general result from the second CTC paper[2]: once the saturation limit is reached, the curvature no longer increases and additional mass enlarges the volume of the saturated region rather than deepening the collapse. Planck Phase Interpretation In the present work, we further identify the saturation value 𝐾max with the Planck curvature scale, 𝐾max=𝐾Pl∼1 ℓPl 4, so that the saturated region corresponds to a Planck-phase vacuum. Because 𝐾Pl is extraordinarily large, the radius 𝑟𝑠 obtained by substituting 𝐾Pl into the 𝑟𝑠3 formula is extremely small. For stellar-mass black holes this yields radii on the order of 𝑟𝑠∼10−23–10−25 m, many orders of magnitude smaller than both the Schwarzschild radius and any physical length scale relevant to astrophysical collapse. Thus, even though the scaling 𝑟𝑠∝𝑀1/3 remains valid, the absolute size of the saturated region is microscopic whenever 𝐾max is identified with the Planck scale. Physical Implications Because the corresponding radius is so small, ordinary stellar or supermassive black holes do not come close to reaching the saturation boundary. Their interior curvature remains many tens of orders of magnitude below 𝐾Pl. Therefore, a Planck-phase core forms only if collapse drives the transport divergence so far inward that it reaches the theoretical limit 𝐾𝑃𝑙. In realistic astrophysical environments this never occurs. That said, if saturation were actually reached, the interior would enter a rigidity-dominated Planck phase in which: • curvature is fixed at 𝐾Pl, • gravitational transport ceases, • the Higgs mechanism and the concept of mass are undefined, • gauge fields cannot propagate, and • quantum oscillatory modes are suppressed. General relativity remains valid only outside this microscopic, saturated region meaning that in realistic astrophysical collapse, the transport field never approaches this saturation limit, so the Planck-phase boundary is never reached. The interior vacuum therefore remains in the ordinary, dynamical phase, and gravity remains fully active throughout the black-hole interior except at the classical singularity predicted by general relativity. 3.2 Cosmogenesis: Outward Loading at Saturation In contrast to black holes, where inward transport loading must climb toward saturation, the early Universe begins already at the saturation boundary. The primordial vacuum is fully outward-loaded, such that ∇𝜇𝐹𝐺𝜇=+𝐷Pl, is the positive extremum of the divergence law. In this regime the vacuum occupies its rigiditydominated Planck phase from the outset. Unlike gravitational collapse, which never attains this limit in practice, cosmogenesis starts precisely where the transport field is saturated. Because the total effective flux load of the Universe is enormous, 𝑀𝑈∼1053 kg, a value consistent with standard cosmological estimates [7], [21], [22], the corresponding saturated radius is not microscopic but macroscopic. Substituting this mass into the saturation relation [2]: 𝑟birth 3 ∼√48 𝐺2 𝑀𝑈 2 𝑐4 𝐾max ,⇒𝑟birth ∝𝑀𝑈 1/3 and identifying the saturation curvature with the Planck value 𝐾max=𝐾Pl, yields 𝑟 birth∼10−6−10−4 m=1μm−100 μm, a finite physical scale at which the early vacuum can support the total outward flux load while remaining at the saturation curvature 𝐾max. This radius is neither zero nor arbitrarily small: it is exactly the size required for a fully saturated Planck-phase vacuum containing the entire cosmic flux content for a given mass. This leads to a contrasting structural picture: • Black holes attempt to reach saturation through inward collapse but never attain it in practice; if they did, their saturated radii would be microscopic. • The Universe, by contrast, begins already at saturation, with a macroscopic radius 𝑟birth determined directly by its total flux load. As expansion proceeds and the vacuum softens, the Planck-phase boundary recedes, the transport field becomes unsaturated, and familiar field-theoretic degrees of freedom, including mass, gauge fields, and quantum fluctuations gradually emerge. Cosmogenesis is therefore the outward evolution of an initially saturated Planck-phase core into a progressively softer vacuum capable of supporting the physics we observe today. 3.3. The Big Start: A Finite-Radius Planck-Phase Beginning In the CTC picture, the Universe does not begin at a singularity. It begins as a finite spherical region of fully saturated, Planck-phase vacuum. In this state: • gravity cannot propagate, • electromagnetism does not exist, • quantum fluctuations are absent, • the Higgs field is uncondensed, • matter and radiation cannot form. Physics, as we know it, has not yet begun. Cosmogenesis starts only when the saturated vacuum softens and the divergence of the transport flux drops below the bound, ∣∇𝜇𝐹𝐺𝜇∣<𝐷Pl. This transition is not explosive. It is a transport-driven unjamming of an extremely stiff, or vacuum relaxation that gradually restores propagating degrees of freedom (DOF) and allows the Universe to enter the dynamical phase described by physics. 4. Phase-Ordered Cosmogenesis: The Sequence of Emergence As the Universe expands from its saturated beginning, the vacuum softens in distinct steps. Each softness threshold corresponds to a characteristic radius at which a new class of physical degrees of freedom becomes possible: 𝑟birth<𝑟𝐺<𝑟𝑇<𝑟𝐻<𝑟𝑆. These radii reflect the order in which the vacuum becomes capable of supporting curvature, waves, quantum behavior, mass, and finally causal cosmological evolution. 4.1. 𝒓𝐛𝐢𝐫𝐭𝐡 — End of Saturation (No Modes Exist) For radii smaller than 𝑟birth, the vacuum remains in its Planck phase. This phase is defined by complete saturation of the transport flux, expressed by ∣∇⋅𝐹𝐺∣=𝐷𝑃𝑙. This condition forces the curvature transport field 𝐹𝐺to maintain its maximal divergence everywhere within the region. A saturated flux cannot support spatial variation. As a result, no curvature degrees of freedom are available. The vacuum in this regime behaves as an ideal medium of infinite rigidity, unable to deform in any direction. The stiffness tensor in this phase is effectively 𝐾𝑖𝑗(𝑟<𝑟𝑃𝑙)=∞ 𝛿𝑖𝑗, and any attempt to introduce a displacement field 𝑢𝑖leads to an unbounded energetic penalty. The Lagrangian density 𝐿𝑃𝑙=1 2𝜌(∂𝑡𝑢𝑖)2−1 2𝐾𝑖𝑗(∇𝑢𝑖)(∇𝑢𝑗), becomes ill defined in the limit 𝐾𝑖𝑗→∞. The medium can neither support gradients nor allow finite strain. In this saturated phase: • no displacements can occur, • no gradients can form, • no oscillations are possible, • no waves can propagate, • no dynamical fields exist. The Planck phase therefore represents a state in which spacetime has no internal degrees of freedom and no capacity to host physical modes of any kind. 4.2 𝒓𝑮: Emergence of Gravity Through Longitudinal Mobility After the radius grows beyond 𝒓𝐏𝐥, the vacuum leaves the fully saturated Planck phase. The divergence bound remains in force, ∣∇⋅𝐹𝐺∣≤𝐷𝑃𝑙, but saturation no longer holds. The stiffness tensor begins to relax from its infinite value, 𝐾𝑖𝑗(𝑟>𝑟𝑃𝑙)<∞, giving the vacuum a small, nonzero capacity for deformation. This relaxation is anisotropic: the radial direction softens first while the angular directions remain locked. This directional asymmetry sets the stage for the first dynamical mode. The radius 𝑟𝐺 thus marks the moment when this partial relaxation becomes strong enough to permit radial motion. Only the longitudinal component of the displacement field 𝑢𝑖=(𝑢𝑟,𝑢𝜃,𝑢𝜙) is released, while the tangential components remain fixed. The vacuum therefore acquires exactly one mechanical degree of freedom (DOF). The stiffness tensor in this interval takes the form 𝐾𝑖𝑗(𝑟𝐺<𝑟<𝑟𝑇)=diag(𝐾𝐿, 0, 0), with a finite longitudinal modulus 𝐾𝐿>0 and vanishing angular terms. The corresponding Lagrangian density is thus: ℒ<𝑟𝑇=1 2𝜌(∂𝑡𝑢𝑟)2−1 2𝐾𝐿(∇𝑢𝑟)2, and contains no terms involving 𝑢𝜃 or 𝑢𝜙. Because the vacuum has no transverse restoring force, the angular components cannot support oscillatory behavior. The only propagating distortion is longitudinal compression of the radial field, corresponding to curvature transport but not to a wave-supporting medium. This regime is therefore characterized by: • a single dynamical direction, • no transverse modes, • no harmonic oscillators, • no possibility of quantization. The field supports curvature response but cannot sustain electromagnetic or quantum-mechanical phenomena. Gravity in this domain is strictly longitudinal: curvature can be redistributed, but not in a form that admits a spectral decomposition or wave propagation. 4.3. 𝒓𝑻: Emergence of Electromagnetism and the Onset of Quantum Fluctuations At the radius 𝑟𝑇the vacuum undergoes a key change: it develops transverse stiffness. This means the vacuum can now support shear-like motion in the two angular directions. The stiffness tensor becomes 𝐾𝑖𝑗(𝑟>𝑟𝑇)=diag(𝐾𝐿,𝐾𝑇,𝐾𝑇),𝐾𝑇>0. Once 𝐾𝑇appears, the transverse components of the displacement field 𝑢⊥=(𝑢𝜃,𝑢𝜙) become dynamical. Their Lagrangian has a kinetic term and an elastic term: 𝐿⊥=1 2𝜌 (∂𝑡𝑢⊥)2−1 2𝐾𝑇(∇𝑆2𝑢⊥)2, where ∇𝑆2is the covariant gradient on the sphere. The first term describes time-varying motion; the second describes angular shear. Varying this action gives the transverse wave equation: 𝜌 ∂𝑡2𝑢⊥−𝐾𝑇 Δ𝑆2𝑢⊥=0. Because 𝑢⊥has two components, it cannot be expanded using scalar spherical harmonics alone. The correct basis is the set of vector spherical harmonics, which provide a complete basis for any vector field on a sphere (see Jackson Classical Electrodynamics[23]; Arfken & Weber[24]; or Hobson–Efstathiou– Lasenby[25]). We write: 𝑢⊥(𝜃,𝜙,𝑡)=∑𝑞𝑙𝑚 (𝑝) 𝑙,𝑚,𝑝 (𝑡) 𝑌𝑙𝑚 (𝑝)(𝜃,𝜙),𝑝=1,2, where • 𝑝=1gives the divergence-free (toroidal) family, • 𝑝=2gives the curl-free (poloidal) family. Each vector spherical harmonic satisfies Δ𝑆2𝑌𝑙𝑚 (𝑝)=−𝑙(𝑙+1)𝑌𝑙𝑚 (𝑝). Substituting into the wave equation shows that every coefficient obeys an independent harmonicoscillator equation: 𝑞󰇘𝑙𝑚 (𝑝)+𝜔𝑙2 𝑞𝑙𝑚 (𝑝)=0,𝜔𝑙2=𝐾𝑇 𝜌 𝑙(𝑙+1). Thus the vacuum at 𝑟𝑇supports an entire spectrum of transverse oscillatory modes. These are the mechanical precursors of electromagnetic waves and the first genuine quantum fluctuations. For each mode, the reduced Lagrangian is simply 𝐿𝑙𝑚 (𝑝)=1 2[𝑞󰇗𝑙𝑚 (𝑝) 2−𝜔𝑙2𝑞𝑙𝑚 (𝑝) 2], with canonical momentum 𝜋𝑙𝑚 (𝑝)=𝑞󰇗𝑙𝑚 (𝑝). Quantization follows in the usual way: [𝑞𝑙𝑚 (𝑝),𝜋𝑙′𝑚′ (𝑝′)]=𝑖ℏ 𝛿𝑙𝑙′𝛿𝑚𝑚′𝛿𝑝𝑝′. Defining creation and annihilation operators gives the standard oscillator Hamiltonian: 𝐻=∑ℏ 𝑙,𝑚,𝑝 𝜔𝑙(𝑎𝑙𝑚 (𝑝)†𝑎𝑙𝑚 (𝑝)+1 2). This is the first radius at which photons, zero-point motion, and vacuum fluctuations exist. Quantization is therefore not added by hand—it appears automatically when the vacuum becomes mechanically able to support transverse oscillatory modes. With these degrees of freedom now present (Figure 1), the Higgs field becomes a well-defined quantum field, even though the vacuum is still in the symmetric electroweak phase and ⟨𝐻⟩=0. Universe at r= rG Universe at r= rT Figure 1 Transition from the 𝑟𝐺to the 𝑟𝑇threshold in the CTC emergence sequence. At radius 𝑟𝐺, the vacuum possesses longitudinal stiffness only, permitting scalar curvature transport but no transverse or oscillatory modes. At radius 𝑟𝑇, the vacuum acquires transverse stiffness, enabling the first quantum fluctuations and the emergence of electromagnetic degrees of freedom. The structured pattern on the right sphere represents one of the allowed transverse spherical modes. 4.4 𝒓𝑯: Birth of Mass (Higgs Condensation) The radius 𝑟𝐻marks the transition from the electroweak-symmetric phase to the broken phase in which the Higgs field acquires a nonzero vacuum expectation value. For all 𝑟𝑇<𝑟<𝑟𝐻, the Higgs exists as a quantum scalar with a symmetric potential and ⟨𝐻⟩=0. At 𝑟=𝑟𝐻, the Higgs condenses and ⟨𝐻⟩=𝑣≠0, giving mass to fermions and gauge bosons through the usual Standard Model mechanism. This marks the first radius at which massive degrees of freedom can exist. Before 𝑟𝐻, all excitations are massless despite the presence of photons and quantum fluctuations introduced at 𝑟𝑇. 4.5 𝒓𝑺: Classical Causality and the Start of FRW Cosmology The radius 𝑟𝑆is reached only when gravity, quantum fields, and mass are all active simultaneously. Only beyond this point does the vacuum support a classical spacetime with meaningful light cones, timelike worldlines, and a well-defined stress–energy tensor. Thus 𝑟𝑆 marks the beginning of the FRW causal regime. For 𝑟>𝑟𝑆, spacetime is described by the standard Einstein–FRW equations and the Universe enters its familiar radiation-, matter-, and dark-energy–dominated eras. For 𝑟<𝑟𝑆, the necessary degrees of freedom for classical cosmology do not yet exist, and the FRW picture cannot be applied. 4.6. r_Λ: Residual Flux and the Emergence of Dark Energy After the Universe has passed the sequence 𝑟Pl<𝑟𝐺<𝑟𝑇<𝑟𝐻<𝑟𝑆, Electromagnetism Quantum Fluctuation Gravity [3] A. H. Guth, “The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,” Phys. Rev. D, vol. 23, pp. 347–356, 1981, doi: 10.1103/PhysRevD.23.347. [4] A. D. Linde, “Chaotic inflation,” Physics Letters B, vol. 129, no. 3, pp. 177–181, Sept. 1983, doi: 10.1016/0370-2693(83)90837-7. [5] V. Mukhanov, Physical Foundations of Cosmology. Oxford: Cambridge University Press, 2005. doi: 10.1017/CBO9780511790553. [6] A. Albrecht and P. J. Steinhardt, “Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking,” Phys. Rev. Lett., vol. 48, no. 17, pp. 1220–1223, Apr. 1982, doi: 10.1103/PhysRevLett.48.1220. [7] Planck Collaboration et al., “Planck 2018 results,” A&A, vol. 641, 2020, doi: 10.1051/00046361/201833910. [8] A. Vilenkin, “Creation of universes from nothing,” Physics Letters B, vol. 117, no. 1, pp. 25–28, Nov. 1982, doi: 10.1016/0370-2693(82)90866-8. [9] J. B. Hartle and S. W. Hawking, “Wave function of the Universe,” Phys. Rev. D, vol. 28, no. 12, pp. 2960–2975, Dec. 1983, doi: 10.1103/PhysRevD.28.2960. [10] A. Linde, “Quantum creation of an open inflationary universe,” Phys. Rev. D, vol. 58, no. 8, p. 083514, Sept. 1998, doi: 10.1103/PhysRevD.58.083514. [11] A. Ashtekar, T. Pawlowski, and P. Singh, “Quantum Nature of the Big Bang,” Phys. Rev. Lett., vol. 96, no. 14, p. 141301, Apr. 2006, doi: 10.1103/PhysRevLett.96.141301. [12] A. Ashtekar and P. Singh, “Loop quantum cosmology: a status report,” Classical and Quantum Gravity, vol. 28, no. 21, p. 213001, Sept. 2011, doi: 10.1088/0264-9381/28/21/213001. [13] J. Khoury, B. A. Ovrut, P. J. Steinhardt, and N. Turok, “Ekpyrotic universe: Colliding branes and the origin of the hot big bang,” Phys. Rev. D, vol. 64, no. 12, p. 123522, Nov. 2001, doi: 10.1103/PhysRevD.64.123522. [14] P. J. Steinhardt and N. Turok, “Cosmic evolution in a cyclic universe,” Phys. Rev. D, vol. 65, no. 12, p. 126003, May 2002, doi: 10.1103/PhysRevD.65.126003. [15] M. Reuter and F. Saueressig, “Quantum Einstein gravity,” New Journal of Physics, vol. 14, no. 5, p. 055022, May 2012, doi: 10.1088/1367-2630/14/5/055022. [16] A. Bonanno and M. Reuter, “Cosmology with self-adjusting vacuum energy density from a renormalization group fixed point,” Physics Letters B, vol. 527, no. 1, pp. 9–17, Feb. 2002, doi: 10.1016/S0370-2693(01)01522-2. [17] R. Penrose, Cycles of Time: An Extraordinary New View of the Universe. Knopf Doubleday Publishing Group, 2011. [Online]. Available: https://books.google.com/books?id=gv8o1XydoCQC [18] M. Eckstein, “Conformal Cyclic Cosmology, gravitational entropy and quantum information,” General Relativity and Gravitation, vol. 55, no. 2, p. 26, Jan. 2023, doi: 10.1007/s10714-023-03070-2. [19] N. Popławski, “Universe in a Black Hole in Einstein-Cartan Gravity,” \apj, vol. 832, no. 2, p. 96, Dec. 2016, doi: 10.3847/0004-637X/832/2/96. [20] 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. [21] Valev, “Estimations of Total Mass and Energy of the Observable Universe,” Physics International, vol. 5, no. 1, pp. 15–20, Jan. 2014, doi: 10.3844/pisp.2014.15.20. [22] D. W. Hogg, “Distance measures in cosmology,” May 1999. [23] J. D. Jackson, Classical Electrodynamics. Wiley, 2021. [Online]. Available: https://books.google.com/books?id=6VV-EAAAQBAJ [24] G. B. Arfken, G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists: A Comprehensive Guide. Elsevier Science, 2013. [Online]. Available: https://books.google.com/books?id=qLFo_Z-PoGIC [25] M. P. Hobson, G. P. Efstathiou, and A. N. Lasenby, General Relativity. 2006. doi: 10.2277/0521829518. [26] Planck Collaboration et al., “Planck 2015 results,” A&A, vol. 594, 2016, doi: 10.1051/00046361/201526681. [27] P. A. R. Ade et al., “Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season,” Phys. Rev. Lett., vol. 127, no. 15, p. 151301, Oct. 2021, doi: 10.1103/PhysRevLett.127.151301. Supplement Materials: Three Fundamental CTC Results and Their Planck Signatures The Curvature–Transport Correspondence links the mechanical response of the vacuum to the structure of primordial perturbations. The divergence law governing curvature transport determines which modes can exist, how they evolve, and which components of curvature become active as the vacuum crosses the thresholds 𝑟Pl<𝑟𝐺<𝑟𝑇<𝑟𝐻<𝑟𝑆. From this structure follow three mathematical results that shape the primordial spectrum. Planck observations reveal the corresponding empirical signatures, without any additional assumptions or parameter tuning. The data simply reflect the emergence properties of the vacuum. S.1 Result 1: Mode–Existence Condition Before the vacuum reaches 𝑟𝑇, transverse stiffness is zero, 𝐺𝑇(𝑡<𝑡𝑇)=0, and no oscillatory solutions exist. This implies the primordial cutoff 𝑘≥𝑘min=1/𝑟𝑇. Planck signature Planck observes: • disappearance of temperature correlations for angles 𝜃≳60∘, • suppression of power at low multipoles ℓ≲20, and • absence of long-wavelength modes. These results come from Planck 2015 XVI (Isotropy and Statistics of the CMB, A&A 594, A16, 2016)[26], which documents the large-angle anomaly and the vanishing correlation function beyond about sixty degrees. The same cutoff forbids any trans-Planckian oscillations in 𝑃(𝑘), consistent with Planck 2018 X (Constraints on Inflation, A&A 641, A10, 2020)[7], which finds no evidence of periodic modulation or logarithmic features. S.2 Result 2: Stiffness–Tilt Relation Between 𝑟𝐺and 𝑟𝑇 the stiffness softens, generating a tilted scalar spectrum, 𝑃(𝑘)∝𝑘𝑛𝑠−1,𝑛𝑠<1, with monotonic negative running, 𝛼𝑠<0. Planck signature Planck 2018 VI (Cosmological Parameters, A&A 641, A6, 2020)[7] reports 𝑛𝑠=0.9649±0.0042,𝛼𝑠≲−0.004, in agreement with the CTC prediction. In ΛCDM these values depend on slow-roll conditions; in CTC they follow directly from stiffness gradients in the vacuum. S.3 Result 3: Tensor-Suppression Condition Before 𝑟𝐺, the vacuum supports only longitudinal curvature. Transverse gravitational modes do not exist, 𝐺𝑇(𝐺)(𝑡<𝑡𝐺)=0, and therefore the primordial tensor spectrum vanishes, 𝑃𝑇(𝑘)=0. Planck signature Joint BICEP/Keck + Planck results give 𝑟<0.036. This consolidated bound comes from the BICEP/Keck 2018 analysis (PRL 127, 151301, 2021)[27], which combines WMAP, Planck, and ground-based data to rule out detectable primordial gravitational waves. Inflation must finely tune its potential to achieve such small 𝑟. In CTC, tensor suppression is a structural consequence of the vacuum’s stiffness ordering. S.4 Combined Interpretation Taken together, the three CTC results provide a unified explanation of the full pattern of Planck observations. The mode-existence cutoff determines which perturbations can ever appear, removing all wavelengths longer than 𝑟𝑇and all sub 𝑟𝑇modes. The stiffness–tilt relation fixes the shape of the scalar spectrum, generating both the observed tilt and the small but persistent negative running. The tensorsuppression condition eliminates primordial gravitational waves at their origin. When these three consequences are viewed as parts of a single emergence sequence 𝑟Pl<𝑟𝐺<𝑟𝑇<𝑟𝐻<𝑟𝑆, the signatures measured by Planck follow naturally. The low power on large angular scales, the smooth scalar spectrum with tilt and negative running, the absence of trans Planckian features, the lack of primordial tensors, and the slightly enhanced lensing amplitude all arise from the same underlying structure of the vacuum. No additional parameters or model adjustments are required. By contrast, the ΛCDM framework must introduce extra assumptions or extensions to account for these features individually. In CTC, they emerge automatically from the mechanical response of the vacuum as it evolves through the stiffness thresholds that mark the beginning of spacetime structure.