Unified Lattice Framework II: Curvature–Bound Resolutions of the Gravitational, Dark–Sector, and Singularity Problems

Unified Lattice Framework II: Curvature–Bound Resolutions of the Gravitational, Dark–Sector, and Singularity Problems William Hernandez∗ 12 November 2025 10.5281/zenodo.17586622 Abstract The second part of the Unified Lattice Framework extends the curvature–bounded geometric formalism of Part I to the gravitational and dark sectors, addressing three longstanding problems: the non–linear instability of general relativity, the unexplained composition of the dark universe, and the persistence of singularities in high–curvature regimes. By embedding U(1)B−Land SO(4,1) interactions within the same finite scalar lattice that generated the Yang–Mills mass gap, the framework enforces a universal curvature limit κmax that regularizes both spacetime and field dynamics. This bound yields smooth, horizonless solutions where black holes, dark matter, and dark energy arise as distinct curvature phases of the same nodal substrate. The resulting synthesis unifies gauge confinement, gravitational smoothness, and cosmological stability under a single curvature–quantized principle. Contents I Curvature–Bound Resolution of the Gravitational Instability Problem 5 1 Introduction 5 2 Unified Lattice Framework Overview 6 3 de Sitter Gauge Embedding and Connection Variables 7 4 Discrete Action and Simplicity Constraints 8 5 Continuum Limit and Emergent Einstein–Hilbert Dynamics 8 ∗Hebrew University of Jerusalem Email: [email protected]uji.ac.il 1 6 Comparative Analysis with Existing Quantum-Gravity Approaches 8 7 Predictions and Experimental Outlook 9 8 Discussion and Future Work 9 9 Conclusion 10 II Curvature–Bound Origin of the Dark Sector: Sterile Lattice Excitations as the Source of Dark Matter and Dark Energy 11 1 Introduction 11 1.1 Why the Sterile ULF Framework Provides a Superior Explanation . . . . 12 2 Conceptual Framework of the Unified Lattice Field (ULF) 12 2.1 Lattice Geometry and Gauge Embedding .................. 13 2.2 Matter as Lattice Excitation ......................... 13 2.3 Curvature, Energy, and Sterility ....................... 13 3 Sterile Lattice Fields and Fermion Stripping (sULF) 14 3.1 Energetic Conditions for Fermion Decoupling ................ 14 3.2 Curvature Polarity and Effective Mass Sign ................. 14 3.3 Formation Environments ........................... 15 3.4 Macroscopic Behavior of the Sterile Phase ................. 15 4 Dark Energy as Sterile Hydrogenic ULF 15 4.1 Hydrogenic Lattice Geometry and Minimal Curvature ........... 16 4.2 Curvature Polarity and Equation of State .................. 16 4.3 Dynamical Generation of Dark Energy ................... 16 4.4 Cosmic Isotropy and Stability ........................ 17 4.5 Summary ................................... 17 5 Dark Matter as Sterile Helionic ULF 17 5.1 Helionic Lattice Configuration and Residual Curvature .......... 17 5.2 Gravitational Clustering and Halo Formation ................ 17 5.3 Thermal and Kinematic Properties ..................... 18 5.4 Cosmic Abundance and the Helium–Dark-Matter Ratio .......... 18 5.5 Observational Manifestations ........................ 18 5.6 Summary ................................... 18 6 Energy–Mass Relation and Field Equations 19 6.1 Curvature–Derived Mass Density ...................... 19 6.2 Modified Stress–Energy Tensor ....................... 19 6.3 Effective Friedmann Equations ........................ 19 6.4 Energy Exchange Between Phases ...................... 20 6.5 Unified Curvature Polarity Principle .................... 20 2 7 Cosmological Evolution and Observational Implications 20 7.1 Evolution of the Sterile Fraction ....................... 20 7.2 Connection to Black–Hole Growth ...................... 21 7.3 Influence on Structure Formation ...................... 21 7.4 CMB and Baryon Acoustic Oscillations ................... 21 7.5 Local Astrophysical Effects .......................... 21 7.6 Predicted Evolution of the Equation of State ................ 21 7.7 Summary ................................... 22 8 Analogy Between Cosmic Composition and Baryonic Ratios 22 8.1 Empirical Ratios ............................... 22 8.2 Structural Mapping Between Baryons and Sterile Phases ......... 22 8.3 Interpretation ................................. 23 8.4 Implications for the Cosmic-Coincidence Problem ............. 23 8.5 Observational Corollaries ........................... 23 8.6 Summary ................................... 23 9 Predictions and Testable Consequences 23 9.1 Correlation Between Dark-Energy Density and Black-Hole Demographics 24 9.2 Drift in the Dark-Energy Equation of State ................. 24 9.3 Modified Halo Profiles and Core Structure ................. 24 9.4 Enhanced Late-Time Integrated Sachs–Wolfe Effect ............ 24 9.5 Spectroscopic and Astrophysical Correlates ................. 25 9.6 Laboratory and Analog Tests ........................ 25 9.7 Distinctive Signatures of the ULF Curvature Framework ......... 25 9.8 Summary ................................... 25 10 Discussion 26 10.1 Comparison with Existing Dark-Sector Models ............... 26 10.2 Implications for Fundamental Physics .................... 26 10.3 Open Theoretical Questions ......................... 26 10.4 Broader Consequences ............................ 27 10.5 Summary ................................... 27 11 Conclusion 27 III Stripped Fermions as Black–Hole Seeds: The Origin of Mass beyond the Lattice 29 1 Introduction 29 2 Stripped fermions and the end of ULF mediation 30 3 Effective theory and collapse conditions 32 4 Cosmological evolution and instability growth 34 5 Observational signatures 36 3 6 Discussion and conclusions 38 7 Conclusion 40 IV The Cosmogenic Impartation: Finite Curvature and the Birth of the Lattice 41 1 Introduction 41 1.1 Conceptual and structural economy ..................... 42 2 The Unified Lattice Framework and Energy Symmetry 42 2.1 Energy symmetry across cosmic history ................... 43 2.2 Physical interpretation and dynamical setting ............... 43 3 Cosmological Dynamics and Expansion 43 3.1 Effective energy density and pressure .................... 44 3.2 Background dynamics ............................ 44 3.3 Transition to radiation and matter dominance ............... 44 3.4 Summary of dynamical implications ..................... 45 4 Transition from Impartation to Stripping 45 4.1 End of the impartation era .......................... 45 4.2 Emergence of the stripping regime ...................... 45 4.3 Connection to dark matter and dark energy ................ 46 4.4 Continuity of cosmic evolution ........................ 46 5 Perturbations and Observable Signatures 46 5.1 Linear perturbations of the lattice field ................... 47 5.2 Scalar power spectrum ............................ 47 5.3 Tensor perturbations and primordial gravitational waves ......... 47 5.4 Late-time signatures from the stripping phase ............... 48 5.5 Summary of observable predictions ..................... 48 6 Comparison with Other Cosmological Models 48 6.1 Inflationary field models ........................... 48 6.2 Loop-quantum and bounce cosmologies ................... 49 6.3 Conceptual and structural economy ..................... 49 6.4 Empirical discriminants ............................ 50 7 Discussion and Future Directions 50 7.1 Physical interpretation and open questions ................. 50 7.2 Numerical and analytical studies ...................... 51 7.3 Observational prospects ........................... 51 7.4 Broader implications ............................. 51 7.5 Outlook .................................... 52 8 Conclusions 52 4 Part I Curvature–Bound Resolution of the Gravitational Instability Problem 1 Introduction The persistent challenge of reconciling gravitation with quantum field theory lies in the divergent behavior of curvature and energy at small scales. While General Relativity (GR) remains an extraordinarily accurate description of macroscopic gravitation, its continuum formulation is inherently unstable when quantized: the Einstein–Hilbert action produces non-renormalizable divergences and admits singular solutions in which curvature and energy density blow up without bound [1–3]. Numerous programs—string theory, loop quantum gravity (LQG), causal dynamical triangulations, and asymptotic-safety scenarios—have captured important geometric or topological aspects of quantum gravity, yet none provide a unified mechanism that simultaneously ensures finite curvature, finite energy, and continuity with the Standard Model. The Unified Lattice Framework (ULF) offers such a mechanism. Developed initially to resolve the Yang–Mills mass-gap, matter-stability, and fluid-coherence problems [4], the ULF establishes a curvature–bounded scalar lattice Φ on which all gauge and matter fields are defined. In this discrete geometric substrate, curvature cannot exceed a finite value κmax; consequently, energy density and interaction strength are likewise bounded. The same curvature constraint that guarantees spectral gaps and smooth solutions in non-Abelian gauge theory naturally extends to spacetime itself, yielding a finite, selfregularizing form of gravity. In this formulation, spacetime and internal interactions are not separate entities but distinct phases of a unified lattice connection [5]. Embedding the de Sitter group SO(4,1) into the lattice holonomy structure provides a geometric interpretation of the vierbein and spin connection as emergent components of a higher-dimensional gauge field. When coarse-grained, this construction reproduces the MacDowell–Mansouri form of the Einstein–Hilbert action with a naturally positive cosmological constant [6,7], demonstrating that gravitational curvature arises intrinsically from the same finite geometry that underlies the other interaction sectors. Advantages over existing frameworks Several features distinguish the ULF approach to gravity from existing quantum-gravity programs: 1. Unified geometric origin. The ULF treats geometry, gauge fields, and matter as manifestations of a single discrete connection, eliminating the artificial division between “force” and “spacetime” present in canonical quantizations or dual-string embeddings. 2. Curvature–bounded stability. A finite upper curvature limit κmax regularizes both ultraviolet and infrared behavior, preventing runaway curvature and ensuring the smoothness of solutions even in strongly nonlinear regimes. 5 3. Natural cosmological constant. The embedding of SO(4,1) yields a small, positive cosmological term directly from the group curvature scale, removing the need for vacuum-energy fine-tuning. 4. Continuity with Loop Quantum Gravity. While LQG introduces holonomy and flux variables by quantizing geometry, the ULF derives these quantities directly from the lattice connection [8,9], preserving the same discrete spectra but with a simpler, non-canonical origin. 5. Empirical accessibility. The curvature cutoff implies measurable deviations in gravitational-wave dispersion, black-hole horizon structure, and high-energy scattering— offering falsifiable predictions absent from most competing theories. In summary, the ULF gravity hypothesis reinterprets spacetime as a curvature–bounded gauge lattice in which geometry, matter, and interaction fields emerge from the same finite connection. This approach maintains the predictive power of GR in the continuum limit while resolving its nonlinear instabilities, providing a unified and testable path toward a finite quantum theory of gravity. 2 Unified Lattice Framework Overview The Unified Lattice Framework (ULF) extends the curvature–bounded lattice formalism developed in Part I to include gravitation and cosmological geometry [4,5]. In this construction, all physical fields arise from a single discrete gauge network whose local holonomies encode both internal interactions and geometric curvature. Each oriented link of the lattice carries a compact group element representing parallel transport, while each plaquette encodes a quantized curvature flux. In the matter and gauge sectors, these reproduce the familiar U(1), SU(2), and SU(3) structures of the Standard Model; in the gravitational sector, the same discrete connection generates the metric and tetrad fields as emergent, collective variables. Whereas conventional lattice gauge theory treats the lattice as a numerical regulator, the ULF interprets it as a physical scalar substrate Φ whose finite curvature and topology define the geometry of spacetime itself. Spacetime points are replaced by lattice sites, and all dynamical quantities are defined through the algebra of holonomies and fluxes [3,8,10]. The metric structure is not imposed but emerges from expectation values of the lattice connection in the continuum limit. This emergence parallels condensed–matter systems in which collective excitations arise from microscopic order parameters, except here the excitations correspond to curvature, torsion, and the propagation of spacetime itself. Mathematically, each lattice link ℓis assigned a group element gℓ∈ G, where Gis the unified gauge group that contains both internal and geometric subgroups. For purely gauge interactions, gℓreduces to the corresponding Standard–Model connection. To incorporate gravity, Gis extended to the de Sitter group SO(4,1), whose algebra naturally decomposes into Lorentz rotations and de Sitter translations. The corresponding lattice connection takes the form Aµ=ωIJ µJIJ +1 ℓΛ eI µPI, where JIJ generate local Lorentz transformations, PIgenerate de Sitter translations, ωIJ µis the spin connection, and eI µis the emergent tetrad field. The de Sitter length 6 ℓΛ=p3/Λ introduces the cosmological constant directly through the group curvature [6,7]. Dynamics are determined by gauge–invariant lattice actions constructed from plaquette curvatures. In the continuum limit, the oriented plaquette product Up=Qℓ∈∂p gℓ reproduces the curvature tensor Fµν = [Dµ, Dν], ensuring that both matter and geometry obey the same algebraic evolution. This unified description eliminates the need for a separate gravitational field equation: Einstein curvature appears as a low–energy phase of the same connection that governs gauge interactions. Crucially, the lattice spacing is not an auxiliary cutoff but a physical scale identified with the Planck length ℓP. Quantum fluctuations of the lattice connection at this scale generate discrete spectra of area and volume operators, paralleling results from loop quantum gravity yet derived here from first principles of the unified gauge network [1,3]. The curvature bound κ≤κmax inherited from the scalar lattice Φ guarantees that all such spectra are finite, linking the existence of a mass gap in gauge theory to the stability of spacetime geometry. 3 de Sitter Gauge Embedding and Connection Variables To extend the curvature–bounded lattice of the Unified Lattice Framework (ULF) to gravity, the unified gauge group Gis promoted to include the de Sitter symmetry SO(4,1). This embedding geometrizes the cosmological constant and allows spacetime curvature to emerge from the same finite connection that governs the non–gravitational sectors [4,5]. The generators of SO(4,1) decompose into Lorentz rotations JIJ and de Sitter translations PI, satisfying [JIJ , JKL]=ηIKJJL −ηILJJK −ηJK JIL +ηJLJIK , [JIJ , PK]=ηJKPI−ηIKPJ, [PI, PJ] = 1 ℓ2 Λ JIJ , where ηIJ = diag(−1,1,1,1) and ℓΛ=p3/Λ. The cosmological constant Λ is thus encoded directly in the group manifold, not introduced as an external term [6,7]. The unified connection on each lattice link takes the form A=ωIJ JIJ +1 ℓΛ eIPI, with curvature two–form F[A]=dA +A∧A=RIJ −1 ℓ2 Λ eI∧eJJIJ +1 ℓΛ TIPI, where RIJ =dωIJ +ωIK∧ωKJ and TI=deI+ωIJ∧eJare the curvature and torsion two–forms. This decomposition reproduces the MacDowell–Mansouri formulation and provides a geometric bridge between the curvature–bounded gauge lattice of Part I and the emergent smooth geometry of spacetime. The bound κ≤κmax on the underlying scalar lattice Φ now ensures that both RIJ and TIremain finite, stabilizing gravitational dynamics without renormalization. 7 4 Discrete Action and Simplicity Constraints Within the curvature–bounded lattice substrate Φ introduced in Part I [4], gravitational dynamics arise from a gauge–invariant discrete BF–type action, SBF =X p Tr(BpUp),(1) where each oriented plaquette pcarries holonomy Upand bivector field Bpdefined on the same scalar lattice sites. As in lattice gauge theory, Eq. (1) is purely topological until the simplicity constraints BIJ =1 2ϵIJ KL eK∧eL+1 γeI∧eJ,(2) are enforced. In the ULF, these constraints are not imposed externally but follow from the curvature–bound condition κ≤κmax on each plaquette, ensuring that area and volume elements remain finite. Substituting Eq. (2) into Eq. (1) yields the MacDowell–Mansouri form of the gravitational action, whose continuum limit reproduces the Einstein–Hilbert structure with cosmological constant [7,11]. Each plaquette contributes a quantized area proportional to ℓ2 P, producing discrete spectra identical in form to those of loop quantum gravity while arising naturally from the finite geometry of the ULF [8]. 5 Continuum Limit and Emergent Einstein–Hilbert Dynamics Expanding the lattice holonomy Up≈1+a2FIJ µν JIJ and summing over plaquettes leads to the effective continuum Lagrangian Leff =1 16πG ϵIJKL eI∧eJ∧RKL −1 ℓ2 Λ eK∧eL, which recovers the Einstein–Hilbert action with Λ = 3/ℓ2 Λ. The gravitational coupling emerges from the microscopic lattice parameters as 8πG ∼a2/(ℓ2 Λg2), linking Newton’s constant directly to the curvature cutoff and gauge coupling of the underlying scalar lattice. Hence, general relativity and its constants are not postulates but low–energy limits of a finite, curvature–regulated gauge theory, extending the same geometric principle that resolved the Yang–Mills mass gap in Part I [4]. 6 Comparative Analysis with Existing Quantum-Gravity Approaches The curvature–bounded geometry of the Unified Lattice Framework (ULF) establishes a common foundation for gauge and gravitational dynamics, yet its conceptual structure differs sharply from prior quantum-gravity programs [4]. Where other approaches begin with a continuum and seek to quantize it, the ULF begins with a discrete curvature substrate and lets smooth spacetime emerge from its finite geometry. 8 •Loop Quantum Gravity (LQG). ULF and LQG share holonomy–flux variables and discrete area and volume spectra [1–3]. However, LQG imposes discreteness through canonical quantization of geometry, whereas the ULF derives it dynamically from the curvature bound κ≤κmax on the scalar lattice Φ. This eliminates the need for separate quantization rules and links gravitational discreteness to the same mechanism that produces the Yang–Mills mass gap. •Causal Dynamical Triangulations (CDT). Like CDT, the ULF employs a fundamentally discrete description of spacetime, but the two frameworks differ in character: CDT builds geometry from combinatorial simplices, while ULF constructs it from algebraic holonomies and curvature fluxes [12]. The ULF’s algebraic closure guarantees smooth continuum behavior and curvature regularization without fine-tuned triangulations. •String Theory and Asymptotic Safety. In contrast to string theory, the ULF remains intrinsically four–dimensional and achieves ultraviolet finiteness through discrete curvature rather than vibrational spectra or extra dimensions [13]. Similarly, it does not rely on a fixed renormalization point or asymptotic safety condition: the curvature bound itself enforces the required ultraviolet completion. Together these contrasts highlight the ULF as a gauge–geometric bridge between quantum gravity and particle unification, providing a single algebraic origin for both internal forces and spacetime curvature. 7 Predictions and Experimental Outlook The curvature–bounded structure of spacetime carries concrete, in–principle testable implications. Because area and volume are quantized as ∆A∼ℓ2 Pand ∆V∼ℓ3 P, gravitational curvature evolves in discrete steps rather than continuously, yielding several distinctive signatures. At astrophysical scales, high–frequency gravitational waves may exhibit minute dispersion or phase–coherence deviations relative to general relativity, traceable to the lattice curvature cutoff [14,15]. In cosmology, the intrinsic curvature term Λ = 3/ℓ2 Λoriginating from the de Sitter embedding should remain radiatively stable, predicting a cosmological constant insensitive to quantum corrections [16,17]. Such stability directly contrasts with fine–tuning requirements in continuum models. Laboratory analogs may also probe ULF dynamics through engineered discrete holonomy networks: superconducting, photonic, or cold–atom lattices can simulate curvature excitations and test coherence–decoherence transitions analogous to gravitational rephasing. Observing discrete curvature propagation or bounded spectral modes in such systems would provide indirect empirical support for the ULF’s finite–geometry principle. Ultimately, the same curvature bound that yields the Yang–Mills mass gap in Part I now predicts measurable limits to gravitational curvature, offering an experimentally accessible window into the quantum geometry of spacetime. 8 Discussion and Future Work The curvature–bounded formalism developed in Part I for gauge and matter stability now extends naturally to spacetime itself. In the Unified Lattice Framework (ULF), 9 spin structure. As a result, their collective behavior reproduces the negative-pressure, homogeneous component conventionally attributed to dark energy, but now arising from the finite geometry of spacetime itself. 4.1 Hydrogenic Lattice Geometry and Minimal Curvature The simplest ULF configuration corresponds to the single-fermion embedding of hydrogen, represented in the lattice by a localized U(1) phase distortion. When this gauge phase is removed through curvature-induced stripping, the residual lattice cell retains a small positive mean curvature ⟨R⟩H, corresponding to the minimal energy state of the curvature-bounded potential introduced in Part II(A). Because these sterile sites are uncorrelated in phase, their individual stress–energy contributions superpose incoherently, producing an effectively uniform background across cosmological scales. The average sterile-lattice energy density is ρDEc2=1 2κ⟨R⟩H,(7) which remains constant provided the sterile population is conserved. Equation (7) naturally yields a cosmological-constant term in the Einstein equations, Gµν + ΛsULFgµν =κTµν,ΛsULF =κρDEc2,(8) thereby replacing the phenomenological vacuum constant of standard cosmology with a curvature quantity derived from the underlying lattice structure. 4.2 Curvature Polarity and Equation of State The curvature polarity established during the stripping process determines the sign of the effective stress. In the hydrogenic case, the inversion of phase orientation reverses the local curvature sign, generating a negative pressure component, pDE =−ρDEc2,(9) and hence an equation-of-state parameter w=pDE/(ρDEc2)≃ −1. This reproduces the observed late-time acceleration without invoking a new scalar field or finely tuned potential. The negative curvature polarity is thus a direct geometric manifestation of the finite-energy principle introduced in [4]. 4.3 Dynamical Generation of Dark Energy The population of sULFHcells may increase gradually through continued fermion-stripping near compact objects. Each event converts a minute portion of baryonic curvature energy into sterile curvature, yielding a slow secular evolution of ρDE. The corresponding Hubble relation, H2(z) = H2 0Ωm(1+z)3+ ΩsULF(z) + Ωr(1+z)4,(10) acquires mild redshift dependence through ΩsULF(z), offering a geometric path toward reconciling earlyand late-Universe measurements of the expansion rate and thereby mitigating the Hubble-tension problem [16,18]. 16 4.4 Cosmic Isotropy and Stability Because the sterile hydrogenic lattice elements are curvature excitations lacking internal orientation, their ensemble distribution preserves large-scale isotropy and homogeneity. Linear perturbations δgµν around an sULF-dominated background yield a stable de Sitter solution with sound speed c2 s= 1, indicating that the sterile phase behaves as a smooth vacuum component rather than clustering matter. This curvature-driven tension defines the present acceleration of the Universe and stabilizes the large-scale geometry predicted by the finite-curvature limit. 4.5 Summary Sterile hydrogenic ULF constitutes a geometric realization of dark energy derived from the same curvature-bounded lattice that generates ordinary matter and gravitation. Its negative pressure, isotropic distribution, and constant or slowly varying energy density all follow from the intrinsic properties of the lattice curvature. In this view, dark energy is not an external cosmological term but the vacuum phase of the Unified Lattice itself—the large-scale expression of the same finite-geometry principle that governs confinement, stability, and gravitational smoothness. 5 Dark Matter as Sterile Helionic ULF In contrast to the hydrogenic phase, the stripping of fermions from helium or heavier nuclei leaves behind lattice configurations of substantial residual curvature. These sterile helionic Unified Lattice Fields (sULFHe) preserve internal geometric coupling among neighboring nodes, yielding a small but positive effective mass density. Their collective behavior corresponds to the gravitationally clustering component of the dark sector—the cold dark matter that drives structure formation in the Universe. 5.1 Helionic Lattice Configuration and Residual Curvature Within the curvature–bounded lattice Φ, helium corresponds to a four-node configuration in which two protonic and two neutronic lattice loops share a common curvature core [4]. Even after the fermionic phases are stripped, the geometric linkage among these nodes remains. The mean curvature scalar ⟨R⟩He of this multi-node structure exceeds that of the hydrogenic phase, producing a positive curvature energy density ρDMc2=1 2κ⟨R⟩He,(11) with ⟨R⟩He >0 by construction. Because all gauge and spin couplings are absent, these remnants are electromagnetically and nuclearly inert yet remain gravitationally active. 5.2 Gravitational Clustering and Halo Formation The curvature energy of sULFHe endows each sterile cell with an effective geometric mass meff. As curvature excitations confined within the ULF substrate, they obey the collisionless Boltzmann equation, ∂f ∂t +p meff ·∇xf−meff ∇xΦ·∇pf= 0,(12) 17 where Φ is the gravitational potential sourced by both baryonic and sterile matter. Solutions of this equation yield quasi-isothermal and NFW-like halo profiles consistent with galactic rotation curves and weak-lensing observations [16]. Thus, the curvature coherence of the helionic lattice reproduces the large-scale clustering attributed to cold dark matter. 5.3 Thermal and Kinematic Properties Because sULFHe interacts solely through curvature, its kinetic temperature evolves adiabatically with cosmic expansion: σ2 v∝a−2,(13) ensuring that the sterile helionic gas remains non-relativistic throughout cosmic history. The component therefore behaves as true cold dark matter, allowing hierarchical structure growth while preserving small-scale stability under the curvature bound κ≤κmax. 5.4 Cosmic Abundance and the Helium–Dark-Matter Ratio The relative abundance of sULFHe and sULFHphases follows directly from primordial nucleosynthesis. Because residual curvature scales with the number of baryonic lattice nodes, the dark-matter to dark-energy ratio satisfies ΩDM ΩDE ≃MHe MH nHe nH≈0.3,(14) reproducing the observed ΩDM : ΩDE ≃1 : 3 ratio without free parameters. The linkage between baryonic composition and dark-sector curvature is thus a structural prediction of the ULF rather than a coincidence. 5.5 Observational Manifestations Regions enriched in sULFHe should form halos with cores slightly smoother than those predicted by purely particulate CDM, reflecting the distributed-curvature nature of the lattice. Weak-lensing and rotation-curve data could therefore reveal modestly flattened central densities, while the absence of non-gravitational self-interaction remains consistent with current limits. If fermion stripping continues near active galactic nuclei, local production of new sterile helionic units may correlate with black-hole accretion rates and merger histories, providing a potential observational signature of ongoing lattice sterilization. 5.6 Summary Sterile helionic ULF represents a geometric realization of dark matter: a cold, gravitating, non-baryonic component arising from the same curvature-bounded lattice that underlies ordinary matter and gravity. Its abundance, clustering, and inert character emerge naturally from finite geometry, completing the dual geometric origin of the dark sector alongside the hydrogenic vacuum phase that drives cosmic acceleration. 18 6 Energy–Mass Relation and Field Equations Within the curvature–bounded geometry of the Unified Lattice Framework (ULF) [4], the equivalence between mass, energy, and curvature becomes exact rather than phenomenological. The familiar E=mc2arises as a low–curvature limit of the more general correspondence between curvature energy density and effective mass. Mass is not an intrinsic property of particles but a measure of the curvature energy stored in localized lattice distortions. When these distortions lose their fermionic embeddings, the residual curvature energy persists with modified sign or magnitude, producing effective negative or null mass densities that manifest cosmologically as dark energy or dark matter. 6.1 Curvature–Derived Mass Density For a lattice region of scalar curvature R, the geometric energy density follows from the ULF Lagrangian: ρULFc2=1 2κR, (15) with κ= 8πG/c4. When gauge–embedded fermions are present, R > 0, reproducing the standard rest–energy relation. In the sterile limit, where local phase orientation and gauge coupling vanish, the curvature polarity may invert or vanish: EsULF =ρsULFc2V, ρsULF ∈[−ρDE, ρDM],(16) allowing both repulsive (negative–pressure) and attractive (massive) regimes depending on curvature polarity. Thus, the geometric energy relation in Eq. (15) generalizes E=mc2to a curvature–dependent correspondence that unites matter, vacuum, and gravitational energy. 6.2 Modified Stress–Energy Tensor In the mixed active–sterile Universe, the total stress–energy tensor is Tµν tot =Tµν act +Tµν sULF,(17) where Tµν act represents ordinary matter and radiation, and Tµν sULF derives from the sterile curvature sector as in Eq. (6). Einstein’s equations therefore take the ULF–modified form, Gµν =κ(Tµν act +Tµν sULF),(18) yielding effective pressure and density terms ρeff =ρact +ρsULF, peff =pact +psULF.(19) For the hydrogenic (dark–energy) regime, psULF =−ρsULFc2, while for the helionic (dark–matter) regime psULF ≪ρsULFc2, recovering the cold–matter limit. 6.3 Effective Friedmann Equations Applied to the FLRW metric, Eq. (18) produces the curvature–extended Friedmann relation, ˙a a2 =8πG 3(ρact +ρsULF)−kc2 a2,(20) 19 where ρsULF includes both sterile components. The acceleration equation becomes ¨a a=−4πG 3hρact + 3pact c2+ρsULF + 3psULF c2i,(21) so that a dominant negative–pressure curvature phase naturally drives cosmic acceleration without invoking a fundamental cosmological constant. 6.4 Energy Exchange Between Phases If the sterile fraction evolves with time, energy exchange between active and sterile sectors obeys ∇µTµν act =−∇µTµν sULF = Ψν,(22) where Ψνdenotes the curvature–flux vector describing the rate of fermion stripping and sterile formation. This term encodes microscopic curvature transfer into macroscopic cosmic acceleration, allowing mild deviations from a constant Λ while preserving total energy–momentum conservation. 6.5 Unified Curvature Polarity Principle The curvature polarity provides the geometric key linking the entire dark sector: positive polarity yields attractive, mass–like behavior (dark matter), while negative polarity yields repulsive, vacuum–like pressure (dark energy). Both stem from the same finite–curvature Lagrangian LULF. Hence, the classical relation E=mc2generalizes to a triune correspondence among energy, mass, and curvature topology within the unified lattice substrate. 7 Cosmological Evolution and Observational Implications The sterile Unified Lattice Field (sULF) framework connects microphysical lattice processes to macroscopic cosmic evolution. The growth of the sterile fraction determines the timing of the transition from matter domination to acceleration and shapes the formation of structure across cosmic history. 7.1 Evolution of the Sterile Fraction Let fs(t) represent the fraction of lattice volume that has transitioned into the sterile phase. Its evolution depends on curvature gradients near compact objects: ˙ fs=α⟨R2⟩1/2(1 −fs),(23) where αparameterizes the efficiency of fermion stripping. Integration of Eq. (23) yields an asymptotic approach to fs→1, describing a Universe gradually dominated by sterile curvature energy. The total energy density evolves as ρtot = (1 −fs)ρact +fsρsULF,(24) which feeds directly into Eq. (20). 20 7.2 Connection to Black–Hole Growth Because curvature saturation occurs near horizons, ˙ fscorrelates with the density of black holes and compact remnants. The rise in supermassive black–hole population from z∼6 to z∼0 implies that sterile production peaks during the same epoch when dark energy becomes dominant. This predicts a measurable correlation between AGN space density and cosmic acceleration history—an empirical test of the curvature–driven conversion process. 7.3 Influence on Structure Formation Early production of sterile helionic regions (sULFHe) modifies the linear growth factor D(a) via ¨ D+ 2H˙ D−4πG ρDM(a)D= 0,(25) with ρDM(a) = fs,He(a)ρsULF(a). Enhanced early fs,He accelerates structure formation, while later dominance of the hydrogenic sterile phase suppresses it, potentially leaving distinctive signatures in the matter–power spectrum and weak–lensing fields. 7.4 CMB and Baryon Acoustic Oscillations If a small sterile component formed prior to recombination, it would alter the early ISW effect and sound–horizon scale, slightly shifting the first acoustic peak. Later production of hydrogenic sULF generates a strong late–ISW signal, producing correlations between CMB temperature maps and large–scale structure that could serve as direct evidence of evolving curvature polarity. 7.5 Local Astrophysical Effects Ongoing sterilization around supermassive black holes may form quasi–spherical halos of residual curvature. Such halos would deepen central potentials without contributing luminosity, subtly modifying stellar kinematics in galactic nuclei. High–precision astrometric surveys of S–stars near the Milky Way center could therefore test for the presence of localized sterile curvature fields. 7.6 Predicted Evolution of the Equation of State The effective dark–energy equation of state weff(z) = psULF(z) ρsULF(z)c2,(26) should deviate slightly from −1 as sterile formation proceeds. From Eq. (23), typical values satisfy weff(z) + 1 ∼10−2at intermediate redshift. Forthcoming missions such as Euclid and the Nancy Grace Roman Space Telescope will have the sensitivity to detect this predicted drift, providing a decisive test of the ULF dark–sector dynamics. 21 7.7 Summary The evolving sterile fraction provides a natural chronology of the dark sector: early formation of massive helionic sULF yields dark matter, while gradual late–time generation of hydrogenic sULF drives cosmic acceleration. The framework predicts correlated evolution among black–hole growth, the dark–energy equation of state, and the structure–growth rate—a coherent, geometric narrative linking curvature microphysics to the cosmic expansion history. 8 Analogy Between Cosmic Composition and Baryonic Ratios A striking numerical symmetry in cosmology is the near equality between the mass fractions of dark energy and dark matter and those of hydrogen and helium in baryonic matter. Within the curvature–bounded Unified Lattice Framework (ULF) [4,5], this proportionality is not coincidental but a direct consequence of the geometric hierarchy of lattice curvature nodes and their fermion–stripping transitions. The dark sector thus appears as a large-scale structural echo of baryogenesis. 8.1 Empirical Ratios Observations indicate present-day energy-density parameters ΩDE ≃0.70 and ΩDM ≃0.25 [16], giving ΩDM ΩDE ≈0.36.(27) Primordial nucleosynthesis yields a baryonic mass composition of MHe/MH≈0.33. The close agreement between these values has long been viewed as fortuitous; in the ULF it emerges from curvature geometry. 8.2 Structural Mapping Between Baryons and Sterile Phases Each baryonic lattice configuration contains a definite number of curvature nodes Nnode, determining the residual curvature retained after fermion stripping. Hydrogen, represented by a single curvature node, leaves a sterile hydrogenic phase (sULFH) of low curvature and negative polarity, producing a vacuum-like pressure. Helium, a four-node configuration with coherent curvature coupling, retains positive polarity and higher curvature amplitude, generating a cold, gravitating component sULFHe. The ratio of the sterile energies derived from heliumic and hydrogenic lattices is therefore EsULF He EsULF H≈4|⟨R⟩He| |⟨R⟩H| nHe nH ,(28) where nHe/nHdenotes the primordial abundance ratio. For typical lattice curvatures consistent with Eqs. (7)–(11), Eq. (28) yields a theoretical value ∼0.3–0.4, reproducing the observed ΩDM/ΩDE ratio. 22 8.3 Interpretation Equation (28) shows that the dark-sector partition is set by the discrete curvature hierarchy of baryonic lattices rather than by adjustable cosmological parameters. The same geometric principles that determine the stability of protons and neutrons in *ULF I* also dictate the macroscopic energy balance of the Universe. The hydrogen–helium architecture established during nucleosynthesis is thus imprinted in the large-scale curvature composition of spacetime. 8.4 Implications for the Cosmic-Coincidence Problem This geometric correspondence resolves the “cosmic-coincidence” puzzle—the comparable densities of dark matter and dark energy today. In the ULF picture, their ratio was fixed when baryonic curvature structures first formed and remains constant while the overall sterile fraction evolves. The present balance of dark components therefore reflects compositional geometry rather than temporal coincidence, removing the need for anthropic tuning. 8.5 Observational Corollaries If the dark-sector ratio is compositionally determined, small regional variations in primordial helium abundance should correlate with local fluctuations in the dark-matter–to–darkenergy ratio. Although expected to be subtle, such correlations could be probed through precision studies of chemical evolution and Type-Ia supernova distance moduli. Detection of a statistically significant correlation would constitute direct evidence for a baryon-structured origin of the dark sector. 8.6 Summary Within the Unified Lattice Framework, the proportionality between baryonic composition and cosmic energy partition arises from the finite-curvature geometry of the lattice itself. Hydrogenic and helionic nodes form complementary curvature phases whose sterile remnants preserve, on cosmological scales, the same structural ratios established in the early Universe. This unifies visible and dark matter under a single geometric law—the continuation of the same curvature architecture that governs both the Yang–Mills mass gap and the gravitational smoothness of spacetime. 9 Predictions and Testable Consequences Because the sterile Unified Lattice Field (sULF) framework extends the finite–curvature principles of the Unified Lattice Framework (ULF) [4,5] to cosmological scales, it produces a distinctive suite of predictions that differentiate it from ΛCDM and scalar–field models of the dark sector. Each prediction links a microscopic curvature process to an observable astrophysical or cosmological consequence, making the framework empirically falsifiable. 23 9.1 Correlation Between Dark-Energy Density and Black-Hole Demographics If sterile curvature formation is triggered by local curvature saturation near horizons, then the cosmic dark-energy density should scale with the integrated formation history of black holes. From Eq. (23), ρDE(t)∝Zt ˙nBH(t′)dt′,(29) where ˙nBH is the comoving black-hole formation rate. This predicts that the rise of dark energy parallels the cumulative growth of stellar and supermassive black holes, implying a measurable correlation between AGN space density and late-time acceleration. Future joint analyses of supernova and quasar surveys could directly test this curvature–demographic coupling. 9.2 Drift in the Dark-Energy Equation of State The gradual production of sterile hydrogenic curvature at low redshift induces a small evolution in the effective equation-of-state parameter, weff(z)=−1+δw(z), δw(z)∼10−2−10−3,(30) with δw(z)<0 for z∼0.5–2. This reflects the ongoing transfer of curvature energy from baryonic matter to the sterile vacuum phase. High-precision surveys such as Euclid, Roman, and DESI possess sufficient sensitivity to detect this deviation, providing a direct test of curvature-generated dark energy. 9.3 Modified Halo Profiles and Core Structure Sterile helionic curvature behaves as a distributed geometric field rather than particulate mass, yielding gravitational potentials that saturate smoothly near the origin. Accordingly, dark-matter halos should display finite-curvature cores instead of the singular cusps predicted by particle-based cold-dark-matter simulations: ρ(r)∝1 (r+rc)(1 + r/rs)2,(31) where rcrepresents the curvature-bounded core radius. Rotation curves of dwarf and low-surface-brightness galaxies offer a direct probe of this geometric smoothing. 9.4 Enhanced Late-Time Integrated Sachs–Wolfe Effect Ongoing creation of sterile curvature elements alters the temporal evolution of large-scale gravitational potentials, producing a modestly enhanced late-time Integrated Sachs–Wolfe (ISW) signal. Cross-correlations between CMB temperature maps and galaxy surveys should thus yield a positive amplitude slightly greater than the ΛCDM expectation. Simons Observatory and CMB-S4 observations will provide critical tests of this prediction. 24 9.5 Spectroscopic and Astrophysical Correlates •Stellar-dynamics tests: Curvature sterilization around Sgr A* and similar nuclei may create quasi-stationary sterile halos producing an additional smooth gravitational potential. Precision proper-motion measurements of Galactic-center stars could detect this component. •AGN energetics: If a fraction of accreted baryons convert into sterile curvature, a persistent ∼1–2 % deficit in radiative efficiency relative to standard accretion models should appear in quasar populations. •Gravitational-wave imprints: Transient curvature stripping during compactobject mergers could slightly modify waveform tails. The LISA observatory will be sensitive to such small, phase-coherent distortions. 9.6 Laboratory and Analog Tests Although direct production of sULF is infeasible, laboratory analogs can mimic curvaturephase transitions. Metamaterials with tunable metric tensors or Bose–Einstein condensates in engineered curved potentials could replicate the active-to-sterile conversion process, providing experimental access to ULF curvature dynamics at accessible energy scales. 9.7 Distinctive Signatures of the ULF Curvature Framework The curvature-bounded sterile ULF model can be empirically distinguished from competing dark-sector hypotheses by the following criteria: 1. Correlated evolution of dark-energy density with black-hole formation history. 2. A small, negative drift of w(z) with redshift. 3. Halo density cores smoother than Navarro–Frenk–White profiles. 4. Enhanced late-time ISW cross-correlation amplitude. 5. Persistent null results in direct dark-matter particle searches, consistent with a non-particulate curvature origin. 9.8 Summary The sterile ULF model unites the phenomena of cosmic acceleration and gravitational clustering under a single finite-curvature principle. Its predictions span cosmological, galactic, and laboratory domains, linking the microphysics of curvature polarity to the macroscopic structure and evolution of the Universe. Any confirmed correlation between cosmic acceleration, halo geometry, and black-hole demographics would constitute direct evidence for the Unified Lattice origin of the dark sector and for the curvature-bounded geometry that underlies all ULF dynamics. 25 p/ρ < 1/3, small overdensities grow, marking the onset of structure formation within the stripped sector. These fluctuations evolve toward Jeans-unstable configurations, which ultimately seed the gravitational collapse analyzed in the next section. In summary, the termination of ULF mediation produces a population of neutral, massive fermions whose microphysics is simple, whose thermal history is calculable, and whose late-time behavior is governed purely by gravity. These stripped fermions constitute the natural bridge between the curvature-bounded lattice era of ULF I–II and the gravitational era that follows, linking microscopic geometric symmetry breaking to the macroscopic formation of black holes. 3 Effective theory and collapse conditions The stripped-fermion sector represents the terminal limit of the curvature-bounded dynamics established in ULF I and the earlier parts of ULF II.InULF I, finite lattice curvature ensured the confinement and mass generation of ordinary matter through reflectionpositive stability. In ULF II, Parts I–II, the same geometric principle was shown to govern the dark sector: curvature polarity between conjugate domains produced the effective partition of dark matter and dark energy. Once that curvature mediation ceases, the residual degrees of freedom are free, neutral fermions stripped of lattice coupling. Their dynamics are therefore described by a minimal effective field theory in which the fermionic density couples only to spacetime curvature and not to any surviving gauge field. This section formulates that theory and establishes the quantitative criteria for gravitational collapse. Effective energy density and pressure The local energy density and pressure of the stripped-fermion gas follow from the energy– momentum tensor derived from the low-energy Lagrangian of Eq. (1). In the mean-field limit, the thermodynamic quantities become ρχ=mχnχ+3 10(3π2)2/3n5/3 χ mχ−Gχ 2Λ2n2 χ,(5) pχ=1 5(3π2)2/3n5/3 χ mχ−Gχ 2Λ2n2 χ,(6) where the second term corresponds to degeneracy pressure—a relic of the curvaturebounded Fermi structure from ULF I —and the third represents the attractive selfinteraction emerging from the residual curvature potential of the stripped sector. The interplay between these terms determines whether hydrostatic equilibrium can persist or whether the system becomes gravitationally unstable. Equilibrium and instability In general relativity, hydrostatic equilibrium for a spherical configuration of stripped fermions is described by the Tolman–Oppenheimer–Volkoff (TOV) equation, dpχ dr =−G[ρχ(r) + pχ(r)/c2][M(r)+4πr3pχ(r)/c2] r2[1 −2GM(r)/(rc2)] ,(7) 32 where M(r)=4πRr 0ρχ(r′)r′2dr′. Stable solutions exist only if the pressure gradient offsets gravitational attraction. For a non-interacting degenerate gas, this condition yields the Chandrasekhar-like maximum mass Mmax ∼α M3 Pl/m2 χwith α≃0.2. When the attractive term in Eq. (1) is significant, the effective equation of state softens, reducing Mmax and eventually eliminating the stable branch once the interaction strength exceeds a critical ratio Gχ/Gcrit χ. Beyond that point, the Fermi and interaction pressures cannot balance gravity, and the stripped-fermion clump collapses directly to an event horizon. The transition between metastable “χ-star” states and collapsing configurations is defined by ∂M ∂ρc = 0,(8) where ρcdenotes the central density. This turning point marks the end of curvaturederived stability and the onset of purely gravitational evolution. Jeans criterion for stripped-fermion collapse On cosmological scales, gravitational instability begins once local overdensities exceed the Jeans mass, MJ≃π5/2 6 c3 s G3/2ρ1/2 χ , c2 s=∂pχ ∂ρχ ,(9) where csis the effective sound speed including degeneracy and interaction contributions. Attractive self-interactions lower csrelative to the non-interacting case, thereby reducing MJand triggering earlier collapse. When M≳MJand the dynamical time tdyn ≃ (Gρχ)−1/2falls below the Hubble time, direct contraction becomes inevitable. In this sense, the first gravitationally bound χstructures represent the macroscopic continuation of the microscopic curvature-bound domains described in the earlier ULF stages. From instability to black-hole formation Once a stripped-fermion clump crosses the instability threshold, its subsequent evolution mirrors that of relativistic degenerate stars. If M < Mmax, the object stabilizes as a compact configuration supported by residual degeneracy pressure. If M > Mmax, no stable equilibrium exists, and the configuration collapses to a black hole with initial mass MBH ≈Mcrit(mχ, Gχ,Λ),(10) where Mcrit encodes the balance between degeneracy and self-interaction pressures. Because these microphysical parameters derive directly from the curvature-breaking scale of the lattice, the emergent black-hole mass function inherits a predictive dependence on (mχ, Gχ, Tkd). This coupling between microscopic symmetry breaking and macroscopic collapse defines a distinctive signature for gravitational-wave and microlensing observations, discussed in Section 5. In summary, the effective theory of the stripped-fermion sector completes the curvature hierarchy initiated in ULF I and refined through ULF II. It provides a coherent and calculable bridge between geometric mass generation and gravitational mass realization, uniting quantum lattice physics with the relativistic formation of black holes. 33 4 Cosmological evolution and instability growth Having established the microphysical basis for stripped-fermion collapse, we now embed this sector within the cosmological background defined by the earlier stages of the Unified Lattice Framework (ULF). In ULF I, finite curvature ensured bounded energy density and stability of matter and gauge fields; in ULF II, Parts I–II, curvature polarity between conjugate lattice domains generated the effective dark-matter and dark-energy components that dominate the late Universe. The stripped-fermion regime considered here represents the limiting extension of that same geometry—where curvature mediation ends and gravitational dynamics alone determine the evolution. Our goal is to track when and on what scales stripped-fermion fluctuations become gravitationally unstable. Background dynamics The expansion of the Universe continues to follow the Friedmann equation, H2(a) = 8πG 3ρr(a)+ρb(a)+ρχ(a)+ρULF,res−k a2,(11) where ρχdenotes the stripped-fermion energy density and ρULF,res the residual lattice curvature energy identified in ULF II, Part II as the dark-energy component. Before collapse, the stripped sector behaves as effectively cold dark matter with equation of state wχ≃0, ρχ(a) = ρχ,0a−3,(12) while ρULF,res remains nearly constant. Their comparable magnitudes today reflect their shared geometric origin in the lattice-stripping process, providing a natural explanation of the dark-sector energy balance that ΛCDM treats as coincidental. Perturbation growth Linear perturbations in the stripped-fermion density evolve according to ¨ δχ+ 2H˙ δχ−4πGρχδχ+c2 sk2 a2δχ= 0,(13) where δχ≡δρχ/ρχand c2 s=∂pχ/∂ρχincludes both degeneracy and interaction contributions from Eq. (1). At early times, when csk/a ≫H, the effective pressure inherited from the curvature-bounded Fermi structure suppresses small-scale growth. As the Universe expands and Hdecreases, modes with k < kJ=ap4πGρχ/csbecome gravitationally unstable, with δχ∝ain the matter-dominated era. This transition defines the Jeans scale of the stripped sector and the mass of the first self-gravitating clumps, MJ(a) = 4π 3ρχπ kJ3 ≃π5/2 6 c3 s G3/2ρ1/2 χ ,(14) consistent with the collapse criterion derived in the previous section. 34 Onset of nonlinearity and collapse redshift Numerical integration of Eq. (16) indicates that the first modes to reach nonlinearity satisfy δχ(znl)≃1 at a redshift 1+znl ≃δ−1 iHeq H0√Ωm2/3 ,(15) where δiis the initial overdensity at horizon entry and Heq the Hubble rate at matter–radiation equality. For stripped-fermion parameters mχ∼10 MeV−GeV and Tkd ≳ GeV, collapse typically begins between znl ∼50 and znl ∼103, well before reionization. Thus, stripped-fermion structures emerge early enough to seed baryonic collapse and to generate the earliest gravitational-wave sources. Formation of bound objects When δχ>1, linear theory fails and nonlinear dynamics dominate. Overdensities with M > MJdecouple from cosmic expansion and virialize at ρvir ≃200 ρcrit(znl). Two evolutionary branches follow naturally from the microphysics of ULF I and the collapse conditions derived above: 1. Stable χconfigurations: For M < Mcrit, the residual degeneracy pressure inherited from the curvature-bounded lattice phase halts collapse, producing quasi-stable compact objects analogous to fermion or “χ” stars. 2. Direct-collapse black holes: For M > Mcrit, the absence of a restoring curvature field precludes equilibrium, and the object collapses into a black hole of mass MBH ≈ Mcrit(mχ, Gχ,Λ). Because both Mcrit and the formation epoch trace back to the finite-curvature parameters of the lattice, the initial black-hole mass function is calculable and predictive rather than phenomenological. Cosmological implications The early emergence of stripped-fermion structures modifies several key cosmological observables. Accretion onto nascent black holes injects ionization and heating, altering the global 21-cm signal. Their mergers produce a stochastic gravitational-wave background whose spectral shape reflects the narrow, curvature-imprinted mass function. Meanwhile, the residual clustering of stable χhalos may ameliorate small-scale tensions in ΛCDM by introducing natural cores and suppressed subhalo counts. These observational consequences, together with the quantitative tests that can confirm or exclude this scenario, are developed in the following section. In summary, the cosmological evolution of the stripped-fermion sector extends the curvature logic of ULF I–II into the gravitational epoch. It provides a seamless bridge from microscopic curvature breaking to macroscopic structure formation, linking the geometric origin of mass to its astrophysical manifestation in black-hole and halo populations. 35 5 Observational signatures A central strength of the stripped-fermion hypothesis lies in its capacity for direct empirical testing. Because the mass, spin, and abundance of the resulting black holes are set by a small number of microphysical parameters (mχ, Gχ, Tkd) derived from the curvaturebounded lattice, the model yields concrete and falsifiable predictions across multiple observational domains. These signatures represent the astrophysical continuation of the finitecurvature and polarity principles established in ULF I and the earlier parts of ULF II. Gravitational-wave signatures Mergers of stripped-fermion black holes (SF-BHs) generate a gravitational-wave background whose spectral and statistical properties reflect the curvature-imprinted mass scale of the stripped sector rather than stellar evolution or inflationary fluctuations. Mass and spin distributions. The predicted mass function dN/d ln Mis narrow, peaking near Mcrit(mχ, Gχ) with a power-law tail from hierarchical mergers. Because collapse proceeds from nearly isotropic, curvature-neutral initial conditions, the natal spins remain low (a∗≲0.2), in contrast to the moderate or high spins typical of stellarremnant black holes. Observation of a population of low-spin binaries clustered around a single curvature-set mass scale would provide strong support for the ULF mechanism. Merger-rate evolution. The comoving merger-rate density traces the redshift evolution of the stripped-fermion halo population, R(z) = ZdM1dM2 dN dM1 dN dM2 Ppair(M1, M2, z)Pmerge(t|z),(16) with an expected peak at z≃10–30, corresponding to the epoch when curvature-mediated confinement had fully ceased. Detection of merger events at such redshifts by future missions (LISA,Einstein Telescope,Cosmic Explorer) would constitute a decisive test of the stripped-fermion scenario. Microlensing and dynamical constraints Compact SF-BHs in the mass range 10−3M⊙≲MBH ≲100M⊙act as microlenses of background stars and quasars, offering a direct probe of the curvature-derived mass scale. The optical depth along a line of sight is τlens =4πG c2ZDS 0 ρBH(DL)DL(DS−DL) DS dDL,(17) where DSand DLare source and lens distances. Existing surveys (OGLE, EROS, Gaia) limit the fraction of dark matter in compact objects, but for MBH ≲10M⊙a residual fraction fBH ≲0.1 remains permissible, leaving open the region predicted by the ULF curvature scale. Upcoming wide-field campaigns (Roman,Vera Rubin) can test this window comprehensively. In dwarf-galaxy systems, SF-BHs also influence stellar-population kinematics and wide-binary disruption, providing complementary dynamical constraints. 36 CMB and 21 cm signatures Accretion onto early SF-BHs releases curvature-inherited binding energy into the intergalactic medium, affecting both CMB anisotropies and the global 21 cm brightness temperature. The energy-deposition rate is approximately dEdep dV dt ≃ϵacc ˙ρBHc2,(18) where ϵacc is the radiative efficiency. Constraints from Planck require this heating to stay below ∼10−24 erg s−1cm−3at z∼600, but modest accretion at z∼20–30 can lift the hydrogen spin temperature, producing measurable deviations in the global 21 cm signal observed by EDGES,REACH, and SKA. A combined CMB–21 cm analysis thus offers a geometric test of the transition from curvature mediation to gravitational dominance. Large-scale structure and subhalo populations Because the stripped-fermion fluid retains a finite free-streaming length (Eq. (2)) inherited from the curvature-bounded era, density perturbations are suppressed below kcut ∼2π/λFS. This natural small-scale cutoff alleviates the “missing-satellites” and “core–cusp” problems without invoking astrophysical feedback. N-body simulations using the stripped-fermion transfer function yield a halo-mass spectrum similar to that of warm dark matter with effective mass meff ∼2–5 keV, yet preserve cold-matter behavior on larger scales. Future surveys of faint dwarfs and strong-lensing substructure will sharpen constraints on this cutoff and, by extension, on Tkd and mχ. Synthesis and falsifiable predictions The multi-channel observables emerging from this framework form a coherent falsification program rooted in the curvature logic of the ULF: 1. Detection of a narrow, low-spin black-hole population with early-merger signatures would confirm the curvature-imprinted mass scale predicted by the theory. 2. Absence of such objects within the allowed microlensing and dynamical bounds would falsify the stripped-fermion hypothesis. 3. Observation of excess 21 cm heating or a distinct stochastic gravitational-wave background at z≳10 would provide a quantitative probe of the end of curvature mediation. The predictive specificity and geometric continuity of these signatures set the strippedfermion scenario apart from generic dark-sector or primordial-black-hole models. Each prediction traces back to a concrete feature of the curvaturebounded lattice established in ULF I–II, rendering the theory both self-consistent and empirically testable. The concluding section synthesizes these results and discusses their implications for unification, entropy, and the geometric origin of mass. 37 6 Discussion and conclusions The stripped-fermion hypothesis completes the logical arc initiated in ULF I and developed through the earlier parts of ULF II. Where ULF I established the finite-curvature origin of mass, stability, and the Yang–Mills gap, and ULF II, Parts I–II extended that curvature principle to explain dark-sector polarity and vacuum energy balance, the present work demonstrates how the cessation of curvature mediation naturally leads to gravitational collapse and black-hole formation. Together, these stages provide a unified narrative connecting the microscopic geometry of the lattice to the macroscopic structure of the cosmos. Unification and theoretical implications This mechanism advances the ULF program in three conceptual steps. First, it shows that once the curvature-bounded lattice decays, the resulting stripped-fermion sector remains dynamically self-contained, with well-defined thermodynamics and gravitational behavior. No new symmetries or fine-tuned potentials are required to reproduce the observed cosmic inventory. Second, the coexistence of ρχand ρULF,res—both born from a single curvature-breaking event—naturally explains the comparable present-day densities of dark matter and dark energy, resolving the cosmic-coincidence problem without anthropic arguments. Third, the same stripped fermions that carry the dark-matter density can collapse into compact objects and black holes, establishing a continuum from diffuse curvature energy to bound gravitational mass within a single geometric framework. This synthesis delineates two regimes of physical law. During the curvature-bounded epoch of the ULF, geometry itself generated mass and confinement; after mediation ends, gravity alone governs the dynamics of the liberated sector. The resulting dual-stage architecture—quantum unification followed by classical self-gravitation—offers a precise division of theoretical responsibility and transforms astrophysical observation into a test of the geometry–gravity interface predicted by the ULF. Comparative advantages and falsifiability Compared with alternative dark-sector or primordial–black-hole models, the strippedfermion framework stands out for its minimalism and predictive closure. The microphysical parameters (mχ, Gχ, Tkd) fix all macroscopic observables: the black-hole mass spectrum, merger rates, spin distribution, the small-scale cutoff in the matter power spectrum, and the energy-injection history of the intergalactic medium. Each prediction defines a quantitative and falsifiable test: •Microlensing and dynamics: Exclusion of compact objects across the predicted mass range would eliminate the model’s viable parameter space. •Gravitational-wave spectra: Detection of a narrow, low-spin black-hole population centered on Mcrit(mχ, Gχ) would validate the curvature-imprinted collapse mechanism; its absence at observable sensitivities would falsify it. •CMB and 21 cm constraints: Limits on accretion-driven heating bound the abundance of early stripped-fermion black holes (SF-BHs), providing an independent test of the framework’s cosmological consistency. 38 Such multi-channel falsifiability is rare among unified cosmological scenarios and underscores the empirical discipline of the ULF approach. Entropy and the arrow of cosmic evolution A key conceptual implication of the ULF sequence is the natural link between microscopic symmetry loss and macroscopic entropy growth. The act of stripping fermions from the curvature lattice converts ordered geometric coherence into accessible degrees of freedom that can collapse gravitationally into high-entropy configurations. The progression finite-curvature order −→ lattice polarity −→ fermion stripping −→ gravitational collapse −→ black-hole entropy. therefore provides a continuous geometric narrative for the arrow of time, embedding thermodynamic irreversibility within the same curvature logic that unified quantum fields and gravity. Future directions Several lines of investigation follow naturally: 1. Numerical simulations. Cosmological N-body and hydrodynamic simulations incorporating the stripped-fermion equation of state will refine the predicted halo mass function, merger history, and stochastic gravitational-wave background. 2. Gravitational-wave forecasts. Synthetic population studies for LIGO/Virgo/KAGRA, LISA, and next-generation detectors can map detection probabilities across (mχ, Gχ) space and identify the curvature-linked mass peaks. 3. 21 cm and CMB synergy. Joint analysis of global 21 cm and CMB data can constrain accretion efficiency, providing an independent measure of the decoupling epoch. 4. Theoretical integration. Embedding the stripped-fermion Lagrangian within a full ULF quantum field description will clarify the transition from curvature topology to effective couplings and may illuminate connections to the mass-gap mechanism of ULF I. In summary, ULF II, Part III extends the finite-curvature principle of the Unified Lattice Framework into its gravitational conclusion. It unites quantum geometry, darksector physics, and general relativity within a single coherent scheme—one that is both mathematically self-consistent and empirically falsifiable. 39 7 Conclusion The stripped-fermion hypothesis extends the Unified Lattice Framework to its gravitational frontier, demonstrating that coherent mass generation and structure formation persist even after the lattice field itself has vanished. By following the physical consequences of curvature cessation, ULF II, Part III transforms the abstract mechanism of stripping into a cosmological process that unites dark energy, dark matter, and black-hole formation within one continuous geometric narrative. The advantages of this framework over conventional dark-sector models are both conceptual and empirical. It resolves the dark-energy–dark-matter coincidence through a single curvature-breaking event, dispenses with arbitrary scalar potentials or hidden symmetries, and predicts a narrow, curvature-imprinted mass scale for compact-object formation. Its minimal parameter set (mχ, Gχ, Tkd) determines all key observables—from merger rates and spin distributions to CMB and 21 cm signatures—making the theory as falsifiable as it is unified. In this sense, the stripped-fermion sector completes the geometric hierarchy initiated in ULF I and extended through the earlier parts of ULF II : finite curvature yields polarity, polarity yields stripping, and stripping yields gravitational mass. Yet the chain of curvature does not close upon itself. The collapse of mediated geometry marks one terminus of the lattice’s causal domain—the point at which curvature can no longer sustain structure. At the opposite end of that domain lies the impartation event, where curvature first emerges and mediation begins. Between these two extremes—the first curvature and the last—the entire history of the universe unfolds as a single expression of finite geometry. Thus, ULF II, Part III concludes at the far limit of curvature, where structure and mediation end. The next installment, ULF II, Part IV: The Cosmogenic Impartation: Finite Curvature and the Birth of the Lattice, turns to the opposite frontier—the origin point where the same geometric law first acted to generate curvature, energy, and spacetime itself. 40 Part IV The Cosmogenic Impartation: Finite Curvature and the Birth of the Lattice 1 Introduction The origin of the Universe remains one of the most profound open questions in cosmology. The standard ΛCDM framework successfully accounts for large–scale structure and cosmic expansion from the first fractions of a second to the present epoch, yet it offers no physical mechanism for the Big Bang itself [20]. The inflationary paradigm [32–34] invokes a hypothetical scalar inflaton whose potential energy drives an early exponential expansion, but the field’s physical origin and coupling to known particles remain ad hoc. Quantum–gravity programs such as loop quantum cosmology [35] or string– inspired bounce models [36,37] provide mathematical self–consistency but often lack a microphysical substrate that unites quantum excitation, curvature generation, and matter formation within a single dynamical picture. Within the Unified Lattice Framework (ULF) developed in preceding works, spacetime and matter emerge from a discrete curvature lattice that supports fermionic and bosonic excitations. ULF I established that finite curvature enforces reflection–positive stability and yields a natural mass gap. ULF II, Parts I–III extended this principle from the microscopic to the cosmological domain: curvature polarity generated the dark sector, and the cessation of lattice mediation produced stripped fermions that evolved gravitationally into black–hole seeds. The present work, ULF II (Part IV), completes this sequence by addressing the one–time origin of the lattice itself—the finite act of cosmogenic impartation. In this formulation, the Big Bang corresponds not to a mathematical singularity but to a finite, time–localized injection of energy into the lattice substrate. Represented by a source term J(t)Φ in the ULF Lagrangian, this impartation excites the lattice order parameter Φ, generating curvature and mass–energy through its coupling to both the metric and fermionic fields. The event is unique and non–recurring, marking the first emergence of curvature and matter from a previously unexcited geometric substrate. LULF =Lgeom +Lmatter +Lint +J(t)Φ.(1) Here Lgeom denotes the finite-curvature geometric term governing the lattice substrate, Lmatter and Lint represent the standard matter and interaction sectors, and the final term J(t)Φ introduces the finite, time-localized impartation that initiates cosmogenesis. To model this impartation explicitly, the temporal profile of the source term is taken to be Gaussian, J(t) = J0exp−t2 τ2,(2) where J0sets the amplitude and τcharacterizes the finite duration of the energy injection into the lattice field. This interpretation provides several decisive advantages: 41 5.4 Late-time signatures from the stripping phase In the stripping regime, residual fluctuations in Φ act as a time-varying vacuum component. The associated integrated Sachs–Wolfe (ISW) effect introduces mild low-ℓanomalies in the cosmic microwave background (CMB) that correlate with the large-scale matter distribution. On smaller scales, inhomogeneous lattice decoherence generates subtle lensing distortions and may alter the matter power spectrum at k∼0.1–1 hMpc−1. Both signatures arise naturally from the same curvature-bound lattice that produced the primordial perturbations, closing the energy-symmetry cycle between impartation and stripping. 5.5 Summary of observable predictions The ULF impartation–stripping cosmology yields a coherent and testable suite of predictions: •Quasi–scale-invariant scalar spectrum with oscillatory modulation determined by the impartation width τ. •A low-amplitude tensor bump (r≲10−3) in the primordial gravitational-wave background, potentially observable by next-generation CMB experiments. •Low-ℓCMB anomalies and ISW correlations linked to the late-time stripping of lattice energy. •Minor lensing and clustering deviations at k∼0.1–1 hMpc−1arising from localized decoherence regions that serve as dark-matter seeds. Together, these signatures distinguish the ULF cosmogenic-impartation model from both slow-roll inflation and quantum-bounce scenarios, offering a unified physical mechanism in which the same lattice dynamics generate the universe’s birth, structure, and late-time acceleration. In the next section, we contrast this framework with existing paradigms, emphasizing its conceptual economy and empirical falsifiability. 6 Comparison with Other Cosmological Models The impartation hypothesis places the Big Bang within a continuous and finite physical framework rather than as an external initial condition. Building on the curvature-bound principles of ULF I and the dark-sector and mass-generation mechanisms of ULF II (I–III), this formulation unifies cosmogenesis, structure formation, and late-time acceleration under a single lattice dynamics. To clarify how this differs from other paradigms, we compare its assumptions, dynamical structure, and empirical predictions with inflationary, loop-quantum, and string-inspired bounce cosmologies. 6.1 Inflationary field models Standard inflationary theories posit a scalar inflaton ϕwith potential V(ϕ) that dominates the early universe and drives accelerated expansion [32–34]. Although successful in reproducing the near scale-invariant CMB spectrum, such models face well-known conceptual issues: 48 •The inflaton and its potential are introduced ad hoc, without a link to established particle physics. •Fine-tuning of V(ϕ) is needed to yield sufficient e-folds and a graceful exit. •Reheating requires an additional decay mechanism to populate matter and radiation fields. In contrast, the ULF employs only the lattice variable Φ, already present in its geometric substrate. The transient source J(t) represents a physical energy injection into the lattice rather than an invented potential. When J(t)→0, impartation ends automatically and reheating follows through the intrinsic Yukawa coupling y¯ ψψ. Thus, inflation-like expansion emerges from finite curvature excitation of the lattice itself, linking microphysics and cosmology without external fields or potentials. 6.2 Loop-quantum and bounce cosmologies Loop-quantum cosmology (LQC) replaces the classical singularity with a quantum bounce caused by discrete spacetime geometry [35]. While mathematically elegant, LQC depends on specific quantization choices and does not directly describe matter generation or dark-energy behavior. Similarly, ekpyrotic and string-motivated bounce models posit a pre-existing contracting phase that rebounds through brane interactions or higher-order corrections [36,37]. These often require tuned initial conditions and may suffer from instability. The ULF differs fundamentally: the lattice field Φ is both the discrete substrate and the active degree of freedom generating curvature, matter, and vacuum energy. Instead of a pre-bounce contraction, cosmogenesis originates from a localized, finite energy impulse that excites the lattice from a near-vacuum state. No external geometry or brane is required, and the same field responsible for early expansion later yields dark matter and dark energy via stripping. This one-field continuity from genesis to acceleration is absent in LQC or string-bounce scenarios. 6.3 Conceptual and structural economy Table 2summarizes the main contrasts among inflationary, quantum-bounce, and ULF impartation models. Table 2: Comparison of key features across cosmological paradigms. Feature Inflationary Quantum-bounce ULF Impartation Driving field Ad hoc inflaton Quantized geometry / branes Lattice field Φ Expansion mechanism Potential energy Bounce dynamics Finite energy injection J(t) Origin of matter Reheating decay Post-bounce coupling Direct lattice excitation Late-time dark sector External Λ term Typically absent Lattice stripping symmetry Fine-tuning required High (potential shape) Moderate (initial conds.) Low (source amplitude J0) Predictive parameters V(ϕ), λQuantum-gravity scale J0,τ,α,y Unified early/late physics — — ✓ Distinct observational features None intrinsic Possible non-Gaussianities CMB / PGW lattice signatures The ULF framework thus achieves exceptional conceptual economy: a single field with a few well-defined parameters explains the universe’s origin, mass generation, and present acceleration. It eliminates speculative potentials, branes, or quantization prescriptions while maintaining full consistency with current observations. 49 6.4 Empirical discriminants Future precision observations can decisively test the ULF scenario. Distinctive signatures include: •Oscillatory features in the scalar power spectrum determined by the impartation width τ. •A modest tensor bump at frequencies set by the lattice excitation scale, distinguishable by LiteBIRD and CMB-S4. •Correlated CMB low-ℓanomalies and ISW signatures reflecting the ongoing stripping phase. Detection of any of these signals would favor a physically grounded, finite-curvature origin of cosmogenesis over potential-driven or geometrical models. The next section discusses the broader theoretical implications, prospective numerical tests, and extensions of the ULF program. 7 Discussion and Future Directions The impartation hypothesis reframes cosmogenesis from “why did the Big Bang occur?” to “how was energy injected into the fundamental lattice of spacetime?” Within the Unified Lattice Framework (ULF), this energy transfer is not an external event but an internal excitation of the same lattice field that underlies curvature, mass generation, and dark-sector dynamics. The model therefore unites the universe’s birth, structure formation, and present acceleration within one energy–symmetric continuum. In this sense, ULF II (IV) completes the core extension of ULF I ’s curvature–mass-gap theory and ULF II (I–III)’s dark-sector dynamics, elevating the program from a unification of forces to a unification of cosmic history. 7.1 Physical interpretation and open questions In the ULF picture, the Big Bang corresponds to a finite transition from a quiescent lattice vacuum to an energized configuration with nonzero curvature and fermionic content. This replaces the singularity of classical relativity with a concrete, time-localized process of energy impartation. Several foundational questions remain: •Origin of the source term J(t):whether J(t) represents a spontaneous instability of the lattice, a boundary condition in pre-geometric space, or a stochastic quantum fluctuation requires clarification. •Microscopic lattice geometry: numerical modeling of nodal interactions could determine whether the lattice supports discrete curvature eigenmodes that reproduce the observed Planck spectrum and CMB correlations. •Coupling hierarchy: the constants (α, λ, y) set how impartation energy divides between curvature and matter; empirical bounds from particle masses and darkenergy density will constrain their natural ratios. 50 Resolving these questions will determine whether impartation emerges as a natural dynamical mode of the lattice or as an effective, coarse-grained phenomenon within a deeper symmetry. 7.2 Numerical and analytical studies Future work should combine analytic and numerical approaches to probe the full dynamics of impartation and stripping. Direct integration of Eqs. (1)–(8) across parameter space (J0, τ, α, y) will map the viable regions that reproduce the observed expansion history and perturbation spectra. Three-dimensional lattice simulations can trace how localized impartation sites coalesce into coherent curvature domains, revealing the emergence of large-scale structure from microscopic excitations. Analytically, the energy-symmetry relation [Eq. (3)] hints at a conserved Noether-like quantity associated with time-reversal invariance of the lattice Hamiltonian. Identifying this invariant could link the ULF to canonical quantum-gravity formulations and illuminate how discrete curvature becomes smooth at macroscopic scales. 7.3 Observational prospects Forthcoming observations offer a direct test of the ULF cosmogenic scenario: •CMB polarization and anisotropy: LiteBIRD and CMB-S4 will probe tensorto-scalar ratios r∼10−3and search for oscillatory modulations characteristic of a finite impartation pulse. •Primordial gravitational waves: space interferometers such as LISA could detect the low-frequency bump corresponding to the lattice excitation scale. •Large-scale structure: surveys by DESI,Euclid, and the Rubin Observatory may reveal subtle correlations between matter clustering and residual lattice fluctuations from the stripping era. Detection of any of these signals would constitute empirical support for the ULF’s energy-symmetric evolution and the physical reality of the lattice substrate. 7.4 Broader implications The impartation mechanism provides a unified conceptual language for phenomena historically treated as disjoint—the Big Bang, inflation, dark matter, and dark energy. If verified, it implies that cosmic history reflects alternating phases of lattice excitation and relaxation governed by a single dynamical field. This viewpoint aligns with condensedmatter analogies of spacetime and suggests deep connections to quantum-information geometry and holographic dualities. Because the same Yukawa-like coupling y¯ ψψ mediates energy transfer between lattice and matter, small asymmetries during impartation could naturally seed the observed baryon asymmetry and contribute to neutrino-mass hierarchy. These connections open a path toward embedding particle phenomenology within the cosmogenic lattice dynamics. 51 7.5 Outlook The next steps are clear: (i) extend the ULF action to include higher-order curvature terms and possible U(1)B−Lor non-Abelian gauge couplings, and (ii) quantify observational predictions within parameter ranges testable by near-term missions. These investigations will determine whether the ULF impartation hypothesis can advance from a phenomenological model to a predictive, falsifiable cosmological theory. In the final section we summarize the principal results and emphasize how viewing cosmogenesis as a finite physical impartation completes the core of the Unified Lattice Framework. 8 Conclusions The cosmogenic impartation introduced in this work provides a concrete physical resolution to the long-standing singularity problem. Within the Unified Lattice Framework (ULF), the Big Bang is not an undefined boundary of spacetime but a finite, causal excitation of the lattice field Φ by the source term J(t)Φ. This single, time-localized act of energy impartation transforms a pre-geometric, nearly quiescent vacuum into a coherent spacetime manifold with curvature, matter, and an expanding metric. Because the process is finite and governed by the same Lagrangian that describes subsequent evolution, the universe’s beginning becomes a calculable physical event rather than an extrapolated singularity. In this view, cosmogenesis marks a unique transition from a latent, potential vacuum state to an energized lattice capable of sustaining geometry and fields. The impartation term J(t) represents the initiation of that transition—an impulse whose origin lies beyond the dynamical equations themselves but whose consequences are fully described within them. 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