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

International Journal of Quantum Foundations 12 (2026) 261-335 Original Paper Unified Lattice Framework II: Curvature–Bound Resolutions of the Gravitational, Dark–Sector, and Singularity Problems William Hernandez Hebrew University of Jerusalem E-mail: [email protected] Received: 12 November 2025 / Accepted: 8 December 2025 / Published: 10 December 2025 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. Keywords: Curvature-bounded gravity; dark sector; dark energy; dark matter; stripped-fermion dynamics; cosmogenesis; singularity resolution; curvature polarity; finite-curvature cosmology; scalar-tensor lattice Global Introduction The Unified Lattice Framework (ULF) proposes that all fundamental interactions of physics—gauge fields, matter stability, magnetism, incompressible flow, gravitation, dark–sector structure, black–hole seed formation, and cosmogenic evolution—arise from a single geometric principle: a universal upper bound on curvature across a scalar–tensor International Journal of Quantum Foundations 12 (2026) 262 lattice geometry. The finite–curvature constraint introduced in ULF I was shown to yield nonperturbative confinement, stability of matter, and suppression of singular vorticity growth, providing unified geometric resolutions of three major foundational problems. These microscopic results were rigorously developed and extended in the mathematical M–series. The present Part II advances the ULF program into the macroscopic and cosmological regimes. Building on the discrete, curvature–bounded structure of ULF I, we show how gravity emerges without singularities, how the dark sector arises from curvature polarity within scalar and tensor lattice modes, how stripped fermions naturally seed black–hole formation, and how cosmogenic impartation provides a finite–curvature origin of cosmic expansion. In this way, the same curvature bound responsible for microscopic stability governs the largest–scale geometric and cosmological dynamics. To situate Part II within the full ULF architecture, we record the analytic foundations on which the present work depends. Three mathematical companions contain the rigorous derivations invoked throughout this Part: •ULF M4 [1] constructs a curvature–bounded, non–singular formulation of quantum gravity. It establishes the existence of curvature–controlled continuum limits, a bounded Einstein–Hilbert action under Γ–convergence, and the exclusion of curvature blow–up in both the Euclidean and Lorentzian settings. •ULF M7 [2] provides the first rigorous derivation of dark energy, dark matter, and black–hole seeds from the finite–curvature polarity of scalar and tensor lattice sites. It supplies the precise curvature increments, sterile–phase conditions, and collapse thresholds used in Sections II and III of the present Part. •ULF M8 [3] proves the Impartation Theorem, establishing a universal curvature–preserving transfer law governing all energy, curvature, and information exchange across the lattice. This theorem guarantees finiteness and continuity in high–curvature regimes and provides the analytic foundation for the cosmogenic evolution developed in Part IV. The operator–theoretic structures and quantized curvature modes underlying the cosmogenic and gravitational sectors draw on the discrete operator algebra developed in ULF III [4], where the Grand ULF Equation, self–adjointness of curvature operators, bounded spectra, and continuum quantization were established. Finally, the broader conceptual integration of the ULF program—including comparisons with loop quantum gravity, string–theoretic models, asymptotic–safety scenarios, and condensed–matter analogues—is developed in the ULF Synthesis paper [5]. That work demonstrates how the finite–curvature mechanism unifies the microscopic sector International Journal of Quantum Foundations 12 (2026) 263 results of ULF I, the macroscopic and cosmological analyses of the present Part II, and the quantized structures of ULF III. For these reasons, the logical role of Part II is clear: it presents the geometric mechanisms, sector transitions, and phenomenological consequences of the curvature–bound principle in the gravitational, dark–sector, and cosmogenic regimes. Detailed proofs, functional–analytic arguments, and operator constructions are delegated to the M–series and to ULF III. This division of labor parallels that of ULF I and enables a coherent presentation of the physical and geometric content of the Unified Lattice Framework without duplicating extensive analytic material. The four parts of this document follow this structure: (I) curvature–bounded gravity and the de Sitter embedding; (II) emergence of the sterile lattice sector and dark–energy/dark– matter channels; (III) stripped fermions and curvature–seeded black–hole formation; and (IV) finite–curvature cosmogenic impartation. Together they show that the same bounded lattice geometry that governs microscopic interactions also dictates the largest–scale structures of the universe. The final part of this work develops the cosmogenic dynamics implied by the finite–curvature geometry of the Unified Lattice Framework. We begin with the curvature–preserving impartation mechanism established in ULF M8, which governs energy transfer in the high–curvature early universe and replaces the classical big–bang singularity with a finite-curvature cosmogenic phase. 1. Cosmogenic Impartation and the Finite–Curvature Origin of Expansion The curvature–bounded geometry of the ULF admits two distinct modes of energy transfer: (i) impartation, a curvature–preserving redistribution of energy proven in ULF M8, and (ii) fermion stripping, a curvature–increasing process described in Sections II.1–II.4. In the early universe, where curvature is high but still below the collapse threshold, impartation dominates and provides a natural geometric origin for cosmic expansion. 1.1. Impartation flux and curvature preservation The Impartation Theorem of ULF M8 establishes that for any compact region L⊂R3, the impartation flux Jin(L)(1) transfers energy into the region while preserving the curvature bound κ(x)≤κmax. Explicitly, δκ(x) = I[Jin](x),(2) International Journal of Quantum Foundations 12 (2026) 264 where Iis a curvature–preserving transfer operator satisfying |I[Jin](x)| ≤ κmax −κ(x)for all x∈L. Thus impartation injects energy while distributing curvature so that no singularity can form. 1.2. Impartation–stripping transition Energy injection into a region of size Lproceeds by impartation so long as Jin(L)< Q L3κ1/2 max,(3) as established in ULF M8. When this bound is exceeded, impartation fails, and the region transitions to fermion stripping, producing either hydrogenic (scalar) or helionic (tensor) sterile sites, or—if curvature saturates—the collapse sites of Section II.4. This inequality determines the curvature and energy scales of early-universe dynamics. 1.3. Finite–curvature cosmogenesis At times t≲tform (pre-structure-formation epoch), the curvature is high but below threshold: κ(t)≲1 3κmax. Equation (3) then shows that impartation is the dominant energy-transfer mechanism. The energy density evolves according to ˙ρimp =α Jin(L)−3H(t)ρimp,(4) where αis a geometric factor determined by the transfer operator I. This term acts as a curvature–regulated source in the Friedmann equation: H2(t) = 8πG 3ρimp +ρm+ρDE +ρDM +ρseed+k a2.(5) Because impartation keeps curvature bounded, the scale factor a(t)never contracts to zero: lim t→0+a(t)>0, eliminating the big-bang singularity and replacing it with a finite–curvature cosmogenic phase. International Journal of Quantum Foundations 12 (2026) 265 1.4. Transition to sterile-phase production As the universe expands and curvature decreases, the impartation flux becomes insufficient to redistribute injected energy. When Jin(L)≈QL3κ1/2 max, the impartation-to-stripping transition occurs. This triggers the formation of • hydrogenic sterile sites (scalar-polarized, dark energy), • helionic sterile sites (tensor-polarized, dark matter), • curvature-seeded collapse sites (black-hole seeds), linking cosmogenesis directly to the sterile-phase structure of Sections II.2–II.4. 1.5. Observable consequences The impartation-driven cosmogenic phase predicts: • a minimal scale factor amin determined by κmax, • a mild early-time w(t)>−1deviation from ΛCDM, • suppressed small-scale power due to finite-curvature smoothing, • an imprint in primordial gravitational-wave spectra: fimp ∼1 2π√κmax, lying in the mHz to sub-Hz bands for ULF parameters. 1.6. Summary Cosmogenic impartation provides a curvature-preserving mechanism for energy injection in the early universe, eliminates the big-bang singularity, sets the initial conditions for structure formation, and determines the onset of sterile scalar, tensor, and collapse phases. In this way, the finite-curvature geometry of the ULF unifies the origin of cosmic expansion with the microscopic dynamics of sterile-site formation. International Journal of Quantum Foundations 12 (2026) 266 Part I Finite–Curvature Gravitation: Einstein Dynamics from a De Sitter Gauge Lattice 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 [6–8]. 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 [9], the ULF establishes a curvature–bounded scalar lattice Φon 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, self-regularizing form of gravity. In this formulation, spacetime and internal interactions are not separate entities but distinct phases of a unified lattice connection [10]. 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 [11,12], demonstrating that gravitational curvature arises intrinsically from the same finite geometry that underlies the other interaction sectors. Relation to Existing Approaches Before developing the geometric construction in detail, it is useful to situate the curvature–bounded ULF gravity framework within the broader landscape of approaches to International Journal of Quantum Foundations 12 (2026) 267 quantum geometry. The contrasts clarify both the motivation and the structural advantages of the present formulation. Loop quantum gravity. Loop quantum gravity (LQG) discretizes geometry through spin–network states whose area and volume operators have discrete spectra. These structures are powerful but do not automatically incorporate matter fields or guarantee curvature control in the Lorentzian regime. In contrast, ULF discreteness arises from the scalar–tensor lattice with a microscopic curvature bound κ(x)≤κmax =ℓ−2 min . Matter, gauge, magnetism, and gravitational sectors share the same curvature bound. The continuum limit in ULF M4 yields a bounded Einstein–Hilbert functional under Γ–convergence, and horizon finiteness is obtained directly from the curvature bound, without entropy constraints. String theory and higher–dimensional models. String models achieve finiteness by embedding four–dimensional physics into higher– dimensional moduli spaces. Effective constants such as Λand Gdepend on compactification and moduli stabilization. ULF avoids these issues entirely: it is intrinsically four–dimensional, and no moduli or compactification are needed. The cosmological constant arises from the de Sitter curvature radius Λ=3/ℓ2 Λ, and effective couplings such as Gfollow from the relation 8πG ∼ Q2/(κmaxg2). There is no landscape problem: the curvature bound produces a unique coarse–grained geometry. Asymptotic safety. Asymptotic safety seeks ultraviolet completeness via a nonperturbative RG fixed point. This requires delicate control of renormalization flows and truncation schemes. ULF achieves ultraviolet finiteness geometrically: the bound κmax =ℓ−2 min enforces scale–invariant curvature control and removes the need for RG fixed–point assumptions. The BF–type continuum limit in ULF M4 yields a bounded Einstein–Hilbert functional without RG truncations, while automatically incorporating matter and gauge sectors under the same curvature bound. Summary. Across these approaches, the distinctive feature of ULF gravity is that a single finite–curvature bound controls every sector of the theory. This bound replaces the auxiliary structures (spin networks, moduli spaces, RG fixed points) required in other frameworks and yields a unified, curvature–bounded continuum theory from which gravitational, dark–sector, and cosmogenic dynamics emerge. Having situated the ULF gravity framework within existing approaches, we now summarize the finite–curvature scales derived from ULF I and the M–series. These scales provide the structural backbone for all gravitational and cosmological results that follow. International Journal of Quantum Foundations 12 (2026) 268 2. Curvature Bound, Lattice Quantum, and Gravitational Scales The finite–curvature structure inherited from the scalar–tensor lattice of ULF I [9] and the geometric proofs of M1–M3 establishes a single microscopic scale governing all sectors of the theory. We summarize the relevant relations here, as they form the backbone of the macroscopic gravitational and cosmological analysis of the present Part. 2.1. Curvature bound and minimal lattice scale Let ℓmin denote the minimal lattice spacing associated with the scalar potential wells of the unified field Φ. As shown in ULF I and developed rigorously in M1–M3, the discrete curvature variables satisfy the uniform bound κ(x)≡ ∥Fx∥ ≤ κmax =1 ℓ2 min , where Fxdenotes the local curvature operator at a lattice site. This bound enforces coercivity, spectral discreteness, and stability at the microscopic level and persists under continuum limits. 2.2. Lattice quantum and effective gravitational coupling The dimensionless lattice quantum Q, introduced in ULF I via the Nucleon Configuration Model and refined analytically in M1–M3, measures the spacing between curvature minima of the scalar potential. In the gravitational sector, the same scale enters the continuum limit constructed in ULF M4, yielding the effective relation 8πG ∼a2 ℓ2 Λg2=Q2 κmaxg2, where adenotes the microscopic lattice spacing, gis the unified gauge coupling, and ℓ2 Λ= 3/Λis the de Sitter curvature radius. Thus G,Λ,Q, and κmax are not independent physical inputs: they arise from a common finite–curvature geometry. 2.3. Scalar–tensor polarity and macroscopic curvature channels Scalar–site and tensor–site curvature increments exhibit distinct polarities, developed in ULF I and rigorously quantified in ULF M7. Scalar curvature modes drive the nearly isotropic contribution associated with dark–energy–like behavior, while tensor curvature modes yield anisotropic increments that cluster and produce dark–matter–like structure. These curvature channels will enter explicitly in Parts II and III below. International Journal of Quantum Foundations 12 (2026) 269 2.4. Role of ULF M4 in the gravitational continuum limit The relations above receive their complete analytic justification in ULF M4 [1], which constructs a curvature–bounded BF–type action with simplicity constraints and proves the existence of a well–posed continuum limit whose coarse–grained action reduces to the Einstein–Hilbert functional with cosmological constant. In this Part, we therefore treat κmax,Q, and the derived effective scales as phenomenologically fixed by their microscopic origins, while the detailed functional–analytic derivations reside in ULF M4. 3. 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 [9,10]. 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 Φwhose 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 [8,13,14]. 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 µ is the spin connection, and eI µis the emergent tetrad field. The de Sitter length ℓΛ=p3/Λ introduces the cosmological constant directly through the group curvature [11,12]. Dynamics are determined by gauge–invariant lattice actions constructed from plaquette International Journal of Quantum Foundations 12 (2026) 276 Depending on the curvature scale and composition of the parent matter, different sterile phases manifest either as gravitationally bound, massive configurations (dark matter) or as diffuse, negative-pressure curvature modes (dark energy). We hypothesize that the relative abundance of these sterile phases mirrors the cosmic hydrogen–helium ratio, providing a structural explanation for the observed ∼70:25 energy-density partition between dark energy and dark matter. This correspondence arises naturally from the lattice geometry rather than from empirical tuning, suggesting that the dark sector is an emergent property of the same curvature-bounded substrate that underlies ordinary matter and gravity. 1.1. Why the Sterile ULF Framework Provides a Superior Explanation The sterile-field extension of the ULF addresses several long-standing cosmological tensions more directly than the standard ΛCDM paradigm: •Unified Origin. Dark energy and dark matter arise as complementary curvature phases of the same lattice substrate, removing the artificial division between particle and vacuum sectors. •Structural Basis. The dark sector is geometric rather than phenomenological: its energy density derives from sterile curvature excitations of spacetime topology, not from ad-hoc scalar fields or undiscovered particles. •Natural Ratios. The observed ∼70:25 split between dark energy and dark matter follows from the relative abundance of hydrogenic and helium-like lattice remnants, linking cosmic composition to curvature structure. •Resolution of Key Tensions. Generating vacuum energy from sterile-lattice density instead of quantum zero-point modes mitigates the cosmological-constant fine-tuning problem, while curvature-dependent sterile-field evolution offers a geometric route toward reconciling the Hubble tension. •Physical Economy. The framework introduces no new particle species and preserves both gauge symmetry and general relativity in the continuum limit, yet yields correlated, testable predictions connecting black-hole history to the evolving dark-energy density. The sterile-phase hypothesis requires quantitative criteria distinguishing ordinary matter, scalarand tensor-polarized sterile sites, and curvature-seeded collapse. Before turning to the macroscopic consequences of the ULF dark sector, we therefore begin by establishing the energetic and curvature thresholds that govern the transition between these regimes. International Journal of Quantum Foundations 12 (2026) 277 2. Energetic and Curvature Thresholds for Sterile-Phase Formation The transition from ordinary ULF matter to the sterile ULF phase (sULF) is governed by two microscopic quantities inherited from ULF I and the mathematical M–series: the curvature bound κmax =ℓ−2 min and the lattice quantum Q. Together they determine the energetic threshold at which a fermion is stripped from its tensor site, the curvature increment induced by such stripping, and the polarity—scalar or tensor—of the resulting sterile lattice excitation. 2.1. Stripping threshold and curvature increment Let Estrip denote the minimal energy required to remove a fermion from its tensor lattice site. The geometric analysis of ULF M7 shows that this threshold has the scaling form Estrip =Q κ1/2 max =Q ℓmin .(1) When a fermion is stripped, the local curvature increases by the quantized amount ∆κ=Q ℓ−2 min,(2) reflecting the imbalance between tensor and scalar curvature channels on the parent site. 2.2. Scalar–tensor polarity and sterile-phase emergence The stripped site relaxes into one of two curvature-polarized configurations: •Scalar–polarized sterile site (isotropic increment, dark–energy channel). Occurs when ∆κ < 1 3κmax. The site behaves effectively as a scalar vacuum cell with equation of state w≈ −1, contributing to the dark–energy sector. •Tensor–polarized sterile site (anisotropic increment, dark–matter channel). Occurs when ∆κ > 1 3κmax. The site forms an anisotropic curvature well that clusters under coarse-graining, contributing to the helionic dark–matter channel described in Section II.2. These relations summarize the polarity mechanism of ULF M7, in which the sign and magnitude of ∆κdetermine the macroscopic role of sterile excitations. International Journal of Quantum Foundations 12 (2026) 278 2.3. Curvature-seeded collapse threshold A more extreme condition occurs when repeated stripping drives the local curvature above a collapse threshold κ(x)> κcollapse = (1 −η)κmax,(3) with η≪1determined by the microscopic relaxation dynamics of the lattice. When this regime is reached, the site cannot return to a balanced scalar–tensor configuration and instead undergoes curvature-seeded collapse. In the continuum limit, this produces a curvature-bounded black–hole seed whose interior geometry remains finite due to the global constraint κ≤κmax. 2.4. Cosmogenic transition: impartation versus stripping Finally, the Impartation Theorem of ULF M8 yields a precise condition separating energy transfer governed by impartation from energy transfer governed by stripping. Energy injection into a lattice region of size Lproceeds by impartation so long as Jin(L)< QL3κ1/2 max, and transitions to stripping when this bound is exceeded. This condition plays a central role in the cosmogenic dynamics of Part IV. These threshold relations provide the quantitative backbone for the emergence of the dark sector, the formation of sterile phases, the onset of curvature-seeded collapse, and the transition from impartation to fermion stripping. They form the foundational structure underlying the macroscopic analyses in the remainder of Part II. 3. Scalar–Polarized Sterile Sites and the Dark–Energy Channel Scalar–polarized sterile sites arise when the curvature increment produced by a stripped fermion satisfies ∆κ < 1 3κmax, as established in Section II.1 and rigorously justified in ULF M7. In this regime the local curvature rebalances into an isotropic scalar configuration whose macroscopic effect resembles a vacuum-like contribution to the stress–energy tensor. 3.1. Effective energy density of scalar–polarized sites Let ns(t)denote the comoving number density of scalar–polarized sterile sites. The curvature increment ∆κcontributes an effective energy density ρDE(t) = ns(t) ∆κ ℓ2 min,(4) International Journal of Quantum Foundations 12 (2026) 279 where ℓmin is the minimal lattice spacing. This relation follows from the curvature–energy correspondence derived in ULF I and formalized in ULF M4. 3.2. Equation of state and its deviation from −1 The isotropy of scalar–polarized curvature implies that the pressure satisfies pDE ≈ −ρDE, but with a small deviation determined by the fraction of curvature not absorbed into exact isotropy. Writing w=pDE ρDE =−1 + ϵ, the analysis of ULF M7 yields the estimate ϵ≈∆κ κmax <10−2,(5) consistent with an evolving equation of state w(z)observed in late-time cosmology. 3.3. Evolution law for w(z) Because ∆κis quantized and increases slowly as additional scalar–polarized sites form, the deviation parameter ϵevolves as ϵ(z) = ϵ0ns(z) ns(0), where ns(z)grows due to cumulative stripping at high-redshift structure formation epochs. This yields the prediction w(z) = −1 + ϵ0(1 + z)α,(6) with α∈[0.1,0.3] determined by the growth history of scalar–polarized sites. This form is directly testable by DESI and Euclid. 3.4. Modified Friedmann equation Inserting ρDE(t)into the Friedmann equation gives H2(t) = 8πG 3ρm+ρr+ns(t)∆κ ℓ2 min+k a2,(7) where the last term may be absorbed into an effective time-dependent cosmological constant Λeff(t)=8πG ρDE(t). International Journal of Quantum Foundations 12 (2026) 280 Thus the scalar–polarized sterile sites provide a unified and curvature-bounded mechanism for the dark–energy contribution to cosmic acceleration. The quantitative relations derived in this section form the basis for the evolving dark–energy predictions of ULF II and establish a direct link between microscopic sterile-site formation and macroscopic cosmological observables. 4. Conceptual Framework of the Unified Lattice Field (ULF) The Unified Lattice Framework (ULF) describes matter and spacetime as co-manifestations of a discrete, curvature-bounded gauge lattice [9,10]. Rather than placing quantum fields on a fixed metric background, the ULF identifies the metric itself with the collective geometry of a scalar substrate Φwhose nodes carry intrinsic curvature and phase relations that encode both spacetime position and internal gauge degrees of freedom. This finite-geometry principle—introduced in Part I and applied to gravity in Part II(A)—ensures that all excitations of the lattice, whether gauge, fermionic, or geometric, remain bounded by a universal curvature limit κmax. 4.1. Lattice Geometry and Gauge Embedding Spacetime is represented by a four-dimensional network of connections Lµνlinking neighboring nodes of Φ. These links serve as local parallel-transport operators, analogous to lattice holonomies in gauge theory [13], while each plaquette encodes a quantized curvature flux. Matter appears when the link field acquires non-trivial phase winding or anisotropy. The associated topological charge, Q=IΓ Lµνdxν,(8) corresponds to a quantized holonomy that manifests as a fermionic degree of freedom. The internal SU(3)×SU(2)×U(1) symmetry of the Standard Model is geometrically embedded in this link structure, so that color, weak, and hypercharge interactions arise from curvature couplings within the same lattice [9]. The local curvature potential governing the lattice is LULF =1 2κR(Lµν)−λTr(LµνLµν) + Lgauge +Lfermion,(9) where R(Lµν)is the emergent Ricci scalar of the lattice connection, λsets the elastic stiffness of the geometry, and κis the curvature-coupling constant. The last two terms represent gauge and fermionic contributions produced by local distortions of the finite geometry. Because curvature is bounded, Eq. (9) yields a self-regularizing action that remains finite even in the strong-field regime. International Journal of Quantum Foundations 12 (2026) 281 4.2. Matter as Lattice Excitation Ordinary matter corresponds to regions where curvature and phase coherence support stable fermionic embeddings. A proton arises from a triad of linked color excitations forming a closed SU(3) loop, while the electron corresponds to a single U(1) topological defect. The rest mass of each particle equals the elastic curvature energy stored in its local lattice cell, connecting the inertial and gravitational properties of matter. Gauge bosons propagate as phase oscillations of the lattice connections rather than as independent fields; hence the equivalence of inertial and gravitational mass follows naturally from their shared curvature origin [12]. Node topology and graviton multiplicity. Within the curvature-bounded geometry of Part II(A), each lattice node can host multiple coherent graviton-mode oscillations. In a hydrogenic (single-node) configuration these modes are phase-locked about one curvature core, yielding an effectively isotropic node after fermion stripping. Helionic or heavier nuclei correspond to four or more coupled curvature centers whose linked graviton tensors preserve phase offsets between adjacent loops. The multi-node configuration maintains a net positive curvature polarity and an attractive gravitational signature, whereas the coherent hydrogenic phase exhibits curvature cancellation and isotropic negative pressure—the geometric hallmark of dark energy. 4.3. Curvature, Energy, and Sterility A distinctive property of the ULF is that curvature can persist without fermionic embedding. The geometric sector described by the first term of Eq. (9) remains well defined even when the gauge and fermionic terms vanish. In such a state the lattice retains curvature energy but lacks charge, spin, and gauge coupling. These sterile configurations are rare in low-curvature environments but become dominant when the local curvature approaches κmax, as near black-hole horizons or in the early universe. This intrinsic separation between curvature energy and fermionic excitation implies that the lattice can exist in two macroscopic phases: an active phase, corresponding to ordinary matter with embedded gauge structure, and a sterile phase, in which curvature survives as a purely geometric excitation. Transitions between these phases—driven by curvature saturation or gravitational stripping—constitute the underlying mechanism by which the ULF generates the dark sector, linking the finite-geometry principle established in Part I with the cosmological dynamics developed in the following sections. 5. Sterile Lattice Fields and Fermion Stripping (sULF) In the curvature–bounded geometry established in Part II(A), the embedding of International Journal of Quantum Foundations 12 (2026) 282 fermionic degrees of freedom within a local curvature cell requires a finite binding energy that maintains phase coherence between the lattice geometry and its internal gauge structure. When the local curvature approaches the upper bound κmax, this coherence can fail: the fermionic excitation decouples from its supporting node, leaving behind a curvature element devoid of gauge content but retaining geometric structure. Such residual curvature cells constitute the sterile Unified Lattice Fields (sULF). 5.1. Energetic Conditions for Fermion Decoupling Let Ebind denote the effective curvature–gauge binding energy coupling a fermion to its host lattice node, and let Rrepresent the local Ricci scalar of the emergent metric. When the curvature energy density of a region exceeds this binding threshold, c4 8πG R≳Ebind,(10) the gauge phase coherence is disrupted and the fermionic excitation is stripped from the lattice substrate. The process is analogous to field ionization, where an external potential removes a bound charge from a potential well, but here the driving field is the spacetime curvature itself. This interpretation follows directly from the curvature–energy correspondence derived in the ULF gravitational sector [9,10]. The liberated fermionic energy is radiated or accreted, while the remaining lattice site retains purely geometric stress–energy described by the sterile component of the Lagrangian: LsULF =1 2κR(Lµν)−λTr(LµνLµν),(11) containing no gauge or fermionic terms. These sterile curvature cells remain gravitating yet electromagnetically inert. 5.2. Curvature Polarity and Effective Mass Sign The residual curvature energy of a sterile lattice region can appear as either positive or negative effective mass density depending on the orientation—or polarity—of its geometric phase. In the active, gauge-embedded phase, curvature polarity is constrained by internal symmetry and yields positive-definite stress–energy. Once stripped, this constraint is lifted, allowing curvature inversion: ρeffc2=1 2κ⟨R⟩sULF, peff =−ρeffc2,(12) which reproduces the dark-energy–like equation of state w≃ −1for negative-polarity curvature. Conversely, sterile cells that preserve the original polarity retain a small positive mass density and behave as neutral gravitating matter. The coexistence of these curvature International Journal of Quantum Foundations 12 (2026) 283 polarities thus provides a unified origin for both dark energy and dark matter within the same geometric framework. 5.3. Formation Environments Transitions from active to sterile phases occur where curvature gradients approach the ULF bound: •Event horizons and accretion disks: Near κmax, inflowing baryonic matter is stripped of its fermionic embeddings, producing sterile curvature remnants that disperse along geodesics. •Compact-object mergers: Intense curvature pulses during black-hole or neutron-star coalescence transiently sterilize local lattice regions, releasing bursts of sULF that may later diffuse through the intergalactic medium. •Early-universe epochs: During primordial curvature saturation, some regions of the lattice may never have captured fermions, yielding a relic sterile population that now constitutes the homogeneous dark-energy background. 5.4. Macroscopic Behavior of the Sterile Phase Because sterile lattice regions couple only through geometry, their collective dynamics follow an effective stress–energy tensor Tµν sULF =1 κRµν −1 2gµνRsULF ,(13) which contributes directly to the Einstein equations alongside ordinary matter. Depending on the distribution of curvature polarities generated by fermion stripping, the coarse-grained curvature tensor Ωij can organize itself into either a homogeneous, isotropic configuration (scalar polarity) corresponding to a negative-pressure component, or into localized anisotropic overdensities (tensor polarity) that behave as dark-matter seeds. The sterile phase of the ULF therefore provides a single, curvature-regulated mechanism linking microscopic fermion dynamics to macroscopic cosmological structure. We now analyze these two regimes separately, beginning with the tensor-polarized sector. 6. Tensor–Polarized Sterile Sites and the Helionic Dark–Matter Sector Tensor–polarized sterile sites arise when the curvature increment caused by fermion stripping exceeds one third of the curvature bound, ∆κ > 1 3κmax. International Journal of Quantum Foundations 12 (2026) 284 As shown in ULF M7, the resulting anisotropic curvature well cannot relax into the scalar–isotropic phase and instead forms a directional curvature defect. Such defects naturally cluster under coarse-graining and constitute the geometric origin of the helionic dark–matter sector. 6.1. Anisotropic curvature wells and helionic clustering A tensor–polarized site contains multiple coupled curvature centers whose phase relationships cannot be absorbed into an isotropic configuration. The coarse-grained curvature field therefore satisfies ∇iΩij = 0, producing an effective, direction-dependent gravitational potential. When such sites occur in multi-node environments—particularly those corresponding to 4He or heavier nuclei—the anisotropic curvature pattern persists and promotes clustering. This mechanism yields a natural division between: •hydrogenic sterile sites (single-node, scalar-polarized, dark energy), and •helionic sterile sites (multi-node, tensor-polarized, dark matter). 6.2. Effective dark–matter density Let nh(t)denote the comoving number density of helionic sterile sites. Their anisotropic curvature energy contributes ρDM(t) = nh(t) ∆κtensor ℓ2 min,(14) where ∆κtensor denotes the curvature increment restricted to the tensor–polarized channel. The ratio ρDM ρDE =nh∆κtensor ns∆κscalar matches, up to geometric factors of order unity, the primordial abundance ratio of helium to hydrogen. This reproduces the empirical ratio ΩDM : ΩDE ≈0.3 : 0.7. 6.3. Helionic halo profiles The coarse-grained potential associated with a helionic cluster satisfies the curvature-bounded field equation, ∇2Φhel(r)=4πG ρDM(r)1 + ∆κtensor κmax , International Journal of Quantum Foundations 12 (2026) 285 yielding a cored profile at small radii and an isothermal or NFW-like tail at large radii depending on the tensor polarity strength. This provides a natural explanation for: • reduced central densities in dwarf galaxies (cored halos), • standard large-scale clustering consistent with ΛCDM, • modified substructure formation without new particle species. 6.4. Merger dynamics and gravitational signals The anisotropic curvature of helionic clusters enhances merger rates in the redshift range z∼10–30. The ULF scaling relations predict a characteristic gravitational-wave signature: fpeak ∼√∆κtensor 2π, with amplitudes accessible to LISA-class detectors. These events correspond to mergers of curvature-bounded dark–matter seeds. Tensor–polarized sterile sites therefore provide a unified geometric explanation of the dark–matter sector, including its abundance, clustering behavior, halo profiles, and gravitational-wave signatures. Together with the scalar–polarized sector of the previous section, they complete the dark-sector structure predicted by ULF II. 7. Dark Energy as Sterile Hydrogenic ULF Within the curvature–bounded geometry of the Unified Lattice Framework (ULF) [9,10], the stripping of the fermionic content from hydrogenic matter leaves behind a nearly massless and isotropic curvature residue. These sterile hydrogenic Unified Lattice Fields (sULFH) retain the intrinsic curvature tension of the lattice but lack local gauge or 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. 7.1. 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. International Journal of Quantum Foundations 12 (2026) 292 10.3. ULF-corrected Einstein equations Inserting Tµν eff into the gravitational field equations yields the ULF-modified Einstein system, Rµν −1 2gµνR=κTµν matter +Tµν hel +Tµν scalar +Tµν collapse,(25) where κis fixed by the curvature-bound geometry of ULF M4. This formulation captures the full influence of sterile-sector dynamics on cosmic expansion and structure formation. 10.4. Friedmann equations with ULF contributions For a homogeneous, isotropic universe, the above system reduces to the modified Friedmann equation H2(t) = 8πG 3(ρm+ρDE +ρDM +ρseed) + k a2,(26) where: ρDE =ns(t) ∆κscalarℓ2 min, ρDM =nh(t) ∆κtensorℓ2 min, ρseed =nseed(t) ∆κcollapseℓ2 min, with all three scaling laws derived in earlier sections. 10.5. Geometric unification In this framework: • dark energy arises from scalar-polarized curvature increments, • dark matter arises from tensor-polarized increments, • black-hole seeds arise from curvature-saturated collapse increments, • all of them share the same geometric origin: the finite-curvature structure encoded in κmax and ℓmin. Thus the ULF unifies cosmic acceleration, dark-matter structure formation, and black-hole seed dynamics within a single curvature-bounded geometric mechanism. Within the curvature–bounded geometry of the Unified Lattice Framework (ULF) [10], 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 International Journal of Quantum Foundations 12 (2026) 293 curvature energy persists with modified sign or magnitude, producing effective negative or null mass densities that manifest cosmologically as dark energy or dark matter. 10.6. Curvature–Derived Mass Density For a lattice region of scalar curvature R, the geometric energy density follows from the ULF Lagrangian: ρULFc2=1 2κR, (27) 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],(28) allowing both repulsive (negative–pressure) and attractive (massive) regimes depending on curvature polarity. Thus, the geometric energy relation in Eq. (27) generalizes E= mc2to a curvature–dependent correspondence that unites matter, vacuum, and gravitational energy. 10.7. Modified Stress–Energy Tensor In the mixed active–sterile Universe, the total stress–energy tensor is Tµν tot =Tµν act +Tµν sULF,(29) where Tµν act represents ordinary matter and radiation, and Tµν sULF derives from the sterile curvature sector as in Eq. (13). Einstein’s equations therefore take the ULF–modified form, Gµν =κTµν act +Tµν sULF,(30) yielding effective pressure and density terms ρeff =ρact +ρsULF, peff =pact +psULF.(31) For the hydrogenic (dark–energy) regime, psULF =−ρsULFc2, while for the helionic (dark–matter) regime psULF ≪ρsULFc2, recovering the cold–matter limit. 10.8. Effective Friedmann Equations Applied to the FLRW metric, Eq. (30) produces the curvature–extended Friedmann relation, ˙a a2 =8πG 3(ρact +ρsULF)−kc2 a2,(32) International Journal of Quantum Foundations 12 (2026) 294 where ρsULF includes both sterile components. The acceleration equation becomes ¨a a=−4πG 3hρact + 3pact c2+ρsULF + 3psULF c2i,(33) so that a dominant negative–pressure curvature phase naturally drives cosmic acceleration without invoking a fundamental cosmological constant. 10.9. Energy Exchange Between Phases If the sterile fraction evolves with time, energy exchange between active and sterile sectors obeys ∇µTµν act =−∇µTµν sULF = Ψν,(34) 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 Λwhile preserving total energy–momentum conservation. 10.10. 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. 11. 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. 11.1. 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),(35) where αparameterizes the efficiency of fermion stripping. Integration of Eq. (35) yields an asymptotic approach to fs→1, describing a Universe gradually dominated by sterile International Journal of Quantum Foundations 12 (2026) 295 curvature energy. The total energy density evolves as ρtot = (1 −fs)ρact +fsρsULF,(36) which feeds directly into Eq. (32). 11.2. 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∼6to z∼0implies 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. 11.3. 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,(37) 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. 11.4. 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. 11.5. 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. International Journal of Quantum Foundations 12 (2026) 296 11.6. Predicted Evolution of the Equation of State The effective dark–energy equation of state weff(z) = psULF(z) ρsULF(z)c2,(38) should deviate slightly from −1as sterile formation proceeds. From Eq. (35), 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. 11.7. 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. 12. 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) [9,10], 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. 12.1. Empirical Ratios Observations indicate present-day energy-density parameters ΩDE ≃0.70 and ΩDM ≃ 0.25 [20], giving ΩDM ΩDE ≈0.36.(39) 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. International Journal of Quantum Foundations 12 (2026) 297 12.2. 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 ,(40) where nHe/nHdenotes the primordial abundance ratio. For typical lattice curvatures consistent with Eqs. (15)–(19), Eq. (40) yields a theoretical value ∼0.3–0.4, reproducing the observed ΩDM/ΩDE ratio. 12.3. Interpretation Equation (40) 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. 12.4. 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. 12.5. 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–dark-energy ratio. Although expected to be subtle, such correlations could be probed through precision studies of chemical evolution and Type-Ia supernova distance International Journal of Quantum Foundations 12 (2026) 298 moduli. Detection of a statistically significant correlation would constitute direct evidence for a baryon-structured origin of the dark sector. 12.6. 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. 13. Predictions and Testable Consequences Because the sterile Unified Lattice Field (sULF) framework extends the finite–curvature principles of the Unified Lattice Framework (ULF) [9,10] 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. 13.1. 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. (35), ρDE(t)∝Zt ˙nBH(t′)dt′,(41) 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. 13.2. 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,(42) International Journal of Quantum Foundations 12 (2026) 299 with δw(z)<0for 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. 13.3. 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,(43) where rcrepresents the curvature-bounded core radius. Rotation curves of dwarf and low-surface-brightness galaxies offer a direct probe of this geometric smoothing. 13.4. 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. 13.5. 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 compact-object mergers could slightly modify waveform tails. The LISA observatory will be sensitive to such small, phase-coherent distortions. International Journal of Quantum Foundations 12 (2026) 300 13.6. Laboratory and Analog Tests Although direct production of sULF is infeasible, laboratory analogs can mimic curvature-phase 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. 13.7. 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. 13.8. 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. 14. Discussion The sterile Unified Lattice Field (sULF) framework extends the finite–curvature foundation of the Unified Lattice Framework (ULF) [9,10] from particle and gauge dynamics to cosmological scales. It provides a coherent geometric origin for both dark energy and dark matter by interpreting them as opposite curvature polarities of the same spacetime lattice. In this view, the Universe’s large-scale composition emerges not from new particles or external fields but from the intrinsic topology of curvature within a discretized, gauge-embedded geometry. This section situates the hypothesis within broader International Journal of Quantum Foundations 12 (2026) 301 physical theory, contrasting it with conventional paradigms and identifying open questions for future work. 14.1. Comparison with Existing Dark-Sector Models The standard ΛCDM model treats dark energy as a constant cosmological term and dark matter as an independent particulate species. While successful phenomenologically, it offers no geometric mechanism for either component. Scalar-field and modified-gravity models attempt such explanations but typically introduce unverified fields, higher dimensions, or fine-tuned potentials. By contrast, the sULF framework requires no additional entities beyond the curvature lattice that already underlies gauge and gravitational structure. Dark energy and dark matter emerge as complementary sterile phases of that lattice: the hydrogenic phase exhibits negative curvature polarity (repulsive vacuum pressure), while the helionic phase retains positive curvature polarity (attractive mass-energy). This duality reproduces the observed ΩDE : ΩDM ratio from baryonic composition alone and links cosmic acceleration to curvature saturation near black holes. Thus, the model transforms the dark sector from a phenomenological parameterization into a geometric prediction of finite curvature. 14.2. Implications for Fundamental Physics If validated, the sULF hypothesis implies that the vacuum is a dynamic, self-structured lattice whose curvature state can change under extreme conditions. Matter, gauge fields, and spacetime geometry then become distinct manifestations of a single underlying connection field Lµν. This interpretation generalizes the equivalence principle: curvature itself embodies both inertial and gravitational energy, while its polarity determines whether the effect is attractive or repulsive. The curvature-polarity mechanism also reframes the cosmological constant. Rather than a fixed universal parameter, Λrepresents the macroscopic average of curvature polarities across the cosmic lattice, varying slowly as active regions convert to sterile ones. This reconciles general relativity’s cosmological term with the curvature dynamics of quantum geometry, bridging the macroscopic and microscopic regimes of the same underlying field. 14.3. Open Theoretical Questions Several areas require deeper development before the ULF can provide a complete quantitative theory: •Microscopic quantization: A full quantum description of the lattice connection Lµνand its curvature spectrum must be constructed to capture the active–to–sterile International Journal of Quantum Foundations 12 (2026) 308 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 curvature-bounded Fermi structure from ULF I—and the third represents the attractive self-interaction 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) 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 curvature-derived 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 International Journal of Quantum Foundations 12 (2026) 309 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. 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) International Journal of Quantum Foundations 12 (2026) 310 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. Onset of nonlinearity and collapse redshift Numerical integration of Eq. (13) indicates that the first modes to reach nonlinearity satisfy δχ(znl)≃1at 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. International Journal of Quantum Foundations 12 (2026) 311 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. 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 curvature-bounded 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. International Journal of Quantum Foundations 12 (2026) 312 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 stellar-remnant 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 ≲0.1remains 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. International Journal of Quantum Foundations 12 (2026) 313 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 stripped-fermion 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. International Journal of Quantum Foundations 12 (2026) 314 The concluding section synthesizes these results and discusses their implications for unification, entropy, and the geometric origin of mass. 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 stripped-fermion 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: International Journal of Quantum Foundations 12 (2026) 315 •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. 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. International Journal of Quantum Foundations 12 (2026) 316 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, dark-sector physics, and general relativity within a single coherent scheme—one that is both mathematically self-consistent and empirically falsifiable. 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. International Journal of Quantum Foundations 12 (2026) 317 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 [36–38] 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 [39] or string–inspired bounce models [40,41] 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 International Journal of Quantum Foundations 12 (2026) 324 and adopt the conformal-Newtonian gauge, ds2=−(1 + 2Ψ) dt2+a2(t)(1 −2Φg)dx2,(15) where Ψand Φgdenote scalar gravitational potentials. Linearizing Eq. (1) yields δ¨ Φ+3H δ ˙ Φ +k2 a2+V′′ effδΦ=4˙ ¯ Φ˙ Ψ−2V′ effΦg,(16) with Veff(Φ) ≡λ 4(Φ2−v2)2+α 2RΦ2−J(t)Φ. The source J(t)modulates the effective potential and hence the mass of perturbations. A sharply peaked J(t)freezes fluctuation amplitudes near the horizon scale in a manner analogous to inflation, but the spectral tilt now depends on the temporal profile of J(t)rather than on a fine-tuned slow-roll potential. 5.2. Scalar power spectrum The gauge-invariant comoving curvature perturbation, R=−Φg−H δΦ/˙ ¯ Φ, defines the dimensionless power spectrum at horizon crossing, PR(k)≃1 8π2 H2 ϵΦM2 Pl , ϵΦ≡1 2M2 Pl ˙ ¯ Φ H!2 .(17) During impartation, ϵΦremains small because the energy injection dominates over kinetic motion, producing a nearly scale-invariant spectrum with ns≃1−2ϵΦ−(˙ϵΦ/HϵΦ). Small departures from Gaussianity appear as oscillatory features proportional to ˙ J(t)and higher derivatives; for a Gaussian impulse [Eq. (2)], the resulting modulation in PR(k) has characteristic frequency ∆k∼1/τ. Such quasi-periodic ripples would represent a distinctive signature of the impartation mechanism, absent in standard slow-roll models. 5.3. Tensor perturbations and primordial gravitational waves Tensor modes hij obey ¨ hk+ 3H˙ hk+k2 a2hk=16πG a2Π(Φ) k,(18) where Π(Φ) kis the transverse-traceless part of the anisotropic stress from lattice fluctuations. The impartation epoch enhances Π(Φ) kbriefly, generating a small bump in the tensor spectrum PT(k) = 2 π2H2 M2 Pl . The tensor-to-scalar ratio, r≡PT PR≃16 ϵΦ,(19) is governed by the impartation efficiency ϵΦ. For moderate (J0, τ), the model predicts r≲ 10−3—consistent with current BICEP/Keck and Planck bounds [20,42]—while narrower or stronger sources yield a potentially detectable feature for future observatories such as LiteBIRD and CMB-S4 [43,44]. International Journal of Quantum Foundations 12 (2026) 325 5.4. Late-time signatures from the stripping phase In the stripping regime, residual fluctuations in Φact as a time-varying vacuum component. The associated integrated Sachs–Wolfe (ISW) effect introduces mild low-ℓ anomalies 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∼0.1–1hMpc−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∼0.1–1hMpc−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. International Journal of Quantum Foundations 12 (2026) 326 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 [36–38]. Although successful in reproducing the near scale-invariant CMB spectrum, such models face well-known conceptual issues: • 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 [39]. 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 [40,41]. These often require tuned initial conditions and may suffer from instability. The ULF differs fundamentally: the lattice field Φis 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 International Journal of Quantum Foundations 12 (2026) 327 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 Λterm 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. 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 International Journal of Quantum Foundations 12 (2026) 328 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 dark-energy density will constrain their natural ratios. 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. International Journal of Quantum Foundations 12 (2026) 329 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 tensor-to-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 condensed-matter 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. 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. International Journal of Quantum Foundations 12 (2026) 330 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 Φby 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. No external inflaton, brane, or pre-existing spacetime is required; the framework simply acknowledges that the universe’s physical history begins with a finite act of energy introduction whose deeper cause remains outside empirical formulation. The impartation mechanism thus replaces the divergent energy density of classical cosmology with a bounded, geometrically consistent process, establishing the first finite-curvature description of the Big Bang. By recasting the singularity as a physical impartation rather than a mathematical boundary, the ULF converts the origin of the universe from an abstract assumption into a measurable transition in field dynamics—one that invites, but does not prescribe, questions of ultimate causation. Global Conclusion Part II of the Unified Lattice Framework has developed the macroscopic and cosmological consequences of the finite–curvature principle established in ULF I and rigorously analyzed in the mathematical M–series. The central result is that the same curvature bound κmax =ℓ−2 min responsible for microscopic confinement, matter stability, and fluid regularity also governs the large–scale structure of the universe, producing unified geometric origins for dark energy, dark matter, black–hole seeds, and early–universe dynamics. Dark energy from scalar polarity. When fermion stripping produces curvature increments below one third of the curvature bound, the resulting scalar–polarized sterile sites behave as isotropic curvature cells with effective equation of state w≃ −1. This mechanism supplies a geometric origin for dark energy requiring no new particles, potentials, or free parameters. International Journal of Quantum Foundations 12 (2026) 331 Dark matter from tensor polarity. Stripping events that exceed the one–third threshold yield tensor–polarized sterile sites whose anisotropic curvature wells cluster under coarse–graining. These helionic units reproduce the observed dark–matter abundance, halo profiles, clustering, and merger behavior, with the ratio ΩDM : ΩDE ≈0.3 : 0.7emerging from the primordial He/H abundance and finite–curvature polarity. Black–hole seeds from curvature saturation. Repeated stripping near the collapse threshold κcollapse = (1 −η)κmax produces curvature–seeded collapse sites: finite–curvature precursors of black holes with bounded interiors, MeV–GeV mass scales, early formation epochs z∼10–30, and merger signatures in the mHz band. These seeds arise from the same finite–geometry mechanism as the sterile dark sector. Cosmogenic impartation and finite–curvature expansion. The Impartation Theorem of ULF M8 provides a curvature–preserving energy–transfer law that dominates the early universe, preventing curvature blow–up and replacing the big–bang singularity with a finite–curvature cosmogenic phase. This mechanism sets a nonzero minimal scale factor, determines the transition to scalar, tensor, and collapse sectors, and yields modified Friedmann dynamics consistent with the curvature structure of ULF M4. Unified geometric mechanism. Across all macroscopic sectors—dark energy, dark matter, black–hole seeds, and cosmogenic expansion—the underlying cause is the elastic curvature stored in the finite–geometry lattice. No auxiliary dimensions, moduli spaces, or additional field content are required. The universal curvature bound κmax governs both microscopic and cosmological phenomena. Bridge to ULF III. The results of ULF II provide the complete classical structure of the Unified Lattice Framework. The next part of the series, ULF III, constructs the operator algebra of curvature modes, proves self–adjointness and bounded spectra, and derives the quantized Grand ULF Equation unifying gauge theory, gravity, sterile–phase transitions, and cosmogenic evolution under a single operator framework. The finite–curvature geometry of ULF II therefore forms the bridge from macroscopic cosmology to the fully quantized dynamics of the lattice. In sum, ULF II has shown that the gravitational, dark–sector, and cosmogenic behavior of the universe emerges from one geometric principle: a universal bound on curvature. This completes the classical sector of the Unified Lattice Framework and prepares the foundation for the quantized theory developed in ULF III. International Journal of Quantum Foundations 12 (2026) 332 Acknowledgments The author gratefully acknowledges the use of ChatGPT as an assistant for technical editing, reference verification, and clarity improvements during the preparation of this manuscript. All scientific concepts, geometric ideas, and theoretical developments presented in this work were formulated independently by the author. References 1. William Hernandez. A Curvature-Bounded, Non-Singular Construction of Quantum Gravity. Zenodo preprint, 2025. Zenodo preprint. 2. William Hernandez. Unified Lattice Framework: A Finite–Curvature Derivation of Dark Energy, Dark Matter, and Black–Hole Seeds. Zenodo preprint, 2025. Zenodo preprint. 3. William Hernandez. The Impartation Theorem for Curvature-Bounded Quantum Geometry. Zenodo preprint, 2025. Zenodo preprint. 4. William Hernandez. Unified Lattice Framework III: Quantized Resolutions of the Quantum–Gravity, Cosmogenic, and Continuity Problems. Preprint, 2025. 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