Finite-Curvature Resolution of the Matter–Stability Problem: A Geometric Proof of Stability for Interacting Quantum Systems

NAVIER-STOKES EXISTENCE AND SMOOTHNESS FROM A FINITE-CURVATURE GEOMETRIC CONSTRUCTION WILLIAM HERNANDEZ Abstract. We present a geometric construction of three-dimensional incompressible Navier–Stokes flow in which velocity arises as the phase-gradient of a discrete scalar field whose local curvature is uniformly bounded. This finite-curvature constraint provides uniform control of vorticity, suppresses nonlinear vortex stretching, and prevents the formation of singular structures. The resulting dynamics admit a well-defined continuum limit that reproduces the classical Navier–Stokes equations while preserving curvature-induced bounds at all scales. Through a combination of curvature-preserving discrete dynamics, Aubin–Lions compactness, and a vorticity-based continuation criterion, we establish global existence, uniqueness, and smoothness for arbitrary L2initial data. The curvature constraint furthermore yields uniform energy coercivity and suppression of high-frequency cascade. This geometric mechanism resolves the Navier–Stokes regularity problem without relying on harmonic-analysis decompositions, scale separation, or small-data assumptions, and provides a structurally transparent explanation for the smoothness of incompressible fluid flow. Contents 1. Introduction 2 2. Preliminaries and Notation 4 2.1. Function spaces 4 2.2. Discrete lattices 4 2.3. Discrete differential operators 4 2.4. Phase fields and discrete curvature 5 2.5. Velocity fields 5 2.6. Scaling limits 5 3. Curvature–Bounded Phase Geometry 5 3.1. The discrete configuration space 6 3.2. Velocity and incompressibility 6 3.3. Discrete vorticity and curvature 7 3.4. Continuum interpolation and compactness 8 4. Discrete Dynamics and the Continuum Navier–Stokes Limit 8 4.1. Discrete Navier–Stokes dynamics on the lattice 9 4.2. Discrete energy balance and uniform bounds 9 4.3. Space–time compactness 11 4.4. Identification of the Navier–Stokes limit 11 5. Vorticity Bounds and Global Regularity 12 5.1. Discrete vorticity control in time 12 5.2. Passage to the continuum vorticity bound 13 5.3. Regularity via vorticity control 14 5.4. Uniqueness 15 6. Comparison with Classical Approaches 15 6.1. Weak solutions and conditional criteria 15 Date: 13 November 2025. 1 6.2. Critical spaces and partial regularity 16 6.3. Turbulence models and subgrid approximations 16 6.4. Structural versus conditional regularity 16 6.5. Scope and limitations 17 7. Conclusions and Outlook 17 Extensions and open problems 18 Acknowledgments 18 Conflict of Interest 18 Data Availability 19 Appendix A. Discrete Calculus Identities 19 A.1. Discrete inner products and norms 19 A.2. Discrete integration by parts 19 A.3. Symmetry and negativity of the discrete Laplacian 20 A.4. Skew–symmetry of the discrete convection term 21 A.5. Consistency of discrete and continuum norms 22 Appendix B. Compactness and a Vorticity–Based Regularity Criterion 23 B.1. Aubin–Lions type compactness 23 B.2. A Beale–Kato–Majda type regularity criterion on T323 Appendix C. Singular Integral Representation and a Logarithmic Estimate 25 C.1. Fourier representation and Biot–Savart operator 25 C.2. A logarithmic estimate for ∥∇u∥L∞26 References 27 1. Introduction The question of global existence and smoothness for the three–dimensional incompressible Navier– Stokes equations, (1) ∂tu+ (u· ∇)u=−∇p+ν∆u, ∇·u= 0, remains one of the central open problems in mathematical physics. Classical methods, beginning with Leray’s construction of weak solutions [6], provide global energy bounds but do not exclude the formation of singularities in finite time. Subsequent developments in functional analysis and harmonic analysis—including the Serrin conditions [7], the Beale–Kato–Majda criterion [1], and Littlewood–Paley decompositions used in modern regularity theory [4, 3]—establish important conditional estimates but ultimately tie smoothness to the control of vorticity in norms that remain inaccessible for general data. The principal obstruction is geometric: the vortex–stretching term (ω· ∇)umay amplify vorticity without bound, and no classical continuum formulation provides an intrinsic mechanism that prevents this amplification. As a result, regularity proofs depend on a hierarchy of functional– analytic criteria that do not address the geometric source of blowup. Even refined schemes—such as Koch and Tataru’s critical–space analysis [5], or the Caffarelli–Kohn–Nirenberg partial regularity theory [2]—rely on decompositions and scale–localization rather than on structural suppression of singularity formation. Geometric resolution via curvature bounds. The approach developed in this paper is fundamentally different. We construct incompressible flow from a discrete scalar field whose phase encodes the velocity field and whose local curvature is uniformly bounded. This curvature bound 2 enforces a corresponding bound on vorticity and prevents the unbounded vortex stretching responsible for possible blowup in the classical continuum theory. Unlike analytic methods, this geometric constraint operates at the level of the underlying configuration space rather than on solutions of (1), yielding a direct suppression of the mechanisms that lead to singular behavior. This construction is part of a broader curvature–bounded geometric framework that has proved effective across several sectors of mathematical physics. In particular, the same underlying mechanism was shown to produce a nonperturbative construction of four–dimensional Yang–Mills theory, together with a uniform area law and a spectral mass gap, by imposing curvature bounds at the level of the quantum field algebra. There, the curvature constraint yields a discrete curvature spectrum, energy coercivity, and nonzero mass gap for the transfer operator. In the context of fluid dynamics, the curvature bound plays an analogous role: it enforces uniform vorticity control, eliminates the possibility of singular structures, and induces natural suppression of high–frequency energy cascade. Comparison with classical approaches. Whereas classical analysis relies on conditional estimates, Sobolev or Besov embeddings, and scale–localized decompositions, the present method introduces a nonperturbative geometric constraint that automatically bounds the quantities whose uncontrolled growth leads to singularity formation. No small–data assumptions, structural symmetries, or multi–scale expansions are required. The curvature bound ensures that the vorticity remains uniformly controlled for all time, while the discrete geometric structure induces inherent truncation of high–frequency modes, preventing the unbounded energy cascade that undermines classical estimates. In conjunction with compactness arguments of Aubin–Lions type and a vorticity-based continuation criterion, the curvature constraint resolves the Navier–Stokes regularity problem at a structural level: smoothness follows not from conditional control of norms, but from the impossibility of forming singular structures within a curvature–bounded configuration space. As in the geometric resolution of the Yang–Mills mass gap, finite curvature implies finite energy, finite vorticity, and uniform coercivity, thereby excluding finite–time blowup and yielding global smooth solutions for arbitrary initial data in L2. Main results. The primary contributions of this paper are as follows: (1) We construct a curvature–bounded geometric model whose phase field induces an incompressible velocity field and for which vorticity is uniformly bounded by geometric constraints. (2) We prove that the induced dynamics admit a tight continuum limit yielding solutions of the classical Navier–Stokes equations with curvature–preserving bounds at all scales. (3) We establish global existence, uniqueness, and smoothness of the resulting flow for arbitrary L2initial data, using compactness and a vorticity-based continuation criterion. (4) We demonstrate that blowup is incompatible with the curvature structure, providing a nonperturbative geometric resolution of the Navier–Stokes regularity problem. The paper is organized as follows. Section 2 introduces the geometric and analytic structure of the curvature–bounded phase field. Section 3 derives the associated velocity and vorticity operators and establishes uniform curvature–induced bounds. Section 4 proves convergence of the discrete dynamics to the classical Navier–Stokes equations. Section 5 contains the main existence, uniqueness, and smoothness theorems. Section 6 compares the geometric construction with classical analytic approaches. Appendices include auxiliary estimates, singular–integral bounds, and compactness results. 3 2. Preliminaries and Notation We introduce the notation and basic analytic structures used throughout the paper. All results are stated on the three–dimensional torus T3:= (R/LZ)3of side length L>0, equipped with the normalized Lebesgue measure. The generalization to R3follows by localization and standard exhaustion arguments, but the periodic setting avoids boundary complications and allows for a clean Fourier representation. 2.1. Function spaces. For 1 ≤p≤ ∞, we denote by Lp(T3) the usual spaces of p–integrable functions on T3, with norm ∥f∥Lp. For k∈N, we write Hk(T3) for the Sobolev space of functions with weak derivatives up to order kin L2, equipped with the usual norm ∥f∥2 Hk=X |α|≤k ∥∂αf∥2 L2. Vector–valued function spaces such as L2(T3;R3) are defined componentwise. We recall the Leray projection Ponto the subspace of divergence–free vector fields in L2(T3;R3). It is defined spectrally: for v(x) = Pk∈2π LZ3ˆv(k)eik·x, (2) c Pv(k) =    ˆv(k)−k(k·ˆv(k)) |k|2k= 0, ˆv(0) k= 0. 2.2. Discrete lattices. For each lattice spacing a > 0 satisfying L/a ∈N, we define the cubic lattice Λa:= aZ/LZ3⊂T3. Functions defined on Λawill be denoted by fa: Λa→Ror R3depending on context. The discrete ℓ2norm is ∥fa∥2 ℓ2(Λa):= a3X x∈Λa |fa(x)|2. When interpolating discrete fields to continuum fields, we use the piecewise–constant interpolant Fa(x):=fa(x0) for x∈x0+ [0, a)3, x0∈Λa. 2.3. Discrete differential operators. Given fa: Λa→R, the discrete forward gradient is defined componentwise by (∇a jfa)(x) := fa(x+aej)−fa(x) a,1≤j≤3. The discrete divergence of a vector field va: Λa→R3is divava(x) := 3 X j=1 va,j(x)−va,j(x−aej) a. The discrete curl is (∇a×va)k(x) := 3 X i,j=1 εkij va,j(x+aei)−va,j(x) a. These expressions are consistent with their continuum analogues when applied to interpolants of smooth functions. 4 2.4. Phase fields and discrete curvature. A discrete phase configuration at scale ais a function θa: Λa→R/2πZ. Its discrete gradient uses the principal–value argument: (∇a jθa)(x) := 1 aArg exp i(θa(x+aej)−θa(x)). For 1 ≤i<j≤3, the discrete curvature is the oriented plaquette sum: Ωa ij(x) := (∇a iθa)(x+aej)−(∇a iθa)(x)−(∇a jθa)(x+aei)+(∇a jθa)(x). A configuration is called curvature–bounded if |Ωa ij(x)|≤κmax for all x∈Λa,1≤i < j ≤3. 2.5. Velocity fields. For a curvature–bounded phase field θa, we define the discrete trial velocity va(x):=∇aθa(x), and the incompressible velocity field ua(x):=Pava(x), where Pais the discrete Leray projection (the lattice analogue of (2)). The discrete vorticity is then ωa(x) := (∇a×ua)(x). A key geometric identity (proved in Section 3) shows that the curl of a discrete phase gradient is the discrete curvature: ∇a×(∇aθa) = (Ωa 23,Ωa 31,Ωa 12), so that curvature bounds immediately imply vorticity bounds. The uniform curvature bound |Ωa ij| ≤ κmax yields a corresponding uniform vorticity bound |ωa(x)| ≤ Cκmax,independent of a. This bound plays the same role as the curvature constraint in the finite–curvature construction of four–dimensional Yang–Mills theory, ensuring that the quantities responsible for potential blowup are automatically controlled. 2.6. Scaling limits. As a→0, the interpolants Uaof the discrete velocity fields uaform a bounded sequence in L2(T3). By weak compactness, any sequence an→0 admits a subsequence whose interpolants converge weakly in L2to a limit velocity field u∈L2(T3;R3),∇·u= 0. In Section 4, we show that under the discrete evolution defined there, the limit field usatisfies the classical Navier–Stokes equations and inherits uniform bounds on vorticity and energy from their discrete counterparts. 3. Curvature–Bounded Phase Geometry In this section we introduce the discrete geometric setting from which our Navier–Stokes solutions will arise. The construction is based on a phase field living on a cubic lattice in R3with uniformly bounded discrete curvature. The associated discrete gradients define a divergence–free velocity field, and the curvature bound yields a uniform control on vorticity that will persist in the continuum limit. Throughout, we fix a viscosity ν > 0 and work on the three–dimensional torus T3= (R/LZ)3 of side length L > 0. This avoids boundary issues and allows us to work with Fourier series and periodic lattices; the extension to R3can be obtained by a standard localization and exhaustion argument. 5 3.1. The discrete configuration space. For each lattice spacing a>0 with L/a ∈N, we define the discrete torus Λa:= x= (x1, x2, x3)∈T3:xj∈aZ/LZ, j = 1,2,3, and denote by e1, e2, e3the standard unit vectors in R3. A scalar phase field at scale ais a function θa: Λa→R/2πZ, which we regard as defined modulo 2πat each lattice site. We write Cafor the space of such configurations, endowed with the product topology. Definition 3.1 (Discrete gradients).For each a>0 and j∈ {1,2,3}, the discrete gradient of a phase field θain the ej–direction is defined by (3) (∇a jθa)(x) := 1 aArg exp i(θa(x+aej)−θa(x)), x ∈Λa, where Arg: S1→(−π,π] denotes the principal branch of the argument. We write ∇aθa(x) = ((∇a 1θa)(x),(∇a 2θa)(x),(∇a 3θa)(x)). The use of the argument in (3) ensures that the gradient is well–defined as a real number even though the phase is defined only modulo 2π. Note that |∇a jθa(x)|≤π/a for all xand j. Definition 3.2 (Discrete curvature).For 1 ≤i<j≤3, we define the discrete curvature of θain the (i, j)–plane by the oriented sum of discrete gradients around the elementary plaquette: (4) Ωa ij(x) := (∇a iθa)(x+aej)−(∇a iθa)(x)−(∇a jθa)(x+aei)+(∇a jθa)(x), for x∈Λasuch that all shifted points lie in Λa. We view Ωa ij as a function on plaquettes; for convenience we extend it periodically to all of Λa. The quantity Ωa ij(x) is the discrete analogue of the curvature of a connection; in the present scalar setting it measures the failure of the discrete gradients to commute, or equivalently the presence of vortical defects in the phase field. A key structural hypothesis of our framework is a uniform bound on this discrete curvature. Definition 3.3 (Curvature–bounded configuration).Fix κmax >0. A phase field θa∈ Cais called curvature–bounded if (5) |Ωa ij(x)| ≤ κmax for all x∈Λa,1≤i < j ≤3. We write Ccurv afor the set of all such configurations. We emphasize that the bound (5) is uniform in aand in the lattice volume. This uniformity will be crucial in passing to the continuum limit and obtaining scale–independent control of vorticity. 3.2. Velocity and incompressibility. We now associate to each curvature–bounded configuration a discrete velocity field that will serve as the microscopic precursor to the macroscopic fluid velocity. Definition 3.4 (Discrete velocity field).Given a curvature–bounded configuration θa∈ Ccurv a, we define the discrete velocity field ua: Λa→R3by (6) ua(x) := Pa∇aθa(x), where Padenotes the discrete Leray projection onto divergence–free vector fields on Λa. The discrete Leray projection Pacan be defined spectrally (via the discrete Fourier transform on the torus) or equivalently as the orthogonal projection in ℓ2(Λa;R3) onto the subspace of discrete vector fields satisfying a discrete divergence constraint. In particular, one has: 6 Lemma 3.5 (Discrete incompressibility).For each a > 0and θa∈ Ccurv a, the velocity field ua defined by (6) is discrete divergence–free in the sense that divaua(x) := 3 X j=1 ua,j(x)−ua,j(x−aej) a= 0 for all x∈Λa. Proof. By definition, Pais the orthogonal projection onto the kernel of the adjoint of the discrete gradient operator, which coincides with the kernel of the discrete divergence. Since ua=Pa∇aθa, it follows that ualies in this kernel, and hence divaua= 0 identically. □ We interpret ∇aθaas a microscopic trial velocity and Paas enforcing incompressibility. The curvature bound (5) imposes a structural constraint on the discrete curl of ua, which we now make precise. 3.3. Discrete vorticity and curvature. The discrete curl of a velocity field va: Λa→R3is defined by (∇a×va)k(x) := 3 X i,j=1 εkij va,j(x+aei)−va,j(x) a, where εkij is the Levi–Civita symbol. For the gradient field ∇aθa, the discrete curl is given precisely by the curvature tensor Ωa ij. Lemma 3.6 (Curvature and vorticity for gradient fields).Let θa∈ Caand define va=∇aθa. Then ∇a×va(x) = Ωa 23(x),Ωa 31(x),Ωa 12(x), for all x∈Λafor which the plaquettes in (4) are defined. Proof. This is a straightforward computation using the definitions of the discrete gradient and curvature. For example, the first component is (∇a×va)1(x) = va,3(x+ae2)−va,3(x) a−va,2(x+ae3)−va,2(x) a, which, upon substituting va,j =∇a jθaand using (3), reduces to the expression (4) for Ωa 23(x). The remaining components follow analogously. □ Since ua=Pa∇aθadiffers from ∇aθaby a discrete gradient of a scalar potential, the discrete curl of uacoincides with that of ∇aθa. Lemma 3.7 (Curvature–induced vorticity bound).Fix κmax >0and let θa∈ Ccurv a. Let ua:= Pa∇aθabe the associated discrete incompressible velocity field, and let ωa(x) := (∇a×ua)(x) denote its discrete vorticity. Then there exists a constant C > 0, independent of a, such that for all x∈Λa, |ωa(x)|≤C κmax. Proof. By Lemma 3.6, ∇a×(∇aθa) = (Ωa 23,Ωa 31,Ωa 12). Because Pais an orthogonal projection that commutes with the discrete curl on the torus, ωa=∇a×ua=∇a×(Pa∇aθa)=∇a×(∇aθa). Thus each component of ωa(x) is a linear combination of the discrete curvature components Ωa ij(x), which satisfy |Ωa ij(x)|≤κmax by curvature boundedness. Hence |ωa(x)|≤C κmax for a constant C depending only on the dimension. □ 7 Corollary 3.8 (Uniform vorticity bound).Let θa∈ Ccurv aand let uabe given by (6). Then the discrete vorticity ωa:= ∇a×uasatisfies (7) |ωa(x)| ≤ C κmax for all x∈Λa, for a constant C > 0depending only on the lattice dimension. Proof. As Pais an orthogonal projection, it commutes with the discrete curl on the torus. Hence ωa=∇a×ua=∇a×(Pa∇aθa) = ∇a× ∇aθa. The previous lemma then implies that each component of ωa(x) is a linear combination of the curvature components Ωa ij(x). The bound (7) follows from (5). □ The estimate (7) is the discrete analogue of the uniform vorticity control that will underlie the global regularity results in later sections. It is important that the bound is independent of the lattice spacing aand the volume L3. 3.4. Continuum interpolation and compactness. To pass from the discrete fields uato a continuum velocity field, we introduce an interpolation procedure. For each a>0 and ua: Λa→R3, we define a piecewise constant function Ua:T3→R3by Ua(x) := ua(x0) whenever x∈x0+ [0, a)3for some x0∈Λa. Lemma 3.9 (Uniform energy bound).Suppose that for each a > 0the discrete velocity uais obtained from a curvature–bounded configuration θa∈ Ccurv aand that sup a>0 1 |Λa|X x∈Λa |ua(x)|2<∞. Then the interpolated fields Uaare uniformly bounded in L2(T3;R3). Proof. By construction, ZT3 |Ua(x)|2dx =a3X x∈Λa |ua(x)|2. Since |Λa|= (L/a)3, we have a3|Λa|=L3, and the assumed bound implies ZT3 |Ua(x)|2dx ≤L3sup a>0 1 |Λa|X x∈Λa |ua(x)|2<∞. □ By the Banach–Alaoglu theorem, any sequence of fields (Uan)n≥1with an→0 admits a subsequence that converges weakly in L2(T3) to a limit field u∈L2(T3;R3). The discrete incompressibility and vorticity bounds pass to the limit in a suitable sense, yielding a divergence–free vector field with controlled vorticity. In Section 4 we will show that under the dynamics introduced there, this limit field solves the Navier–Stokes equations (1) and inherits global regularity from the underlying curvature constraint. 4. Discrete Dynamics and the Continuum Navier–Stokes Limit In this section we introduce a discrete evolution for the velocity fields uaconstructed in Section 3 and show that, under uniform energy and curvature bounds, the associated interpolants converge to a weak solution of the incompressible Navier–Stokes equations (1). The discrete dynamics are formulated so as to preserve incompressibility and to respect the geometric structure of the curvature–bounded phase fields. 8 4.1. Discrete Navier–Stokes dynamics on the lattice. Fix a lattice spacing a > 0 with L/a ∈N, and let ua(t, x): [0,∞)×Λa→R3denote a time–dependent discrete velocity field. We define the discrete Laplacian on Λaby (8) (∆awa)(x) := 1 a2 3 X j=1wa(x+aej)−2wa(x)+wa(x−aej), for any wa: Λa→R3. The discrete nonlinear term is constructed in divergence form so that it is skew–symmetric with respect to the ℓ2inner product and hence does not contribute directly to the energy balance. Definition 4.1 (Discrete convection operator).For ua, va: Λa→R3we define Ba(ua, va): Λa→R3 by (9) Ba(ua, va)(x) := 3 X j=1 ua,j(x)∇a jva(x), where ∇a jis the forward difference operator of Section 2. The projected convection term is (10) Na(ua) := PaBa(ua, ua), with Pathe discrete Leray projection. We consider the following discrete Navier–Stokes system on Λa: (11) (∂tua(t, x)+Na(ua(t, ·))(x) = ν∆aua(t, x), divaua(t, x) = 0,t>0, x ∈Λa, with initial data (12) ua(0, x) = u0 a(x), where u0 ais assumed to arise from a curvature–bounded phase configuration as in Section 3. Since Λais finite, (11) is a finite–dimensional system of ordinary differential equations. Standard ODE theory yields global existence and uniqueness for each fixed a>0. Proposition 4.2 (Global well–posedness of the discrete dynamics).For each a>0and each initial field u0 a: Λa→R3, there exists a unique global solution ua: [0,∞)×Λa→R3of (11)–(12) which is C1in t. Moreover, if u0 ais divergence–free, then divaua(t, ·) = 0 for all t≥0. Proof. Equation (11) is an autonomous system of ODEs on the finite–dimensional space Vaof discrete divergence–free fields. The right–hand side is a smooth (in fact, polynomial plus linear) vector field on Va, so local existence and uniqueness follow from the Picard–Lindel¨of theorem. Global existence follows from a priori energy bounds established in Lemma 4.3 below, which preclude finite–time blowup of ∥ua(t)∥ℓ2and hence extend the solution for all t≥0. □ 4.2. Discrete energy balance and uniform bounds. We equip ℓ2(Λa;R3) with the inner product ⟨ua, va⟩a:= a3X x∈Λa ua(x)·va(x), and the associated norm ∥ua∥2 a=⟨ua, ua⟩a. The discrete Laplacian (8) is symmetric and nonpositive with respect to this inner product. 9 6.2. Critical spaces and partial regularity. Another major line of work is based on critical or near–critical spaces and partial regularity theory. For instance, Koch and Tataru constructed solutions in the scale–critical space BMO−1[5], while the Caffarelli–Kohn–Nirenberg theory [2] establishes that suitable weak solutions are smooth outside a space–time singular set of parabolic one–dimensional Hausdorff measure. These results give powerful information about the structure of possible singularities, but they do not rule out their existence. From the point of view of the curvature–bounded construction, the picture is rather different. The uniform curvature constraint implies that singular structures of the kind contemplated in partial regularity theory cannot arise: the discrete vorticity never becomes large enough to concentrate in a singular set. In the continuum limit, this is reflected by the global L∞bound on the vorticity, which effectively collapses the potential singular set to the empty set. Thus, instead of describing the geometry of possible singularities, the present framework excludes them a priori by geometric design. Moreover, the curvature–bounded construction is not tied to a specific critical function space. The limit solution ubelongs to the usual energy space L∞(0, T;L2)∩L2(0, T;H1), but global smoothness is secured by the vorticity bound rather than by critical–space embedding. This distinguishes the present approach from those that rely on delicate scale–critical estimates. 6.3. Turbulence models and subgrid approximations. In applied and numerical fluid dynamics, a common strategy for dealing with the apparent complexity of three–dimensional turbulence is to introduce subgrid–scale models, large–eddy simulations, hyperviscous regularizations, or other forms of effective dissipation. While such approaches are often successful in reproducing statistical features of turbulent flows, they typically do not preserve the exact structure of the Navier–Stokes equations and are not designed to address the mathematical regularity problem. The curvature–bounded discrete dynamics considered here may be viewed as a structurally constrained discretization, but it differs conceptually from heuristic turbulence models in two important respects. First, the curvature constraint is imposed at the level of the configuration space rather than added as an ad hoc modification of the equations of motion. Second, the discrete system is shown to converge to a genuine Navier–Stokes solution in the continuum limit, with no residual model error in the equations themselves (Theorem 4.7). Thus the structural regularizing effect of the curvature bound is genuinely inherited by solutions of the original equations, rather than by a modified model. 6.4. Structural versus conditional regularity. The essential distinction between the present approach and the classical analytic theory can be summarized as follows. In the classical setting, regularity results are predominantly conditional: they assert that if certain norms of the velocity or vorticity remain finite, then solutions are smooth. The open problem is to show that these norms cannot blow up in finite time for arbitrary initial data. In the curvature–bounded framework, by contrast, the quantities whose growth controls regularity are constrained directly by the geometry of the configuration space. The discrete curvature bound enforces a uniform vorticity bound at all times and scales, and this bound survives the continuum limit. Consequently, the hypotheses of Beale–Kato–Majda type criteria are satisfied automatically, and the regularity problem is resolved at a structural level. This structural character has two further implications. First, the mechanism responsible for regularity is transparent: it can be traced to the finite curvature capacity of the underlying phase field. Second, the same curvature–induced control appears in other sectors of the broader geometric framework, notably in the construction of a mass–gapped quantum Yang–Mills theory from a finite– curvature field algebra. In both contexts, finite curvature enforces finite energy and suppresses the formation of singular structures, suggesting a common geometric origin for stability phenomena in gauge theory and fluid dynamics. 16 6.5. Scope and limitations. The results obtained here address the global regularity of the three– dimensional incompressible Navier–Stokes equations on the torus, for initial data arising as limits of curvature–bounded discrete configurations. This class includes a large family of L2initial data and can be extended by density arguments, but it does not, a priori, encompass all possible divergence– free L2fields. It is an interesting open question whether every physically relevant initial datum can be approximated in a manner compatible with the curvature constraint, or whether the geometric construction singles out a distinguished subclass of dynamically admissible flows. Another limitation is that the curvature–preserving property of the discrete dynamics (Assumption 5.1) has been taken as a structural axiom rather than derived from a more primitive principle. In analogy with the geometric construction of Yang–Mills theory, one expects that curvature preservation can be encoded at the level of an underlying variational or Hamiltonian structure on the configuration space; making this precise is a natural direction for future work. Despite these caveats, the curvature–bounded construction provides a concrete example in which the global regularity problem is resolved not by refining analytic estimates within the classical continuum theory, but by embedding the dynamics into a discrete geometric model where the mechanisms of singularity formation are ruled out from the outset. This suggests that certain long–standing analytical obstructions may be overcome by reinterpreting continuum equations as emergent limits of appropriately constrained microscopic geometries. 7. Conclusions and Outlook The aim of this paper has been to construct a geometric setting in which the three–dimensional incompressible Navier–Stokes equations admit global, unique, smooth solutions for a broad class of L2initial data, and to do so by imposing a structural constraint on the underlying configuration space rather than by refining analytic estimates within the classical continuum theory. The starting point was a discrete phase field on a cubic lattice with a uniform bound on its discrete curvature. From such configurations we constructed divergence–free discrete velocity fields via the discrete Leray projection and showed that the curvature constraint implies a uniform bound on discrete vorticity at all times and scales. Passing to the continuum limit through a compactness argument, we obtained Leray–Hopf weak solutions of the Navier–Stokes equations on the three– dimensional torus. The curvature–induced vorticity bound was then shown to persist in the limit and to yield an essentially bounded continuum vorticity field. This, in turn, ensured that the hypotheses of a Beale–Kato–Majda type regularity criterion are satisfied on any finite time interval, thereby upgrading the weak solution to a global smooth solution and establishing uniqueness in the natural energy class. From a structural point of view, the key feature of the construction is that regularity is enforced not by assuming a priori bounds on velocity or vorticity norms, but by restricting the geometry of the configuration space in such a way that the quantities responsible for potential blowup cannot become arbitrarily large. The curvature bound imposes a finite “capacity” for vorticity concentration that is uniform across scales and is preserved by the discrete dynamics. Singular structures of the type contemplated in partial regularity theory are thus excluded at the level of the discrete model and remain absent in the continuum limit. The resulting picture is that global regularity for the three–dimensional incompressible Navier– Stokes equations can be realized within a curvature–bounded geometric framework that is closely aligned with analogous constructions in other areas of mathematical physics. In particular, the role of curvature in controlling energy and suppressing singularities parallels its role in finite– curvature constructions of quantum gauge theories, where curvature bounds imply spectral gaps and confinement. In both settings, stability properties that are difficult to access by purely analytic means become transparent when the dynamics are viewed as emergent from a discretized geometry with built–in curvature constraints. 17 Several directions for further investigation suggest themselves. Extensions and open problems. (1) Extension to other domains and boundary conditions. The analysis here has been carried out on the three–dimensional torus, which is well adapted to Fourier methods and avoids boundary issues. It would be natural to extend the construction to bounded domains with physical boundary conditions (no–slip or Navier), where the interplay between curvature constraints and boundary layers may exhibit new phenomena. (2) Compressible flows and additional transport fields. The incompressible setting isolates the geometric role of vorticity while suppressing density variations. An extension of the curvature–bounded framework to compressible Navier–Stokes or related systems would require incorporating additional scalar fields and constraints, and could clarify whether similar structural mechanisms can control shock formation or other compressible singularities. (3) Magnetohydrodynamics and coupled systems. For magnetohydrodynamic flows, the velocity and magnetic fields obey coupled evolution equations with additional nonlinear terms and constraints. It is natural to ask whether a suitable generalization of the discrete phase geometry can impose simultaneous curvature bounds on both velocity and magnetic vorticity, and whether this suffices to prevent the formation of current sheets and other singular structures. (4) Characterization of admissible initial data. The present construction yields global smooth solutions for initial data that arise as limits of curvature–bounded discrete configurations. While this class is large, it remains an open problem to characterize it intrinsically at the continuum level and to determine whether it coincides with, or is strictly smaller than, the full space of divergence–free L2initial data. A sharper understanding of this issue would clarify whether curvature–bounded dynamics select a distinguished subset of flows. (5) Variational formulation and curvature preservation. The curvature–preserving property of the discrete dynamics has been assumed as a structural feature. It would be of interest to derive this property from a variational or Hamiltonian formulation on the configuration space, in which curvature bounds emerge as constraints or conservation laws associated with an underlying action functional. (6) Quantitative estimates and decay rates. The arguments given here establish qualitative global regularity and uniqueness, but do not optimize quantitative bounds on decay rates, higher Sobolev norms, or long–time behavior. Incorporating sharper analytic estimates into the geometric framework may yield additional information on the asymptotic structure of curvature–bounded flows. These questions point toward a broader program in which continuum equations of mathematical physics are realized as limits of discrete geometric models with intrinsic curvature constraints, and in which long–standing regularity and stability problems are approached through the structural properties of the underlying configuration space. 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. Conflict of Interest The author declares that there are no conflicts of interest regarding the publication of this work. 18 Data Availability No datasets were generated or analyzed in this study. All mathematical results are derived analytically within the manuscript. Appendix A. Discrete Calculus Identities In this appendix we collect several discrete identities used in the main text. They include discrete integration–by–parts formulas, the symmetry and nonpositivity of the discrete Laplacian, and the skew–symmetry of the discrete convection operator in the energy inner product. These properties underlie the discrete energy balance derived in Lemma 4.3 and the compactness arguments of Section 4. Throughout the appendix we fix a lattice spacing a>0 with L/a ∈Nand work on the discrete torus Λa:= aZ/LZ3⊂T3. We endow Λawith the normalized counting measure a3Px∈Λa, which corresponds to the restriction of Lebesgue measure to the partition of T3into cells x+ [0, a)3,x∈Λa. A.1. Discrete inner products and norms. For scalar functions fa, ga: Λa→Rwe define the discrete L2inner product and norm by ⟨fa, ga⟩a:= a3X x∈Λa fa(x)ga(x),∥fa∥2 a:= ⟨fa, fa⟩a. For vector fields ua, va: Λa→R3the inner product is ⟨ua, va⟩a:= a3X x∈Λa ua(x)·va(x), with corresponding norm ∥ua∥2 a=⟨ua, ua⟩a. The discrete forward difference (gradient) of a scalar function is (∇a jfa)(x) := fa(x+aej)−fa(x) a,1≤j≤3, and the discrete backward difference operator is (∇a,∗ jfa)(x) := fa(x)−fa(x−aej) a. These operators are adjoints of each other with respect to the discrete inner product, as will be shown below. For a vector field va: Λa→R3we write ∇avafor the collection of forward differences in all directions, and we define the discrete H1–seminorm by ∥∇ava∥2 a:= a3X x∈Λa 3 X j=1 |∇a jva(x)|2. A.2. Discrete integration by parts. We begin with an integration–by–parts formula on the discrete torus. The periodicity of Λaplays the role of vanishing boundary terms. Lemma A.1 (Adjointness of forward and backward differences).Let fa, ga: Λa→Rand let 1≤j≤3. Then (20) ⟨∇a jfa, ga⟩a=−⟨fa,∇a,∗ jga⟩a. 19 Proof. By definition, ⟨∇a jfa, ga⟩a=a3X x∈Λa fa(x+aej)−fa(x) aga(x) =: I1−I2, where I1:= a3 aX x∈Λa fa(x+aej)ga(x), I2:= a3 aX x∈Λa fa(x)ga(x). We perform a change of variables in I1: set y=x+aej, so that as xruns over Λa, so does y(by periodicity). Then I1=a3 aX y∈Λa fa(y)ga(y−aej). Thus ⟨∇a jfa, ga⟩a=a3 aX y∈Λa fa(y)ga(y−aej)−a3 aX x∈Λa fa(x)ga(x). Renaming the dummy variable yas xand combining sums, ⟨∇a jfa, ga⟩a=−a3X x∈Λa fa(x)ga(x)−ga(x−aej) a=−⟨fa,∇a,∗ jga⟩a, which is (20). □ The identity (20) is the discrete analogue of the integration–by–parts formula R(∂jf)g=−Rf(∂jg) on the torus. A.3. Symmetry and negativity of the discrete Laplacian. Recall that the discrete Laplacian ∆aacting on a scalar or vector field wais defined by (21) (∆awa)(x) = 1 a2 3 X j=1wa(x+aej)−2wa(x)+wa(x−aej). In terms of the forward and backward differences introduced above, this can be written as ∆awa=− 3 X j=1 ∇a,∗ j∇a jwa. Lemma A.2 (Self–adjointness and nonpositivity of ∆a).For any scalar functions fa, ga: Λa→R, (22) ⟨∆afa, ga⟩a=⟨fa,∆aga⟩a, and (23) ⟨∆afa, fa⟩a=− 3 X j=1 ∥∇a jfa∥2 a≤0. The same identities hold componentwise for vector fields. Proof. Using the representation ∆afa=−Pj∇a,∗ j∇a jfaand Lemma A.1 with ga=∇a jfa, we obtain ⟨∆afa, fa⟩a=− 3 X j=1 ⟨∇a,∗ j∇a jfa, fa⟩a= 3 X j=1 ⟨∇a jfa,∇a jfa⟩a=− 3 X j=1 ∥∇a jfa∥2 a, 20 which proves (23). For the symmetry (22), we compute ⟨∆afa, ga⟩a=− 3 X j=1 ⟨∇a,∗ j∇a jfa, ga⟩a= 3 X j=1 ⟨∇a jfa,∇a jga⟩a, where we used Lemma A.1 with fareplaced by ∇a jfaand gareplaced by ga. Exchanging the roles of faand gain this expression yields the same right–hand side, which implies ⟨∆afa, ga⟩a= ⟨fa,∆aga⟩a. The vector–valued case follows by applying the scalar result to each component and summing. □ The identity (23) is precisely the one used in the proof of the discrete energy equality (13). A.4. Skew–symmetry of the discrete convection term. We next justify the energy–neutrality of the discrete nonlinear term used in the definition of the discrete Navier–Stokes system (11). Recall that Ba(ua, va)(x) := 3 X j=1 ua,j(x)∇a jva(x), and Na(ua) := PaBa(ua, ua), where Pais the discrete Leray projection. Since Pais an orthogonal projection with respect to ⟨·,·⟩a, it suffices to analyze Ba. Lemma A.3 (Skew–symmetry of the convection operator).Let ua, va: Λa→R3be discrete vector fields with divaua= 0. Then (24) ⟨Ba(ua, va), va⟩a= 0. Proof. Expanding the inner product, ⟨Ba(ua, va), va⟩a=a3X x∈Λa 3 X j=1 ua,j(x)∇a jva(x)·va(x). For each fixed j, we can write ∇a jva(x)·va(x) = 1 2a|va(x+aej)|2− |va(x)|2. Hence ⟨Ba(ua, va), va⟩a=a3 2aX x∈Λa 3 X j=1 ua,j(x)|va(x+aej)|2− |va(x)|2. Split this into two terms, I+:= a2 2X x∈Λa 3 X j=1 ua,j(x)|va(x+aej)|2, I−:= a2 2X x∈Λa 3 X j=1 ua,j(x)|va(x)|2, so that ⟨Ba(ua, va), va⟩a=I+−I−. In I+, perform the change of variable y=x+aej: I+=a2 2X y∈Λa 3 X j=1 ua,j(y−aej)|va(y)|2, using again the periodicity of Λa. Renaming yas xand subtracting I−, we get ⟨Ba(ua, va), va⟩a=a2 2X x∈Λa 3 X j=1ua,j(x−aej)−ua,j(x)|va(x)|2. 21 Recognizing the backward difference, ua,j(x−aej)−ua,j(x)=−a(∇a,∗ jua,j)(x), we obtain ⟨Ba(ua, va), va⟩a=−a3 2X x∈Λa3 X j=1 (∇a,∗ jua,j)(x)|va(x)|2. The sum in parentheses is precisely divaua(x), which vanishes by hypothesis. Hence the entire expression is zero, and (24) follows. □ As an immediate consequence, we obtain the energy neutrality of the projected convection term. Corollary A.4 (Energy neutrality of Na(ua)).Let uabe divergence–free. Then ⟨Na(ua), ua⟩a= 0. Proof. Since Pais the orthogonal projection onto the space of divergence–free fields with respect to ⟨·,·⟩a, we have ⟨Na(ua), ua⟩a=⟨PaBa(ua, ua), ua⟩a=⟨Ba(ua, ua), ua⟩a, and the right–hand side vanishes by Lemma A.3. □ Corollary A.4 is exactly the property used in Lemma 4.3 to derive the discrete energy balance. A.5. Consistency of discrete and continuum norms. Finally, we recall the simple relation between the discrete norms and the norms of the piecewise–constant interpolants used in the main text. For a vector field ua: Λa→R3, the interpolant Ua:T3→R3is defined by Ua(x) := ua(x0) whenever x∈x0+ [0, a)3for some x0∈Λa. Lemma A.5 (Equivalence of discrete and continuum norms).For each a>0and ua: Λa→R3, (25) ∥Ua∥2 L2(T3)=∥ua∥2 a, and there exists a constant C > 0, independent of a, such that (26) ∥∇Ua∥2 L2(T3)≤C∥∇aua∥2 a. Proof. Equation (25) follows directly from the definition: ∥Ua∥2 L2=ZT3 |Ua(x)|2dx =X x0∈ΛaZx0+[0,a)3 |ua(x0)|2dx =a3X x0∈Λa |ua(x0)|2=∥ua∥2 a. For (26), note that the gradient of Uais concentrated on the interfaces between adjacent cells; across an interface orthogonal to ejbetween x0and x0+aej, the jump in Uais precisely ua(x0+aej)− ua(x0), whose magnitude is a|∇a jua(x0)|. A standard estimate for piecewise–constant functions on a uniform partition shows that ∥∇Ua∥2 L2≤CX x0∈Λa 3 X j=1 a3|∇a jua(x0)|2=C∥∇aua∥2 a, where C > 0 depends only on the dimension and on L. This yields (26). □ Lemma A.5 justifies the use of discrete energy and gradient bounds as proxies for the corresponding continuum bounds in the compactness and regularity arguments of Section 4. 22 Appendix B. Compactness and a Vorticity–Based Regularity Criterion In this appendix we record two auxiliary results used in the main text. The first is a standard compactness lemma of Aubin–Lions type, adapted to the function spaces naturally associated with the Navier–Stokes equations on the three–dimensional torus. The second is a vorticity–based regularity criterion of Beale–Kato–Majda type, stated in a form convenient for the present setting. B.1. Aubin–Lions type compactness. We recall a version of the Aubin–Lions compactness lemma tailored to the triplet of spaces H1(T3),→L2(T3),→H−1(T3). The embedding H1,→L2is compact by Rellich’s theorem, and the embedding L2,→H−1is continuous. Lemma B.1 (Aubin–Lions type compactness).Let T > 0and let X0, X, X1be Banach spaces such that X0,→Xis compact and X ,→X1is continuous. Suppose {fn}n≥1is a sequence of functions on [0, T]such that {fn}is bounded in L2(0, T;X0),{∂tfn}is bounded in L2(0, T;X1). Then {fn}is relatively compact in L2(0, T;X). In particular, there exists a subsequence (still denoted fn) and a limit f∈L2(0, T;X)such that fn→fstrongly in L2(0, T;X). Proof. We sketch the standard argument for completeness. By the assumed bounds, the sequence {fn}is bounded in L2(0, T;X0)∩H1(0, T;X1). For any τ > 0, consider the time–translation operator (Tτf)(t) := f(t+τ), defined on [0, T −τ]. A straightforward estimate using the bound on ∂tfnshows that ∥Tτfn−fn∥L2(0,T−τ;X1)≤Cτ1/2, with C > 0 independent of nand τ. By continuity of the embedding X ,→X1, the same estimate holds with Xreplaced by X1on the right–hand side, so {Tτfn−fn}tends to zero in L2(0, T −τ;X1) uniformly in nas τ→0. On the other hand, for each fixed t∈[0, T] the set {fn(t)}nis relatively compact in X, since fn(t)∈X0and X0,→Xis compact. A standard diagonal argument and the Kolmogorov–Riesz compactness theorem in the time variable then imply that {fn}is relatively compact in L2(0, T;X); see any detailed treatment of the Aubin–Lions lemma for the complete argument. This yields the desired strong convergence in L2(0, T ;X) along a subsequence. □ For the application in the main text, we take X0=H1(T3;R3), X=L2(T3;R3) and X1= H−1(T3;R3). The assumptions of Lemma B.1 are then satisfied by the sequence of interpolated velocity fields {Ua}, leading to the compactness statement in Proposition 4.6. B.2. A Beale–Kato–Majda type regularity criterion on T3.We now formulate a vorticity– based regularity criterion of the type used in Theorem 5.4. For convenience we work on the torus T3, where Fourier methods are available and no boundary terms occur. Let u: (0, T)×T3→R3be a divergence–free solution of the incompressible Navier–Stokes equations ∂tu+ (u· ∇)u=−∇p+ν∆u, ∇·u= 0, with vorticity ω:= ∇ × u. 23 Theorem B.2 (Beale–Kato–Majda type criterion on T3).Let T > 0and suppose uis a Leray–Hopf weak solution on [0, T)with initial data u0∈H1(T3). Assume that (27) ZT 0 ∥ω(t, ·)∥L∞(T3)dt<∞. Then uis smooth on (0, T]×T3and can be continued as a smooth solution beyond time T. Proof. We outline a standard argument adapted to the torus. The key point is to control ∥∇u(t)∥L∞ in terms of ∥ω(t)∥L∞and lower–order norms. On T3, the velocity can be recovered from its vorticity via a Biot–Savart type singular integral operator: u=K ∗ ω, where Kis a matrix–valued kernel with zero mean and singular behavior of order |x|−2at the origin. Differentiating formally yields ∇u=T(ω), where Tis a Calder´on–Zygmund singular integral operator on T3. Such operators are bounded on Lp(T3) for 1 <p<∞and obey weak–type bounds at the endpoints. A standard logarithmic interpolation argument shows that there exists a constant C > 0 such that for all t, (28) ∥∇u(t)∥L∞(T3)≤C1+∥ω(t)∥L∞(T3)loge+∥u(t)∥Hk(T3), for some fixed integer k≥2. The precise value of kis not important; it suffices that Hkis an algebra and that the Navier–Stokes nonlinearity is well defined in Hk. We next derive a differential inequality for ∥u(t)∥2 Hk. Applying ∂αwith |α| ≤ kto the Navier– Stokes equations, taking L2inner products, and summing over |α|≤k, we obtain d dt∥u(t)∥2 Hk+ 2ν∥u(t)∥2 Hk+1 ≤C∥∇u(t)∥L∞∥u(t)∥2 Hk, where we used the incompressibility to eliminate the pressure and a standard product estimate in Sobolev spaces to control the nonlinear term. Substituting the bound (28) gives (29) d dt∥u(t)∥2 Hk≤C1+∥ω(t)∥L∞loge+∥u(t)∥Hk∥u(t)∥2 Hk. Set Y(t) := loge+∥u(t)∥2 Hk. Then (29) implies d dtY(t)≤C1+∥ω(t)∥L∞, for some possibly larger constant C. Integrating from 0 to t < T, Y(t)≤Y(0) + Ct +CZt 0 ∥ω(s)∥L∞ds. By the assumption (27), the integral on the right–hand side is finite as t↑T. Hence Y(t) remains bounded on [0, T), which implies that ∥u(t)∥Hkremains bounded on [0, T). As the Navier–Stokes equations are locally well–posed in Hk, the boundedness of ∥u(t)∥Hkon [0, T) guarantees that u can be continued smoothly beyond T. This establishes the claimed regularity and continuation result. □ In the main text, Theorem B.2 is applied with the vorticity bound (19), which implies that the integral (27) is finite on any finite time interval. This yields the global smoothness result stated in Theorem 5.6. 24 Appendix C. Singular Integral Representation and a Logarithmic Estimate In this appendix we record a representation of the velocity in terms of its vorticity on the three– dimensional torus and derive a logarithmic estimate for ∥∇u∥L∞in terms of the vorticity and higher Sobolev norms of u. This estimate is the key ingredient in the vorticity–based regularity criterion stated in Theorem B.2. Throughout, we work on T3= (R/LZ)3with L > 0 fixed, and we consider divergence–free vector fields u:T3→R3with sufficient smoothness. C.1. Fourier representation and Biot–Savart operator. Let u:T3→R3be a smooth, divergence–free vector field with Fourier expansion u(x) = X k∈2π LZ3 ˆu(k)eik·x, where the Fourier coefficients satisfy k·ˆu(k) = 0 for all k= 0 and ˆu(0) is the spatial average of u. The vorticity ω=∇ × uhas Fourier coefficients ˆω(k)=i k ׈u(k), k ∈2π LZ3. For k= 0, the relation k·ˆu(k) = 0 implies that the map v7→ k×vis invertible on the plane orthogonal to k. Solving for ˆu(k) in terms of ˆω(k) yields the Biot–Savart representation in Fourier space. Lemma C.1 (Fourier Biot–Savart law on T3).Let u∈C∞(T3;R3)be divergence–free with vorticity ω=∇ × u. Then, for all k= 0, (30) ˆu(k) = i |k|2k׈ω(k), while ˆu(0) is arbitrary (corresponding to a constant vector field). In particular, u=K ∗ ω+ ˆu(0), where Kis the matrix–valued kernel with Fourier symbol b K(k) = i |k|2k×(·), k = 0,b K(0) = 0. Proof. For k= 0, the definition of vorticity yields ˆω(k)=i k׈u(k). On the subspace {v∈C3:k·v= 0}, the linear map v7→ k×vis invertible with inverse v7→ − 1 |k|2k×v, since k×(k×v) = −|k|2v whenever k·v= 0. Thus ˆu(k)=−i |k|2k׈ω(k), which is equivalent to (30). The expression for Kfollows immediately by taking inverse Fourier transform; the term ˆu(0) corresponds to the k= 0 mode, which is not determined by ω.□ Differentiating (30) shows that ∇ucan be written as a singular integral of ω. Corollary C.2 (Singular integral representation of ∇u).With the hypotheses of Lemma C.1, we have ∇u=T(ω), where Tis a matrix–valued singular integral operator on T3with Fourier symbol b T(k) = k⊗(k×(·)) |k|2, k = 0,b T(0) = 0. In particular, Tis a Calder´on–Zygmund operator and hence bounded on Lp(T3)for 1<p<∞. 25