On Structural Stability:
Atemporal Persistence of Navier–Stokes Flow
in Nonlinear Continua

1 Introduction

The three-dimensional Navier–Stokes equations describe one of the most fundamental nonlinear systems in mathematical physics [2],

Despite their compact form and extensive use, it remains unknown [3] whether smooth divergence-free initial data always generate globally smooth solutions or whether finite-time singularities can occur. The classical problem is therefore one of temporal continuation: given \(u_0\), does the trajectory \(u(t)\) remain regular for all time?

Let \(\Phi _{\mathrm {NS}}\) denote the global regularity proposition that every admissible smooth initial state admits global smooth continuation. In its strongest form, CJM–Navier asks whether this proposition can be structurally discriminated as

without requiring the future trajectory to be sequentially generated as the route to that judgment. The central question is therefore not whether time disappears from fluid dynamics, but whether the global admissibility of regular continuation must itself be discovered through time.

This viewpoint extends the Changbal Jump Machine (CJM) framework and the atemporal complexity class O(J) developed in our previous studies [1], following applications to structural criticality, form, and capacity. Navier–Stokes presents a distinct challenge because the system is continuous and evolutionary [2]. Physical evolution remains temporal. Atemporality instead refers to a separation between temporal generation and structural discrimination: the flow evolves through \(u(x,t)\), while CJM asks whether the admissibility of its continuation can be judged from structural organization without sequentially deriving the future trajectory.

The present work does not yet perform the final TRUE/FALSE discrimination of \(\Phi _{\mathrm {NS}}\). It investigates an operational precursor to that stronger objective. We ask whether an instantaneous flow state contains an observable organization associated with the structural possibility of regular continuation. We refer to the persistence and reorganization of this possibility space as structural persistence.

To examine this question, we introduce CJM–Navier. Independently generated three-dimensional flow snapshots are converted into finite structural representations using vortex-stretching and viscous-control information and are evaluated through a fixed CJM core. The response is examined through both scalar structural diagnostics and the organization of its admissibility texture. A Taylor–Green baseline is supplemented by small divergence-free symmetry-breaking perturbations to test whether the observed organization persists, reorganizes, or exhibits enhanced structural susceptibility under changes of the supplied flow state.

No proof of global regularity or construction of singularity is claimed. The purpose is to test a more fundamental possibility: whether Navier–Stokes regularity may admit a structural description that is distinguishable from the temporal generation of the trajectory itself. The flow evolves in time; CJM asks whether its global admissibility must also be discovered through time.

2 The Atemporal Navier–Stokes Question

The classical Navier–Stokes regularity problem asks whether a smooth solution can be continued for all time [4]. Its usual formulation is temporal: starting from an initial condition \(u_0\), one follows

and asks whether regularity survives throughout the evolution.

CJM–Navier begins from a different question. If \(\Phi _{\mathrm {NS}}\) denotes the proposition of global smooth continuation for all admissible smooth initial data, must the future trajectory be sequentially generated before the truth of that proposition can be discriminated? In its strongest form, the atemporal formulation is

The distinction is therefore between temporal generation and structural discrimination. The Navier–Stokes equations remain evolutionary, but CJM asks whether the admissibility of global regular continuation must itself be derived through the ordered construction of the entire future trajectory.

The present study does not yet perform this final global discrimination. Instead, it examines an operational precursor at the level of an instantaneous flow state. Each supplied configuration \(u(x,t_0)\) is associated with a finite regular-continuation admissibility structure,

representing the organization compatible with continued regular motion under the adopted structural encoding. The object \(\mathcal {A}_{\mathrm {reg}}\) is not a classical solution space or regularity criterion [5]; it is a finite operational surrogate for whether the encoded organization remains broad, concentrated, fragmented, persistent, or reorganized.

The corresponding CJM probe may be written schematically as

where \(J\) denotes the fixed CJM discrimination output. A snapshot may be generated through conventional temporal integration, but once supplied its CJM evaluation requires neither preceding nor future CJM states. Thus,

may be evaluated without knowledge of \(J_{i-1}\) or \(J_{i+1}\).

Atemporality therefore does not remove physical time from the fluid equations and does not imply computation beyond the Turing framework. Its defining claim is narrower and deeper: structural admissibility need not necessarily be obtained by sequentially generating the trajectory whose continuation is being judged. Navier–Stokes evolves in time; CJM asks whether the admissibility of that evolution must also be discovered through time.

3 Atemporal Structural Probe

The ultimate CJM–Navier question concerns the structural admissibility of the global regularity proposition \(\Phi _{\mathrm {NS}}\). The present experiment addresses a preceding operational question: whether a finite representation of an instantaneous three-dimensional flow can exhibit persistent structural organization without requiring its subsequent trajectory to be supplied to the CJM discrimination stage.

To examine this possibility numerically, we apply the CJM probe to three-dimensional Taylor–Green flow [6] on the periodic domain \([0,2\pi ]^3\), with

Reference states are generated using a pseudo-spectral incompressible Navier–Stokes solver on a \(20^3\) grid with \(2/3\)-rule dealiasing [7] and SSP-RK3 integration [8]. Independent cases span \(Re=50,75,\ldots ,400\), with \(\nu =1/Re\), reference time \(t_0=4\), and \(\Delta t=0.02\). The common reference time permits comparison of structural responses across changing Reynolds-number conditions. Temporal integration is used only to generate each supplied snapshot \(u_{Re}(x,t_0)\); no subsequent trajectory information enters the CJM discrimination stage.

Conceptually, each snapshot is treated as a finite global flow configuration,

where \(\mathcal {G}_{\mathrm {flow}}\) denotes the adopted structural representation of the instantaneous flow. The fields \(P_\omega \) and \(D_\omega \) encode vortex-stretching and viscous-control information, while \(x_1\) and \(x_2\) provide the finite logic-like interface used by the CJM core.

Each independently generated representation is then passed through the same fixed CJM core,

No CJM output from another Reynolds-number state is required. The sweep therefore consists of independent structural discriminations rather than a sequential CJM trajectory. This distinction provides the operational atemporal component of the present probe: physical states are generated in time, but their CJM discrimination does not propagate from one flow state to the next.

Figure 3(a) shows a rapid decline in both the viscous-control fraction and encoded AND3 density as \(Re\) increases. Within the adopted representation, this indicates a systematic reduction of instantaneous structural control. The two quantities are constructed differently, yet their concurrent decline establishes a common structural trend across the Reynolds-number sweep.

Figure 3(b) retains information that is lost in a scalar summary. Each column is the independently discriminated vector

Persistent spatial bands remain visible across the scan while their relative intensity and distribution reorganize at higher \(Re\). Because the response is independently median-centered and normalized within each snapshot, the relevant comparison is the organization of the texture rather than an absolute energy difference between columns.

The central observation is therefore the coexistence of two effects: instantaneous structural control declines, while an organized CJM response does not disappear uniformly. Instead, it persists in a reorganized form. This persistence is the experimental object considered here. It is not identified with mathematical regularity itself, nor is any Reynolds number designated as a Changbal threshold or singular boundary.

The result should therefore be read in relation to the larger atemporal question introduced above. The experiment does not yet discriminate \(\Phi _{\mathrm {NS}}\) as TRUE or FALSE. It tests whether structural organization relevant to such a future discrimination can remain observable in a supplied instantaneous state before the subsequent trajectory is known.

4 CJM–Navier: An Optional Structural Filter

The common CJM architecture is organized conceptually as [1]

Within this architecture, the preprocessing stage provides a modular position for problem-specific structural representation. Earlier applications used this position for Hodge and BSD filters; here the same principle is extended to nonlinear flow through a Navier Filter. The filter remains optional and may be bypassed through the identity operator,

where \(I\) denotes direct passage to the downstream CJM stages. The Navier Filter therefore neither defines nor modifies the common CJM core. Its role is to construct a finite Navier-specific structural interface through which an evolutionary flow state can enter the common discrimination architecture.

For the present implementation, an instantaneous flow state \(u(x,t_0)\) is treated as a supplied global configuration. We compute the vorticity and strain tensor [9],

and construct two structural quantities,

Here \(P_{\omega }\) represents local vortex-stretching production [10], while \(D_{\omega }\) is used as a positive viscous structural proxy. It is not the exact pointwise diffusive contribution; on a periodic domain,

as follows from the standard periodic enstrophy balance [2].

These continuous quantities are mapped into logic-like structural literals,

and

Thus \(x_1\) marks regions of positive vortex stretching, while \(x_2\) marks locations where the viscous proxy is at least comparable to the positive stretching contribution. In this sense,

provides the operational structural lift used in the present study. The pair \((x_1,x_2)\) is not itself a proof-theoretic reduction of the full Navier–Stokes regularity proposition. Rather, it is the finite interface through which flow organization is exposed to the common CJM discrimination layer.

After this Navier-specific transformation, the downstream CJM core is kept fixed. For the AND3 interaction used here,

where \(i-1\) denotes an adjacent structural element within the same snapshot, not a preceding time state. The distinction is essential: spatial adjacency is encoded inside one supplied configuration, whereas temporal succession is not used as an input to the CJM core.

The fixed median-centered CJM contrast map then produces

The full vector is retained because its internal organization is central to the present analysis. Scalar summaries are defined by

and the periodic median-crossing fragmentation rate

These scalar quantities remain secondary diagnostics. The principal object is the organization of the full admissibility texture and how that organization persists or reorganizes under changes in the supplied flow state.

The operational CJM–Navier path is therefore summarized as

Each Reynolds-number state is evaluated independently, without requiring a preceding or future CJM state. This snapshot-level construction is not the final discrimination of the global proposition \(\Phi _{\mathrm {NS}}\); it provides a finite structural pathway toward that stronger objective. The Navier Filter remains an optional problem-specific lens, while the common CJM core and its atemporal discrimination principle remain unchanged.

5 Robustness and Structural Susceptibility

To test whether the structure in Figure 3 depends only on exact Taylor–Green symmetry, we introduce small divergence-free perturbations while keeping the solver, structural encoding, Reynolds-number conditions, and fixed CJM core unchanged,

Two independent symmetry-breaking perturbation modes are examined. A \(3\%\) RMS perturbation using the first mode across the full Reynolds-number sweep preserves the principal CJM organization, with a median baseline–perturbed correlation of

The high median correlation indicates that the observed admissibility organization is not destroyed by a small departure from exact Taylor–Green symmetry. At the same time, the response is not completely rigid: localized reconfiguration remains visible at particular Reynolds-number conditions.

A focused test is therefore performed with both perturbation modes over \(Re=275\)–\(375\) and amplitudes of \(1\%\)–\(3\%\). We define the finite structural susceptibility

so that larger \(\chi _J\) indicates stronger reorganization of the fixed-core CJM texture relative to the unperturbed baseline.

Figure 3 compares this structural response with the corresponding physical displacement of the flow field. Panel (a) shows that \(\chi _J\) varies nonuniformly with Reynolds number and depends on both perturbation mode and amplitude. Mode 1 exhibits a pronounced localized increase near \(Re\approx 325\), whereas Mode 2, particularly at \(3\%\), develops a broader increase beginning near the same region and continuing toward higher \(Re\). Panel (b), in contrast, shows that the relative physical flow-field displacement changes smoothly and nearly monotonically across the same Reynolds-number interval. The contrast is therefore not simply a consequence of a larger imposed perturbation at one particular Reynolds number.

Schematically,

The two perturbation modes do not identify a unique critical Reynolds number. Instead, they indicate a mode-dependent regime of enhanced structural susceptibility emerging in the higher-\(Re\) portion of the present sweep. This is consistent with the coexistence of persistence and selective reorganization: the admissibility texture remains recognizable under small perturbations, while its internal organization can become substantially more sensitive under particular flow conditions.

Part of this sensitivity may arise from threshold crossings in the binary definitions of \(x_1\) and \(x_2\). The observed susceptibility is therefore interpreted as a property of the present finite CJM probe, not as evidence of a Navier–Stokes singularity, turbulence transition, or mathematically distinguished Reynolds number.

6 Structural Interpretation and Limitations

Figures 3 and 4 together show two related tendencies. As Reynolds number increases, the selected viscous-control fraction declines, while the fixed-core CJM organization persists, reorganizes, and becomes selectively more sensitive to small perturbations. Schematically,

This interpretation is structural rather than dynamical. Structural persistence does not imply that the flow becomes static or that all of its organization is preserved. It refers to the continued detectability of an organized admissibility texture as the selected instantaneous control measures decrease [10]. The perturbation tests further show that this organization can remain recognizable while undergoing nonuniform reconfiguration.

The observed persistence and susceptibility are not identified with singularity, finite-time blow-up, turbulence, instability, or loss of regularity [5]. In particular, the perturbation experiments do not establish a unique critical Reynolds number. The present result is instead the coexistence of declining structural control, persistent organization, and enhanced sensitivity within a finite CJM representation.

The probe remains limited by its \(20^3\) resolution, finite Reynolds-number interval, fixed reference time \(t_0=4\), small number of perturbation modes, and direction-dependent spatial encoding. Taylor–Green flow is known to develop increasingly complex small-scale structure through three-dimensional vortex stretching [11]. Higher resolutions, additional flow states and times, independent perturbations, and equivalent \(x\)-, \(y\)-, and \(z\)-oriented encodings are therefore required to test numerical, structural, and rotational robustness.

The binary definitions of \(x_1\) and \(x_2\) also introduce sharp encoding boundaries, so part of the measured susceptibility may arise from threshold crossings rather than from a distinct physical transition. Likewise, the independently normalized CJM texture represents within-snapshot organization rather than a common absolute energy scale.

Finally, the reference snapshots are generated through conventional temporal integration. Atemporality applies only to the subsequent structural discrimination: the CJM judgment does not require sequential propagation from preceding to future CJM states. It therefore neither removes physical time from Navier–Stokes dynamics nor implies hypercomputation. The Navier Filter remains an optional finite structural interface, and the present probe remains an operational precursor rather than the final TRUE/FALSE discrimination of \(\Phi _{\mathrm {NS}}\).

7 Conclusion

This paper reconsidered the three-dimensional Navier–Stokes regularity problem as a question of atemporal structural discrimination. The ultimate CJM–Navier question is whether global smooth continuation can be discriminated as structurally admissible or inadmissible without requiring the future trajectory to be sequentially generated as the route to that judgment [4]. Physical evolution remains temporal, but structural discrimination is conceptually separated from trajectory generation.

The present study does not yet perform this final TRUE/FALSE discrimination. Instead, it examines a finite operational precursor. In the three-dimensional Taylor–Green probe [6], increasing Reynolds number is accompanied by declining instantaneous structural control, while the fixed-core CJM response retains an organized but reorganized admissibility texture. The principal observation is thus the coexistence of reduced control with persistent structural organization.

The perturbation experiments extend this result. A \(3\%\) Mode 1 symmetry-breaking perturbation retains a median baseline–perturbed correlation of \(0.9745\), indicating substantial robustness. At the same time, Mode 1 shows a localized increase in susceptibility near \(Re\approx 325\), whereas Mode 2 develops a broader higher-\(Re\) response. The sensitivity is therefore mode-dependent and does not identify a unique critical Reynolds number.

Across the same interval, the physical velocity-field displacement changes smoothly, while the CJM susceptibility varies nonuniformly. Within the present representation,

The result therefore supports a combined picture of structural persistence and structural susceptibility, rather than evidence for a new physical transition [11]. CJM–Navier is accordingly interpreted as an optional problem-specific structural interface within the common CJM architecture [1], not as a replacement for the Navier–Stokes equations or a direct regularity criterion.

Important limitations remain. The present calculations use a \(20^3\) grid, a finite Reynolds-number interval, a fixed sampling time, two perturbation modes, and a direction-dependent binary encoding. Threshold crossings in \(x_1\) and \(x_2\) may also contribute to the measured susceptibility. No claim of global smoothness, finite-time singularity, or critical Reynolds number is therefore made [12]. Higher resolution, alternative flow states, spatial encodings, and broader perturbation tests remain necessary.

The deeper question remains whether Navier–Stokes regularity can ultimately be characterized not only through the temporal survival of a trajectory, but through the structural admissibility of that continuation before the trajectory is known. The flow evolves in time; CJM asks whether the truth of its global continuation must also be reached through time.

References

Appendix 1: Python code for Figure 1

def make_spectral_grid(N): 
 
    k = np.fft.fftfreq(N, d=1.0 / N) 
 
    KX, KY, KZ = np.meshgrid( 
        k, k, k, 
        indexing="ij" 
    ) 
 
    K2 = ( 
        KX**2 
        + 
        KY**2 
        + 
        KZ**2 
    ) 
 
    K2_SAFE = K2.copy() 
    K2_SAFE[0, 0, 0] = 1.0 
 
    K_MAG = np.sqrt(K2) 
 
    K_CUT = N // 3 
 
# Full code available at the links below (Colab, GitHub, project website).

Appendix 2: Navier Filter Pseudocode

 
PROCEDURE step1_input_navier_filter(input_instance): 
 
    # Interpret the input according to its current physical or logical form 
    IF input_instance is FlowSnapshot u(x,t0): 
        U <- ExtractInstantaneousFlowState(u(x,t0)) 
        Sigma_N <- BuildNavierStructural 
        Representation(U) 
 
    ELSE IF input_instance is NavierIndicatorProfile Theta_N: 
        Sigma_N <- LiftToNavierStructural 
        Representation(Theta_N) 
 
    ELSE IF input_instance is Navier3SATFormula F: 
        Sigma_N <- LiftToNavierStructural 
        Representation(F) 
 
    ELSE IF input_instance is SATReducedInstance F: 
        Sigma_N <- LiftToNavierStructural 
        Representation(F) 
 
# Full code available at the links below (Colab, GitHub, project website).

Appendix 3: Python code for Figure 3

 
def make_spectral_grid(N): 
    k = np.fft.fftfreq(N, d=1.0 / N) 
    KX, KY, KZ = np.meshgrid(k, k, k, indexing="ij") 
 
    K2 = KX**2 + KY**2 + KZ**2 
    K2_SAFE = K2.copy() 
    K2_SAFE[0, 0, 0] = 1.0 
 
    K_CUT = N // 3 
    DEALIAS = ( 
        (np.abs(KX) <= K_CUT) 
        & (np.abs(KY) <= K_CUT) 
        & (np.abs(KZ) <= K_CUT) 
    ) 
 
    return KX, KY, KZ, K2, K2_SAFE, DEALIAS 
 
 
def project_divergence_free(u_hat, KX, KY, KZ, K2_SAFE): 
    k_dot_u = KX * u_hat[0] + KY * u_hat[1] + KZ * u_hat[2] 
 
    out = u_hat.copy() 
    out[0] -= KX * k_dot_u / K2_SAFE 
    out[1] -= KY * k_dot_u / K2_SAFE 
    out[2] -= KZ * k_dot_u / K2_SAFE 
 
    out[:, 0, 0, 0] = 0.0 
    return out 
 
 
# Full code available at the links below (Colab, GitHub, project website).