Keywords. Navier–Stokes equations; suitable weak solutions; local regularity; Caffarelli–Kohn–Nirenberg theory; pressure defect; Reynolds stress; energy flux; defect cascade; observability; scale-critical regularity.

2020 Mathematics Subject Classification. 35Q30; 35B65; 35B45; 76D05.

1 Introduction↩︎

This paper is organized around the following Clay-adjacent reduction principle: \[\boxed{ \text{potential Navier--Stokes singularities are reduced to the existence of an} }\] \[\boxed{ \text{NS-realizable, cleaned, scale-critical, combined-invisible defect cascade.} }\] The statement is deliberately not a proof of global Navier–Stokes regularity, and it is not a singular-solution construction. It is a structural reduction theorem. If a point fails to enter the Caffarelli–Kohn–Nirenberg smallness regime at all sufficiently small dyadic scales, then in the present framework the obstruction must have a very special form. It must survive dyadic coarse graining, separation of active and harmonic pressure, finite-window localization, pressure testing, flux testing, positive energy testing, and adjoint trace testing.

The clean one-sentence conclusion is:

Potential Navier–Stokes singularities are reduced to the existence of an NS-realizable combined-invisible defect cascade.

Thus the role of the paper is not to announce that singularities have been excluded. Rather, it compresses the possible shape of a singular branch from a vague compactness failure into a highly constrained scale-critical object.

The theorem spine↩︎

Let \[r_n=2^{-n}\] be dyadic scales around a potential singular point. A non-CKN branch means that \[C(r_n)+D(r_n)> \varepsilon_{\mathrm{CKN}} \qquad\text{for all sufficiently large } n.\] From such a branch one attempts to extract NS-realizable coarse-grained defect packages \[D_{n,\ell_n} = (U_{n,\ell_n},P_{n,\ell_n};P^{\mathrm{act}}_{n,\ell_n},P^{\mathrm{har}}_{n,\ell_n},R_{n,\ell_n},\Pi_{n,\ell_n}),\] with \[R_{n,\ell_n} = S_{\ell_n}(u^{(n)}\otimes u^{(n)}) -U_{n,\ell_n}\otimes U_{n,\ell_n}, \qquad \Pi_{n,\ell_n}=-R_{n,\ell_n}:\nabla U_{n,\ell_n}.\] These objects are not formal defects. They are produced by Navier–Stokes rescaling and coarse graining, and hence carry the exact momentum identity, the pressure-compatibility relation, the active/harmonic pressure split, the Reynolds covariance constraint, and the local energy-flux identity.

In each finite window \[W_n=(n,\ell_n,\Lambda_n,\chi_n,s_n),\] we test the cleaned defect by the combined observation map \[O^{\mathrm{comb}}_{W_n} = (O^P_{W_n},O^F_{W_n},O^E_{W_n},O^T_{W_n}),\] consisting of active pressure, flux, energy, and adjoint-trace channels. The finite-window hierarchy is \[\text{pressure} \Longrightarrow \text{flux} \Longrightarrow \text{energy} \Longrightarrow \text{adjoint trace}.\] Active-pressure-visible defects are not phantoms. Active-pressure-null defects are tested by flux, while the local harmonic pressure component is retained in the momentum and energy channels rather than quotiented out. Pressure–flux invisible defects enter the finite-dimensional kernel \(K_W^{\mathrm{PF}}\). Positive NS-realizable directions inside this kernel are then tested by the energy channel. Residual pressure–flux–energy invisible directions enter \(K_W^{\mathrm{PFE}}\) and are finally tested by selected-time trace-cost duality. The final fixed-window residual is the projected pressure–flux–energy–trace obstruction space \[\mathcal{T}^{\mathrm{PFET}}_{W,\mathrm{NS}}.\]

The moving-window theorem has the following form: \[\boxed{ \begin{gather} \text{dyadic defect extraction + observable depletion}\\ \text{+ finite-window NS-realizability exclusion}\\ \text{+ moving-window growth control}\\ \Longrightarrow \text{no indefinitely persistent non-CKN dyadic branch.} \end{gather}}\] Equivalently, \[\boxed{ \begin{gather} \text{controlled observability + no NS-realizable invisible cascade}\\ \Longrightarrow \text{CKN scale} \Longrightarrow \text{local regularity.} \end{gather}}\] The contrapositive is the sharper structural statement: \[\boxed{ \text{if no CKN scale occurs, then either moving-window observability is non-effective} }\] \[\boxed{ \text{or there exists an NS-realizable combined-invisible defect cascade.} }\]

Main contributions↩︎

The first contribution is the shift from a one-component or single-channel viewpoint to a full three-dimensional defect-cascade problem. The object of study is not a special smallness assumption on one velocity component, but the scale-critical structure that any potential singular branch must carry.

The second contribution is the NS-realizability filter. A formal finite-dimensional kernel is not automatically a Navier–Stokes obstruction. It becomes dangerous only after intersection with the closure of the range of actual NS-derived coarse-grained packages: \[d\in \overline{\operatorname{Range}\mathcal{R}^{\mathrm{NS}}_W}.\] This distinction prevents model phantoms from being mistaken for genuine Navier–Stokes obstructions.

The third contribution is the combined observability mechanism. Pressure, flux, energy, and trace are not treated as independent decorations. They are tied together by the coarse momentum identity, pressure compatibility, local harmonic pressure work, positive covariance, the flux identity, and the local energy inequality. A defect invisible to one channel is forced into a smaller residual class tested by the next channel.

The fourth contribution is the explicit failure object. A genuine moving-window failure must produce a sequence \[(W_n,d_n,y_n)\] such that \(d_n\) is NS-realizable, \(\|d_n\|\ge c_0\), \(\|y_n\|=1\), \[\mathcal{O}_n^*(y_n)\to0, \qquad |\langle d_n,y_n\rangle|\ge c_1>0.\] This is the NS-realizable invisible moving-window defect cascade. A regularity route must rule it out. A counterexample route must construct it while preserving the coarse momentum equation, pressure compatibility, \(R\ge0\), the flux identity, and the local energy inequality.

The fifth contribution is the reduction of future work to four theorem targets: \[\begin{gather} \boxed{\text{dyadic defect extraction}}, \qquad \boxed{\text{observable depletion}},\\[2mm] \boxed{\text{finite-window NS-realizability exclusion}}, \qquad \boxed{\text{moving-window growth control}}. \end{gather}\] Proving these targets in an intrinsic window class would give a local regularity theorem. Constructing an NS-realizable failure of one of them would identify a genuine obstruction to this route.

From invisible defects to critical recurrence↩︎

The combined-invisible cascade language is useful because it cleans away formal artifacts and identifies the residual obstruction left by pressure, flux, energy, and trace observations. However, it should not be viewed as the final explanatory language. A possible singularity would have to do more than hide from a chosen observation map: it would have to reproduce a non-CKN pattern across infinitely many scales. This motivates the critical recurrence viewpoint developed later in the manuscript.

In this viewpoint the rescaled sequence \[u^{(n)}(x,t)=r_nu(r_nx,r_n^2t), \qquad p^{(n)}(x,t)=r_n^2p(r_nx,r_n^2t)\] is regarded as an orbit in a critical state space. A persistent non-CKN branch is then a recurrent bad orbit. The question becomes: what sustains this recurrence? We isolate three possible scale-critical sustaining mechanisms: vortex-stretching production, active pressure work, and interscale energy flux. The programmatic part of the paper formulates diagnostics for these mechanisms and explains how they refine the earlier defect-observability framework.

1.1 Relation to existing literature and novelty of the formulation↩︎

The framework should be viewed as a synthesis of several established themes in the local theory of the Navier–Stokes equations, rather than as an isolated replacement for them. The weak-solution background begins with the Leray–Hopf theory [1], [2]. The local suitable-weak-solution and partial-regularity endpoint used here comes from the work of Scheffer, Caffarelli–Kohn–Nirenberg, and Lin [3]–[6]; related refinements and alternative proofs include Struwe, Ladyzhenskaya–Seregin, Choe–Lewis, Vasseur, and Seregin’s monograph [7]–[11]. In this theory, regularity is obtained once a scale-invariant local quantity enters a universal smallness regime. The present paper keeps this CKN endpoint but changes the object studied before smallness occurs: instead of tracking only the size of a norm, it tracks the equation-generated defect package which remains along a non-CKN dyadic branch.

A second related direction is local \(\varepsilon\)-regularity, pressure-sensitive regularity criteria, and quantitative regularity near possible singularities. Interior criteria and one-scale criteria appear in work of Gustafson–Kang–Tsai, Seregin, Guevara–Phuc, and Albritton–Barker–Prange [12]–[15]. Barker and Prange [16] show that, under critical or Type-I hypotheses, potential singularities force quantitative concentration behavior and yield bounds on regularity mechanisms. This line of work is close in spirit to the present reduction because both approaches ask what must persist if a regular scale has not yet appeared. The difference is that the present paper does not formulate the obstruction only as concentration of a critical norm. It further decomposes the obstruction into pressure, flux, energy, and trace channels, and then asks whether the remaining invisible direction is realized by actual Navier–Stokes dyadic defect packages.

A third related direction is the large literature on critical regularity criteria and critical-space obstruction mechanisms. The classical Prodi–Serrin criteria and their refinements form part of this background [17]–[20]. The endpoint and critical-element viewpoints are represented by Escauriaza–Seregin–Šverák, Kenig–Koch, Gallagher–Koch–Planchon, and Jia–Šverák [21]–[24]. The moving-window defect cascade in the present paper plays an analogous structural role, but it is not an ancient solution, a self-similar profile, or a compact critical element. It is a sequence of cleaned finite-window defect directions extracted from coarse-grained Navier–Stokes packages, together with dual directions invisible to the combined observation map.

A fourth point of contact is one-component and anisotropic regularity theory. Results of Kukavica–Ziane, Zhou–Pokorný, Cao–Titi, Chemin–Zhang, Chemin–Zhang–Zhang, Kukavica–Rusin–Ziane, Han–Lei–Li–Zhao, and Kang–Nguyen show how partial information about one component, one direction, or one entry of the gradient can force regularity under suitable critical hypotheses [25]–[34]. The present manuscript does not assume one-component smallness. Its use of “visible” and “invisible” directions grew out of the same lesson: a possible singular branch should be studied through the structure that remains after the known regularizing channels have been tested. The finite-scale harmonic-pressure viewpoint in [35] is related to the pressure-cleaning convention used here, but the present paper shifts the emphasis from one-component approximation to full defect-cascade realizability.

A fifth source of motivation is the defect-measure and energy-flux viewpoint. Constantin–E–Titi [36], Eyink [37], and Duchon–Robert [38] show that coarse graining and commutator fluxes provide a natural way to measure anomalous energy transfer for weak solutions. Leslie and Shvydkoy [39] use an energy measure to quantify possible failure of energy equality near blow-up. The quantity \[\Pi_{n,\ell}=-R_{n,\ell}:\nabla U_{n,\ell}\] used here belongs to this circle of ideas. The additional point in the present paper is that energy flux is not treated as a standalone defect. It is combined with pressure compatibility, positive covariance, local energy balance, and selected-time trace duality.

The pressure side of the framework is also tied to the regularity theory of local pressure. The decomposition into an active part generated by the local quadratic source and a retained harmonic part is consistent with pressure regularity and local pressure analyses of Sohr–von Wahl, Seregin–Šverák, Seregin, and Wolf [11], [40]–[42]. This is why the paper quotients only the genuine pressure gauge, namely addition of a function of time, and not the whole local harmonic pressure component.

Finally, Tao’s averaged Navier–Stokes blow-up construction [43] gives an important cautionary lesson. It shows that energy cancellation and harmonic-analysis-type estimates alone do not capture all the nonlinear structure needed for the true Navier–Stokes problem. This is one reason that the present framework insists on NS-realizability: a formal kernel vector or a formal defect envelope is not regarded as a genuine obstruction unless it lies in the finite-window closure of defect packages generated by actual suitable weak solutions.

Thus the main contribution of this paper is not the individual existence of coarse-grained stresses, pressure decompositions, energy fluxes, compactness alternatives, finite-dimensional observability estimates, or parabolic adjoint maps. These ingredients all have predecessors, including standard compactness tools such as Simon’s compactness theorem [44] and linear parabolic theory for adjoint evolutions [45], [46]. The new formulation is the combined reduction \[\boxed{ \begin{gather} \text{non-CKN dyadic branch} \\ \Longrightarrow \\ \text{non-effective moving-window observability} \\ \text{or NS-realizable combined-invisible defect cascade.} \end{gather}}\] In this formulation, the remaining obstruction is required to be simultaneously scale-critical, cleaned of gauge and localization artifacts, invisible to active pressure, flux, energy, and adjoint trace observations, and compatible with the Navier–Stokes coarse momentum, pressure, flux, positivity, and local-energy structures.

Honest status↩︎

This paper does not prove the global Navier–Stokes regularity conjecture. It does not prove that all potential singularities are impossible. It also does not construct a singular solution.

The proved content is a structural reduction and a set of finite-window model verifications. The reduction is designed to make the remaining obstruction precise. A possible singularity, if it survives the combined observability mechanism, must be much more organized than an arbitrary concentration of energy or pressure. It must be an NS-realizable scale-critical defect cascade that remains invisible to energy, active pressure, flux, and adjoint trace observations while preserving the coarse-grained Navier–Stokes identities, pressure compatibility, Reynolds stress structure, and local energy inequality.

Thus the broad question “Can a Navier–Stokes singularity occur?” is transformed into the more structured question “Can there exist an NS-realizable invisible moving-window defect cascade?” This is the central object identified in the paper.

Equivalently, at the conceptual level, the output of the paper is the following reduction: \[\boxed{ \begin{gather} \text{non-CKN dyadic branch}\\ \Longrightarrow\\ \text{non-effective observability}\\ \text{or NS-realizable invisible defect cascade.} \end{gather}}\] Thus a positive route must prove dyadic defect extraction, observable depletion, finite-window NS-realizability exclusion, and moving-window growth control. A counterexample route must construct a cleaned, NS-realizable, scale-critical cascade invisible to active pressure, flux, energy, and adjoint trace while preserving the coarse-grained Navier–Stokes identities and the local energy inequality.

2 Dyadic coarse-grained defect packages↩︎

The purpose of this section is to define the scale-local objects used throughout the paper. We do not assume smallness of any velocity component, nor do we compare the solution with a lower-dimensional shadow class. Instead, to each suitable weak solution and each dyadic scale we associate a coarse-grained Navier–Stokes system with an exact Reynolds defect, an exact pressure compatibility law, and an exact coarse-grained energy-flux identity. These objects form the first version of the scale-critical defect package.

2.1 Suitable weak solutions and scale-invariant quantities↩︎

Let \[Q_r(z_0)=B_r(x_0)\times(t_0-r^2,t_0), \qquad z_0=(x_0,t_0),\] and write \(Q_r=Q_r(0,0)\). We consider suitable weak solutions \((u,p)\) of \[\partial_t u-\Delta u+\nabla\cdot(u\otimes u)+\nabla p=0, \qquad \nabla\cdot u=0 \label{eq:NS}\tag{1}\] in a parabolic cylinder. Thus \[u\in L_t^\infty L_x^2\cap L_t^2H_x^1, \qquad p\in L^{3/2},\] the equations hold distributionally, and the local energy inequality holds in the Caffarelli–Kohn–Nirenberg sense [1]–[3], [5].

For \(z_0=(x_0,t_0)\), define the scale-invariant quantities \[A(z_0,r) := \operatorname*{ess\,sup}_{t_0-r^2<t<t_0} \frac{1}{r}\int_{B_r(x_0)} |u(x,t)|^2\,dx,\] \[E(z_0,r) := \frac{1}{r}\int_{Q_r(z_0)} |\nabla u|^2\,dx\,dt,\] \[C(z_0,r) := \frac{1}{r^2}\int_{Q_r(z_0)} |u|^3\,dx\,dt,\] and \[D(z_0,r) := \frac{1}{r^2}\int_{Q_r(z_0)} |p-(p)_{B_r(x_0)}(t)|^{3/2}\,dx\,dt.\] We also write \[\Phi(z_0,r)=A(z_0,r)+E(z_0,r)+C(z_0,r)+D(z_0,r), \qquad \Psi(z_0,r)=C(z_0,r)+D(z_0,r).\] At the origin we suppress \(z_0\) from the notation.

The Caffarelli–Kohn–Nirenberg criterion and its later variants give a universal constant \(\varepsilon_{\rm CKN}>0\) such that if \[\Psi(z_0,r)\le \varepsilon_{\rm CKN},\] then \(u\) is regular in a smaller cylinder [5], [6], [10], [12], [13], [15]. Hence a potential singular point is one at which no sufficiently small scale is known to enter the CKN regime. The role of the defect package is to record what remains at those scales.

2.2 Dyadic rescaling near a potential singular point↩︎

Fix a point \(z_0=(x_0,t_0)\). After translation we may assume \(z_0=(0,0)\). Let \[r_n=2^{-n},\qquad n\ge0.\] The Navier–Stokes rescaling at scale \(r_n\) is \[u^{(n)}(x,t)=r_nu(r_nx,r_n^2t), \qquad p^{(n)}(x,t)=r_n^2p(r_nx,r_n^2t).\] Then \((u^{(n)},p^{(n)})\) is again a suitable weak solution in the rescaled cylinder, and \[A_{u^{(n)}}(1)=A_u(r_n), \qquad E_{u^{(n)}}(1)=E_u(r_n),\] \[C_{u^{(n)}}(1)=C_u(r_n), \qquad D_{p^{(n)}}(1)=D_p(r_n).\] Thus each dyadic scale near \(z_0\) is converted into a unit-scale problem.

We do not assume a uniform bound on \(\Phi(r_n)\) at this stage. If a sequence of scales has \(\sup_n\Phi(r_n)<\infty\), then the corresponding packages are uniformly scale-critical. If \(\Phi(r_n)\to\infty\), that growth is itself part of the concentration profile. The formalism below is valid in either case, with constants depending on the scale under consideration.

2.3 Interior coarse graining↩︎

Let \(\rho\in C_c^\infty(\mathbb{R}^3\times\mathbb{R})\) be a nonnegative parabolic mollifier with \[\int_{\mathbb{R}^3\times\mathbb{R}}\rho(x,t)\,dx\,dt=1.\] For \(0<\ell<1\), set \[\rho_\ell(x,t)=\ell^{-5}\rho(x/\ell,t/\ell^2),\] and define \[S_\ell f=\rho_\ell*f.\] All identities below are understood in an interior cylinder \(Q_{\rm obs}\Subset Q_{\rm prep}\), with \(\ell\) smaller than the parabolic distance from \(Q_{\rm obs}\) to \(\partial Q_{\rm prep}\). Thus convolution commutes with derivatives in \(Q_{\rm obs}\). Local cutoffs and the commutators they generate are treated later; this coarse-graining viewpoint is parallel to the commutator-flux formalism used in energy-conservation and anomalous-dissipation problems [36]–[38].

For the rescaled solution \((u^{(n)},p^{(n)})\), define \[U_{n,\ell}=S_\ell u^{(n)}, \qquad P_{n,\ell}=S_\ell p^{(n)}.\] When no confusion is possible, we write simply \(U=U_{n,\ell}\) and \(P=P_{n,\ell}\). Since \(S_\ell\) commutes with spatial derivatives in the interior, \[\nabla\cdot U=0.\]

2.4 The Reynolds defect↩︎

The coarse-grained nonlinear flux is not equal to the nonlinear flux of the coarse-grained velocity. The difference is the Reynolds defect \[R_{n,\ell} := S_\ell(u^{(n)}\otimes u^{(n)}) -U_{n,\ell}\otimes U_{n,\ell}. \label{eq:Reynolds-defect}\tag{2}\] Equivalently, \[S_\ell(u^{(n)}\otimes u^{(n)}) =U_{n,\ell}\otimes U_{n,\ell}+R_{n,\ell}.\] The tensor \(R_{n,\ell}\) is symmetric and, because the mollifier is nonnegative, nonnegative as a quadratic form: \[R_{n,\ell}(x,t)\ge0.\] Indeed, for every vector \(a\in\mathbb{R}^3\), \[a_iR_{n,\ell,ij}a_j =S_\ell\bigl((a\cdot u^{(n)})^2\bigr) -\bigl(S_\ell(a\cdot u^{(n)})\bigr)^2 \ge0,\] by Jensen’s inequality. In particular, \[\kappa_{n,\ell}:=\frac{1}{2}\operatorname{tr}R_{n,\ell}\ge0\] is the unresolved kinetic energy density.

2.5 Coarse-grained Navier–Stokes identity↩︎

Applying \(S_\ell\) to the Navier–Stokes equations for \((u^{(n)},p^{(n)})\) gives \[\partial_t U-\Delta U+ \nabla\cdot S_\ell(u^{(n)}\otimes u^{(n)})+ \nabla P=0.\] Using \[S_\ell(u^{(n)}\otimes u^{(n)})=U\otimes U+R,\] we obtain the exact coarse-grained momentum identity \[\partial_t U-\Delta U+ \nabla\cdot(U\otimes U)+\nabla P = -\nabla\cdot R, \qquad \nabla\cdot U=0. \label{eq:coarse-momentum}\tag{3}\] Thus the coarse-grained velocity \(U\) solves Navier–Stokes with an internal stress forcing \(-\nabla\cdot R\). This forcing is not arbitrary; it is generated by the unresolved part of the same Navier–Stokes solution.

2.6 Pressure compatibility, active pressure, and harmonic pressure↩︎

Taking divergence in 3 and using \(\nabla\cdot U=0\), we get the pressure compatibility law \[-\Delta P = \partial_i\partial_j \bigl(U_iU_j+R_{ij}\bigr). \label{eq:coarse-pressure-poisson}\tag{4}\] Equivalently, \[-\Delta P = \partial_i\partial_jS_\ell(u_i^{(n)}u_j^{(n)}).\]

There are two different pressure issues which must be kept separate. The true Navier–Stokes pressure gauge is the addition of a function of time, since such a term has zero spatial gradient. A spatially harmonic local pressure component is not a gauge: its gradient enters the momentum equation and its pressure work enters the local energy identity. Therefore the local harmonic pressure component is not quotiented out of the defect package. This convention follows the standard local-pressure distinction between the source-generated part and the harmonic remainder [11], [40]–[42].

In a local cylinder \(Q\), let \[\mathcal{H}(Q)= \{h\in L^{3/2}(Q):\Delta h(\cdot,t)=0 \text{ in the spatial ball for a.e. }t\}\] and let \[\mathcal{H}_0(Q)=\{a(t):a\in L^{3/2}_{\rm loc}\}\] be the true pressure-gauge subspace. A finite-window pressure cleaning means a fixed bounded choice of an active representative \[P^{\mathrm{act}}=\mathsf P_Q^{\mathrm{act}}P, \qquad P^{\mathrm{har}}=P-P^{\mathrm{act}},\] such that \[-\Delta P^{\mathrm{act}} = \partial_i\partial_j(U_iU_j+R_{ij})\] in the chosen local representative and \(\Delta P^{\mathrm{har}}=0\) in the observation ball. The component \(P^{\mathrm{act}}\) is the pressure generated by the active quadratic source. The component \(P^{\mathrm{har}}\) records the harmonic local pressure field and is retained in the momentum and energy identities.

This pressure compatibility law is the first place where the Navier–Stokes structure imposes a constraint among the objects in the defect package. The tensor \(R\) cannot be chosen independently of \(P\); its double divergence contributes directly to the active pressure source, while the harmonic pressure component remains part of the local dynamics.

2.7 Coarse-grained energy flux↩︎

Let \[e_U=\frac{1}{2}|U|^2.\] Taking the scalar product of 3 with \(U\), using \(\nabla\cdot U=0\), and writing transport terms in divergence form gives \[\partial_t e_U -\Delta e_U +|\nabla U|^2 + \nabla\cdot \left[ (e_U+P)U+RU \right] = R:\nabla U. \label{eq:coarse-energy}\tag{5}\] Here \((RU)_j=R_{ij}U_i\). We define the coarse-grained energy flux by \[\Pi_{n,\ell} := -R_{n,\ell}:\nabla U_{n,\ell}. \label{eq:energy-flux}\tag{6}\] With this sign convention, 5 becomes \[\partial_t e_U -\Delta e_U +|\nabla U|^2 + \nabla\cdot \left[ (e_U+P)U+RU \right] = -\Pi_{n,\ell}. \label{eq:coarse-energy-flux}\tag{7}\] Thus \(\Pi_{n,\ell}\) measures the energy exchanged between the resolved velocity and the unresolved stress. It is the natural local flux quantity associated with a scale-critical cascade.

2.8 Scale-critical bounds for the defect package↩︎

The objects above inherit scale-critical bounds from the original solution. These estimates are not smallness statements. Their role is to show that the package is controlled at the same scaling as the Navier–Stokes equation.

2.9 The NS-realizable dyadic defect package↩︎

We now collect the preceding objects into a single definition.

The package satisfies the three structural identities \[\partial_t U-\Delta U+ \nabla\cdot(U\otimes U)+\nabla P = -\nabla\cdot R,\] \[-\Delta P = \partial_i\partial_j(U_iU_j+R_{ij}), \qquad P=P^{\mathrm{act}}+P^{\mathrm{har}}, \qquad \Delta P^{\mathrm{har}}=0 \quad\text{locally},\] and \[\partial_t\frac{1}{2}|U|^2 -\Delta\frac{1}{2}|U|^2 +|\nabla U|^2 + \nabla\cdot\left[ \left(\frac{1}{2}|U|^2+P\right)U+RU \right] = -\Pi.\] These identities show that \(U\), \(P\), \(R\), and \(\Pi\) are not independent observables. The Reynolds defect determines the active pressure source, the local harmonic pressure component contributes to momentum and pressure work, and the same Reynolds defect interacts with the resolved strain to produce the energy flux.

2.10 Potential singularities and scale-critical defect cascades↩︎

The preceding construction applies at every point. It becomes significant near a potential singular point. Suppose \(z_0\) is not known to be regular. If no dyadic scale \(r_n\) satisfies the CKN smallness condition \[\Psi(z_0,r_n)\le \varepsilon_{\rm CKN},\] then the sequence of packages \[\left\{\mathfrak D_{n,\ell_n}(u,p;z_0)\right\}_{n\ge n_0}\] records the scale-critical information that persists along the possible singular cascade.

The central problem may now be stated informally:

Can an NS-realizable sequence of dyadic defect packages remain scale-critical while escaping all combined observation channels?

Equivalently, if all nonzero NS-realizable dyadic defects are detected by pressure, flux, energy, or adjoint trace observations with summable constants, then a CKN scale should eventually appear. Conversely, if no CKN scale appears, then one should expect to extract a scale-critical defect cascade that is NS-realizable and invisible to the combined observability mechanism.

This section constructs the objects and proves their exact identities. The next step is to define the relevant observation maps acting on \(\mathfrak D_{n,\ell}\).

3 Combined observability maps↩︎

In the previous section we associated to each dyadic scale and each coarse-graining length an NS-realizable defect package \[\mathfrak D_{n,\ell}= \left(U_{n,\ell},P_{n,\ell};P^{\mathrm{act}}_{n,\ell},P^{\mathrm{har}}_{n,\ell},R_{n,\ell},\Pi_{n,\ell}\right).\] We now specify what it means for such a package, or for a normalized defect direction extracted from it, to be observed. The purpose of this section is structural: we define finite observation windows, the active defect quotient, four observation channels, the combined observability map, and the corresponding notion of an invisible defect direction. No uniform or summable observability estimate is asserted here.

The four observation channels are \[\text{energy}, \qquad \text{active pressure}, \qquad \text{flux}, \qquad \text{adjoint trace}.\] Together they form the combined observability map. A scale-critical defect cascade is dangerous only if, after all exact gauge and exact null directions have been removed, and after perturbative localization, CKN-small, and harmonic-pressure leakage errors have been accounted for, it remains invisible to this combined map.

3.1 Finite observation windows↩︎

A finite observation window is a tuple \[W=(n,\ell,\Lambda,\chi,s_*),\] where \(n\) is the dyadic scale index, \(0<\ell<1\) is the coarse-graining length in the rescaled unit cylinder, \(\Lambda\) is a finite active frequency or finite-dimensional test window, \(\chi\in C_c^\infty(Q_{\rm prep})\) is a cutoff with \(\chi\equiv1\) on \(Q_{\rm obs}\Subset Q_{\rm prep}\), and \(s_*\in(-1,0)\) is a selected time when selected-time or adjoint-trace observation is used.

The role of \(\Lambda\) is not necessarily Fourier-theoretic. In a periodic model it may be a finite Fourier set. In a localized cylinder it may be a finite-dimensional space of test functions, wave packets, Stokes eigenfunctions, localized Fourier packets, or adjoint test modes. We only require that all spaces associated with \(W\) be finite-dimensional.

Let \[\Pi_W^{\rm proj}\] denote the finite-window projection acting on the dyadic defect package. It extracts the part of \[(U,P;P^{\mathrm{act}},P^{\mathrm{har}},R,\Pi)\] visible at the window \(W\). We write \[\Pi_W^{\rm proj}\mathfrak D_{n,\ell} = \left(U_W,P_W;P_W^{\mathrm{act}},P_W^{\mathrm{har}},R_W,\Pi_W^{\rm flux}\right).\]

3.2 The constrained active defect quotient↩︎

The raw finite-window variables contain two different kinds of removable pieces. Exact null directions may be quotiented. Perturbative effects, such as cutoff leakage, CKN-small pieces, finite-window tails, and harmonic-pressure leakage into channels that do not observe it, are not quotiented; they are recorded as errors in the corresponding estimates.

Let \(\mathcal{D}_W^{\rm raw}\) be the finite-dimensional space of projected variables \[\dot{\mathfrak D}_W=(\dot{U},\dot{P};\dot{P}^{\mathrm{act}},\dot{P}^{\mathrm{har}},\dot{R},\dot{\Pi})\] supported in the window \(W\). The dot notation indicates a finite-window defect direction, not necessarily a full nonlinear package. We first restrict to the constrained finite-window tangent space \(\mathcal{Z}_W\subset\mathcal{D}_W^{\rm raw}\) consisting of directions satisfying, in the projected window sense, \[\nabla\cdot \dot{U}=0,\] \[\partial_t\dot{U}-\Delta\dot{U} +\nabla\cdot(\dot{U}\otimes U+U\otimes\dot{U}) +\nabla\dot{P} = -\nabla\cdot\dot{R},\] \[-\Delta\dot{P} = \partial_i\partial_j \bigl(\dot{U}_iU_j+U_i\dot{U}_j+\dot{R}_{ij}\bigr), \qquad \dot{P}=\dot{P}^{\mathrm{act}}+\dot{P}^{\mathrm{har}}, \qquad \Delta\dot{P}^{\mathrm{har}}=0 \quad\text{locally},\] and \[\dot{\Pi}=-\dot{R}:\nabla U-R:\nabla\dot{U}.\] Thus pressure compatibility and the flux identity are constraints on admissible defect directions, not identities imposed after quotienting arbitrary raw variables.

The exact quotient-null subspace is denoted by \[\mathcal{G}_W^{\rm ex} = \mathcal{G}_W^{\rm div} + \mathcal{G}_W^{\rm time}.\] Here \(\mathcal{G}_W^{\rm div}\) denotes exact Leray-projection null directions already removed by the divergence-free projection, and \(\mathcal{G}_W^{\rm time}\) denotes additions of functions of time to the pressure. These are the pressure-gauge directions with zero spatial gradient. Spatially harmonic pressure functions are not placed in \(\mathcal{G}_W^{\rm ex}\).

The cleaned active defect space is \[Y_W=\mathcal{Z}_W/\mathcal{G}_W^{\rm ex}. \label{eq:active-defect-space}\tag{8}\] We denote the cleaned class of \(\dot{\mathfrak D}_W\) by \([\dot{\mathfrak D}_W]\in Y_W\). The norm on \(Y_W\) is fixed by choosing finite-window representatives orthogonal to \(\mathcal{G}_W^{\rm ex}\): \[\left\|[\dot{\mathfrak D}_W]\right\|_{Y_W} = \inf_{\mathfrak g\in\mathcal{G}_W^{\rm ex}} \left\|\dot{\mathfrak D}_W+\mathfrak g\right\|_{\mathcal{Z}_W}.\]

The perturbative sectors \[\mathcal{N}_W^{\rm loc},\qquad \mathcal{N}_W^{\rm ckn},\qquad \mathcal{N}_W^{\rm har},\qquad \mathcal{N}_W^{\rm tail}\] are not quotient-null spaces. Their contributions enter estimates through bounds of the form \[\|\widetilde{\mathcal{O}}_W n\| \le \varepsilon_W\|n\|\] for the relevant observation channel. This distinction is essential: a small direction is not the same thing as a zero direction.

Thus the object observed by the combined map is not the entire solution and not a raw pressure quotient. It is the constrained, cleaned active defect direction \(d_W\), with local harmonic pressure retained unless a specific observation channel explicitly projects it away.

3.3 Energy observation↩︎

The energy channel records the part of the defect visible through the local energy balance. Let \[\Theta_W^E=\operatorname{span}\{\varphi_1,\ldots,\varphi_{N_E}\} \subset C_c^\infty(Q_{\rm obs})\] be a finite-dimensional space of energy test functions. For a smooth coarse package \[\mathfrak D_W=(U,P;P^{\mathrm{act}},P^{\mathrm{har}},R,\Pi), \qquad P=P^{\mathrm{act}}+P^{\mathrm{har}},\] define the coarse energy functional \[\mathscr E_{\mathfrak D_W}(\varphi) = \iint \left[ -e_U(\partial_t+\Delta)\varphi -\left((e_U+P)U+RU\right)\cdot\nabla\varphi +\left(|\nabla U|^2+\Pi\right)\varphi \right]\,dx\,dt,\] where \(e_U=\frac{1}{2}|U|^2\). For an exact smooth coarse-grained solution this quantity vanishes by the coarse-grained energy identity. For a defect direction, its linearization detects how the local energy balance changes in that direction.

We define the energy observation map \[\mathcal{O}_W^E:Y_W\longrightarrow E_W\] by \[\mathcal{O}_W^E(d) = \left( D\mathscr E_{\mathfrak D_W}[\dot{\mathfrak D}_W](\varphi_j) \right)_{j=1}^{N_E}, \label{eq:energy-observation}\tag{9}\] where \(\dot{\mathfrak D}_W\) is any representative of \(d\). The definition is independent of the representative after quotienting by the exact null space \(\mathcal{G}_W^{\rm ex}\). In particular, adding a function of time to \(P\) contributes \[-\iint a(t)U\cdot\nabla\varphi\,dx\,dt=0\] after integration by parts and \(\nabla\cdot U=0\). The target space is \(E_W\simeq\mathbb{R}^{N_E}\) with a fixed Euclidean norm.

3.4 Active pressure observation↩︎

The pressure channel records the non-harmonic part of the pressure generated by the active quadratic source. It does not declare the local harmonic pressure component to be physically irrelevant; that component remains available to the momentum, energy, and trace channels.

Let \[\Theta_W^P=\operatorname{span}\{\psi_1,\ldots,\psi_{N_P}\}\] be a finite-dimensional space of pressure test functions chosen for the active pressure representative. Equivalently, \(\Theta_W^P\) is taken in the dual of the active pressure range produced by the chosen pressure-cleaning operator.

For a constrained defect direction \[d=[\dot{U},\dot{P};\dot{P}^{\mathrm{act}},\dot{P}^{\mathrm{har}},\dot{R},\dot{\Pi}]\in Y_W,\] define \[\mathcal{O}_W^P d = \left( \left\langle \dot{P}^{\mathrm{act}},\psi_j\right\rangle \right)_{j=1}^{N_P}.\] Because \(d\in\mathcal{Z}_W\), the active representative may equivalently be expressed through the linearized active pressure source: \[-\Delta \dot{P}^{\mathrm{act}} = \partial_i\partial_j \left( \dot{U}_iU_j+U_i\dot{U}_j+\dot{R}_{ij} \right)\] in the chosen active pressure representative. Therefore \[\mathcal{O}_W^P d = \left( \left\langle \mathsf P_W^{\mathrm{act}}(-\Delta)^{-1}\partial_i\partial_j (\dot{U}_iU_j+U_i\dot{U}_j+\dot{R}_{ij}), \psi_j \right\rangle \right)_{j=1}^{N_P},\] with all inverse and projection operators understood inside the fixed finite-window pressure cleaning. This representation is not asserted for arbitrary raw directions outside \(\mathcal{Z}_W\).

3.5 Flux observation↩︎

The flux channel records the part of the defect visible through interscale energy transfer. Recall that \[\Pi=-R:\nabla U.\] For a constrained defect direction \[d=[\dot{U},\dot{P};\dot{P}^{\mathrm{act}},\dot{P}^{\mathrm{har}},\dot{R},\dot{\Pi}]\in Y_W,\] the linearized flux identity is \[\dot{\Pi} = -\dot{R}:\nabla U-R:\nabla\dot{U}. \label{eq:linearized-flux}\tag{10}\] Let \[\Theta_W^F=\operatorname{span}\{\eta_1,\ldots,\eta_{N_F}\} \subset C_c^\infty(Q_{\rm obs})\] be a finite-dimensional space of flux test functions. Define \[\mathcal{O}_W^F d = \left( \iint \dot{\Pi}\,\eta_j\,dx\,dt \right)_{j=1}^{N_F}.\] Equivalently, \[\mathcal{O}_W^F d = -\left( \iint \left( \dot{R}:\nabla U+R:\nabla\dot{U} \right)\eta_j\,dx\,dt \right)_{j=1}^{N_F}.\] The flux observation is the most direct channel for detecting a cascade. If a defect is invisible to pressure but produces nonzero interscale transfer, then it is not a true invisible defect.

3.6 Adjoint trace observation↩︎

The adjoint trace channel measures whether the defect direction can be seen by selected-time perturbations. Let \(H_W\) be the finite-dimensional selected-time trace correction space. Elements of \(H_W\) are divergence-free vector fields on the selected time slice \(t=s_*\), localized in the observation ball and projected to the window \(\Lambda\).

Let \[A_W:H_W\longrightarrow Y_W\] be the active trace-defect map. Given \(\xi\in H_W\), one evolves \(\xi\) through the linearized coarse-grained Navier–Stokes system around \(U\), with pressure determined through the active/harmonic pressure split, and records the resulting active defect direction in \(Y_W\). Thus \(A_W\xi\) is the active defect produced by the selected-time trace correction \(\xi\).

The dual map \[A_W^*:Y_W^*\longrightarrow H_W\] is defined by \[\left\langle A_W\xi,y\right\rangle_{Y_W,Y_W^*} = \left\langle \xi,A_W^*y\right\rangle_{H_W}.\] Equivalently, \(A_W^*y\) is obtained by solving the backward adjoint linearized Navier–Stokes equation with forcing \(y\) in the active defect quotient and reading off its trace at \(t=s_*\). This finite-window construction is a localized, finite-dimensional version of standard parabolic duality and linearized parabolic theory [45], [46].

3.7 Primal and dual combined maps↩︎

The primal combined observability map is \[\mathcal{O}_W^{\rm comb}:Y_W \longrightarrow E_W\oplus P_W\oplus F_W\oplus T_W,\] where \[\mathcal{O}_W^{\rm comb}d = \left( \mathcal{O}_W^E d, \mathcal{O}_W^P d, \mathcal{O}_W^F d, \mathcal{O}_W^T d \right). \label{eq:primal-combined-map}\tag{11}\] Here \(\mathcal{O}_W^T\) is the primal selected-time trace observation \[\mathcal{O}_W^T d=\Pi_W^T\dot{U}(s_*),\] where \(\Pi_W^T\) projects the selected-time velocity perturbation onto the finite trace window.

The primal observability constant is \[M_W^{\rm pr} = \sup_{0\ne d\in Y_W} \frac{\left\|d\right\|_{Y_W}}{ \left\|\mathcal{O}_W^Ed\right\|_{E_W} +\left\|\mathcal{O}_W^Pd\right\|_{P_W} +\left\|\mathcal{O}_W^Fd\right\|_{F_W} +\left\|\mathcal{O}_W^Td\right\|_{T_W} }.\] If the denominator vanishes for some nonzero \(d\), we set \(M_W^{\rm pr}=+\infty\).

For trace-cost and phantom analysis, the dual formulation is often more important. Write \[J_W^P:Y_W^*\to P_W^*, \qquad J_W^F:Y_W^*\to F_W^*, \qquad J_W^E:Y_W^*\to E_W^*\] for the dual observation components. Thus \(J_W^Py\) is the part of the dual direction tested by the active pressure channel, \(J_W^Fy\) is the part tested by the flux channel, and \(J_W^Ey\) is the part tested by the energy channel.

The dual combined observability map is \[\mathfrak O_W^*:Y_W^* \longrightarrow H_W\oplus E_W^*\oplus P_W^*\oplus F_W^* \label{eq:dual-combined-map}\tag{12}\] defined by \[\mathfrak O_W^*y = \left( A_W^*y, J_W^Ey, J_W^Py, J_W^Fy \right). \label{eq:dual-map-components}\tag{13}\] The corresponding dual combined observability constant is \[M_W^* = \sup_{0\ne y\in Y_W^*} \frac{\left\|y\right\|_{Y_W^*}}{ \left\|A_W^*y\right\|_{H_W} +\left\|J_W^Ey\right\|_{E_W^*} +\left\|J_W^Py\right\|_{P_W^*} +\left\|J_W^Fy\right\|_{F_W^*} }.\] If the denominator vanishes for some nonzero \(y\), then \(M_W^*=+\infty\).

3.8 Invisible directions and true defect phantoms↩︎

A true defect phantom is not merely a vector in one of these kernels. It must also be realized by the Navier–Stokes equation.

Similarly, a dual true phantom is a nonzero \(y\in\mathcal{K}_W^*\) that has nontrivial pairing with an NS-realizable defect: \[|\left\langle d_W,y\right\rangle| \ge c\left\|d_W\right\|_{Y_W}\left\|y\right\|_{Y_W^*}\] along a sequence of windows, after all gauge and localization artifacts have been removed.

3.9 Finite-dimensional equivalence↩︎

Because \(Y_W\) and \(Y_W^*\) are finite-dimensional, absence of invisible directions is equivalent to a finite observability constant.

3.10 Moving-window combined observability↩︎

The fixed-window criterion is not enough for a regularity theorem. A potential singularity may move through windows as the scale changes. Let \[W_n=(n,\ell_n,\Lambda_n,\chi_n,s_n)\] be a moving sequence of windows. We say that the sequence has controlled combined observability if there exists a majorant \(M_n\) such that \[M_{W_n}^*\le M_n\] and the sequence \(M_n\) is absorbable by the scale gains available in a later CKN-selection argument.

The exact meaning of absorbability depends on the final optimization. Two model cases are polynomial growth in the window frequency \(N_n\), \[M_n\le C N_n^p,\] and single-exponential parabolic growth, \[M_n\le C N_n^p\exp(CN_n^q).\] In the latter case one chooses \(N_n\) logarithmically relative to the small scale parameter so that the exponential loss remains absorbable.

Such a cascade is the full three-dimensional analogue of a true phantom cascade. It is not merely a failure of one observation map. It is a scale-moving sequence of NS-realizable defects that remains invisible to energy, active pressure, flux, and adjoint trace observations simultaneously.

4 Finite-window pressure–flux/energy reduction↩︎

We first state the global logical consequence of the estimates proved later.

5 Clean periodic pressure–flux coercivity↩︎

We now isolate the first finite-window theorem. The purpose is deliberately modest. We do not claim that pressure and flux observe every Navier–Stokes defect in the full local problem. We prove only the clean periodic finite-window statement: on nonzero periodic modes, active pressure sources are fully visible, and joint pressure–flux invisibility is equivalent to membership in a finite-dimensional kernel.

6 Perturbative localization of the pressure–flux map↩︎

We now pass from the clean periodic finite-window model to a localized finite-window model. The purpose of this section is to prove that the clean pressure–flux estimate is stable under a perturbative cutoff. This step does not use the local energy inequality and does not yet prove depletion. It says only that localization, when perturbative at the chosen window, does not create a new pressure–flux invisible direction.

Let \[Q_{\rm obs}\Subset Q_{\rm prep}\] be two parabolic cylinders. Let \[\chi_R(x,t)=\chi(x/R,t/R^2)\] be a slowly varying cutoff, where \(\chi\in C_c^\infty\) is fixed and equals \(1\) near the origin. We assume \(R\) is large enough so that \(\chi_R\equiv1\) on \(Q_{\rm obs}\). The derivatives satisfy \[|\nabla_x^a\partial_t^b\chi_R| \le C_{a,b}R^{-a-2b}. \label{eq:cutoff-derivative-bound}\tag{20}\]

Fix a finite active window \(\Lambda\) and the clean periodic space \(Z_\Lambda\). The localized pressure–flux observation map is denoted by \[\mathcal{O}^{\mathrm{PF}}_{\Lambda,R}z=(\mathsf B_{\Lambda,R}z,\mathsf F_{\Lambda,R}z).\] Here \(\mathsf B_{\Lambda,R}\) is the localized active pressure-source observation in the chosen active representative, with retained harmonic pressure terms estimated separately, and \(\mathsf F_{\Lambda,R}\) is the localized flux observation generated by 10 . We write \[\mathcal{O}^{\mathrm{PF}}_{\Lambda,R} = \mathcal{O}^{\mathrm{PF}}_\Lambda+\mathcal{C}^{\mathrm{PF}}_{\Lambda,R}, \qquad \mathcal{C}^{\mathrm{PF}}_{\Lambda,R}:=\mathcal{O}^{\mathrm{PF}}_{\Lambda,R}-\mathcal{O}^{\mathrm{PF}}_\Lambda. \label{eq:localization-commutator}\tag{21}\]

7 Pressure–flux budget certificates↩︎

We now take the first analytic step beyond finite-dimensional observability. The goal is to connect a nonzero pressure–flux observation to a positive local budget. This is weaker than a full dyadic depletion theorem, but it is the correct first estimate: it shows that a pressure–flux visible NS-realizable defect cannot be cost-free.

Throughout this section \(W=(n,\ell,\Lambda,\chi,s_*)\) is a clean periodic or perturbatively localized finite window. Let \[D_W=(U_W,P_W;P_W^{\mathrm{act}},P_W^{\mathrm{har}},R_W,\Pi_W)\] be the finite-window projection of an NS-realizable dyadic coarse-grained package. Recall that \[\Pi_W=-R_W:\nabla U_W\] and that the full pressure satisfies compatibility while the active component is selected by the chosen active/harmonic split: \[-\Delta P_W = \partial_i\partial_j(U_{W,i}U_{W,j}+R_{W,ij}).\]

7.1 The local pressure–flux budget↩︎

Let \(P^a_W\) denote the active pressure representative selected by the finite-window pressure-cleaning operator. In a clean periodic window this is simply the zero-mean pressure supported on the active modes. In a localized window it is the chosen active representative; the retained harmonic pressure component is not discarded but contributes to the harmonic-work error when relevant.

Define the pressure–flux budget density by \[d\mathfrak b^{\mathrm{PF}}_W := |P^a_W|^{3/2}\,dx\,dt + |\Pi_W|\,dx\,dt.\] For a finite window \(W\), set \[\mathfrak B^{\mathrm{PF}}_W(D_W) := \int_{Q_W}|P^a_W|^{3/2}\,dx\,dt + \int_{Q_W}|\Pi_W|\,dx\,dt. \label{eq:local-pf-budget}\tag{26}\] Here \(Q_W\) denotes the physical support of the window. In a localized window, one may insert the cutoff \(\chi\) in both integrals; this changes only finite-window constants.

7.2 From local budget certificate to dyadic depletion↩︎

The previous lemma proves that a pressure–flux visible defect has a positive local pressure–flux budget. To turn this into dyadic depletion, one needs a geometric or telescoping principle saying that the local budget measured in \(W_n\) is paid by a decrease of a dyadic quantity \(B_n\).

8 Controlled packing of pressure–flux budgets↩︎

We now prove a first budget-compatible selection lemma. The result is not yet the full intrinsic Navier–Stokes depletion theorem. Instead, it gives a controlled packing mechanism: if the selected pressure–flux windows have a finite weighted total local budget, then the weighted tail of these local budgets is a genuine dyadic budget and telescopes exactly.

Let \[W_n=(n,\ell_n,\Lambda_n,\chi_n,s_n)\] be a moving sequence of clean or perturbatively localized finite windows associated with dyadic scales \(r_n=2^{-n}\). For each window, let \[D_{W_n}=(U_{W_n},P_{W_n};P^{\mathrm{act}}_{W_n},P^{\mathrm{har}}_{W_n},R_{W_n},\Pi_{W_n})\] be the finite-window projection of the NS-realizable dyadic package. Define \[\mathfrak B^{\mathrm{PF}}_{W_n} := \int_{Q_{W_n}}|P^a_{W_n}|^{3/2}\,dx\,dt + \int_{Q_{W_n}}|\Pi_{W_n}|\,dx\,dt. \label{eq:packing-local-budget}\tag{35}\]

8.1 A sufficient Carleson packing condition↩︎

The following finite-overlap argument is valid when the selected window budgets are restrictions of one common pressure–flux measure. This distinction matters: along a moving dyadic branch the packages at different scales are, in general, different rescaled objects. A Tonelli argument may be used only after all windows have been pulled back to a common physical cylinder and dominated by a single finite measure.

9 Intrinsic energy–pressure budget for pressure–flux activity↩︎

The preceding tail-budget construction gives a rigorous telescoping budget, but the budget is generated from the selected pressure–flux observations themselves. We now formulate a more intrinsic version. The goal is to relate pressure–flux activity to a local Navier–Stokes energy–pressure envelope.

The local energy inequality does not give a sign-definite monotone quantity at every scale. Transport terms, pressure work, harmonic pressure leakage, and cutoff errors may all enter. The correct statement is therefore a budget inequality with explicitly isolated work and localization errors.

Let \(z_0=(0,0)\), \(r_n=2^{-n}\), and \[Q_n:=Q_{r_n}(z_0).\] Choose smooth parabolic cutoffs \(\phi_n\in C_c^\infty(Q_n)\) such that \[0\le\phi_n\le1, \qquad \phi_n\equiv1 \text{ on } Q_{n+1},\] and \[|\nabla\phi_n|\lesssim r_n^{-1}, \qquad |\partial_t\phi_n|+|\Delta\phi_n|\lesssim r_n^{-2}. \label{eq:dyadic-cutoff-bounds}\tag{41}\] Define the local energy–pressure envelope \[\begin{align} \mathcal{E}_n :=&\; r_n^{-1}\operatorname*{ess\,sup}_{-r_n^2<t<0} \int_{B_{r_n}}|u(x,t)|^2\phi_n(x,t)\,dx + r_n^{-1}\int_{Q_n}|\nabla u|^2\phi_n\,dx\,dt \notag\\ &+ r_n^{-2}\int_{Q_n}|p-(p)_{B_{r_n}}(t)|^{3/2}\phi_n\,dx\,dt. \label{eq:energy-pressure-envelope} \end{align}\tag{42}\] This quantity is scale-invariant up to harmless cutoff constants.

9.1 A local-energy work identity↩︎

The following identity explains the origin of 43 . It is an exact coarse-grained local energy balance.

9.2 Forward signed pressure–flux depletion↩︎

The sign in 45 is important. With our convention \(\Pi=-R:\nabla U\), only the forward signed integral \(\int \Pi\varphi\) can be paid by the same decreasing energy–pressure envelope without adding a new positive dissipation cost. Negative signed flux (backscatter) must be treated as a separate mechanism or placed into an oscillation/backscatter error.

9.3 Pressure activity and the pressure envelope↩︎

10 Energy positivity on the pressure–flux kernel↩︎

We now analyze residual directions that survive the pressure–flux channels. The previous sections show that pressure–flux visible defects consume a local budget. Therefore the next possible obstruction must lie in the pressure–flux kernel \[K^{\mathrm{PF}}_W= \ker O^P_W\cap\ker O^F_W.\] The purpose of this section is to show that a large part of this kernel is still visible through the energy channel, provided the direction is NS-realizable in the positive-covariance sense.

The important point is that a formal linear defect may have sign-changing stress variation, but an actual Reynolds covariance generated by coarse graining satisfies \[R_{n,\ell}\ge0, \qquad \kappa_{n,\ell}:=\frac{1}{2}\operatorname{tr}R_{n,\ell}\ge0.\] This positivity is a genuine NS-realizability constraint.

10.1 The positive energy observation↩︎

Fix a finite window \(W\). Let \[z=(\dot{U},\dot{R})\in K^{\mathrm{PF}}_W\] be a pressure–flux invisible active direction. The corresponding pressure variation \(\dot{P}=\dot{P}^{\mathrm{act}}+\dot{P}^{\mathrm{har}}\) is determined by the constrained pressure compatibility relation \[-\Delta\dot{P} = \partial_i\partial_j \bigl(U_i\dot{U}_j+\dot{U}_iU_j+\dot{R}_{ij}\bigr).\] Since \(z\in K^{\mathrm{PF}}_W\), this active pressure source is invisible in the chosen pressure window and the linearized flux observation also vanishes.

Let \[\Theta^{\rm tr}_W=\{\theta_1,\dots,\theta_{N_{\rm tr}}\}\] be nonnegative selected-time kinetic tests, and let \[\Theta^{\rm bulk}_W=\{\zeta_1,\dots,\zeta_{N_b}\}\] be nonnegative bulk tests supported in the observation region. We define the positive energy seminorm \[\begin{align} Q^E_W(z) :=& \left( \sum_{j=1}^{N_{\rm tr}} \int \theta_j(x)|\dot{U}(x,s_*)|^2\,dx \right)^{1/2} \notag\\ &+ \left( \sum_{j=1}^{N_b} \iint \zeta_j|\nabla\dot{U}|^2\,dx\,dt \right)^{1/2} + \sum_{j=1}^{N_b} \iint \zeta_j\frac{1}{2}\operatorname{tr}\dot{R}\,dx\,dt, \label{eq:positive-energy-seminorm} \end{align}\tag{53}\] whenever \(\dot{R}\ge0\). The last term is used only on the positive covariance cone. For a general sign-changing formal variation, it is only a linear functional and cannot be treated as positive.

We also keep the linearized energy-balance observation. For a test function \(\phi\in C_c^\infty(Q_W)\), define \[\begin{align} L^E_Wz(\phi) = \iint \Big[{}& -(U\cdot\dot{U})(\partial_t+\Delta)\phi \notag\\ &- \Big( (U\cdot\dot{U}+\dot{P})U + \Big(\frac{1}{2}|U|^2+P\Big)\dot{U} + \dot{R}\,U + R\dot{U} \Big)\cdot\nabla\phi \notag\\ &+ \bigl(2\nabla U:\nabla\dot{U}+\dot{\Pi}\bigr)\phi \Big]dx\,dt, \label{eq:linearized-energy-observation} \end{align}\tag{54}\] where \(\dot{\Pi}=-\dot{R}:\nabla U-R:\nabla\dot{U}\). The finite-dimensional energy observation is \[O^E_Wz = \bigl(L^E_Wz(\phi_1),\dots,L^E_Wz(\phi_{N_E}),Q^E_W(z)\bigr).\]

For sign-changing formal variations one must not use \(Q^E_W\) as a positive seminorm. Instead, fix a finite-dimensional linear energy observation \[\mathcal{E}^{\rm lin}_W:K^{\mathrm{PF}}_W\longrightarrow E^{\rm lin}_W\] consisting of the linearized energy-balance tests \(L^E_Wz(\phi_j)\), together with any chosen linear kinetic-trace, dissipation, and signed covariance trace coefficients. The positive quantity \(Q^E_W\) is reserved for the positive covariance cone \(\dot{R}\ge0\).

11 Trace-cost exactification on the pressure–flux–energy kernel↩︎

We now treat the residual directions that survive pressure, flux, and energy. After the previous reductions, the only remaining finite-window obstruction lies in the pressure–flux–energy kernel \[K^{PFE}_W = K^{PF}_W\cap \ker O^E_W .\] A residual in \(K^{PFE}_W\) is invisible to the primitive channels, but it may still be removable by a selected-time trace correction. The purpose of this subsection is to prove the finite-dimensional trace-cost alternative: \[\text{trace-aligned residual} \quad\Longrightarrow\quad \text{controlled trace cost},\] while failure of trace alignment produces a left-singular invisible residual.

11.1 The trace-defect map↩︎

Let \(W=(n,\ell,\Lambda,\chi,s_*)\) be a finite observation window. Let \[Y_W\] be the cleaned active defect space, and let \[H_W\] be the finite-dimensional selected-time trace correction space. Elements of \(H_W\) are divergence-free finite-window velocity corrections at time \(s=s_*\).

The active trace-defect map is \[A_W:H_W\longrightarrow Y_W .\] Given \(\xi\in H_W\), \(A_W\xi\) is the cleaned active defect generated by evolving the selected-time correction through the linearized coarse-grained Navier–Stokes system around the resolved background.

The adjoint map is \[A_W^*:Y_W^*\longrightarrow H_W,\] defined by \[\langle A_W\xi,y\rangle_{Y_W,Y_W^*} = \langle \xi,A_W^*y\rangle_{H_W}.\] A dual direction \(y\in Y_W^*\) with \[A_W^*y=0\] is invisible to selected-time trace corrections.

Let \[\iota_W:K^{PFE}_W\hookrightarrow Y_W\] denote the inclusion of the pressure–flux–energy residual kernel into the active defect space. For \[z\in K^{PFE}_W,\] write \[g=\iota_W z\in Y_W\] for the active residual.

11.2 Trace cost and its dual formula↩︎

11.3 Trace alignment↩︎

Small singular values of \(A_W\) are not dangerous by themselves. They are dangerous only if the residual \(g\) has a non-negligible component in the corresponding left singular directions.

11.4 The pressure–flux–energy–trace obstruction↩︎

11.5 NS-realizable PFET obstruction↩︎

A finite-window trace obstruction is still only formal. It becomes relevant to Navier–Stokes only if it is generated by actual dyadic defect packages.

Let \[\mathcal{R}^{NS,T}_W\] be the map sending an NS-realizable dyadic package to its cleaned residual in \(Y_W\) after the pressure–flux–energy reductions. Define the NS-realizable PFET obstruction space by \[\mathcal{T}_{W,\mathrm{NS}}^{\mathrm{PFET}} := \Pi_W^\perp\, \iota_W \left( K_W^{\mathrm{PFE}} \cap \overline{\operatorname{Range}\mathcal{R}^{NS,T}_W} \right) \subset (\operatorname{Range}A_W)^\perp . \label{eq:ns-pfet-obstruction-space}\tag{60}\] The closure is taken in the finite-dimensional topology.

11.6 Moving-window trace failure↩︎

The previous results are fixed-window statements. A singular branch may still escape through a moving sequence of windows whose trace maps have degenerating singular values. We isolate the precise failure object.

12 Moving-window growth control and the defect-cascade alternative↩︎

We now assemble the finite-window hierarchy into a scale-by-scale alternative. The result is a conditional reduction theorem. It does not prove unconditional regularity and it does not construct a singular solution. Its purpose is to state precisely what must persist if a point fails to enter the Caffarelli–Kohn–Nirenberg smallness regime at all sufficiently small dyadic scales.

A technical point is important. A defect extraction theorem may produce nontrivial defects only on a subsequence of dyadic scales. One cannot then sum one-step inequalities \(B_n-B_{n+1}\) over all integers unless the inequalities hold for all sufficiently large \(n\). The formulation below therefore uses an extraction index set and a matching selected-interval budget. The full-sequence case is recovered by taking the extraction set to be all large integers.

12.1 Non-CKN dyadic branches↩︎

Let \((u,p)\) be a suitable weak solution in \(Q_1\), and fix \[z_0=(0,0).\] Let \[r_n=2^{-n}, \qquad \Psi(r)=C(r)+D(r).\] Let \(\varepsilon_{\rm CKN}>0\) be a Caffarelli–Kohn–Nirenberg smallness constant.

For each dyadic scale \(r_n\), choose a coarse-graining length \(\ell_n\), and construct the NS-realizable package \[\mathfrak D_{n,\ell_n} = (U_{n,\ell_n},P_{n,\ell_n};P^{\mathrm{act}}_{n,\ell_n},P^{\mathrm{har}}_{n,\ell_n},R_{n,\ell_n},\Pi_{n,\ell_n}).\] Let \[W_n=(n,\ell_n,\Lambda_n,\chi_n,s_n)\] be a moving finite observation window and set \(Y_n:=Y_{W_n}\). The cleaned finite-window projection of the dyadic package is \[d_n=[\Pi_{W_n}\mathfrak D_{n,\ell_n}]\in Y_n .\]

12.2 Combined observed strength↩︎

For a moving window \(W_n\), define the combined primal observed strength by \[\mathsf O_n(d) := \|O^P_nd\|+ \|O^F_nd\|+ \|O^E_nd\|+ \|O^T_nd\|. \label{eq:combined-primal-strength}\tag{70}\] The corresponding dual observed strength is \[\mathsf O_n^*(y) := \|J^P_ny\|+ \|J^F_ny\|+ \|J^E_ny\|+ \|A_n^*y\|_{H_n}. \label{eq:combined-dual-strength}\tag{71}\] Let \(Y_n^{\mathrm{NS}}\subset Y_n\) denote the finite-dimensional closure of NS-realizable cleaned defect directions in the window \(W_n\). The branch observability constant is \[M_n := \sup\left\{ \frac{\|d\|_{Y_n}}{\mathsf O_n(d)}: 0\ne d\in Y_n^{\mathrm{NS}} \right\}, \label{eq:branch-observability-constant}\tag{72}\] with the convention \(M_n=+\infty\) if the denominator vanishes for some nonzero \(d\in Y_n^{\mathrm{NS}}\). Whenever \(M_n<\infty\), every normalized NS-realizable defect satisfies \[\mathsf O_n(\widehat d)\ge M_n^{-1}. \label{eq:observability-lower-bound}\tag{73}\]

12.3 Observable depletion on selected intervals↩︎

The finite-window hierarchy becomes useful across scales only if observed defects consume a finite dyadic budget on the same index set on which the defects are extracted.

12.4 Effective moving-window observability↩︎

12.5 Growth profiles and logarithmically admissible windows↩︎

A typical controlled growth profile has the form \[M_n\le C N_n^a\exp(CN_n^b), \label{eq:growth-profile}\tag{77}\] where \(N_n\) is the effective size of the active window \(\Lambda_n\).

12.6 The main moving-window alternative↩︎

12.7 Controlled-regime corollary↩︎

12.8 Failure objects↩︎

The moving-window alternative leaves a precise list of possible failure mechanisms.

12.8.0.1 Failure of dyadic defect extraction.

The non-CKN quantity may fail to produce a nontrivial cleaned active defect after quotienting only exact pressure-gauge/null directions and after accounting for harmonic-pressure leakage, localization leakage, CKN-small components, and finite-window tails as errors.

12.8.0.2 Failure of selected-interval depletion.

A defect may be visible in the finite-window hierarchy, but its observation may fail to deplete a genuine Navier–Stokes budget on the same index set on which the defect is extracted. This includes signed flux cancellation, non-summable cutoff work, pressure leakage, or the absence of a valid interval budget for a sparse subsequence.

12.8.0.3 Finite-window trace obstruction.

For some fixed window \(W\), the projected obstruction space \[\mathcal{T}_{W,\mathrm{NS}}^{\mathrm{PFET}}\] may be nonzero. This is the corrected finite-window PFET obstruction.

12.8.0.4 Residual left-singular invisible cascade.

After pressure, flux, and energy are removed, an NS-realizable residual may pair with dual directions satisfying \[A_n^*y_n\to0, \qquad J_n^Py_n\to0, \qquad J_n^Fy_n\to0, \qquad J_n^Ey_n\to0.\] This is the trace-cost failure mechanism.

12.8.0.5 Non-effective moving-window growth.

Every fixed window may be observable, but the constants \(M_n\) may grow so fast on the extracted branch that \[\sum_{k}\lambda_{n_k}M_{n_k}^{-q}<\infty.\]

12.9 Interpretation↩︎

The theorem converts the local regularity problem into a precise defect-cascade question. If no CKN scale appears, then after dyadic rescaling, coarse graining, exact pressure-gauge cleaning, active/harmonic pressure separation, pressure–flux testing, energy testing, and trace-cost testing, one of two things must happen: \[\text{the moving-window constants are not absorbable on the extracted branch,}\] or \[\text{there exists an NS-realizable combined-invisible defect cascade.}\] Thus the remaining obstruction is required to satisfy NS-realizability, scale-criticality, exact pressure-gauge cleaning, retained harmonic-pressure control, pressure invisibility, flux invisibility, energy invisibility, and trace invisibility simultaneously.

13 Model regimes and verification of controlled observability↩︎

The moving-window alternative is a reduction theorem. It becomes useful when its finite-window observability hypotheses can be verified in concrete regimes. This section records two model settings: the clean periodic finite-window regime and the perturbatively localized finite-window regime. The conclusion is not an unconditional Navier–Stokes regularity theorem. Rather, the conclusion is that ordinary active pressure, flux, energy, and trace obstructions reduce to explicit finite-dimensional kernels or projected trace-obstruction spaces.

13.1 Clean periodic finite-window regime↩︎

We work on the periodic box \[\mathbb{T}^3=\mathbb{R}^3/(2\pi\mathbb{Z})^3.\] Let \[\Lambda\subset \mathbb{Z}^3\setminus\{0\}\] be a finite symmetric set of output modes. The exclusion of the zero mode removes the only periodic pressure gauge, namely spatial constants. In a localized cylinder the analogue is not quotienting by all spatially harmonic pressures; the local harmonic component must be retained or estimated as leakage because it can enter momentum and energy through its gradient and pressure work.

A clean periodic finite-window defect direction is a tuple \[z=(\dot{U},\dot{R})\] with Fourier support in \(\Lambda\), where \[\nabla\cdot\dot{U}=0, \qquad \dot{R}=\dot{R}^T.\] The pressure variation is determined by \[-\Delta\dot{P} = \partial_i\partial_j (U_i\dot{U}_j+\dot{U}_iU_j+\dot{R}_{ij}). \label{eq:periodic-pressure-variation}\tag{82}\] We write the active pressure source as \[b_z = \partial_i\partial_j (U_i\dot{U}_j+\dot{U}_iU_j+\dot{R}_{ij}).\] The flux variation is \[\dot{\Pi}_z=-\dot{R}:\nabla U-R:\nabla\dot{U}.\] The energy observation is the chosen finite-dimensional linear energy map from 26, together with the positive energy tests on the positive covariance cone. The adjoint trace map is \[A^*_\Lambda:Y_\Lambda^*\to H_\Lambda.\]

13.2 Periodic pressure source visibility↩︎

13.3 The clean finite-window kernel hierarchy↩︎

Define \[K^P_\Lambda=\{z:b_z=0\}.\] Define the pressure–flux kernel \[K^{\mathrm{PF}}_\Lambda = \{z\in K^P_\Lambda:\dot{\Pi}_z=0\text{ in the selected flux window}\}.\] Define the pressure–flux–energy kernel by the chosen linear energy observation: \[K^{\mathrm{PFE}}_\Lambda = \ker\left(\mathcal{E}^{\rm lin}_\Lambda|_{K^{\mathrm{PF}}_\Lambda}\right).\] Let \[\iota_\Lambda:K^{\mathrm{PFE}}_\Lambda\hookrightarrow Y_\Lambda\] be inclusion and let \(\Pi_\Lambda^\perp\) be the orthogonal projection onto \((\operatorname{Range}A_\Lambda)^\perp\). The clean trace obstruction space is \[\mathcal{T}^{\mathrm{PFET}}_\Lambda := \Pi_\Lambda^\perp\iota_\Lambda(K^{\mathrm{PFE}}_\Lambda). \label{eq:clean-trace-obstruction}\tag{84}\]

13.4 Clean finite-window combined observability↩︎

13.5 Perturbatively localized finite-window regime↩︎

Let \(\chi_R\) be a slowly varying cutoff, \[\chi_R(x,t)=\chi(x/R,t/R^2),\] and let \(O_{\Lambda,R}\) denote the localized combined observation map. Write \[O_{\Lambda,R}=O_\Lambda^{\rm per}+C_{\Lambda,R}, \label{eq:localized-combined-map-decomp}\tag{87}\] where \(O_\Lambda^{\rm per}\) is the clean periodic principal map and \(C_{\Lambda,R}\) is the localization commutator. The commutator includes cutoff leakage, localized pressure inverse error, boundary leakage, energy-flux cutoff terms, and trace-window projection error.

13.6 Nonperturbative localization↩︎

For a fixed localized finite window \((\Lambda,\chi)\), let \[K^{\rm comb}_{\Lambda,\chi}=\ker O_{\Lambda,\chi}\subset Y_{\Lambda,\chi}.\] Then exactly one of the following holds: either \(K^{\rm comb}_{\Lambda,\chi}=0\) and a finite observability constant exists by compactness, or \(K^{\rm comb}_{\Lambda,\chi}\ne0\) and every localized combined-invisible direction lies in this finite-dimensional kernel. A vector in this kernel is not automatically a Navier–Stokes obstruction; it may be caused by a poor cutoff, harmonic pressure leakage, boundary cancellation, or a non-NS-realizable finite-dimensional artifact.

13.7 Active/harmonic pressure separation in the localized regime↩︎

Localized pressure is not determined by its local quadratic source alone. The correct localized structure is \[P=P^{\mathrm{act}}+P^{\mathrm{har}}, \qquad \Delta P^{\mathrm{har}}=0 \quad\text{in the observation ball},\] with only functions of time treated as true pressure gauge. The active pressure channel may be defined after projecting to \(P^{\mathrm{act}}\), but the harmonic component is retained in the finite-window package and contributes to the energy and trace channels or to an explicitly estimated harmonic-pressure error.

Let \(\mathcal{G}^{\rm ex}_{\Lambda,\chi}\) be the exact null sector, consisting of the true pressure-gauge directions and any exact Leray-projection null directions. The cleaned localized active space is \[\widehat Y_{\Lambda,\chi} = \mathcal{Z}_{\Lambda,\chi}/\mathcal{G}^{\rm ex}_{\Lambda,\chi}.\] The localization and harmonic-pressure leakage sectors are denoted \[\mathcal{N}^{\mathrm{loc}}_{\Lambda,\chi}, \qquad \mathcal{N}^{\mathrm{har}}_{\Lambda,\chi}.\] They are not divided out unless a specific observation channel annihilates them exactly. In the usual localized regime they enter perturbative estimates, for instance \[\|\widetilde{\mathcal{O}}_{\Lambda,\chi} n\| \le \varepsilon_{\Lambda,\chi}\|n\|, \qquad n\in \mathcal{N}^{\mathrm{loc}}_{\Lambda,\chi}+\mathcal{N}^{\mathrm{har}}_{\Lambda,\chi}.\] By 2, the cleaned observation map is well-defined on \(\widehat Y_{\Lambda,\chi}\) because only exact null directions are quotiented. If the cleaned localized kernel is zero and the perturbative leakage is small enough, finite-dimensional compactness and stability give a finite cleaned observability constant.

13.8 Controlled moving-window regime↩︎

Let \[W_n=(n,\ell_n,\Lambda_n,\chi_n,s_n)\] be a moving sequence of cleaned localized finite windows, and let \(N_n=N_{\Lambda_n}\) denote the frequency size of the window. Suppose \[M_n\le C N_n^p\exp(CN_n^b) \label{eq:model-moving-growth-bound}\tag{90}\] for constants \(C,p,b>0\). On an extraction set \(I=\{n_k\}\), the moving family is controlled if \[\sum_k \lambda_{n_k}N_{n_k}^{-pq}\exp(-qC N_{n_k}^b)=+\infty. \label{eq:model-effective-divergence}\tag{91}\]

13.9 What remains outside the controlled regimes↩︎

The controlled regimes leave the following genuine escape mechanisms:

1. a formal finite-window kernel whose projected trace obstruction \(\mathcal{T}^{\mathrm{PFET}}_{\Lambda,\mathrm{NS}}\) intersects the NS-realizable range;

2. nonperturbative localization too large relative to the clean observability constant;

3. uncontrolled retained harmonic pressure work or leakage;

4. moving-window amplification so fast that \(\sum_k\lambda_{n_k}M_{n_k}^{-q}<\infty\);

5. residual left-singular alignment with dual directions invisible to pressure, flux, energy, and adjoint trace;

6. failure of observable depletion on the selected extraction intervals.

13.10 Summary of the model verification↩︎

Clean periodic active windows reduce combined invisibility to explicit finite-dimensional kernels and the projected trace obstruction \(\mathcal{T}^{\mathrm{PFET}}_\Lambda\). Perturbative localization preserves clean observability. Nonperturbative localization leaves a finite-dimensional kernel. Moving windows require an effective growth series on the extraction set. Thus a singular branch in the controlled regimes must either force non-effective constants or produce an NS-realizable combined-invisible cascade.

14 Critical recurrence program and sustaining diagnostics↩︎

The preceding sections formulate a combined-observability reduction: a persistent non-CKN branch must either pass through non-effective moving-window constants or produce an NS-realizable combined-invisible defect cascade. This section records a more mechanism-oriented program. Its purpose is not to replace the reduction theorem, but to explain what such a cascade would have to do dynamically. A genuine singular branch would have to rebuild a scale-critical bad pattern at smaller and smaller scales. We therefore ask which mechanisms could sustain this recurrence.

The three candidate mechanisms are \[\text{vortex stretching}, \qquad \text{active pressure work}, \qquad \text{coarse-grained interscale flux}.\] The point of the present section is to define scale-invariant diagnostics for these mechanisms which are natural in the rescaled unit cylinder and meaningful for suitable weak solutions after mollification where necessary.

14.1 Renormalized orbit and critical recurrence↩︎

Let \(z_0=(0,0)\) be a candidate singular point, and set \[r_n=2^{-n}.\] The renormalized Navier–Stokes states are \[u^{(n)}(x,t)=r_nu(r_nx,r_n^2t), \qquad p^{(n)}(x,t)=r_n^2p(r_nx,r_n^2t).\] All diagnostics below are defined on the fixed unit cylinder \(Q_1\). This convention is important: scaling correctness is built into the definitions, rather than imposed by hand through powers of \(r_n\).

A natural critical state space is \[\mathcal{X}_{\mathrm{crit}} = \left\{ (v,q): v\in L^3(Q_1),\quad q-(q)_{B_1}(t)\in L^{3/2}(Q_1) \right\},\] possibly refined by the suitable weak solution local energy topology. Define the CKN-good set \[\mathcal{G}_{\mathrm{CKN}} = \left\{ (v,q)\in\mathcal{X}_{\mathrm{crit}}: \exists\rho\in(0,1) \text{ such that } \Psi_{v,q}(\rho)\le\varepsilon_{\mathrm{CKN}} \right\},\] and the CKN-bad set \(\mathcal{B}_{\mathrm{CKN}}=\mathcal{X}_{\mathrm{crit}}\setminus\mathcal{G}_{\mathrm{CKN}}\).

14.2 Admissible cutoffs and relative filters↩︎

Fix a cutoff \[\chi\in C_c^\infty(Q_1), \qquad 0\le\chi\le1, \qquad \chi\equiv1 \text{ on } Q_{1/2}.\] Also fix a nonnegative parabolic mollifier \(\rho\in C_c^\infty(\mathbb{R}^3\times\mathbb{R})\) with \[\int_{\mathbb{R}^3\times\mathbb{R}}\rho(x,t)\,dx\,dt=1,\] and define \[\rho_\ell(x,t)=\ell^{-5}\rho(x/\ell,t/\ell^2), \qquad S_\ell f=\rho_\ell*f.\] Here \(\ell\in(0,\ell_0)\) is a relative filter scale. In physical variables the corresponding filter scale is \(\ell r_n\). The parameter \(\ell\) therefore resolves subscale structure relative to the current blow-up scale, not an absolute length scale.

14.3 Flux diagnostic↩︎

For the renormalized solution define \[U_{n,\ell}=S_\ell u^{(n)},\] \[R_{n,\ell} = S_\ell(u^{(n)}\otimes u^{(n)}) -U_{n,\ell}\otimes U_{n,\ell},\] and \[\Pi_{n,\ell} = -R_{n,\ell}:\nabla U_{n,\ell}.\] Since \(u^{(n)}\in L^3\), the tensor \(u^{(n)}\otimes u^{(n)}\) belongs to \(L^{3/2}\), and for fixed \(\ell>0\) the smoothed objects are smooth. Hence \(\Pi_{n,\ell}\in L^1(Q_1)\).

Define the absolute flux activity \[\mathcal{F}^{\mathrm{abs}}_{n,\ell} = \int_{Q_1}\chi\,|\Pi_{n,\ell}|\,dx\,dt,\] and the signed flux activity \[\mathcal{F}^{\mathrm{sgn}}_{n,\ell} = \left| \int_{Q_1}\chi\,\Pi_{n,\ell}\,dx\,dt \right|.\] The flux coherence ratio is \[\Gamma_{n,\ell} = \frac{\mathcal{F}^{\mathrm{sgn}}_{n,\ell}}{\mathcal{F}^{\mathrm{abs}}_{n,\ell}+\varepsilon}, \qquad \varepsilon>0.\] The parameter \(\varepsilon\) only prevents division by zero and may later be sent to zero.

The distinction is essential. Local energy identities naturally control signed flux, while a cascade strength is closer to absolute flux. Thus a flux-mediated recurrent branch is either sign-coherent, in which case one may hope to deplete a finite energy budget, or oscillatory, in which case the obstruction is a cancellation mechanism not detected by signed energy balance alone.

14.4 Pressure diagnostic↩︎

For suitable weak solutions one has \(p^{(n)}\in L^{3/2}\) and \(u^{(n)}\in L^3\). Therefore the pressure work term appearing in the local energy inequality is integrable. Define the total pressure work \[\mathcal{P}^{\mathrm{tot}}_{n} = \left| \int_{Q_1} \bigl(p^{(n)}-(p^{(n)})_{B_1}(t)\bigr) u^{(n)}\cdot\nabla\chi\,dx\,dt \right|.\] This quantity is invariant under adding a function of time to the pressure, because \(\nabla\cdot u^{(n)}=0\).

To distinguish active pressure from harmonic leakage, choose \(\eta\in C_c^\infty(B_1)\) with \(\eta\equiv1\) on \(B_{3/4}\), and define the local active pressure by \[p^{\mathrm{act}}_n = \mathcal{R}_i\mathcal{R}_j \bigl(\eta u^{(n)}_i u^{(n)}_j\bigr),\] where \(\mathcal{R}_i\) denotes the Riesz transform. Since \(u^{(n)}_i u^{(n)}_j\in L^{3/2}\), we have \(p^{\mathrm{act}}_n\in L^{3/2}\). Define \[\mathcal{P}^{\mathrm{act}}_{n} = \left| \int_{Q_1} p^{\mathrm{act}}_n u^{(n)}\cdot\nabla\chi\,dx\,dt \right|.\] The remaining pressure is denoted \[p^{\mathrm{har}}_n=p^{(n)}-p^{\mathrm{act}}_n,\] modulo functions of time, and is spatially harmonic in the interior region where \(\eta\equiv1\). The corresponding harmonic pressure leakage is \[\mathcal{P}^{\mathrm{har}}_{n} = \left| \int_{Q_1} p^{\mathrm{har}}_n u^{(n)}\cdot\nabla\chi\,dx\,dt \right|.\] The quantity most relevant to a local recurrence mechanism is \(\mathcal{P}^{\mathrm{act}}_n\), because it is generated by the local quadratic source rather than by the retained harmonic pressure component or far-field leakage.

14.5 Resolved vortex-stretching diagnostic↩︎

Let \[\Omega_{n,\ell}=\nabla\times U_{n,\ell}, \qquad S_{n,\ell}=\frac{1}{2}\bigl(\nabla U_{n,\ell}+\nabla U_{n,\ell}^T\bigr).\] For a suitable weak solution the raw stretching term \(\omega\cdot S\omega\) is not a stable \(L^1\) object, since \(\omega\in L^2\) and \(S\in L^2\). The filtered diagnostic is therefore used.

Define the positive resolved stretching production \[\mathcal{V}^+_{n,\ell} = \int_{Q_1} \chi \bigl(\Omega_{n,\ell}\cdot S_{n,\ell}\Omega_{n,\ell}\bigr)_+ \,dx\,dt.\] Define also the alignment ratio \[\mathfrak A_{n,\ell} = \frac{ \int_{Q_1} \chi \bigl(\Omega_{n,\ell}\cdot S_{n,\ell}\Omega_{n,\ell}\bigr)_+ \,dx\,dt }{ \int_{Q_1}\chi |S_{n,\ell}|\,|\Omega_{n,\ell}|^2\,dx\,dt+\varepsilon }.\] Thus \(\mathcal{V}^+_{n,\ell}\) measures the amount of positive resolved stretching, while \(\mathfrak A_{n,\ell}\) measures the degree of geometric alignment between vorticity and expanding strain.

There is also a subgrid vorticity-transfer term. The filtered velocity satisfies \[\partial_t U_{n,\ell}-\Delta U_{n,\ell} +\nabla\cdot(U_{n,\ell}\otimes U_{n,\ell}) +\nabla P_{n,\ell} = -\nabla\cdot R_{n,\ell}.\] Taking curl gives, schematically, \[\partial_t\Omega_{n,\ell}-\Delta\Omega_{n,\ell} +U_{n,\ell}\cdot\nabla\Omega_{n,\ell} = \Omega_{n,\ell}\cdot\nabla U_{n,\ell} -\nabla\times\nabla\cdot R_{n,\ell}.\] Hence unresolved scales may contribute to resolved vorticity growth. Define \[\mathcal{V}^{\mathrm{sg}}_{n,\ell} = \left| \int_{Q_1} \chi\, \Omega_{n,\ell}\cdot (\nabla\times\nabla\cdot R_{n,\ell}) \,dx\,dt \right|.\] Equivalently, after integration by parts, one may use \[\mathcal{V}^{\mathrm{sg}}_{n,\ell} = \left| \int_{Q_1} R_{n,\ell}: \nabla\bigl(\nabla\times(\chi\Omega_{n,\ell})\bigr) \,dx\,dt \right|.\] The full filtered vorticity-sustaining diagnostic is \[\mathcal{V}_{n,\ell} = \mathcal{V}^+_{n,\ell} + \mathcal{V}^{\mathrm{sg}}_{n,\ell}.\] For a first, simpler version of the program one may work only with \(\mathcal{V}^+_{n,\ell}\) and treat \(\mathcal{V}^{\mathrm{sg}}_{n,\ell}\) as an error or as a separate unresolved mechanism.

14.6 Scaling and suitability↩︎

The preceding definitions are natural for three reasons. First, they are made on the rescaled unit cylinder, so they are invariant under the Navier–Stokes scaling. Written back in physical variables, they correspond schematically to \[\mathcal{F}^{\mathrm{abs}}_{n,\ell} \sim r_n^{-1}\int_{Q_{r_n}} |\Pi_{\ell r_n}|\,dx\,dt,\] \[\mathcal{P}_n \sim r_n^{-1}\int_{Q_{r_n}} p\,u\cdot\nabla\chi_{r_n}\,dx\,dt,\] and \[\mathcal{V}_{n,\ell} \sim r_n\int_{Q_{r_n}} (\omega_{\ell r_n}\cdot S_{\ell r_n}\omega_{\ell r_n})_+\,dx\,dt,\] with the appropriate filtered fields.

Second, they are meaningful for suitable weak solutions. Flux and pressure work use only \(u\in L^3\), \(p\in L^{3/2}\), and smoothing. The vorticity-stretching diagnostic is filtered precisely because the raw product \(\omega\cdot S\omega\) is not naturally integrable at suitable-weak regularity.

Third, they separate mechanisms. \(\mathcal{F}^{\mathrm{abs}}_{n,\ell}\) measures interscale activity, \(\Gamma_{n,\ell}\) measures signed coherence, \(\mathcal{P}^{\mathrm{act}}_n\) measures active pressure work, and \(\mathcal{V}_{n,\ell}\) measures resolved stretching plus subgrid vorticity transfer.

14.7 Critical recurrence diagnostic conjecture↩︎

The conjecture should be read as a necessary-mechanism statement, not as a regularity theorem. It says that a recurrent bad branch cannot remain bad without at least one sustaining mechanism. If true, the local regularity problem can be reformulated as follows: determine whether persistent vortex stretching, active pressure focusing, or interscale flux can continue indefinitely without producing a CKN scale or depleting a finite local budget.

14.8 Programmatic trichotomy↩︎

Assuming the diagnostic conjecture, any critical recurrent branch should fall into one of three mechanism classes.

1. Stretching-dominated recurrence: \[\limsup_n \mathcal{V}_{n,\ell_n}>0.\] The problem is whether persistent vortex-stretching alignment can be compatible with diffusion, pressure projection, and known vorticity-direction regularity mechanisms.

2. Pressure-dominated recurrence: \[\limsup_n \mathcal{P}^{\mathrm{act}}_n>0.\] The problem is whether active pressure can refocus strain and sustain recurrence, rather than merely redistributing or dispersing energy.

3. Flux-dominated recurrence: \[\limsup_n \mathcal{F}^{\mathrm{abs}}_{n,\ell_n}>0.\] This case splits into sign-coherent flux recurrence and oscillatory flux-cancellation recurrence according to the behavior of \(\Gamma_{n,\ell_n}\).

14.9 Relation to the observability framework↩︎

The earlier defect-observability framework remains useful, but it should be interpreted as a secondary language. The defect package \[D_{n,\ell} = (U_{n,\ell},P_{n,\ell};P^{\mathrm{act}}_{n,\ell},P^{\mathrm{har}}_{n,\ell},R_{n,\ell},\Pi_{n,\ell})\] records the coarse-grained manifestation of the renormalized orbit. Combined observability asks whether this package is visible to pressure, flux, energy, and trace. The critical recurrence program asks a more mechanism-oriented question: \[\boxed{ \text{what sustains the recurrence of bad scales?} }\] Pressure–flux observability becomes a tool for detecting flux-sustained recurrence. Energy positivity becomes a tool for excluding covariance-supported hidden recurrence. Trace-cost language should be invoked only after vortex stretching, active pressure work, and interscale flux have failed to explain the recurrence.

14.10 Revised research roadmap↩︎

The program suggests the following order of attack.

1. Define robust diagnostics \(\mathcal{V}_{n,\ell}\), \(\mathcal{P}^{\mathrm{act}}_n\), \(\mathcal{F}^{\mathrm{abs}}_{n,\ell}\), and \(\Gamma_{n,\ell}\) with precise admissibility requirements on \(\chi\), \(\eta\), and \(S_\ell\).

2. Prove a weak necessary-mechanism theorem: a persistent non-CKN branch forces at least one of the diagnostics to remain nontrivial along a subsequence.

3. Analyze stretching-dominated recurrence through vorticity-direction geometry, strain concentration, and possible alignment rigidity.

4. Analyze pressure-dominated recurrence by separating active pressure from harmonic leakage and by estimating pressure work in the local energy inequality.

5. Analyze flux-dominated recurrence by separating sign-coherent flux from oscillatory flux cancellation. A sign-coherent flux cascade should be tested against local energy depletion; an oscillatory cascade should be tested against pressure compatibility and Reynolds covariance positivity.

6. Reinterpret any remaining mechanism as an NS-realizable invisible defect cascade in the earlier combined-observability framework.

This section is deliberately programmatic. Its purpose is to turn the final obstruction from a formal invisible defect into a dynamical question about the persistence of scale-critical sustaining mechanisms.

15 Concrete theorem targets and the resulting research program↩︎

The previous sections reduce the local regularity problem to a precise moving-window alternative. A non-CKN dyadic branch can persist only if the combined observability constants are not effective on the extracted branch, the selected-interval depletion mechanism fails, or an NS-realizable invisible defect cascade exists. This section formulates the concrete theorem targets needed to turn the reduction into a positive regularity route.

The statements in this section are targets, not established theorems of the present manuscript.

15.1 Target I: Dyadic defect extraction↩︎

The first target is to justify the bridge between failure of CKN smallness and the existence of nontrivial cleaned dyadic defects.

The main difficulty is to prevent the entire non-CKN quantity from disappearing into the exact time-dependent pressure gauge, retained harmonic-pressure leakage, pure localization leakage, already CKN-small components, or finite-window truncation tails. A useful first version may allow \[|d_{n_k}|_{Y_{W_{n_k}}}\ge c_0-\operatorname{Tail}(W_{n_k})\] and then choose windows so that the tail is smaller than \(c_0/2\).

15.2 Target II: Observable depletion↩︎

The second target is to prove that visible defects consume a finite dyadic budget on the same index set on which they are extracted.

This target is where the local energy inequality becomes structural. It must measure the cost of visible defect production across scales. In the pressure–flux case, the forward sign-coherent flux condition from 16 is one possible route; backscatter or oscillatory flux requires a separate budget or an explicit error term.

15.3 Target III: finite-window NS-realizability exclusion↩︎

The finite-window hierarchy leaves formal kernels and, after the trace step, a projected obstruction space. A formal object becomes relevant to Navier–Stokes only after intersection with the finite-window closure of NS-realizable packages.

There are two possible routes. The first is algebraic: compute the finite-window symbols for pressure, flux, energy, and trace and prove that any common formal obstruction violates the coarse Navier–Stokes identities. The second is variational: use Reynolds covariance positivity, pressure compatibility, and the local energy inequality to show that any nonzero NS-realizable obstruction must carry pressure, flux, energy, or trace signal.

15.4 Target IV: moving-window growth bounds↩︎

Even if every fixed window is observable, a potential singularity may escape by moving through windows whose constants deteriorate.

The expected sources of growth are finite-dimensional quotient conditioning, localized pressure inverse constants, cutoff commutators, backward parabolic adjoint amplification, and trace-cost singular values. The goal is to keep these losses polynomial or single-exponential in the active window size.

15.5 A first conditional main theorem↩︎

The four targets combine into a conditional local regularity theorem.

15.6 Counterexample route↩︎

A counterexample to this route would have to construct a moving-window sequence \[(W_{n_k},d_{n_k},y_{n_k})\] such that \[d_{n_k}\in \overline{\operatorname{Range}\mathcal{R}^{\mathrm{NS}}_{W_{n_k}}}, \qquad |d_{n_k}|_{Y_{W_{n_k}}}\ge c_0, \qquad |y_{n_k}|_{Y_{W_{n_k}}^*}=1,\] \[\mathsf O_{n_k}^*(y_{n_k})\to0, \qquad |\langle d_{n_k},y_{n_k}\rangle|\ge c_1.\] It must also preserve the coarse momentum identity, pressure compatibility, Reynolds covariance positivity, the flux identity, and the local energy inequality. Thus the failure object is a highly organized scale-critical NS-realizable invisible cascade, not an arbitrary concentration defect.

15.7 First feasible theorem: pressure–flux depletion in controlled windows↩︎

A first feasible theorem target is a pressure–flux depletion statement in controlled windows.

15.8 Suggested order of attack↩︎

A conservative order of attack is:

1. prove clean periodic pressure–flux coercivity;

2. prove perturbative localization stability;

3. prove pressure–flux depletion for NS-realizable packages under forward sign coherence;

4. analyze the pressure–flux kernel by positive energy on the covariance cone and by linear energy observations off the cone;

5. prove trace alignment or identify residual left-singular cascades;

6. estimate moving-window growth constants;

7. upgrade sparse extraction plus interval depletion, or prove full-sequence extraction.

15.9 Conclusion of the program section↩︎

The framework converts the local Navier–Stokes regularity problem into a compatibility-and-observability problem. Proving the aligned targets would give a CKN scale. Any failure of the route must produce an NS-realizable, cleaned, non-summable, moving-window invisible defect cascade, or a precise failure of selected-interval depletion or moving-window effectiveness.

16 Conclusion and next steps↩︎

The framework developed here changes the focus from a single preferred observable to a compatibility system of observables. A Navier–Stokes defect relevant to the regularity problem must satisfy all of the following constraints simultaneously:

1. it must be produced by an actual suitable weak solution through dyadic rescaling and coarse graining;

2. it must satisfy the exact coarse momentum identity;

3. its pressure must obey the active pressure compatibility law with the retained harmonic pressure component kept outside the true time-dependent gauge;

4. its Reynolds stress must generate the exact energy flux;

5. it must avoid detection by pressure, flux, energy, and selected-time adjoint trace observations;

6. it must survive trace-cost exactification by aligning with trace-invisible or nearly trace-invisible left singular directions;

7. it must persist through a moving sequence of windows with non-effective observability constants or non-summable amplification.

The added theorem-targets section makes the status of the paper explicit. The manuscript does not prove an unconditional regularity theorem. It identifies four concrete analytic tasks: dyadic defect extraction, observable depletion, finite-window NS-realizability exclusion, and moving-window growth control. If these targets are proved in a controlled window class, the moving-window alternative yields a CKN scale. If the route fails, the failure must occur through a much more structured object than an arbitrary compactness defect.

The critical recurrence program added above refines the meaning of this remaining obstruction. A combined-invisible defect cascade should not be regarded merely as an algebraic kernel. If it is genuinely Navier–Stokes-realizable, it must be sustained dynamically across scales. The proposed diagnostics \(\mathcal{V}_{n,\ell}\), \(\mathcal{P}^{\mathrm{act}}_n\), \(\mathcal{F}^{\mathrm{abs}}_{n,\ell}\), and \(\Gamma_{n,\ell}\) are meant to identify whether the sustaining mechanism is vortex stretching, active pressure work, sign-coherent flux, or oscillatory flux cancellation.

The main obstruction statement is therefore sharpened as follows. If a non-CKN dyadic branch persists, then either visible defects deplete a finite local budget, or the branch produces an NS-realizable invisible defect cascade. After pressure, flux, and energy have been removed, the remaining obstruction is a residual left-singular cascade: a sequence of NS-generated residuals pairing nontrivially with dual directions whose adjoint trace observations degenerate.

This gives a precise target for both directions. A positive route must prove that the finite-window kernels do not intersect the NS-realizable range with non-summable moving-window amplification, and must verify the depletion inequality for the chosen observations. A counterexample route must construct a cleaned, NS-realizable, scale-critical cascade invisible to active pressure, flux, energy, and adjoint trace while preserving pressure compatibility, the local energy inequality, and the Reynolds covariance structure.

In the controlled regimes analyzed above, ordinary active pressure defects are visible, perturbative localization is stable, and nonperturbative localization leaves only a finite-dimensional kernel. Thus the hard remaining question is no longer simply whether compactness can fail. It is whether the Navier–Stokes equations can realize a moving-window, scale-critical, combined-invisible defect cascade.

References↩︎