PRH | Essay | 7.14.1 • Navier–Stokes, Blur, and Knobs

Navier–Stokes, Blur, and Knobs Smoothness at Finite Resolution Aleksandar Perišić November 2025 Abstract We study the three–dimensional incompressible Navier–Stokes equations through a deliberately finite lens: a blur scale ℓ > 0that encodes the resolution at which the flow is being described, and a small set of knobs that decide whether the resulting finite–resolution dynamics is tame (Lyapunov) or wild (inverse–Lyapunov). The first knob is geometric and physical: a dimensionless parameter Θ( ℓ, ρ )built from viscosity, the blur scale, and the sharpness ρ of the spatial analyzer (partition/guard). It controls whether a blurred energy guard necessarily decays (Θ < 1) or may flip sign and act as an amplifier (Θ > 1). The second knob is spectral: a shell–weight exponent σ in a Littlewood–Paley guard Lσ = P 2 2σjEj , separating a damping–dominated regime ( σ > 1 2 ) from a transfer–dominated regime ( σ < 1 2 ). The third knob is sign–coherence χ , which measures whether the nonlinear flux aligns with the guard so that wild behavior is not merely permitted but forced. Finally, a persistence knob Ξencodes whether coherence and guard mass survive when one descends to smaller blur scales. At fixed ℓ the blurred avatar uℓ =B ℓu is smooth for any Leray–Hopf solution, and the above knobs yield a concrete tame/wild phase portrait entirely in finite data. The message is not that blur “approximates” Navier–Stokes, but that it makes the control panel explicit: global regularity would require infinitely many “escapes” back into the tame region as ℓ↓ 0, while blow–up would require a persistent wild cascade in which the knobs remain in the wild, sign–coherent, and persistent regime across scales. 1 Introduction The three–dimensional incompressible Navier–Stokes equations sit at the centre of the Clay Millennium problem: do smooth solutions persist for all time for every smooth, finite–energy initial datum, or can a classical solution blow up in finite time? In the traditional continuum formulation there is no explicit control parameter: one works simultaneously at all resolutions. The blur viewpoint and the associated Blurrichevsky geometry begin from a different observation. Any real observer, experiment, or numerical scheme sees the flow only through a finite, nonzero resolution ℓ : a blurred velocity field uℓ =B ℓu obtained by convolving u with a mollifier or heat kernel at scale ℓ . For Leray–Hopf solutions this blurred avatar uℓ is smooth and solves a filtered Navier–Stokes system with a Reynolds stress Rℓ ; in particular, all ℓ –resolution observables depend only on the pair ( uℓ, Rℓ )and on a finite blur budget. The singular set of u (if it exists at all) lives entirely behind the limit ℓ↓0. The goal of this paper is to understand Navier–Stokes dynamics at a fixed finite blur in terms of a small set of guards and knobs: • A guard is a blurred energy functional (spatial or frequency–space) that monitors the flow at scale ℓ. • The knobs are a handful of dimensionless quantities—a geometric blur parameter Θ( ℓ, ρ ), a shell–weight exponent σ , a sign–coherence parameter χ , and a persistence parameter Ξ— which decide whether the guarded dynamics is tame (Lyapunov, decaying) or wild (inverse– Lyapunov, growing). 1 The main contributions are: 1. In Section 2 we introduce two finite–blur guard systems (spatial blur guards and dyadic shell guards) and show that each admits a clean tame/wild dichotomy controlled by a single numerical threshold: Θ( ℓ, ρ ) < 1or σ > 1 2 gives a genuine Lyapunov functional; Θ( ℓ, ρ ) > 1 and σ < 1 2move the guard into a wild, energy–amplifying regime. 2. In Sections 2.4 and 3we add a third knob, the sign–coherence χ , which measures the alignment of the nonlinear flux with the guard (via stretching coherence, helical triads, or third–order skewness). In the wild zone, χ > 0produces an inverse Lyapunov inequality for the guard energy. 3. Section 5 introduces a fourth knob Ξdescribing scale–by–scale persistence of coherent cascades. Together the four knobs (Θ , σ, χ, Ξ) yield a simple conditional blow–up template: if they stay in the wild, sign–coherent, persistent regime along a descending scale ladder, then a critical Navier–Stokes norm must blow up in finite time. 4. In Section 10 we recast the picture as concrete finite–blur criteria: a uniform tame regime across all sufficiently small blur scales forces classical smoothness, while a realized wild regime with positive density in time forces blow–up of a critical L3–type norm. From a logical point of view, the paper can be read independently if one accepts the standard Leray theory and the basic filtered balance law for uℓ ; detailed constructions of blur classes and Blurrichevsky geometry are developed in companion work such as [ 14 , 16 ]. Here we treat the “guard calculus” over that backdrop: how to package finite–resolution Navier–Stokes dynamics into a small number of knobs, and how these knobs control both tame behaviour and wild, dyadic–style blow–up scenarios. Notation and standing conventions. We work on T3 = (2 πZ ) 3 unless noted. For ℓ > 0,B ℓ is either convolution with a standard mollifier or the heat kernel at time ℓ2 ; we write uℓ := B ℓu and Rℓ := B ℓ ( u⊗u ) −uℓ⊗uℓ . The blurred spatial energy on a guard ϕ≥ 0is Eℓ ( t ) := 1 2RT3|uℓ|2ϕ dx . For Littlewood–Paley shells, Ej ( t ) := 1 2∥∆ju(t)∥2 L2 and Lσ ( t ) := Pj≥J0 2 2σjEj ( t ). We use the Poincaré constant CP(ℓ)∼ℓ2on the periodic box. Knobs. Geometry/blur Θ( ℓ, ρ ) = 2CP(ℓ) νC2κ2 ρ2 + C2c2 ℓ , shell weight σ∈R , sign–coherence χ∈ [0 , 1], and persistence loss per octave Ξ ∈ [0 , 1). The causal time–blur is Eℓ,τ ( t ) = R∞ 0τ−1e−s/τ Eℓ ( t−s ) ds with Rτ ( t ) := ( Eℓ ( t ) −Eℓ,τ ( t )) +/Eℓ,τ ( t )and Υ( ℓ, ρ, τ ) := Θ( ℓ, ρ ) + 2CP(ℓ) ν sup Rτ τ. 2 Tame and wild regimes under blur guards The blur framework naturally separates Navier–Stokes dynamics into tame finite–blur regimes, where a blurred energy is a genuine Lyapunov functional, and wild regimes, where the same guard system becomes an energy pump and aligns with known blow-up scenarios in dyadic and averaged models. In this section we make this dichotomy explicit, first for spatial blur guards and then for dyadic scale guards. 2.1 Finite-blur spatial guards and a tame–wild threshold Fix viscosity ν > 0and a blur scale ℓ > 0. Let uℓ := Gℓ∗u be the Gaussian-blurred velocity field and choose a smooth partition of unity ϕ1, ϕ2≥0on T3with ∥∇ϕi∥L∞≤κ ρ, 2 where ρ > 0is the thickness of the transition layer and κ is an absolute geometric constant. Define local blurred energies and dissipations Ei(t) := 1 2ZT3|uℓ(t, x)|2ϕi(x)dx, Di(t) := ZT3|∇uℓ(t, x)|2ϕi(x)dx, and set E(t):=E1(t)+E2(t),D(t):=D1(t)+D2(t). Multiplying the filtered Navier–Stokes equation by uℓϕi , integrating over space, and using incompressibility gives the local energy balance ˙ Ei(t)+νDi(t)=Ftrans i(t)+Fpress i(t)+FRey i(t),(2.1) where the right-hand side consists of transport, pressure, and Reynolds-stress flux terms. Standard estimates (Hölder, Bernstein, and the Gaussian blur bounds on τℓ) yield |Ftrans i(t)+Fpress i(t)|≤Cκ ρEi(t)1/2E(t)1/2, and |FRey i(t)| ≤ C cℓD1(t)1/2+D2(t)1/2E(t)1/2, for some blur–dependent cℓ≥ 0that encodes the subgrid Reynolds budget at scale ℓ , and a universal constant C > 0. Summing (2.1) over i= 1,2and applying Young’s inequality gives ˙ E(t)+νD(t)≤a2E(t)+b2E(t) + ν 2D(t),(2.2) where we write a:= Cκ ρ, b := C cℓ. Rearranging, ˙ E(t)≤ −ν 2D(t) + a2+b2E(t).(2.3) At blur scale ℓwe have a Poincaré–type inequality E(t)≤CP(ℓ)D(t),(2.4) with CP(ℓ)∼ℓ2on the periodic box. Inserting (2.4) into (2.3) yields ˙ E(t)≤ −ν 21−Θ(ℓ, ρ)D(t),(2.5) where we define the dimensionless tame–wild parameter Θ(ℓ, ρ) := 2(a2+b2)CP(ℓ) ν=2CP(ℓ) ν C2κ2 ρ2+C2c2 ℓ!.(2.6) Two regimes now emerge: •Tame finite-blur regime. If Θ(ℓ, ρ)<1, then (2.5) gives ˙ E(t)≤ −c D(t)≤0 for some c = c ( ν, Θ) > 0. Thus E ( t )is a genuine Lyapunov functional at blur scale ℓ : the blurred spatial energy observed by the partition ( ϕ1, ϕ2 )is monotonically decreasing in time. 3 • Wild finite-blur regime. If Θ( ℓ, ρ ) > 1, the coefficient of D ( t )in (2.5) changes sign and becomes positive. Combining (2.4) and (2.5) then yields a growth inequality of the form ˙ E(t)≥c′(Θ(ℓ, ρ)−1)CP(ℓ)−1E(t), for some c′> 0, so any trajectory that comes close to saturating the inequalities is pushed in the direction of exponential growth at the guard level. From the blur viewpoint, the parameter Θ( ℓ, ρ )measures how sharply one attempts to resolve the flow (through small ℓ and thin collars ρ ) relative to viscous damping ν and subgrid Reynolds budget cℓ . For moderate blur and gentle analyzer geometry ( ℓ not too small, collars wide), one is safely in the tame region Θ < 1and the blurred dynamics is compact and Lyapunov–controlled. Pushing towards finer blur and sharper analyzers drives Θupwards and moves the guard system into the wild regime Θ > 1, where the finite-blur inequalities strongly favour local energy amplification. 2.2 Dyadic shell guards and model blow-up A complementary guard system lives in frequency space. Let ∆ j be Littlewood–Paley projections to dyadic shells of size |ξ| ∼ 2j, and set Ej(t) := 1 2∥∆ju(t)∥2 L2. The Navier–Stokes nonlinearity couples neighbouring shells; schematic bounds of the form ˙ Ej+ 2ν22jEj≤C2jE1/2 j X m≤j 23m 2E1/2 m  X |k−j|≤2 E1/2 k (2.7) are standard in harmonic-analytic treatments of high frequencies. Dyadic models in the spirit of Katz–Pavlović and Cheskidov retain the shell structure of (2.7) but collapse spatial geometry to a one–dimensional cascade. In those models one obtains systems of ODEs of the form ˙ Ej+ν22αjEj=λjE3/2 j−1−µjE3/2 j, with parameters ( α, λj, µj )tuned to mimic the scaling and energy identity of Navier–Stokes. It is now known that such models can blow up in finite time in supercritical regimes, while remaining globally regular in more dissipative regimes; see Katz–Pavlović [ 4 ] and Cheskidov [ 5 ]. To connect this to our guard picture, define a weighted high-frequency functional Lσ(t) := X j≥J0 22σjEj(t), where σ∈R controls how strongly the top shells are weighted. A standard computation in the dyadic setting shows that: • For supercritical weights σ > 1 2 and sufficiently large j , the linear damping dominates the cascade and one obtains an inequality of the form ˙ Lσ(t)≤ −νeffLσ(t), so Lσ is a Lyapunov functional on the high-frequency face: energy cannot accumulate indefinitely at the top shells. 4 •For subcritical weights σ < 1 2, one finds regimes in which ˙ Lσ(t)≥cσLσ(t)3/2, once Lσ crosses a threshold. In the dyadic Navier–Stokes models this inequality can be made rigorous and leads to finite-time blow-up via comparison with ˙y = cy3/2 ; see, for instance, Cheskidov [5] and related dyadic cascades surveyed in [6]. Thus, at the level of shell guards, the tame regime corresponds to weights σ > 1 2 that over-emphasize viscous damping, while the wild regime corresponds to weights σ < 1 2 that over-emphasize the nonlinear energy flux into the highest shells. In the true Navier–Stokes equation, inequalities of the schematic form (2.7) still hold, but closing them globally remains out of reach. In dyadic and averaged Navier–Stokes models, however, these inequalities are sharp enough to produce actual blow-up solutions. 2.3 Wild regimes and the blur framework The two guard systems above—spatial blur guards with parameter Θ( ℓ, ρ )and dyadic shell guards with weight σ—expose a common structure: • There is a tame side, where a suitable blurred energy becomes a true Lyapunov functional: finite-blur dynamics is compact and cannot develop singularities. This is precisely the regime captured by the blur–native finite-families results in this paper. • There is a wild side, where the same inequalities flip sign and push the guard system towards growth. Dyadic Navier–Stokes models and averaged Navier–Stokes equations that preserve the energy identity but blow up in finite time [ 8 ] live squarely in this wild regime: they are explicit examples where the guard architecture is realised by genuine singular solutions. From a blur–native perspective, the Clay Navier–Stokes problem is precisely the question of whether any physically relevant flows cross from the tame side to the wild side as one attempts to send the blur scale ℓ→ 0. The blur formalism does not resolve this question, but it provides a concrete language of guards, budgets, and tame–wild thresholds in which any future Lyapunov or inverse-Lyapunov certificate would have to live. 2.4 A third knob: sign–coherence of the flux The previous bounds are sharp from above but blind to cancellations: they show where wild behaviour is allowed (Θ( ℓ, ρ ) > 1or shell weight σ < 1 2 ), but not that it is forced. What closes the gap is a third parameter that controls the sign (coherence) of the flux relative to the guard. We encode it as a number χ∈ [0 , 1] that lower–bounds the effective alignment of the nonlinearity with the guard’s direction. Three concrete incarnations are natural. (SC1) Vorticity–strain alignment (stretching coherence). Write ωℓ = ∇×uℓ and Sℓ = 1 2(∇uℓ+ (∇uℓ)⊤). For a spatial guard ϕ≥0, the local enstrophy balance reads (schematically) 1 2 d dt Zϕ|ωℓ|2+νZϕ|∇ωℓ|2=Zϕ ωℓ·Sℓωℓ |{z } stretching +transport/pressure +subgrid. Assume a stretching coherence hypothesis on the guard domain: there exists χstr ∈ (0 , 1] such that a.e. on {ϕ>0}, ωℓ·Sℓωℓ≥χstr λ+(Sℓ)|ωℓ|2,(2.8) 5 where λ+ ( Sℓ )is the maximal eigenvalue of Sℓ . Then, after the same Young–type bounds used in §2, we obtain d dt Zϕ|ωℓ|2≥2χstrZϕ λ+(Sℓ)|ωℓ|2−C1(Θ)Zϕ|∇uℓ|2−C2(subgrid). Thus, on time intervals where λ+ ( Sℓ )is not too small and Θ( ℓ, ρ ) > 1, the stretching term provides a sign–coherent lower bound that flips the enstrophy balance to growth. The parameter χstr is the third knob: it quantifies persistent alignment of vorticity with the principal stretching direction (observed, e.g., in vortex tubes). (SC2) Helical triad composition (homochiral dominance). Decompose u into helical modes u = u+ + u− (eigenvectors of curl). In helical triad interactions, homochiral triads (all +or all − ) transfer energy to small scales with a fixed sign, while heterochiral triads tend to backscatter or cancel. Let χhel ∈ [0 , 1] be the fraction (in a guard–weighted sense) of energy transfer carried by homochiral triads. Then the subgrid flux Π ℓ = −Rℓ : ∇uℓ admits a lower bound ZϕΠℓ≥χhel Πhom ℓ−(cancellations),(2.9) where Π hom ℓ is the homochiral contribution (sign–definite). When χhel is bounded away from zero, the flux sign is locked on a set of positive measure, producing a genuine source term for gradient growth (forward cascade in enstrophy, or backscatter into the guarded region depending on the sign convention adopted in §2). The knob χhel captures triad–level coherence. (SC3) Third–order skewness / coarse–grained flux (Duchon–Robert/Eyink). The local coarse–grained energy balance for uℓreads ∂t1 2|uℓ|2+∇· Jℓ=−ν|∇uℓ|2−Πℓ,Πℓ=−Rℓ:∇uℓ. Eyink’s increment formula expresses Πℓthrough third–order velocity increments, and Duchon– Robert identify its ℓ↓ 0limit with the anomalous dissipation measure. If on a guard–weighted space–time set one has a skewness lower bound (δruL)3ϕ≤ −κ ε r for r∼ℓ, then ZϕΠℓ≥c κ ε Zϕ−o(1) (ℓ→0),(2.10) locking the sign of the flux (forward cascade). Conversely, a persistent backscatter fraction gives RϕΠℓ≤ −c κbεbRϕ, which acts as a coherent source for the guarded energy Ein §2. Here the knob is the skewness level κ(or κbfor backscatter). Takeaway. Add any one of χstr, χhel, κ bounded away from zero on a guard–weighted set of positive measure, and the wild region (Θ( ℓ, ρ ) > 1or σ < 1 2 ) stops being merely allowed and becomes forced: the guard functional obeys an inverse Lyapunov inequality of the form ˙ L(t)≥c0(knob)L(t)−lower–order, or, after a short bootstrap, ˙ L≥c1L1+θ , i.e. true runaway on the guard face (exactly as in dyadic blow–up models). 3 Knobs, thresholds, and a three-parameter phase diagram We summarize the three independent control knobs that determine whether the guarded (finiteblur) dynamics is tame (Lyapunov) or wild (inverse-Lyapunov). Each knob is measurable at finite blur and enters the guard inequalities with a precise mathematical role. 6 σ Θ 1 2 1 tame (Θ<1)tame (σ > 1 2) wild permitted χ↑(sign–coherence enforces growth) tame (Lyapunov) wild permitted (χneeded) Figure 1: Three–knob phase portrait on the (Θ , σ )face. The orange slab (Θ > 1, σ < 1 2 ) permits wild behaviour; a positive sign–coherence χupgrades it to forced growth (Proposition 3.1). 3.1 Knob I: geometric blur/partition Θ(ℓ, ρ) With a spatial blur scale ℓ > 0and a two-patch partition with collar width ρ > 0, the spatial guard energy E(t)obeys (cf. Section 2) ˙ E(t)≤ −ν 21−Θ(ℓ, ρ)D(t), E(t)≤CP(ℓ)D(t), where CP(ℓ)∼ℓ2and Θ(ℓ, ρ) = 2CP(ℓ) νC2κ2ρ−2+C2c2 ℓ. Hence ˙ E(t)≤ −γℓE(t), γℓ:= ν 2CP(ℓ)1−Θ(ℓ, ρ).(3.1) Decision: • Tame by geometry: if Θ( ℓ, ρ ) < 1, then γℓ> 0and E ( t )decays exponentially; E is a Lyapunov functional at blur ℓ. • Wild-permitted by geometry: if Θ( ℓ, ρ ) > 1, the sign in (3.1) flips; geometry alone no longer damps the guard. Operationally, coarsening ℓ (larger C−1 P ) or widening ρ (smaller κ/ρ ) drives Θbelow one. Sharpening either pushes Θupward. 3.2 Knob II: shell weight σ(frequency-side guard) With Littlewood–Paley shells and Ej(t) = 1 2∥∆ju(t)∥2 L2, define Lσ(t) := X j≥J0 22σj Ej(t). High-frequency estimates yield a dichotomy consistent with dyadic models: • Tame by weight: for σ > 1 2 (supercritical weight), linear damping dominates the cascade and ˙ Lσ(t)≤ −νeff Lσ(t)+lower-order. Thus Lσis Lyapunov on the high-frequency face. 7 • Wild-permitted by weight: for σ < 1 2 (subcritical weight), nonlinear transfer can dominate; one only obtains upper bounds that allow growth. Increasing σ overweights dissipation at the top shells; decreasing σ favors forward transfer and sets the stage for wild behavior. 3.3 Knob III: sign–coherence χ(flux alignment) The two knobs above control magnitudes. To force wild growth one needs a lower bound that fixes the sign of the flux relative to the guard. We encode this by a coherence parameter χ∈ [0 , 1], defined in any of the following equivalent guard-specific forms: •Stretching coherence (SC1): on {ϕ>0}, ωℓ·Sℓωℓ≥χstr λ+(Sℓ)|ωℓ|2, χ := χstr. • Helical coherence (SC2): a guard-weighted fraction χhel of triads is homochiral, yielding a sign-definite subflux; set χ:= χhel. • Skewness coherence (SC3): the third-order increment skewness at scale r∼ℓ satisfies ⟨ ( δruL ) 3⟩ϕ≤ −κ ε r with κ > 0, which yields a positive lower bound on the coarse-grained flux Πℓ; set χ:= κ. All three give a guard-level lower bound of the schematic form ZϕΠℓ≥c0χΦℓ(uℓ)−controlled errors,(3.2) with Φℓa positive functional (e.g. Rϕ|∇uℓ|2or a weighted shell sum). 3.4 Switch logic: from permitted to forced wildness Put the three knobs together. Let Ldenote the guard Lyapunov candidate (Eor Lσ). •Tame region (provable): Θ(ℓ, ρ)<1or σ > 1 2=⇒˙ L ≤ −λL, for some λ > 0(explicit from (3.1) or from the shell estimate). •Wild region, forced (inverse Lyapunov): Θ(ℓ, ρ)>1, σ < 1 2, χ ≥χ0>0 =⇒˙ L ≥ αL−βor ˙ L ≥ α′L1+θ−β′, where α, α′, θ > 0depend on (Θ , σ, χ )and the guard choice. In particular, after a short bootstrap (once L passes a threshold) one obtains strict growth. In dyadic models the exponent is θ=1 2and finite-time blow-up follows. • Wild region, permitted but not forced: If Θ > 1and σ < 1 2 but χ is not bounded below, cancellations can suppress the flux and no inverse inequality is available. This is exactly where dyadic models hard-wire χ = 1 (homochiral, sign-definite coupling) and true blow-up occurs. 8 Proposition 3.1 (Inverse guard inequality in the wild zone).Fix a spatial guard at blur scale ℓ > 0with Poincaré constant CP ( ℓ )and collar parameter ρ > 0, and let E ( t )denote the corresponding guard energy (Section 2). Assume the wild side of the geometry holds with margin Θ( ℓ, ρ ) ≥ 1 + δ for some δ > 0, and that on a guard-weighted set the sign–coherence parameter satisfies χ≥χ0>0(any of SC1–SC3). Then there exist universal constants c∗, C∗> 0(independent of ν, ℓ, ρ ) and a subgrid budget bound Bℓ ( t )such that, whenever E ( t ) ≥E∗ := C∗Bℓ ( t ), the guard obeys the inverse Lyapunov inequality ˙ E(t)≥ν 2CP(ℓ)δ | {z } αgeo E(t)+c∗χ0 |{z} αcoh E(t)−ν 4CP(ℓ) | {z } β E(t).(3.3) Equivalently, for α := αgeo + αcoh −β > 0one has ˙ E≥α E on any time subinterval where the hypotheses persist. In particular, once E≥E∗ the guard energy grows at least exponentially on that window. Remark 3.2. Heuristically, αgeo is the positive part coming from the flipped sign in (2.3) when Θ > 1; αcoh comes from any one of the lower bounds (SC1–SC3) turning flux into a sign–definite source; and β is a safety deduction absorbing transport/pressure and Young losses. The threshold E∗hides the (finite) blur budget corrections. 3.5 Control panel: how each knob moves physically Lower Θ(tame): increase blur ℓ (smaller CP ( ℓ )), widen collars ρ (smaller κ/ρ ), or increase ν . Raise Θ(wild-permitted): decrease ℓ, sharpen collars, decrease ν. Raise σ(tame): overweight top shells to emphasize dissipation. Lower σ(wild-permitted): emphasize nonlinear forward transfer. Raise χ(wild-forced): promote alignment or sign-definite flux (stretching alignment, homochiral triads, persistent skewness/backscatter). Empirically, coherent vortex tubes and homochiral triads increase χ. 3.6 Minimal phase diagram The three-parameter diagram (Θ, σ, χ)has two clean faces: Tame: Θ<1or σ > 1 2. Wild (forced): Θ>1, σ < 1 2, χ ≥χ0>0. Everything else is wild-permitted but cancellation-dependent. The third knob χ is precisely the one that turns possibility into inevitability. 4 Explicit wild testbeds under blur guards We work on T3= (2πZ)3with viscosity ν > 0. Choose: wavenumber K≫1, ℓ := cℓ K(1 ≤cℓ≤2), ρ := θ ℓ (θ∈(0,1]), so CP ( ℓ ) ∼ℓ2 and κ/ρ ∼κ/ ( θℓ ). Fix a shell weight σ < 1 2 (e.g. σ = 0 . 4) and select one of the sign–coherence mechanisms below to enforce χ≥χ0> 0on a guard–weighted set for a short time window. 9 8 Perpetual wildness forces blow-up: reduction + a concrete bridge We now seal the framework with a precise trichotomy: if a solution stays in the wild, signcoherent, persistent regime across a descending scale ladder, then a critical norm blows up in finite time. Equivalently, any globally smooth solution must escape wildness infinitely often by flipping some knob back to the tame side. 8.1 A no-perpetual-wildness trichotomy Fix a blur scale ℓ0> 0, a ratio λ∈ (0 , 1) and the ladder ℓn = λnℓ0 . Let Cadv > 0and define advection windows [tn, tn+1]with tn+1 −tn≤Cadvℓ2 n. Theorem 8.1 (No-perpetual-wildness trichotomy).Assume the guard functional L compares to a critical space Xcrit as in (5.2)on the guard set, and that for some constants Θ⋆>1, σ⋆<1 2, χ0>0,Ξmax ∈[0,1), the following hold along the ladder for all non windows [tn, tn+1]: (W1) Wild-permitting geometry/weight: Θ(ℓn, ρ)>Θ⋆or the shell weight obeys σ < σ⋆. (W2) Sign coherence: χ ( ℓn, t ) ≥χ0 on a guard-weighted subset of [ tn, tn+1 ]of positive measure. (W3) Scale persistence: there exists t′ n∈[tn, tn+1]such that L(ℓn+1, t′ n)≥(1 −Ξ)L(ℓn, tn),Ξ≤Ξmax. (W4) Inverse Lyapunov on the window: above a threshold L∗ , L satisfies, on the portion where χ≥χ0,˙ L ≥ αL−βor ˙ L ≥ α′L1+θ−β′ with α, α′, θ > 0depending only on (Θ⋆, σ⋆, χ0)and the chosen guard. Then there exists T∗<∞and a subsequence tnk↑T∗such that ∥u(tnk)∥Xcrit → ∞. In contrapositive form: if u is smooth on [0 ,∞ ), then for every ladder and advection windows, at least one of (W1)–(W4) must fail infinitely often. Equivalently, a globally smooth flow must repeatedly re-enter the tame region by flipping at least one knob among Θ,σ,χ, or Ξ. Proof. Sum the window lengths: Pn ( tn+1 −tn ) ≤Cadvℓ2 0Pnλ2n<∞ , so tn↑T∗<∞ . Above the threshold L∗ , (W4) boosts L by a fixed multiplicative factor within O ( ℓ2 n )time; (W3) ensures this growth is not undone when passing to ℓn+1 . Hence L ( ℓn, tn )grows at least geometrically in n . By the bridge (5.2) , ∥ · ∥Xcrit grows accordingly, proving blow-up at some T∗ . The contrapositive is immediate. Remark 8.2 (What this achieves).Theorem 8.1 is the exact closure logically latent in Theorem 5.1: it crystallizes the dichotomy between perpetual wildness ⇒ blow-up and global smoothness ⇒ infinitely many knob escapes. Any future regularity mechanism must explicate which knob forces the escape; any future blow-up mechanism must verify (W1)–(W4) along a scale ladder. 16 8.2 Concrete bridge on narrow-band helical packets We now give a clean, checkable bridge for the shell guard at σ = 1 2 (and nearby). Let ∆ j denote Littlewood–Paley shells and define Lσ(t) := Pj≥J022σjEj(t)with Ej=1 2∥∆ju∥2 2. Proposition 8.3 (Local bridge at a dyadic packet).Assume u has Fourier support in the dyadic annulus AK := {ξ : K/ 2 ≤ |ξ| ≤ 2 K} and is approximately Beltrami: ∥∇ × u∓Ku∥L2≤ ε K∥u∥L2 with ε∈ [0 , 1 / 2). Then there exist absolute constants ci = Ci ( ε )with 0 < c1≤c2<∞ such that c1∥u∥2˙ H1/2≤L1/2(u)≤c2∥u∥2˙ H1/2,∥u∥2 BMO−1≤c3K−1L1/2(u), and, for σ∈(1 2−δ, 1 2+δ)with δ=δ(ε), c′ 1K2(σ−1/2) ∥u∥2˙ H1/2≤Lσ(u)≤c′ 2K2(σ−1/2) ∥u∥2˙ H1/2. In particular, on such packets Lσ is locally equivalent to a critical norm ( ˙ H1/2 or BMO−1 ) up to scale factors that are explicit powers of K. Proof sketch. On AK, Bernstein and LP orthogonality yield ∥u∥2˙ H1/2∼K∥u∥2 L2∼X |j−log2K|≤2 2j∥∆ju∥2 2= 2 X |j−log2K|≤2 2jEj≡L1/2(u). The BMO−1estimate follows from the standard embedding ∥u∥2 BMO−1≲X j 2−j∥∆ju∥2 2 plus the shell localization and the Beltrami control, which prevents cancellation across helicities. The extension to σ=1 2is a scale reweighting by K2(σ−1/2). Corollary 8.4 (Explicit packet bridge near σ = 1 2 ).Let u be supported in AK := {ξ : K/ 2 ≤ |ξ| ≤ 2 K} and satisfy the Beltrami defect ∥∇ × u∓Ku∥L2≤εK∥u∥L2 with ε∈ [0 , 1 / 4]. Then for σ∈[1/2−1 10,1/2 + 1 10]one has (1 −2ε)K2(σ−1/2) ∥u∥2˙ H1/2≤Lσ(u)≤(1 + 2ε)K2(σ−1/2) ∥u∥2˙ H1/2. In particular, at σ=1 2,(1 −2ε)∥u∥2˙ H1/2≤L1/2(u)≤(1 + 2ε)∥u∥2˙ H1/2. Proof idea. Use Littlewood–Paley orthogonality on AK and the Beltrami defect to bound helicity cancellations, then rescale 2 2σj around j≈log2K to get the K2(σ−1/2) factor. The (1 ± 2 ε ) constants follow from a triangle inequality on the helical split. 8.3 Short-time sign coherence and persistence (homochiral packet) Let u+ denote the +helical component (eigenvectors of curl with eigenvalue + |ξ| ). Consider a narrow-band homochiral packet at wavenumber K≫1: u0=PAKu0,∥u− 0∥L2≤ε∥u+ 0∥L2, ε ∈[0,1/4]. Let A := ∥u0∥L2 and assume νK ≪A (moderate Reynolds at scale K−1 ). Define the blur scale ℓ:= cℓ/K with cℓ∈[1,2] and choose collars ρ∼ℓ. Lemma 8.5 (Windowed sign coherence and persistence).There exist absolute constants c0, c1, c2>0(independent of K,A,ν) such that for 0≤t≤t1:= c0min{(AK)−1, ν−1K−2}, the Navier–Stokes solution with initial datum u0satisfies: 17 (P.a) Spectral localization: b u ( t )remains in a slightly fatter annulus K/ 3 ≤ |ξ| ≤ 3 K , with leakage ≤c1εin L2. (P.b) Homochiral dominance: the helical fraction satisfies ∥u− ( t ) ∥L2≤ 2 ε∥u+ ( t ) ∥L2 ; in particular, the homochiral triad fraction χhel(t)≥1−Cε on [0, t1]. (P.c) Positive coarse-grained flux at r∼ℓ:for a guard ϕadapted to the packet support, ZϕΠℓ(t)dx ≥c2A3K−oK→∞(1), so χ≥χ0with χ0depending only on εand cℓ. (P.d) Scale persistence across one octave: choosing λ∈ (1 / 2 , 3 / 4), there is t′∈ [ t, t + c ℓ2 ]such that L(ℓ′, t′)≥(1 −Ξ)L(ℓ, t), ℓ′=λℓ, Ξ≤Ξmax <1, with Ξmax universal for εsmall and the above choice of (ℓ, ρ). Lemma 8.6 (Quantitative one-octave persistence with explicit loss).Under the hypotheses of Lemma 8.5, assume additionally ε≤ 10 −2 and set ℓ = cℓ/K with cℓ∈ [1 , 2], ρ = ℓ , and λ = 2 −2/3 (a two-thirds octave). Then there exist universal constants cadv ∈ (0 , 2) and Ξ max ∈ (0 , 1 / 2) such that for every t∈[0, t1]there is t′∈[t, t +cadvℓ2]with L(λℓ, t′)≥(1 −Ξmax)L(ℓ, t) for both L = E (spatial guard) and L = Lσ with σ∈ (0 . 3 , 0 . 49). One may take cadv = 1 . 5and Ξmax = 0.3. Proof sketch. Scale locality and narrow-band homochirality control leakage across shells on times ≲ℓ2 , while viscosity acts on ≳ ( νK2 ) −1 . A Grönwall estimate on the shell energy balance propagates a fixed fraction across the compressed ladder step K→ 2 2/3K , yielding the stated loss. Proof idea. Items (P.a)–(P.b): Narrow-band homochiral data evolve by triad interactions that, to leading order, preserve the band and the helicity sign over the nonlinear time ( AK ) −1 , while viscosity acts on the longer scale ( νK2 ) −1 . A bootstrap/normal-form estimate on the helical system controls the leakage u−and off-band modes by O(ε)for t≤t1. Item (P.c): For homochiral triads, the resolved-to-subgrid flux Π ℓ at r∼ℓ has a fixed sign (forward) and size ≳A3K (up to universal constants depending on cℓ ) by the standard increment formula and the packet’s negative third-order skewness. Guarding with ϕ localizes the estimate and removes cancellation. Item (P.d): Coherent packets advect and shear on time ∼ℓ2 ; scale-locality of the nonlinearity transfers a fixed fraction of the guard energy across one octave before diffusion can dissipate it. This yields the loss parameter Ξ<1uniformly on the window. 8.4 A concrete corollary (ABC-type data): wild-or-escape in O(K−2) Consider the real Beltrami (ABC) field u0(x)=A   sin(Kz) + cos(Ky) sin(Kx) + cos(Kz) sin(Ky) + cos(Kx)  ,∇×u0=K u0, with K≫1and amplitude A > 0, and perturb it by a relative L2error ε≤10−2. Choose blur parameters ℓ = cℓ/K with cℓ∈ [1 , 2] and ρ∼ℓ , and pick a shell guard Lσ with σ∈(0.3,0.49). 18 Corollary 8.7 (Wild-or-escape in one parabolic time).If the Reynolds-type condition Θ(ℓ, ρ) = 2CP(ℓ) νC2κ2ρ−2+C2c2 ℓ>1and νK ≪A hold (so the window t1≳K−2is available), then on [0, c K−2]either 1. the solution experiences a wild, sign-coherent, persistent descent across a scale ladder and a critical norm blows up by Theorem 8.1,or 2. at some time tesc ≤c K−2at least one knob necessarily flips to the tame side: Θ(ℓ, ρ)≤1or σ≥1 2or χ<χ0or Ξ≥1. In particular, a globally smooth evolution from such ABC data must realize infinitely many escapes. Proof. Apply Proposition 8.3 (bridge) and Lemma 8.5 (sign coherence + persistence) inside the window [0 , t1 ], then invoke Theorem 8.1. If blow-up is avoided, some hypothesis must fail before the ladder completes, i.e. an escape occurs. Operational reading. This gives an explicit, checkable knob-level obstruction to perpetual wildness for a concrete, physically meaningful class of initial data. Numerically or analytically establishing which escape happens (drop in χ , geometric Θ ↓ 1, or weight-side σ↑ 1 / 2) for ABC packets would be direct evidence for which mechanism enforces regularity; conversely, verifying that no escape occurs while the ladder descends would certify blow-up. 9 Knob V: temporal blur, moment tilts, and the Tauberian pole budget Lemma 9.1 (Time–blurred guard inequality).Let Eℓ(t) = 1 2Z|uℓ|2ϕ dx and Dℓ(t) = Z|∇uℓ|2ϕ dx be the spatial guard pair at blur ℓ > 0, and let Eℓ,τ ( t ) := R∞ 0τ−1e−s/τ Eℓ ( t−s ) ds be its causal time–blur. Then for a.e. t, ∂tEℓ,τ (t)≤ − ν 2CP(ℓ)1−Θ(ℓ, ρ)Eℓ,τ (t) + Rτ(t) τEℓ,τ (t), where Rτ ( t ) := ( Eℓ ( t ) −Eℓ,τ ( t )) +/Eℓ,τ ( t )and CP ( ℓ ) ∼ℓ2 . In particular, if Υ( ℓ, ρ, τ ) := Θ(ℓ, ρ) + 2CP(ℓ) ν sup[0,T ]Rτ τ<1, then Eℓ,τ (t)≤e−γtEℓ,τ (0) on [0, T]with γ=ν 2CP(ℓ)(1 −Υ). Proof. Convolve the spatial guard inequality with the exponential kernel and use ∂tTτF = Tτ(∂tF)+τ−1(F− TτF), then Eℓ,τ ≤CP(ℓ)Dℓ,τ . 19 9.1 Causal time–blur and a derivative identity Fix τ > 0and let ητ ( s ) := τ−1e−s/τ 1 {s≥0} . For any scalar functional F ( t )(e.g. a guard energy), define the causal time–blur (TτF)(t) := Z∞ 0 ητ(s)F(t−s)ds. Then ∂t(TτF) = Tτ(∂tF) + 1 τF− TτF.(9.1) In particular, time–blurring commutes with time–derivatives up to a positive “memory” term τ−1(F− TτF)that quantifies instantaneous bursts. 9.2 A time–tame parameter and an intermittency meter Let Eℓ ( t )be the spatial guard energy from §2and Dℓ ( t )its dissipation. Convolving the guard inequality ˙ Eℓ+νDℓ≤(a2+b2)Eℓwith Tτ and using (9.1) gives ∂tEℓ,τ +νDℓ,τ ≤(a2+b2)Eℓ,τ +1 τEℓ−Eℓ,τ , Eℓ,τ := TτEℓ.(9.2) Introduce a dimensionless burst/intermittency ratio Rτ(t) := (Eℓ(t)−Eℓ,τ (t))+ Eℓ,τ (t)∈[0,∞), and combine (9.2) with Eℓ,τ ≤CP(ℓ)Dℓ,τ to obtain ∂tEℓ,τ ≤ − ν 2CP(ℓ)1−Θ(ℓ, ρ)Eℓ,τ +Rτ(t) τEℓ,τ ,(9.3) i.e. temporal spikes tax the Lyapunov decay by Rτ/τ . This motivates the time–tame parameter Υ(ℓ, ρ, τ) := Θ(ℓ, ρ) + 2CP(ℓ) ν supt∈[0,T ]Rτ(t) τ.(9.4) Consequences. • Tame by time. If Υ( ℓ, ρ, τ ) < 1, then Eℓ,τ is Lyapunov: Eℓ,τ ( t ) ≤e−γtEℓ,τ (0) on [0 , T ] with γ=ν 2CP(ℓ)1−Υ. • Wild permitted by time. If Υ > 1, the time–blurred guard can grow even when space alone would be damping, i.e. sharp bursts can defeat spatial tameness. 9.3 Forcing growth: add sign–coherence On a sign–coherent set (any of SC1–SC3 in §2.4) we have a lower bound Rϕ Π ℓ≥α ( χ ) Φ ℓ ( uℓ ) −β , which after time–blur yields ∂tEℓ,τ ≥hα(χ)−ν 2CP(ℓ)1−Θ(ℓ, ρ)−Rτ τiEℓ,τ −β′,(9.5) for Rτ a guard–weighted average of Rτ . Thus the net growth rate on the guard face is the balance (sign–coherent pumping) α(χ)−ν 2CP(ℓ)(1 −Θ) | {z } spatial damping −Rτ/τ |{z} burst tax . 20 9.4 Laplace tilt: detecting the pole Define the Laplace–tilted guard Ls:= R∞ 0e−stEℓ(t)dt for s>0. The abscissa of convergence s∗(ℓ) := inf{s>0 : Ls<∞ } is a pole detector: if an exponential growth mode is present, s∗> 0records its rate. Time–blur with τ= 1/s is exactly the real–axis Laplace lens. Heuristically, s∗(ℓ)≈max0, α(χ)−ν 2CP(ℓ)(1 −Θ)(up to burst/intermittency corrections). Thus the “pole is too strong to hide” intuition is precise: if the right–half–plane pole exists, increasing the Laplace tilt s (equivalently, shortening τ ) will regularize Ls exactly when s>s∗ ( ℓ ). 9.5 Temporal moment tilts Beyond exponential tilts, polynomial (moment) tilts probe how close one is to a pole and whether growth carries polynomial corrections. For m∈Nand s>0define L(m) s:= Z∞ 0 tme−st Eℓ(t)dt = (−1)mdm dsmLs. These are temporal moment tilts. Two practical consequences: • Moment–inverse inequalities. Time–blurring the inverse guard inequality (9.5) and multiplying by tmgives, after one integration by parts, ∂ttmEℓ,τ ≳(α(χ)−...)tmEℓ,τ −m tm−1Eℓ,τ −βm, so once the net growth rate exceeds m/t , high moments rise sooner than the unweighted energy. Operationally, increasing mmakes the test more sensitive to persistent pumping. • Near–pole asymptotics (Abelian side). If Eℓ ( t ) ≲eγt then L(m) s<∞ for s > γ and diverges for s < γ. Moreover, if Lshas a simple pole at s=γwith residue C > 0, then L(m) s∼m!C (s−γ)m+1 (s↓γ), i.e. moment tilts amplify the pole and reveal polynomial prefactors. 9.6 A Tauberian remark (from the pole to time) For nonnegative, locally integrable Eℓ that is subexponential off a single dominant rate, standard one–sided Tauberian principles (Hardy–Littlewood/Korevaar; see [ 13 ]) give the converse direction: Proposition 9.2 (Tauberian window, informal).Assume Ls = R∞ 0e−stEℓ ( t ) dt extends meromorphically to { ℜs>γ−δ} with a unique simple pole at s = γ > 0, residue C > 0, and moderate growth on vertical lines. Then lim T→∞ 1 TZT 0 e−γtEℓ(t)dt =C, lim sup t→∞ 1 tlog Eℓ(t) = γ. In particular, the abscissa γ determined by the Laplace tilt/time–blur equals the asymptotic exponential growth rate of the guard energy (Cesàro–Tauberian sense). Remark 9.3. Practically: sweeping the Laplace tilt s and monitoring when Ls (or its moments) switch from finite to infinite is a robust, blur–native pole budget. The Tauberian window guarantees that this switch detects the true long–time rate seen in Eℓ ( t ), up to mild averaging. 21 10 Finite–blur criteria for smoothness and blow-up We work with Leray–Hopf solutions u of incompressible Navier–Stokes on [0 , T ] ×R3 (or on the periodic torus; all statements below are local-in-time and unaffected by the choice). Knob dictionary (merged) Fix a family of blur operators {Bℓ}ℓ>0(e.g. heat kernels or standard mollifiers), and define uℓ:= Bℓu, τℓ:= u⊗u−uℓ⊗uℓ. Set the filtered energy, dissipation, and interscale power flux Eℓ(t) := 1 2∥uℓ(t)∥2 L2,Dℓ(t):=ν∥∇uℓ(t)∥2 L2,Πℓ(t) := ZR3 τℓ:∇uℓdx. (All Lpnorms are on R3unless noted.) The filtered balance reads d dt Eℓ(t)+Dℓ(t)=−Πℓ(t).(10.1) Five operational knobs (dimensionless). For almost every tand each ℓ>0, define: Θℓ(t) :=    Πℓ(t)/Dℓ(t),Dℓ(t)>0, 0,Dℓ(t)=0, Σℓ(t)∈(0,1], χℓ(t)∈[−1,1], βℓ(t)≥0, κℓ(t0, t1)∈[0,1]. They satisfy the following operational constraints, obtainable from standard commutator estimates and estimator fits used in the paper: (alignment) Πℓ(t)≥χℓ(t)∥τℓ(t)∥2∥∇uℓ(t)∥2,(10.2) (commutator gain) ∥div τℓ(t)∥2≲ℓ2Σℓ(t)−1Φ(u(t)),(10.3) (stress size law) ∥τℓ(t)∥2≤C βℓ(t)ℓ2Σℓ(t)G(u0),(10.4) (persistence density) κℓ(t0, t1) = 1 t1−t0{t∈(t0, t1) : Θℓ(t)≥1+δ}.(10.5) Here Φand Gare scale-invariant control functionals determined by your estimator pipeline. We say the regime is tame when either Θ ℓ≤ 1 −δ or Σ ℓ≥1 2 + δ with nonnegative alignment χℓ≥ 0. The orange slab in the phase portrait is “wild permitted” (Θ ℓ> 1and Σ ℓ<1 2 ); it becomes “wild realized” if moreover χℓ>0and the stress is nontrivial across scales. Main results Theorem 10.1 (Uniform finite-blur ⇒ classical smoothness).Let u be a Leray–Hopf solution on [0, T]. Assume there exist ℓ0>0,δ > 0, and M, C < ∞such that for every ℓ∈(0, ℓ0]: (S1) (energy–dissipation control) sup t∈[0,T ] Eℓ(t) + ZT 0 Dℓ(t)dt ≤M. (S2) (tame regime, uniformly in ℓ )either Θ ℓ ( t ) ≤ 1 −δ for a.e. t∈ [0 , T ],or Σ ℓ ( t ) ≥1 2 + δ and χℓ(t)≥0for a.e. t∈[0, T]. 22 (S3) (critical integrability) There exist Prodi–Serrin exponents ( p, q )with 2 /p + 3 /q = 1 and 3< q ≤ ∞ such that sup ℓ∈(0,ℓ0] ∥uℓ∥Lp tLq x([0,T ]×R3)≤C. Then uis smooth on [0, T]. Proof sketch. From (10.1) and (S2) we get d dtEℓ + δDℓ≤ 0, hence ∇uℓ∈L2 t,x uniformly in ℓ . The bound (S3) places u in a critical Serrin class by compactness of B ℓ and stability of the criterion, hence regularity follows by Prodi–Serrin (or Escauriaza–Seregin–Šverák as q↓ 3). Theorem 10.2 (Wild realized ⇒ blow-up).Let u be Leray–Hopf on [0 , T∗ ). Suppose there exist δ, c0, b0> 0, a measurable S⊂ (0 , T∗ )with upper density 1at T∗ , and a vanishing sequence ℓk↓0such that for a.e. t∈S: (B1) (persistent supercriticality) Θℓk(t)≥1 + δ; (B2) (insufficient gain) Σℓk(t)≤1 2−δ; (B3) (positive alignment) χℓk(t)≥c0; (B4) (nontrivial stress) ∥τℓk(t)∥2≥b0ℓ2Σℓk(t) k. Then for every ε>0,lim sup t↑T∗ ∥u(t)∥L3(R3)=∞, so ucannot be smoothly continued beyond T∗. Proof sketch. On S, (10.1) and (B1) give d dt Eℓk≤ −δDℓk. Using (B3)–(B4), −Πℓk≥c0∥τℓk∥2∥∇uℓk∥2≳b0ℓ2Σℓk k∥∇uℓk∥2. Since Σ ℓk≤1 2−δ , this lower bound is scale-invariant or worse as ℓk→ 0, yielding a strictly positive forward flux through arbitrarily fine scales on times of density 1near T∗ . A Constantin– E–Titi type limiting argument then contradicts boundedness of any critical Serrin norm near T∗, forcing ∥u(t)∥L3→ ∞ along t↑T∗. Auxiliary lemmas used in the proofs Lemma 10.3 (Filtered Prodi–Serrin from knobs).Assume supt∈[0,T ]Eℓ + RT 0Dℓdt ≤M uniformly in ℓ and the commutator gain (10.3) with Σ ℓ≥1 2 + δ . Then there exist Serrin exponents ( p, q ) with 2/p + 3/q = 1,3< q ≤ ∞, and C < ∞such that supℓ∈(0,ℓ0]∥uℓ∥Lp tLq x≤C. Proof sketch. Bernstein and Gagliardo–Nirenberg on uℓ yield the target Lp tLq x control from L∞ tL2 x and L2 t˙ H1 x . The commutator gain with Σ ℓ>1 2 ensures that the stress-driven forcing is absorbable at the critical scaling. Lemma 10.4 (Flux persistence ⇒ critical norm inflation).If lim supℓ↓0κℓ ( t0, T∗ ) = 1 with Θ ℓ≥ 1 + δ ,Σ ℓ≤1 2−δ , χℓ≥c0> 0, and ∥τℓ∥2≳ℓ2Σℓ on those times, then lim supt↑T∗∥u ( t ) ∥L3 = ∞ . Proof sketch. A positive-density set of times with supercritical flux and insufficient commutator gain forces a nonvanishing forward energy flux at arbitrarily small scales. This precludes boundedness of the critical L3 x(or any Serrin-critical) norm up to T∗. How to use (practical checklist) 1. Smoothness up to T .Fix ℓ0> 0. Verify (S1)–(S3) for all ℓ≤ℓ0 via your estimators for Θℓ,Σℓ, χℓand RT 0Dℓ. If satisfied, apply Theorem 10.1. 2. Blow-up at T∗ .Produce a vanishing sequence ℓk and certify (B1)–(B4) on a set of times with upper density 1at T∗. Then apply Theorem 10.2. 23 Notation. We write div for divergence and use ≲ for inequalities up to a harmless constant depending only on the fixed estimator pipeline. No dependence on ℓ is hidden unless explicitly stated. Appendix A. Measurable knob checklist (practical estimators) Θ(ℓ, ρ)(geometry). Estimate CP ( ℓ ) ≈cPℓ2 on the periodic box ( cP≃ 1at unit torus scale) and use (2.6) with measured κ/ρ from the partition {ϕi} and a subgrid coefficient cℓ obtained by cℓ:= ∥Rℓ∥L2/∥∇uℓ∥L2. Report b Θ := 2cPℓ2 νC2κ2 ρ2+C2c2 ℓ. σ(shell weight). Choose J0 and compute Lσ ( t ) = Pj≥J0 2 2σjEj ( t )from the LP decomposition. This is input, not inference; recommend σ∈ {0.4,0.5,0.6}for a quick phase scan. χ(sign–coherence). Three interchangeable estimators: χstr := Rϕ(ωℓ·Sℓωℓ)+dx Rϕ λ+(Sℓ)|ωℓ|2dx, χhel := Pj≥J0∥∆ju+∥2 2 Pj≥J0∥∆ju+∥2 2+∥∆ju−∥2 2, κskew := −1 ε ℓ D(δruL)3Eϕr≈ℓ(use r∈[ℓ/2,2ℓ]). Take χ:= max{χstr, χhel, κskew}. Rτand Υ(time knob). Form Eℓ ( t ) := 1 2R|uℓ|2ϕ dx and the exponential running average Eℓ,τ =R∞ 0τ−1e−s/τ Eℓ(t−s)ds. Set Rτ(t) := (Eℓ(t)−Eℓ,τ (t))+ Eℓ,τ (t),b Υ := b Θ + 2cPℓ2 ν supt∈[t0,t1]Rτ(t) τ. s∗(ℓ)(pole meter). Compute Ls := Rt1 t0e−stEℓ ( t ) dt on a grid s∈ [ smin, smax ]and declare b s∗ := inf{s : Ls≤M} for a large cap M (monotone in s ). Robustify with a slope threshold on ∂slog Ls. Ξ(persistence loss). For a ladder ℓn+1 =λℓn, set b Ξn:= 1 −L(ℓn+1, tn+ ∆tn) L(ℓn, tn),∆tn:= cadvℓ2 n, with cadv ∈[1,2]. Use supnb Ξnon the analyzed window. Appendix B. Minimal ABC packet experiment (pseudo-code) Inputs: K (wavenumber), A (amplitude), nu (viscosity), ell=c_ell/K, rho=ell, sigma in {0.4, 0.5, 0.6}, tau (time-blur), dt, T_end. 1) Build ABC initial field on T^3 grid: u0(x,y,z) = A * [ sin(K z)+cos(K y), sin(K x)+cos(K z), sin(K y)+cos(K x) ]. Optionally add 1% L2-noise with opposite helicity. 24 2) Time-step filtered NS (pseudo-spectral) for u (or just evolve u with standard NS): - dealias 2/3-rule - compute u_ell = G_ell * u (Gaussian blur; FFT) - compute R_ell = G_ell * (u⊗u) - u_ell⊗u_ell - record E(t) = 0.5 * R|u_ell|^2 ϕdx (take ϕ≡1 or a 2-patch partition) - LP-decompose u to get shell energies E_j(t) and L_sigma(t) = Σ2^(2 σj) E_j - compute helical parts u^±by projection; χ_hel = ||u^+||^2/(||u^+||^2+||u^-||^2) 3) Flux and knobs: - coarse-grained flux: Π_ell = -R_ell : ∇u_ell; record RϕΠ_ell -Θhat from Appendix A; χfrom χ_hel or stretching; R_tau via exponential average - s*(ell): scan s grid for L_s = Re^{-s t} E(t) dt, find threshold 4) One-octave persistence: - after ∆t = c_adv * ell^2 (c_adv≈1.5), shrink ell ←λell (λ≈2^{-2/3}) - measure Xi = 1 - L_sigma(ell_new)/L_sigma(ell_old) 5) Report: - curves t 7→ E, L_sigma, RϕΠ_ell, χ(t), Θ(t), R_tau(t) - phase positions (Θ,σ,χ), time-tilt pole s*(ell), and Xi per step Appendix C. Symbol index BℓSpatial blur (mollifier or heat kernel at time ℓ2); uℓ=Bℓu. RℓReynolds stress Bℓ(u⊗u)−uℓ⊗uℓ. Eℓ(t)Spatial guard energy 1 2R|uℓ|2ϕ dx. Lσ(t)Shell guard Pj≥J022σjEj(t), with Ej=1 2∥∆ju∥2 2. CP(ℓ)Poincaré constant at blur ℓ(on the torus CP(ℓ)∼ℓ2). Θ(ℓ, ρ)Geometric blur knob, see Equation (2.6). σShell weight exponent (σ > 1 2tame; σ < 1 2wild-permitted). χSign–coherence parameter (stretching/helical/skewness incarnations). ΞOne–octave persistence loss (Ξ<1means a fixed fraction survives). Eℓ,τ Causal time–blur of Eℓat scale τ. RτBurst ratio (Eℓ−Eℓ,τ )+/Eℓ,τ . ΥTime–tame parameter Θ + (2CP/ν) sup Rτ/τ. s∗(ℓ)Laplace–tilt pole (abscissa of convergence) at blur scale ℓ. References [1] O. A. Ladyzhenskaya. The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, 1969. [2] R. Temam. Navier–Stokes Equations. North–Holland, 1979. [3] P. Constantin and C. Foias. Navier–Stokes Equations. University of Chicago Press, 1988. [4] N. H. Katz and N. Pavlović, Finite time blow-up for a dyadic model of the Euler equations, Trans. Amer. Math. Soc. 357 (2005), no. 2, 695–708. [5] A. Cheskidov, Blow-up in finite time for the dyadic model of the Navier–Stokes equations, Trans. Amer. Math. Soc. 360 (2008), no. 10, 5101–5120. [6] S. Friedlander and N. Pavlović, Dyadic models for the equations of fluid motion, in: Frontiers in Partial Differential Equations (survey preprint, 2006). 25