Non-unique vanishing-viscosity selection for planar Euler

9 October 2026

Abstract
For every finite p > 2, we construct one real compactly supported zero-circulation initial
vorticity in L1 (R2 )∩Lp (R2 ) whose unforced Navier–Stokes evolutions have two distinct strong
vanishing-viscosity accumulation paths. Both limits are renormalized unforced Euler solutions
with the same strong initial trace. The construction uses a fixed infinite concentric stack,
a smooth unstable fourfold profile, actual positive scalar envelopes, and a finite nonlinear
expansion retaining the interacting harmonic histories. Discounted source history is controlled
through the complete prospective receiver stop, and exact-cap normalization yields a depthuniform outgoing reset. Protected first crossings, phase degree and a sparse phase assignment
then separate the two limits. The analytical preliminaries, finite-sector spectral scope, sourcereference comparison and uniform parameter order are included in the proof.

Contents
1 Introduction
1.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Background and analytical setting . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Construction and proof strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Logical dependencies and organization . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Frequently used notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3
3
3
4
5
6

2 Preliminaries
2.1 Equations, solution classes, and symmetries . . . . . . . . . . . . . . . . . . . . . .
2.2 Heat and Sobolev estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Classical Euler continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 The classical viscous solution and its dependence on data . . . . . . . . . . . . . .
2.5 Scalar comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Physical and local colour normalization . . . . . . . . . . . . . . . . . . . . . . . .
2.7 A fixed passive colour is determined by the total Euler path . . . . . . . . . . . . .
2.8 Protected first caps and phase shooting . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Compactness for one fixed compact zero-circulation datum . . . . . . . . . . . . . .

6
6
7
8
9
11
12
13
16
19

3 The fixed profile, datum and isolated activation
3.1 An isolated nonlinear active stage for a fixed smooth profile . . . . . . . . . . . . .
3.1.1 The fixed profile and its positive angular spectrum . . . . . . . . . . . . . .
3.1.2 An exactly neutral smooth seed . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 The finite seed–viscosity expansion . . . . . . . . . . . . . . . . . . . . . . .
3.1.4 Nonlinear closure in the actual vorticity norm . . . . . . . . . . . . . . . . .

22
22
23
26
26
28

1

3.1.5

Whole-moment control and the cap . . . . . . . . . . . . . . . . . . . . . . . 29

4 The canonical ancient orbit and signed coupling
4.1 The canonical ancient profile of the isolated smooth activation . . . . . . . . . . .
4.1.1 A high Sobolev error estimate without derivative loss . . . . . . . . . . . .
4.1.2 Finite homological coefficients and compactness . . . . . . . . . . . . . . . .
4.1.3 Uniqueness of the ancient subsequential limit . . . . . . . . . . . . . . . . .
4.1.4 Forward extension and the whole-moment asymptotics . . . . . . . . . . . .
4.2 The smooth receiver’s protected coordinate and transfer coefficient . . . . . . . . .
4.2.1 Exact heat shielding and the remaining coupled terms . . . . . . . . . . . .
4.2.2 A nonlinear receiver theorem for the concrete canonical input . . . . . . . .

31
31
31
33
34
35
38
42
43

5 Actual cores, localization and the cold packet
5.1 Actual viscous outer cores and the local harmonic field . . . . . . . . . . . . . . . .
5.2 A quantitative parabolic localization estimate . . . . . . . . . . . . . . . . . . . . .
5.3 Nonlinear inner-tail suppression through a receiver window . . . . . . . . . . . . .
5.3.1 A Gaussian-energy estimate for a globally weak foreign field . . . . . . . . .
5.3.2 Comparison with the actual local foreign field . . . . . . . . . . . . . . . . .
5.3.3 Physical normalization, the available power margin, and scope . . . . . . .

45
45
49
51
51
55
58

6 The actual inactive prefix
6.1 Fixed seeds: separate initial and forced prefix contributions . . . . . . . . . . . . .
6.2 Actual inactive densities and a large amplitude ratio . . . . . . . . . . . . . . . . .
6.3 An actual packet bound closes the inactive density stop . . . . . . . . . . . . . . .

59
59
61
66

7 Actual source activation and discounted history
7.1 A global weighted foreign-field estimate for the adjoint error . . . . . . . . . . . . .
7.2 Actual postcap source moments under a stopped-parent core . . . . . . . . . . . .
7.3 The actual deepest selected source . . . . . . . . . . . . . . . . . . . . . . . . . . .

69
69
70
73

8 Finite nonlinear expansions for retained actual forcing
8.1 Finite actual forcing: density closure and exact-cap reset . . . . . . . . . . . . . . .
8.1.1 Spaces, operators and the derivative reserve . . . . . . . . . . . . . . . . . .
8.1.2 The finite recursion and its exact residual . . . . . . . . . . . . . . . . . . .
8.1.3 Comparison with the actual foreign velocity . . . . . . . . . . . . . . . . . .
8.1.4 A normalized finite polynomial loses the forcing waveform . . . . . . . . . .
8.1.5 Exact protected-coordinate matching . . . . . . . . . . . . . . . . . . . . . .
8.1.6 State reset and the separate instantaneous cap condition . . . . . . . . . . .

86
86
87
88
90
91
92
93

9 Transmission to an actual receiver
94
9.1 An actual receiver and the loss of source memory . . . . . . . . . . . . . . . . . . . 94
9.1.1 Discounted histories of an arbitrary prepared actual child . . . . . . . . . . 95
10 Uniform descent and fixed-datum phase selection
103
10.1 Uniform constants, the descending induction, and phase selection . . . . . . . . . . 103
10.1.1 Proof of the non-unique-selection theorem . . . . . . . . . . . . . . . . . . . 107

2

1

Introduction

The vanishing-viscosity approximation of an incompressible flow combines an existence procedure
with a selection question. For each positive viscosity the two-dimensional Navier–Stokes equation
has a unique global vorticity evolution from an integrable initial density. As viscosity tends to zero,
compactness can produce Euler solutions, and renormalization can upgrade weak convergence to
strong convergence. These facts do not by themselves determine whether all viscosities select the
same Euler path. The distinction is relevant when the initial vorticity lies in L1 (R2 ) ∩ Lp (R2 ) for
a finite p > 2.
We study the unforced planar equation
∂t ω ν + (K ∗ ω ν ) · ∇ω ν = ν∆ω ν ,

K(x) =

x⊥
.
2π|x|2

The initial condition in our construction is fixed once and for all; only ν varies. Zero circulation
is imposed to place its Biot–Savart velocity in the finite-energy class. The strong topology used
throughout is the sum norm on Xp = L1 ∩ Lp , and the Euler solutions have their strong initial
trace in this space. The definitions of distributional and renormalized solutions are collected in
Section 2.

1.1

Main result

Theorem 1.1 (Non-unique selection from one fixed datum).
For every 2 < p < ∞ there exist
R
1
2
p
2
a real compactly supported ω0 ∈ L (R ) ∩ L (R ) with R2 ω0 = 0, a finite time Tf > 0, two
sequences νn+ ↓ 0 and νn− ↓ 0, and two distinct renormalized solutions ω + , ω − of the unforced
Euler equation such that
ω ± ∈ C([0, Tf ]; Xp ),

ω ± (0) = ω0 ,

±

lim ∥ω νn − ω ± ∥C([0,Tf ];Xp ) = 0.

n→∞

Here ω ν is the unique unforced Navier–Stokes solution with viscosity ν and the same initial condition ω0 for every ν > 0. Distinctness means that the two paths differ on [0, Tf ]. Consequently
this fixed-datum family does not converge as ν ↓ 0.
The theorem asserts non-unique accumulation points in a strong vorticity topology. It does not
assert that every Euler solution from this datum can be selected by viscosity, or prescribe the total
set of accumulation points. The velocity convergence along each of the displayed subsequences is
also strong in C([0, Tf ]; L∞ ), by Lemma 2.6.

1.2

Background and analytical setting

The global integrable-vorticity Navier–Stokes theory used below is stated precisely in Proposition 2.4; its existence input is Gallay–Wayne [1, Theorem 2.2]. Smooth Euler evolution and the
Biot–Savart formulation are part of the classical planar theory; see Majda–Bertozzi [3]. We give
the kernel estimates and the comparison arguments needed by the construction, including the
low-frequency cancellation supplied by zero circulation.
The treatment of transported densities follows the renormalization framework of DiPerna–
Lions [2]. In the present range p > 2, the commutator estimates for the limiting total vorticity
and for bounded passive colours are proved directly, together with their endpoint traces. For
comparison, Ciampa–Crippa–Spirito [4, Theorem 3.2] prove strong convergence to an identified
3

renormalized inviscid limit under their stated approximation hypotheses. That theorem concerns
convergence to a limit already identified along the relevant family, and does not assert uniqueness
of selection. The fixed rough datum used here is handled by the self-contained compactness proof
in Theorem 2.12, rather than by an unstated application of that approximation theorem.
The auxiliary tools in Section 2 are distinguished from the cascade estimates in the later
sections. In particular, compactness and uniqueness of a passive colour for a prescribed Euler
velocity are established analytical inputs. They do not imply uniqueness of the nonlinear Euler
path.

1.3

Construction and proof strategy

The datum is an infinite concentric stack of copies of one smooth radial profile W with zero
circulation, perturbed by neutral fourfold seeds:
ω0 (x) =

X





Aj W (x/Rj ) + αΦϑj (x/Rj ) ,

Aj = β j−1 ,

Rj = q j−1 .

(1)

j≥1

The fixed parameters obey β > 1, 0 < q < 1 and βq 2/p < 1. The series converges absolutely in Xp
and has one common compact support. Each seed has zero circulation and zero fourth complex
moment. Its phase is retained as a free circle parameter. The profile, smoothing length, scales,
amplitude and eventually all seed phases are fixed independently of viscosity.
The radial profile has a simple unstable pair in angular sectors ±4. Its linear analysis is
modal. Every nonlinear approximation uses only a specified finite set of angular sectors and a
finite derivative hierarchy; the actual remainder is controlled by a scalar transport comparison
in Xp . No spectral claim about an infinite angular direct sum is needed. Finite expansions also
produce a canonical smooth ancient Euler orbit and all its finite jets. The same orbit is used at
every receiving scale.
For viscosity
2
νN = q 3/2 AN RN
= q 3/2 (βq 2 )N −1 ,
scale N is the deepest selected source. Smaller scales form a diffusing packet. A signed fourthmoment estimate makes its impulse smaller than the active seed. Larger scales remain inactive
while the selected source prepares. Their actual core and harmonic estimates are closed using
positive scalar envelopes for the complete interacting Navier–Stokes velocity. Thus the packet
force is estimated from the actual solution, rather than imposed as an external forcing assumption.
The main small exponents are explicit. We take 6 < K < 643/100, α = q K and δ =
[log(1/q)]−4 . The selected source reaches its first δ cap in normalized time (K/σ) log(1/q) to
leading order. Its fourth moment transmits an exterior field of order q 6 to the next larger colour.
A finite nonlinear recursion retains the actual harmonic coefficient histories. Normalization at the
receiver’s exact cap removes the incoming waveform from its outgoing finite jet, with a uniform
intrinsic remainder.
Source history is controlled in three pieces. Before its shrinking cap, a cap-relative exponential envelope controls the discounted past. From that cap to one fixed reference state, one-sided
comparison with the canonical orbit controls the short advance, without relating the reference
amplitude to a later expansion threshold. After reference time, positive-envelope moments and a
separate Gaussian estimate at the parent radius bound the discounted future up to the parent’s
provisional stop. An upper input bound is obtained before choosing the receiver’s stop constant.

4

Only after reaching a deterministic prospective cap horizon is the nonzero signed transfer coefficient used. This order avoids a circular use of a future coefficient to justify the interval on which
it is defined.
The actual stage estimates give uniformly protected first caps through an arbitrarily deep finite
descent. Protection comprises an early strict margin and a positive radial slope in a specified late
interval around the first crossing. It supplies continuity of the cap time and phase under seed
homotopies. At a fixed positive viscosity, scaling all other seeds down to zero leaves a degree-one
rotation in the selected seed phase. Homotopy invariance then makes the outer cap phase map
surjective.
Finally, phases are assigned at a sparse sequence of depths so that the outer cap phases
alternate between two separated values. The summable Xp tail of the datum and continuous
dependence at each fixed positive viscosity preserve previously selected outputs. Strong compactness gives a limit along each parity. If the two total Euler paths were equal, the unique
bounded passive outer colour transported by their common velocity would give the same first-cap
phase. The two selected phases exclude that possibility.

1.4

Logical dependencies and organization

Table 1 records the dependencies used in the final proof. Conditional estimates are applied only
after their actual hypotheses have been supplied. The passage from one prepared child to its
parent is separated from the final induction over the depth.
Result
Kernel, transport and
compactness (Section 2)
Profile and isolated
activation (Section 3)
Canonical orbit and signed
coupling (Section 4)
Cores, cold tails and inactive
prefix (Sections 5–6)
Source and reference history
(Section 7)
Finite forced recursion
(Section 8)
Actual receiving stage
(Section 9)
Uniform induction and phase
selection (Section 10)

Required inputs
Fixed Xp datum; bounded
Sobolev limiting velocity
Smooth fixed profile; finite
modal spectrum
Isolated activation and finite jets
Actual scalar contractions and
positive envelopes
Prepared child; actual older
control; provisional parent stop
Actual finite harmonic histories
and derivative reserve
Prepared-child history and finite
recursion
Selected source; receiver;
compactness; passive uniqueness

Role in the proof
Strong limits and recovery of
fixed passive colours
Selected seed, growing coordinate
and finite expansions
One common reference orbit and
a nonzero transform
Unconditional older control and
small deep impulse
Uniform input norm and
reference reset
Exact-cap jet independent of the
incoming waveform
Protected cap and invariant
outgoing preparation
Theorem 1.1

Table 1: Dependency map for the fixed-datum selection theorem.
Section 2 collects the classical analytical inputs and proves the compactness and protectedcap tools in the form used here. Sections 3 and 4 construct the fixed profile and canonical
orbit. Sections 5 and 6 control the actual surrounding colours. Section 7 proves the source
estimates. Sections 8 and 9 establish the nonlinear transmission step. Section 10 makes all
constants independent of depth and proves Theorem 1.1.

5

1.5

Frequently used notation

Symbol
Xp , X
K, Jx
ω ν , wj
Aj , Rj , κj
W , Wϵ
Φϑ , α
λ = σ + iτ
Prad , Bm
Tθ
Tadj , A
L, δ
Ωref , sq
νN , Tf

Meaning
L1 ∩ Lp with sum norm; p > 2 fixed
Planar Biot–Savart kernel; Jx = x⊥
Total vorticity; its actual scalar colour at depth j
Amplitude, radius and normalized viscosity ν/(Aj Rj2 )
The fixed smoothed zero-circulation radial profile
Neutral fourfold seed of phase ϑ; its amplitude q K
Unstable eigenvalue of the fixed profile,
σ>0
R
Angular average; entire moment z̄ m f when absolutely integrable
Scalar rotation f (r, φ + θ/4)
R
Bounded growing adjoint test; A(f ) = Tadj f
log(1/q) and its shrinking cap L−4
Fixed canonical orbit; its exact δ-cap time
Viscosity selecting depth N ; final common physical time

Table 2: Notation used throughout. Local normalized clocks are declared when introduced.

2

Preliminaries

2.1

Equations, solution classes, and symmetries

All spatial Lebesgue and Sobolev spaces are over R2 . For 2 < p < ∞ set
Xp = L1 ∩ Lp ,

∥f ∥Xp = ∥f ∥1 + ∥f ∥p .

The notation X means Xp with the exponent fixed in the statement. Vector norms use the
Euclidean pointwise norm. The norm on C([0, T ]; X) is the supremum of the X norm, and its
elements have strongly continuous representatives. The physical vorticity equation is
∂t ω ν + div(uν ω ν ) = ν∆ω ν ,

uν = K ∗ ω ν ,

K(x) =

x⊥
,
2π|x|2

x⊥ = (−x2 , x1 ).

(2)

The velocity is divergence free, so its divergence-form drift agrees with uν · ∇ω ν whenever the
latter expression is defined. Setting ν = 0 gives the Euler vorticity equation.
A weak Euler solution with datum ω0 on [0, T ] is ω ∈ L∞ ([0, T ]; Xp ), with u = K ∗ ω, such
that
Z Z
Z
T

ω(∂t φ + u · ∇φ) +

ω0 φ(0) = 0

0

φ ∈ Cc∞ ([0, T ) × R2 ) .


Its weak initial trace is the trace specified by this identity. It is renormalized if the same identity
holds with ω, ω0 replaced by b(ω), b(ω0 ) for every b ∈ C 1 (R) with bounded derivative and b(0) = 0.
Norm conservation is obtained by admissible truncations of powers; it is not part of the definition
of a weak solution. All weak and renormalized identities involving a prescribed interior or final
time use the time representatives established in their proofs.
Identify x with z = x1 + ix2 . For each integer m ≥ 0 define
Z

Mm (f ) =

|x|m |f (x)| dx,

Z

Bm (f ) =

z̄ m f (x) dx

when Mm (f ) < ∞.

Thus M0 (f ) = ∥f ∥1 and B0 (f ) is the circulation. An entire moment is used only after its
absolute moment has been established. For a fixed smooth compact radial cutoff χ, the cut
6

moment χ(|x|)z̄ m f (x) dx is defined for every f ∈ L1 . Write Rα for the counterclockwise planar
rotation through angle α, and define angular averaging by
R

(Prad f )(x) =

Z 2π

1
2π

f (Rα x) dα.
0

The notations Prad and a locally specified P denote this same operator. Fourfold symmetry means
f (Rπ/2 x) = f (x) almost everywhere.
The pattern rotation is the scalar action
(Tθ f )(x) = f (Rθ/4 x),

(Tθ f )(r, φ) = f (r, φ + θ/4).

It corresponds to physical rotation of the pattern through −θ/4. For arbitrary functions θ is a
real parameter; on the fourfold subspace the action is 2π-periodic and hence defines a phase-circle
action. Its vector action is the pushforward
(Tθ V )(x) = R−θ/4 V (Rθ/4 x).
Both actions are isometries in the full-plane Lebesgue and Sobolev norms, commute with the
Laplacian, and satisfy
K ∗ (Tθ f ) = Tθ (K ∗ f ),

Tθ (V · ∇f ) = (Tθ V ) · ∇(Tθ f ).

These identities follow by change of variables and K(Rα x) = Rα K(x). For every finite absolute
moment,
Bm (Tθ f ) = eimθ/4 Bm (f ),
B4 (Tθ f ) = eiθ B4 (f ).
In particular, a fourfold function has Bm (f ) = 0 whenever m is not divisible by four and the
moment is defined.

2.2

Heat and Sobolev estimates
2

Lemma 2.1 (Heat estimates). Let Hτ (x) = (4πτ )−1 e−|x| /(4τ ) and Sτ f = Hτ ∗ f for τ > 0, with
S0 the identity. For exponents 1 ≤ r ≤ s ≤ ∞ and each integer a ≥ 0,
∥Da Sτ f ∥s ≤ Ca,r,s τ −a/2−(1/r−1/s) ∥f ∥r .
In particular Sτ is a contraction in every Lr , and
∥∇Sτ f ∥X ≤ Cτ −1/2 ∥f ∥X .
If f, ∆f, . . . , ∆N +1 f ∈ Lr , then
Sτ f −

N
X
τl
l=0

l!

∆l f

≤
r

τ N +1
∥∆N +1 f ∥r .
(N + 1)!

For smooth compact profiles this also holds after taking any fixed number of derivatives or replacing Lr by H k . If f is supported in F and dist(E, F ) ≥ d > 0, then
2

∥1E Da Sτ f ∥r ≤ Ca τ −a/2 e−d /(8τ ) ∥f ∥r
At τ = 0 the separated-support value is zero directly.
7

(1 ≤ r ≤ ∞).

Proof. Scaling the Gaussian and applying Young’s convolution inequality give the first estimates.
Repeated integration of ∂τ Sτ f = Sτ ∆f gives the Taylor remainder
τ N +1
N!

Z 1

(1 − s)N Ssτ ∆N +1 f ds;

0

contraction gives its bound. In the off-diagonal estimate only |x − y| ≥ d contributes. The L1
2
norm of the corresponding Gaussian derivative tail is at most Ca τ −a/2 e−d /(8τ ) , after absorbing
its polynomial factor into half the exponential. Young’s inequality finishes the proof.
Lemma 2.2 (Sobolev energy and Nash estimates). For every integer k ≥ 3, H k ⊂ W 1,∞ and
∥f g∥H k ≤ Ck ∥f ∥H k ∥g∥H k .
The Biot–Savart derivative satisfies ∥∇K ∗ g∥H j ≤ C∥g∥H j for every integer j ≥ 0. For smooth
bounded v and g ∈ H k ,
X

∥Da (v · ∇g) − v · ∇Da g∥2 ≤ Ck ∥∇v∥∞ ∥g∥H k + ∥∇v∥H k−1 ∥∇g∥∞ .


|a|≤k

If v is divergence free and ∂t g + v · ∇g = κ∆g + F , κ ≥ 0, its H k energy estimate follows by
pairing each differentiated equation with the corresponding derivative of g; the undifferentiated
transport has zero pairing and diffusion has nonpositive pairing. Finally, for g ∈ L1 ∩ H 1 ,
∥g∥42 ≤ C∥g∥21 ∥∇g∥22 .
Proof. Fourier inversion and Cauchy–Schwarz give the stated embeddings; the integrability of
(1 + |ξ|2 )−k for k > 1 also gives the Sobolev product bound by Fourier convolution. The multiplier of ∇K∗ is bounded, so Plancherel gives its H j estimate. Leibniz’s rule gives the transport
commutator. Its endpoint terms put one derivative in L∞ and the other factor in L2 ; the intermediate terms are bounded by Sobolev interpolation between these endpoints in dimension two.
Integration by parts gives the energy assertion, with expanding cutoffs justified by bounded v
and integrability of |Da g|2 . For Nash’s inequality split the Fourier integral at radius ρ:
∥g∥22 ≤ Cρ2 ∥g∥21 + Cρ−2 ∥∇g∥22 .
Optimizing ρ and squaring proves the last assertion; zero norms are handled by approximation.

2.3

Classical Euler continuation

Proposition 2.3 (Finite-Sobolev planar Euler evolution). Let k ≥ 3 be an integer, and let ω0 be
real, compactly supported, in H k , and have zero circulation. Then there is a unique global Euler
solution in the finite-Sobolev class
ω ∈ C([0, ∞); H k ) ∩ C 1 ([0, ∞); H k−1 ),

u = K ∗ ω ∈ C([0, ∞); H k+1 ) ∩ C 1 ([0, ∞); H k ).

Its vorticity is transported by the volume-preserving classical flow, so its Lr norms, 1 ≤ r ≤ ∞,
and its circulation are preserved. Its velocity energy is conserved. If supp ω0 ⊂ BR0 , then
∥u(t)∥∞ ≤ V,

supp ω(t) ⊂ BR0 +V t ,

V = C(∥ω0 ∥1 + ∥ω0 ∥∞ ).

All Sobolev norms in the statement are bounded on each finite time interval. Uniqueness here is
in the indicated finite-Sobolev class.
8

Proof. Compact support gives ω0 ∈ L1 , and zero circulation gives u0 ∈ L2 by Lemma 2.6. The
order-zero derivative multiplier in Lemma 2.2 then gives u0 ∈ H k+1 . The classical finite-Sobolev
local Euler construction is obtained by regularization and the energy commutator estimates; see
[3] for this standard local construction. For G(t) = ∥u(t)∥H k+1 , its velocity energy estimate is
d+
G ≤ Ck ∥∇u∥∞ G.
dt
Indeed, the divergence-free transport and pressure have zero energy pairing, and the commutator
estimate of Lemma 2.2 at order k + 1 bounds the remaining terms by Ck ∥∇u∥∞ G2 . In particular
G′ ≤ Ck G2 gives a local existence time controlled by G(0), and the same construction can restart
while G stays bounded. The difference of two such solutions satisfies the L2 energy inequality
with coefficient ∥∇u∥∞ , giving uniqueness.
The local solution has a C 1 spatial flow. Its Jacobian is one, and the scalar vorticity is
constant along trajectories. Thus m = ∥ω0 ∥1 and M = ∥ω0 ∥∞ bound the same norms throughout
its existence interval. The near–far split of Biot–Savart gives ∥u∥∞ ≤ C(m + M ) and hence the
displayed support bound.
We give the continuation estimate explicitly. The off-origin kernel of ∇K∗ has size C|y|−2
and zero angular mean; a possible local multiple of ω contributes at most CM . For 0 < ρ < 1,
cancellation in the ball of radius ρ, the intermediate annuli, and the exterior region give
∥∇u∥∞ ≤ C [ω]C 1/2 ρ1/2 + M (1 + log(1/ρ)) + m .


Sobolev embedding bounds [ω]C 1/2 by Ck G. Take ρ = (2 + G)−2 to obtain


∥∇u∥∞ ≤ Ck,m,M 1 + log(2 + G) .
Thus Y = log(2 + G) satisfies Y ′ ≤ Ck,m,M (1 + Y ). Its integrating-factor bound prevents finitetime growth of G to infinity, so the local solution extends globally. The vorticity and velocity
equations give the asserted time-derivative regularity. Their strong Sobolev time continuity is
part of the local construction and is preserved under restart.

2.4

The classical viscous solution and its dependence on data

Proposition 2.4 (Planar Navier–Stokes inputs). For ν > 0 and real ω0 ∈ L1 , equation (2) has
a unique global mild solution in
C([0, ∞); L1 ) ∩ C((0, ∞); L∞ ).
It preserves circulation and is smooth at positive times. For 1 ≤ r ≤ ∞,
∥ω ν (t)∥r ≤ Cr (νt)−(1−1/r) ∥ω0 ∥1

(t > 0).

Whenever ω0 ∈ Lr , the solution also satisfies the Lr contraction ∥ω ν (t)∥r ≤ ∥ω0 ∥r . If ω0 ∈ Xp ,
then ω ν ∈ C([0, ∞); Xp ), and on each finite interval the solution map is locally Lipschitz in Xp
for this fixed ν. For ω0 ∈ Xp , if additionally u0 = K ∗ ω0 ∈ L2 , its velocity satisfies
ν

∥u (t)∥22 + 2ν

Z t
0

∥∇uν (s)∥22 ds ≤ ∥u0 ∥22 .

9

For ω0 ∈ Xp and its bounded drift uν , a prescribed colour f0 ∈ L1 ∩ Lr , 1 < r < ∞, has a
unique mild path f ∈ Ct (L1 ∩ Lr ) solving
∂t f + div(uν f ) = ν∆f,

f (0) = f0 .

It preserves its integral, its L1 and Lr norms contract, and positivity and order are preserved, and
an initial L∞ bound contracts as well. At fixed viscosity these colour paths depend continuously
on their initial colours and on the total initial datum in the indicated norms.
Proof. The global L1 existence, uniqueness, positive-time L∞ continuity, circulation, and smoothing estimate at viscosity one are Theorem 2.2 of Gallay–Wayne [1, Theorem 2.2]. To obtain
viscosity ν, let Ω solve that equation with datum ω0 /ν and set
ω ν (t, x) = νΩ(νt, x),

uν (t, x) = ν(K ∗ Ω)(νt, x).

Substitution proves (2) and its asserted estimates. The mild equation is
ω ν (t) = Sνt ω0 −

Z t
0

∇ · Sν(t−s) (K ∗ ω ν (s))ω ν (s) ds.




Positive-time smoothness follows by heat bootstrap on intervals bounded away from zero.
For smooth approximations, multiply the scalar equation by the derivative of a convex regularization of |s|r and use expanding cutoffs. Divergence freedom removes transport, and diffusion
is nonpositive. For 1 < r < ∞ this yields
4ν(r − 1)
∥ω (t)∥rr +
ν

r

Z t
0

∥∇|ω ν (s)|r/2 ∥22 ds ≤ ∥ω0 ∥rr .

Absolute-value regularizations give the L1 contraction; the maximum principle, or passage r → ∞,
gives the L∞ contraction. These estimates pass through initial-data approximation. The same
arguments give contraction and order preservation for the linear colour equation.
For Xp data, Lemmas 2.6 and 2.1 make the mild equation a contraction on a bounded Ct Xp
ball for a sufficiently short interval. The scalar contractions prevent loss of this bound, giving
continuation and strong Xp continuity from zero. If two paths lie in an Xp ball of radius M , their
difference d(t) obeys, after a restart at any time a,
Cp M
d(t) ≤ d(a) + √
ν

Z t

(t − s)−1/2 d(s) ds.

a

p

Choose a length ℓ > 0 with 2Cp M ℓ/ν ≤ 1/2. On each such interval the supremum of d is at
most 2d(a). Iteration over [0, T ] proves
sup d(t) ≤ Cν,T,M d(0).
t≤T

This is a fixed-viscosity estimate.
The linear colour mild equation has the same integrable heat-gradient kernel. If Y = L1 ∩ Lr ,
two colour paths satisfy on a short restarted interval
C
df (t) ≤ df (a) + √
ν

Z t
a

(t − s)−1/2 V df (s) + Mf ∥u1 (s) − u2 (s)∥∞ ds,


10

where V bounds the drifts and Mf bounds the second colour in Y . Iteration and the total-path
estimate prove the asserted colour continuity and well-posedness. The convex scalar calculation
justifies all its contractions and comparison statements by approximation.
Under the additional finite-energy assumption, mollify u0 and its vorticity together. Smooth
finite-energy solutions satisfy the velocity energy identity by divergence freedom and integration
by parts. Their uniform L2 bound and dissipation pass to the mild solution by weak lower
semicontinuity, using the already established initial-data continuity. This proves the stated energy
inequality.
For a countable decomposition ω0 = l fl,0 with l ∥fl,0 ∥Xp < ∞, define every colour by the
linear equation with the actual common velocity uν . Contraction gives absolute convergence of
P
T ]; Xp ). Its sum solves that same linear equation with datum ω0 , so linear uniquel fl in C([0,
P
ness gives l fl = ω ν . These are the actual colours used in the interaction estimates. Positive
envelopes are defined by the same linear equation with nonnegative initial data; comparison gives
|fl | ≤ Pl when Pl (0) = |fl,0 |.
P

2.5

P

Scalar comparison

Lemma 2.5 (Scalar and positive-majorant comparison). Let v be smooth and divergence free,
with at most linear growth: |v(t, x)| ≤ C(t)(1 + |x|), where C is integrable on the time interval.
A smooth solution of
∂t z + div(vz) = κ∆z + F,
κ ≥ 0,
r
1 r
satisfies, whenever z ∈ L∞
t L and F ∈ Lt L ,

∥z(t)∥r ≤ ∥z(a)∥r +

Z t

∥F (s)∥r ds,

1 ≤ r ≤ ∞.

a

These estimates and positivity extend to the standard mild viscous solutions by approximation. If
P solves the same scalar equation with datum |z(a)| and source |F |, then |z| ≤ P . Consequently,
for X = Xp , an error equation with source −(K ∗ z) · ∇W∗ + R satisfies
∥z(t)∥X ≤ ∥z(a)∥X +

Z t



Cp ∥∇W∗ (s)∥X ∥z(s)∥X + ∥R(s)∥X ds.

a

For a finite nonnegative system, suppose
xi (t) ≤ bi eλt +

Z t
0



X
eλ(t−s)  kij (s)xj (s) + Fi (s) ds,
j

with bi , Fi , kij ≥ 0, locally integrable coefficients, and continuous xi ≥ 0. If Y ′ = (λI + K)Y + F ,
P
Y (0) = b, then x ≤ Y . For fixed weights ρi > 0, put r(s) = maxi j kij (s)ρj /ρi . Then
max
i

Rt
xi (t)
bi
≤ eλt+ 0 r max +
i ρi
ρi

Z t
0

Rt

eλ(t−s)+ s r max
i

Fi (s)
ds.
ρi

Proof. Convex regularization gives the local inequality ∂t |z| + div(v|z|)R ≤ κ∆|z| + |F |. On a
cutoff annulus of radius R ≥ 1, the drift flux for |z|r is at most Cr C(t) |x|>R |z(t, x)|r . Its time
integral tends to zero by dominated convergence. Thus expanding cutoffs give the global L1
estimate and positive envelope comparison even with this linear growth. Regularized powers and
11

Hölder’s inequality give the finite-r estimates. For r = ∞, use the maximum principle with a
barrier ϵA(t)(1 + |x|2 ), choosing the positive factor A to dominate the integrable drift growth and
diffusion, and then send ϵ ↓ 0. Sum the estimates at r = 1, p and use Lemma 2.6 for the errorequation conclusion. For the nonnegative system, iterate its Volterra inequality. The remainder
after n iterations tends to zero on bounded time intervals by the factorial bound for an iterated
integrable kernel. The resulting positive series is the solution Y . Its weighted maximum obeys the
scalar inequality with coefficient λ + r(s); the integrating-factor formula gives the last bound.
For completeness, a divergence-free drift with the same at-most-linear growth gives its stan2
dard viscous scalar evolution the L1 → L∞ bound
p C/(κ(t − a)), κ > 0. The L energy identity
1
2
and Nash’s inequality give L → L norm C/ κ(t − a). The backward adjoint has the same
estimate because its drift is also divergence free. Duality gives L2 → L∞ ; composition at the
midpoint proves the claim. Initial-data approximation justifies it for the mild colour paths used
here.

2.6

Physical and local colour normalization

Let ωjν (tphys , x) denote a physical colour transported and diffused by the common total velocity
uν . At vorticity amplitude Aj > 0 and length Rj > 0, define
sj = Aj tphys ,

y = x/Rj ,

ν
wj (sj , y) = A−1
j ωj (sj /Aj , Rj y),

Uj (sj , y) = (Aj Rj )−1 uν (sj /Aj , Rj y),

κj =

ν
.
Aj Rj2

The normalized colour satisfies
∂sj wj + divy (Uj wj ) = κj ∆y wj .
The self-induced part of Uj is K ∗ wj ; its other parts are all foreign colours expressed in this same
(l)
receiving frame. If wl is colour l in its own frame, its expression in frame j is
Al (l)
(j)
w
wl (sj , y) =
Aj l

!

Al
Rj
sj ,
y .
Aj
Rl

Thus its own clock is sl = (Al /Aj )sj . Translated clocks sj = Aj (tphys − t∗ ) have the same
diffusivity. At corresponding times the norm, velocity, and moment scalings are
2/r

∥ωjν ∥r = Aj Rj ∥wj ∥r ,

∥K ∗ ωjν ∥∞ = Aj Rj ∥K ∗ wj ∥∞ ,

Bm (ωjν ) = Aj Rjm+2 Bm (wj ).

The last equality is asserted only when the absolute moment is finite. Statements in a local
frame use these normalized quantities; field bounds in physical variables include their displayed
amplitude and length factors.
Lemma 2.6 (The velocity estimates used below). Let p > 2, α = 1 − 2/p, and w ∈ L1 (R2 ) ∩
Lp (R2 ). Then
∥K ∗ w∥∞ + ∥K ∗ w∥p ≤ Cp (∥w∥1 + ∥w∥p ),

[K ∗ w]C α ≤ Cp ∥w∥p ,

If w has compact support and zero integral, then K ∗ w ∈ L2 .
12

∥∇K ∗ w∥p ≤ Cp ∥w∥p .

′

Proof. For the supremum norm split K at radius one: its near part is in Lp since p′ < 2, and
its far part is bounded. For the Lp norm the near part is in L1 and the far part is in Lp ; apply
Young’s inequality. For a velocity difference at points a distance d apart, the two radius-2d disks
contribute at most Cp ∥w∥p dα . On their complement use |K(x − z) − K(y − z)| ≤ Cd|x − z|−2
and Hölder’s inequality; the same power dα follows.
We recall a direct proof of the derivative bound to specify the analytic input. Each component
T of ∇K∗ has a Fourier multiplier bounded in absolute value, and hence is bounded on L2 by
Plancherel. Its off-origin kernel satisfies |t(x)| ≤ C|x|−2 , |∇t(x)| ≤ C|x|−3 and has zero angular
average; a possible multiple of the identity is harmless. For f ∈ L1 ∩ L2 decompose at height λ
P
into g + bQ using maximal dyadic squares with average of |f | larger than λ. The squares are
disjoint, and
∥g∥∞ ≤ 4λ,

∥g∥1 ≤ ∥f ∥1 ,

X

|Q| ≤ ∥f ∥1 /λ,

Z

bQ = 0,

X

∥bQ ∥1 ≤ 2∥f ∥1 .

The L2 bound controls the level set of T g by C∥f ∥1 /λ. Cancellation and the kernel derivative
estimate give
Z
R2 \5Q

|T bQ (x)| dx ≤

Z

|bQ (y)|

Z
R2 \5Q

Q

|t(x − y) − t(x − cQ )| dx dy ≤ C∥bQ ∥1 .

∞
The last inner integral is bounded by C|y − cQ | Cℓ(Q)
r−3 r dr. This proves weak type (1, 1),
first for finite sums and then by their L2 convergence. Splitting f above and below a height
proportional to λ consequently gives

R

"
−1

|{|T f | > λ}| ≤ C λ

Z

|f | + λ

|f |>cλ

−2

Z

#
2

|f |

.

|f |≤cλ

Integration against sλs−1 dλ, with 1 < s < 2, and Fubini bound the Ls norm: the two integrals
produce finite factors (s − 1)−1 and (2 − s)−1 . The adjoint has the same bounds, so duality
with s = p′ proves the required Lp estimate. Density identifies the result with the distributional
derivative of K ∗ w.
Finally, compact support
gives a finite first absolute moment. At low frequencies zero circulaR
b
tion implies |w(ξ)|
≤ |ξ| |x||w(x)| dx. At high frequencies use w ∈ L2 , which follows by interpolation between L1 and Lp . The Fourier multiplier iξ ⊥ /|ξ|2 and Plancherel prove K ∗ w ∈ L2 .

2.7

A fixed passive colour is determined by the total Euler path

The next result allows signed smooth colours as well as indicator colours. It therefore does not
require a patch reference profile.
Proposition 2.7 (Recovery of fixed passive colours). Let u be bounded and divergence free on
1,p
[0, T ] × R2 , with u ∈ L1 ([0, T ]; Wloc
) for some p > 1. For each compactly supported f0 ∈ L1 ∩ L∞
there is at most one distributional solution of
∂t f + div(uf ) = 0,

f (0) = f0 ,

in L∞ ([0, T ]; L1 ∩ L∞ ) with its weak initial trace. Every such solution is renormalized and has a
representative in C([0, T ]; L1 ∩ L2 ); its L1 and L2 norms are conserved.

13

Suppose that the bounded measurable drifts un are divergence free, un → u uniformly on
[0, T ] × R2 , supn ∥un ∥∞ ≤ V , and νn → 0, with 0 ≤ νn ≤ 1. Let fn ∈ C([0, T ]; L1 ∩ L2 ) ∩
L∞ ([0, T ]; L∞ ) solve
∂t fn + div(un fn ) = νn ∆fn ,
fn (0) = f0 ,
in the weak sense, with the L1 , L2 , and L∞ contraction bounds at every time. Then the transport
limit f is unique and
fn −→ f in C([0, T ]; L1 ∩ L2 ).
This includes transported colours with νn = 0 under the same continuity, weak-equation, and
contraction hypotheses.
Proof. We first prove renormalization, including the prescribed trace. The transport equation
and the uniform L2 bound give a weakly continuous L2 representative: compact smooth pairings
are absolutely continuous in time, and density extends their continuity to all L2 tests. Its values
at every time retain the L1 and L∞ bounds, by approaching that time through bounded slices,
taking weak limits on finite balls, and then enlarging the balls.
Renormalization at the prescribed trace. Extend the solution to −1 < t < 0 by fe(t) = f0
e(t) = 0. The weak initial-trace identity gives
and the drift by u
efe) = 0 on (−1, T ) × R2 .
∂t fe + div(u

Thus there is no distributional jump at zero, and the extended drift has the same boundedness,
divergence, and local Sobolev properties. For a spatial mollifier, write feε = fe ∗ ηε . Its equation is
e · ∇feε = rε ,
∂t feε + u

Z

rε (t, x) =

e(t, x) − u
e(t, x − y)]fe(t, x − y) dy.
∇ηε (y) · [u

On a compact spatial set the difference-quotient estimate bounds its L1 norm by
e(t)∥p ∥fe(t)∥p′
C∥∇u

on a slightly larger set. To see this, express the velocity difference as the integral of its gradient along the segment and apply Hölder’s inequality. Approximate the drift by divergence-free
1,p
spatial convolutions in L1t Wloc
. For a smooth approximating drift, divergence freedom allows
e
subtraction of f (t, x) inside the commutator integral. Translation continuity of fe then gives convergence to zero in local L1t,x . The difference-quotient bound controls the drift-approximation
error. Consequently rε → 0 locally in L1 .
For β ∈ C 1 with bounded derivative, the chain rule for the mollified equation and the commutator convergence give

eβ(fe) = 0.
∂t β(fe) + div u
This identity has the prescribed renormalized trace β(f0 ) at zero: its compact smooth pairings
have continuous representatives, and on negative times they equal their pairings with β(f0 ).
There are also renormalized traces at every prescribed time. Indeed, for each fixed ε the mollified
equation is absolutely continuous in time in local spatial norms. Integrate its renormalized
equation between any fixed s, t ∈ [0, T ]. At both endpoints spatial approximate identities give
fε (s) → f (s) and fε (t) → f (t) strongly in L2 on finite balls; the Lipschitz function β passes those
endpoint terms in local L1 . The commutator error tends to zero. This proves the renormalized
identity between every such pair of times, including the endpoints.
14

For β(0) = 0, expanding spatial cutoffs yield
Z

Z

β(f (t)) =

(0 ≤ t ≤ T ),

β(f0 )

because the flux error is at most C∥u∥∞ R−1 ∥β ′ ∥∞ supt ∥f (t)∥1 times the length of the time
interval. Choose a bounded-derivative
extension of s2 from the bounded range of f to obtain
√
conservation of ∥f (t)∥2 . Using s2 + ϵ2 −ϵ and sending ϵ ↓ 0 gives conservation of ∥f (t)∥1 . Apply
the same square argument to the difference of two solutions with zero weak initial trace to prove
uniqueness. Weak L2 continuity and conservation of the square norm give strong L2 continuity
at every time.
Renormalization by absolute value also gives finite propagation. If supp f0 ⊂ BR0 and ∥u∥∞ ≤
V , test the equation for |f | against a smooth nondecreasing function ψ(|x| − R0 − V t) which
vanishes for nonpositive arguments. Its material derivative is nonpositive. Expanding spatial
1
cutoffs justify this bounded exterior test using |f | ∈ L∞
t L . The initial pairing is zero, so the
nonnegative pairing remains zero. Letting the transition width of ψ tend to zero proves
supp f (t) ⊂ BR0 +V t

(0 ≤ t ≤ T ).

Local L1 continuity follows from strong L2 continuity, and this common support ball gives continuity in global L1 .
For the approximating sequence, the weak equation gives for compact smooth φ
d
dt

Z



φfn ≤ ∥f0 ∥1 V ∥∇φ∥∞ + ∥∆φ∥∞ .

The contraction bounds, a countable dense set of L2 tests, and this equicontinuity give weak L2
convergence uniformly in time along a subsequence. The limit has the L1 and L∞ bounds by
local lower semicontinuity. Uniform convergence of the drifts and νn → 0 pass the weak equation
to ∂t f + div(uf ) = 0 with initial trace f0 . Uniqueness identifies every subsequential limit.
Suppose uniform strong L2 convergence fails, and select bad times tn → t. Uniform weak
convergence gives fn (tn ) ⇀ f (t) in L2 , while contraction and renormalization imply
lim sup ∥fn (tn )∥2 ≤ ∥f0 ∥2 = ∥f (t)∥2 .
n

Uniform convexity of L2 gives strong convergence to f (t), and strong time continuity of f contradicts the choice of the times. Thus convergence is uniform strongly in L2 .
For the L1 upgrade set m = ∥f0 ∥1 and choose a fixed ball BR containing the support of f (t)
for all t ≤ T . Conservation of the limiting L1 norm and contraction of the approximating norms
give
Z
Z
Z
c
BR

|fn (t)| ≤ m −

|fn (t)| ≤

BR

|fn (t) − f (t)|.

BR

Consequently
sup ∥fn (t) − f (t)∥1 ≤ 2|BR |1/2 sup ∥fn (t) − f (t)∥2 −→ 0.
t≤T

t≤T

The assumed strong continuity of the approximating paths makes these uniform estimates convergence in the asserted Ct spaces.

15

For p > 2, Lemma 2.6 gives the useful bound
∥K ∗ h∥∞ ≤ Cp (∥h∥1 + ∥h∥p ) :

(3)

strong convergence of total vorticity paths in Ct (L1 ∩Lp ) therefore implies uniform convergence of
their velocities. Whenever the limiting velocities have the Sobolev regularity in Proposition 2.7,
that proposition applies to every fixed bounded compactly supported initial colour. Equal total
Euler paths then have equal recovered fixed colours.
Fix finitely many bounded compactly supported initial colours f1,0 , . . . , fk,0 , complex coefficients cl , and complex tests ψl ∈ Cb2 (R2 ), where the function and its derivatives through order
two are bounded. These choices define the one fixed functional
au (t) =

k
X

Z

cl

ψl (x)fl [u](t, x) dx.

(4)

l=1

Expanding spatial cutoffs in the weak equation justify these bounded noncompact tests: the new
drift and diffusion errors are bounded by C∥f0 ∥1 (V /R + ν/R + ν/R2 ) and vanish. In particular
d
dt

Z



ψl fl ≤ ∥fl,0 ∥1 V ∥∇ψl ∥∞ + ν∥∆ψl ∥∞ .

Thus the observable is uniformly Lipschitz for bounded drifts and 0 ≤ ν ≤ 1, and strong Ct L1
convergence makes its pairings converge uniformly. Proposition 2.7 therefore makes it continuous
under the strong total-path convergence specified above. For degree shooting we use tests with
the covariance
ψl (Rθ/4 x) = e−iθ ψl (x),
where R is the planar rotation matrix. This includes a fixed bounded adjoint-coordinate test and
the compact moment tests ψl (x) = Φl (x) = χl (|x|)z̄ 4 with fixed smooth radial compact cutoffs.
These compact tests cover one smooth outer colour or several patch colours. To recover uncut
fourth moments in an Euler limit, choose their cutoffs equal to one on the common support ball
supplied by finite propagation.

2.8

Protected first caps and phase shooting

We use the first crossing of one fixed observable. Its continuity is controlled by a strict early
margin and a positive radial slope in a fixed neighbourhood of that crossing.
Lemma 2.8 (Continuity of a protected first-cap phase). Fix T, δ, c, e, h, L > 0, with 2e < T . Let
a : [0, T ] → C be L-Lipschitz, and define
τ (a) = inf{t ∈ [0, T ] : |a(t)| ≥ δ},

P(a) = a(τ (a))/δ,

where the infimum of the empty set is +∞ and P is defined when the cap exists. Suppose
|a(0)| ≤ δ − h,

e ≤ τ (a) ≤ T − e,

|a(t)| ≤ δ − h

(0 ≤ t ≤ τ (a) − e),

|a(t)| − |a(s)| ≥ c(t − s)

(τ (a) − e ≤ s < t ≤ τ (a) + e).

For any continuous b : [0, T ] → C with ε = ∥a − b∥∞ < min(h/2, ce/4), its first cap exists and
|τ (a) − τ (b)| ≤ 2ε/c,

|P(a) − P(b)| ≤ δ −1 (1 + 2L/c)ε.
16

Families satisfying the displayed conditions with common constants retain those conditions under
uniform convergence. Hence P is continuous on this protected class. The lower-slope assertion
is confined to the displayed interval around the first crossing.
Proof. Continuity and the strict initial margin give |a(τ (a))| = δ. If ε = 0 the conclusions
are immediate. Suppose ε > 0. Before τ (a) − e the strict margin excludes a cap of b. On
[τ (a) − e, τ (a)], the radial lower-slope inequality gives
|a(t)| ≤ δ − c(τ (a) − t).
Thus b cannot cap before τ (a) − 2ε/c. At τ (a) + 2ε/c its modulus is at least δ + ε, so it has
capped by that time. Both comparison times lie inside the protected interval and inside [0, T ].
This proves the time estimate. Both first-cap values have modulus δ, and
|a(τ (a)) − b(τ (b))| ≤ L|τ (a) − τ (b)| + ∥a − b∥∞
proves the phase estimate.
For a uniformly convergent protected sequence, extract a convergent subsequence of cap times.
The common Lipschitz bound, early margin, and integrated radial lower-slope inequality pass to
the limit; for interval endpoints first pass on strict interior intervals and then use continuity. The
limiting time has modulus δ, every earlier time has strictly smaller modulus, and it is the first cap.
The limiting path satisfies the same protection conditions. This applies to every subsequence,
giving the closure and continuity assertions.
Corollary 2.9 (Separation of total inviscid paths). Consider one fixed total initial datum and
fixed bounded compactly supported initial colours. On both subsequences use the same initial
colours fl,0 , coefficients cl , tests ψl , and cap level δ, so that (4) defines one common functional.
Suppose two Navier–Stokes subsequences converge strongly in Ct (L1 ∩ Lp ), p > 2, their limits
satisfy the velocity regularity of Proposition 2.7, and the approximating observables have first
caps protected by common constants on [0, T ]. If the two limiting first-cap phases of this common
functional differ, the total Euler paths differ. The cap times may differ.
Proof. Proposition 2.7 and Lemma 2.8 pass the fixed colours, observable paths, and first-cap
phases to the limit. Equal total paths would have equal velocities and recovered colours, hence
equal observable paths and first-cap phases. This contradicts the assumed phase separation.
Proposition 2.10 (Conditional degree shooting). Fix ν > 0 and an active index N ≥ 2. Suppose
a continuous coloured initial-data family is indexed by (r, θ) ∈ [0, 1] × S1 : the active seed at N has
phase θ, and all other seed amplitudes are multiplied by r. Assume the total and passive-colour
solution paths at this viscosity depend continuously on these parameters. Use fixed coefficients,
tests ψl ∈ Cb2 satisfying ψl (Rθ/4 x) = e−iθ ψl (x), and cap level in (4), and assume the observable
paths have first caps protected by common constants from Lemma 2.8.
At r = 0, assume the full coloured initial datum at phase θ is the spatial rotation through
angle −θ/4 of the datum at phase zero. This covariance holds when only the active seed rotates
and all background profiles and other initial colours are radial. Then the output first-cap phase
map at r = 1 is surjective onto S1 .
Proof. The first-cap phase map is continuous by Lemma 2.8. Rotational covariance of the full
equation and the fixed tests give
a0,θ (t) = eiθ a0,0 (t),

P(a0,θ ) = eiθ P(a0,0 ).
17

Its degree is one. To see directly that degree is invariant along the homotopy, lift its continuous
argument on [0, 1] × [0, 2π]. The endpoint difference is 2πd(r) for an integer d(r); continuity
makes this integer constant. A circle map omitting a point has a periodic continuous argument
and degree zero. The map at r = 1 therefore cannot omit a point.
Fixed-positive-viscosity continuity follows from the mild equations. In X = L1 ∩ Lp , p > 2,
(3) makes (K ∗ w)w locally Lipschitz on bounded X balls. If two total paths are bounded there
by M , their difference satisfies
Cp M
d(t) ≤ d(0) + √
ν

Z t

(t − s)−1/2 d(s) ds.

0

Take short intervals on which the integral coefficient is at most 1/2 and iterate to obtain continuity
for each fixed ν. The passive mild equation has the same heat-gradient kernel; its additional source
is bounded by the uniform drift difference times the bounded colour norm. It gives continuity of
colours in Ct (L1 ∩ L2 ) when their initial data vary in L1 ∩ L2 . For indicator colours, continuity in
symmetric-difference measure of the initial domains supplies precisely this initial-data continuity.
Uniform cap protection across the homotopy remains a substantive coupled PDE hypothesis.
Proposition 2.11 (Conditional sparse-phase construction). Fix p > 2 and βq 2/p < 1. Let a
family of infinitely seeded, zero-circulation compact initial data be indexed by a phase at every
depth. Assume its defining series has a summable bound in X = L1 ∩ Lp uniform in all phases,
that changing only the phase at depth N changes the datum by at most ηN → 0 uniformly in
the other phases, and that depths N ≥ 2 leave the fixed outer initial colours unchanged. Fix the
coefficients, tests, and cap level of one outer observable (4) throughout the family.
Suppose νN ↓ 0, and at every depth N ≥ 2, for every choice of the other phases, Proposition 2.10 applies at νN with common cap-protection constants on [0, T ]. Suppose also that for
each datum in the family its Navier–Stokes paths are relatively compact in C([0, T ]; X) as ν ↓ 0,
and every limit satisfies the velocity hypothesis of Proposition 2.7. Then one single datum in this
family has two distinct vanishing-viscosity accumulation paths on [0, T ].
Proof. Start with default phases at all depths. At step k, choose a new depth Nk > Nk−1 , Nk ≥ 2,
with νNk < νNk−1 /2 when k > 1, and
ηNk < 2−k min(1, r1 , . . . , rk−1 ),
where at k = 1 the minimum is 1. Shoot its phase to give first-cap output (−1)k at νNk , and call
the resulting datum gk . Fixed-positive-viscosity continuity gives rk > 0 such that an admissible
datum within rk of gk changes that output by less than 1/4. All later admissible phase choices
have protected caps by the common hypothesis.
The summable changes give a limit g∞ ∈ X. Each coordinate is changed at most once, and
uniform absolute convergence of the defining series identifies the limit with the resulting phase
sequence in the assumed family. For every k,
∥g∞ − gk ∥X ≤

X

2−l rk < rk .

l>k

Thus its selected-viscosity output lies within 1/4 of (−1)k . Relative compactness supplies a
limiting path along each parity. Colour convergence and protected-cap continuity put their firstcap phases in the disjoint closed 1/4-neighbourhoods of +1 and −1. Corollary 2.9 makes the
18

total paths different. Circulation passes by strong L1 convergence. Compact support follows
from membership of g∞ in the assumed compact-data phase family; a common support ball, as
in the scaled stack construction, also gives it directly.
The proposition is conditional on uniform cap protection and phase shooting for the actual coupled system. Every intermediate datum retains the complete infinite background, and
compactness alone supplies no cap or phase-shooting estimate.

2.9

Compactness for one fixed compact zero-circulation datum

We prove the strong compactness used in phase separation for one fixed initial datum. The
Navier–Stokes inputs are its global solution, strongly continuous vorticity paths, the L1 and Lp
contraction bounds, and the finite-energy inequality. They follow from smooth approximation of
the datum and do not require uniqueness of its Euler evolution.
Theorem 2.12 (Strong vanishing-viscosity precompactness). Let 2 < p < ∞ and let one fixed
real initial vorticity satisfy
ω0 ∈ L1 ∩ Lp ,

supp ω0 ⊂ BR0 ,

Z

ω0 = 0.

For every sequence νn ↓ 0, the unforced planar Navier–Stokes vorticities with this datum have a
subsequence converging strongly in C([0, T ]; L1 ∩Lp ) for each fixed T . Every limit is a renormalized
Euler solution with initial vorticity ω0 . Its velocity is bounded and belongs to L∞ ([0, T ]; W 1,p ).
At every time its vorticity preserves its L1 and Lp norms and circulation, and
supp ω(t) ⊂ BR0 +V t ,

V = Cp (∥ω0 ∥1 + ∥ω0 ∥p ).

In particular the compactness and Sobolev-velocity hypotheses for fixed-colour first-cap separation
hold for this class.
Proof. Set m = ∥ω0 ∥1 , P = ∥ω0 ∥p , and E = ∥K ∗ ω0 ∥2 , finite by Lemma 2.6. Discarding finitely
many indices ensures 0 < νn ≤ 1. Vorticity contraction and the kinetic-energy inequality give
∥ωn (t)∥1 ≤ m,

∥ωn (t)∥p ≤ P,

∥un (t)∥2 ≤ E,

∥un (t)∥∞ ≤ V,

V = Cp (m + P ).

(5)

Lemma 2.6 also bounds [un (t)]C α , ∥un (t)∥p , and ∥∇un (t)∥p , where α = 1 − 2/p > 0.
Local velocity compactness. Project the velocity equation onto divergence-free fields. The
projection has a bounded matrix Fourier multiplier and is bounded on every H s . Since ∥un ⊗
un ∥2 ≤ V E, the equation implies
∥∂t un ∥H −2 ≤ CE(V + 1).
For a spatial mollifier ηh , 0 < h ≤ 1, its H 2 norm is at most Ch−3 in two dimensions. Spatial
Hölder continuity and H −2 –H 2 duality therefore give
∥un (t) − un (s)∥∞ ≤ Chα + Ch−3 |t − s|.
Take h = |t − s|1/(3+α) for |t − s| ≤ 1. Arzelà–Ascoli on compact cylinders and diagonal extraction
give
un −→ u locally uniformly on [0, T ] × R2 .
19

The limiting field is bounded and divergence free. Weak passage of the global velocity and
1,p .
gradient bounds yields u ∈ L∞
t W
Weak vorticity compactness. For compact smooth φ, the scalar equation gives
d
dt

Z



φωn ≤ m V ∥∇φ∥∞ + ∥∆φ∥∞ .
′

A countable dense set of compact smooth tests in Lp , the uniform Lp bound, and Arzelà–Ascoli
produce a weakly continuous Lp path ω with
Z

sup

′

for every φ ∈ Lp .

φ(ωn (t) − ω(t)) −→ 0

(6)

t≤T

It has ω(0) = ω0 . At every time its L1 norm is at most m, by lower semicontinuity on finite
balls and then enlargement. Local uniform convergence of un and weak convergence of ωn pass
the drift product in compact smooth tests. The diffusion term is bounded by νn P ∥∆φ∥p′ and
vanishes. Thus
∂t ω + div(uω) = 0,
ω(0) = ω0 .
We will identify u = K ∗ ω after obtaining strong convergence.
Renormalization and endpoint traces. Spatial mollification gives
∂t ωε + u · ∇ωε = rε ,
where divergence freedom allows the expression
Z

rε (t, x) =

∇ηε (y) · [u(t, x) − u(t, x − y)][ω(t, x − y) − ω(t, x)] dy.

Sobolev difference quotients and Hölder’s inequality yield
∥rε (t)∥p/2 ≤ C∥∇u(t)∥p sup ∥ω(t, · − y) − ω(t)∥p .
|y|≤ε
p/2

Translation continuity and dominated convergence imply rε → 0 in L1t Lx , hence locally in
space-time L1 because p/2 > 1. For each C 1 function β with bounded derivative the chain rule
for the mollified equation therefore passes to


∂t β(ω) + div uβ(ω) = 0.
This equation has the renormalized trace β(ω(t)) at every prescribed time t, including zero.
Indeed, the spatially mollified weakly continuous path is absolutely continuous in local spatial
norms. Integrate its renormalized equation between any fixed s, t. At each endpoint the spatial
approximate-identity property gives ωε (s) → ω(s) and ωε (t) → ω(t) strongly in Lp . The Lipschitz
function β passes these endpoint terms in local L1 , and the commutator error vanishes. This
proves the renormalized identity with the indicated traces between every pair of times.
For β(0) = 0, use cutoffs equal to one on BR and vanishing outside B2R . Their flux error is
at most CT V m∥β ′ ∥∞ /R. Sending R → ∞ gives
Z

Z

β(ω(t)) =

β(ω0 )

20

(0 ≤ t ≤ T ).

To recover the Lp norm use the convex tangent truncations
(

βM (s) =

|s|p ,
|s| ≤ M,
p−1
p
pM
|s| − (p − 1)M , |s| > M.

They are C 1 with bounded derivative, nonnegative, and increasing to |s|p : on |s| > M their
derivative in M is p(p − 1)M p−2 (|s| − M ) ≥ 0. Monotone convergence proves
∥ω(t)∥p = ∥ω0 ∥p

(0 ≤ t ≤ T ).

(7)

√
The functions s2 + ϵ2 − ϵ followed by ϵ ↓ 0 give conservation of ∥ω(t)∥1 . Taking β(s) = s gives
conservation of circulation.
The renormalized equation for |ω| gives finite propagation exactly by an exterior test ψ(|x| −
R0 −V t) with nonnegative derivative and zero initial pairing. Its material derivative is nonpositive,
and its bounded exterior test is justified by expanding cutoffs and the global L1 bound. Sending
the transition width to zero yields supp ω(t) ⊂ BR0 +V t at every time.
Uniform strong convergence. For p ≥ 2, Clarkson’s inequality reads
f +g p
f − g p ∥f ∥pp + ∥g∥pp
+
≤
.
2
2
2
p
p
Weak time continuity and the constant norm (7) imply ω ∈ Ct Lp by this inequality and lower
semicontinuity. If uniform strong Lp convergence failed, choose bad times tn → t. Equation (6)
gives ωn (tn ) ⇀ ω(t), while contraction gives
lim sup ∥ωn (tn )∥p ≤ P = ∥ω(t)∥p .
n

The same inequality gives strong convergence to ω(t); continuity of ω contradicts the bad times.
Thus convergence is strong in Ct Lp .
Choose R > R0 + V T . The conserved limiting L1 norm and the approximating contraction
imply the tail estimate
Z
c
BR

|ωn (t)| ≤ m −

Z

|ωn (t)| ≤

BR

Z

|ωn (t) − ω(t)|.

(8)

BR

Consequently
sup ∥ωn (t) − ω(t)∥1 ≤ 2|BR |1−1/p sup ∥ωn (t) − ω(t)∥p −→ 0.
t≤T

t≤T

The common support ball also makes the limiting path continuous in L1 . This proves strong
convergence in Ct (L1 ∩ Lp ).
Finally Lemma 2.6 gives uniform global convergence un = K ∗ ωn → K ∗ ω. The previously
obtained local limit is therefore u = K ∗ω. The transport equation is the Euler vorticity equation,
with all the stated regularity and conservation properties. For one subsequence on all finite
windows, perform the extractions on T = 1, 2, . . . and take a diagonal subsequence.
Corollary 2.13. For the compact zero-circulation initial-data class of Theorem 2.12, the relative
compactness as ν ↓ 0 and Sobolev-velocity assumptions in Proposition 2.11 hold automatically.
The remaining substantive hypotheses are uniform protected caps and phase shooting for the actual
coupled system.
21

3

The fixed profile, datum and isolated activation

Lemma 3.1 (Admissibility of a fixed infinite smooth stack). Let W be a fixed real smooth compactly supported radial density with zero integral, and let Φθ be a bounded smooth family with
common compact support, zero integral, and B4 (Φθ ) = 0. Suppose Φθ is fourfold symmetric. Fix
α > 0, Aj = A1 β j−1 , Rj = R1 q j−1 , and A1 , R1 > 0, β > 1, 0 < q < 1, and βq 2/p < 1, where
2 < p < ∞. For any phases θj , the single datum
ω0 (x) =

X

Aj [W (x/Rj ) + αΦθj (x/Rj )]

j≥1

converges absolutely in L1 ∩Lp , has compact support and zero circulation, and satisfies B4 (ω0 ) = 0.
2 + R2/p ), which tends
Changing only phase θN changes its L1 ∩ Lp norm by at most CαAN (RN
N
to zero.
2/r

Proof. For either summand its Lr norm is at most CAj Rj , uniformly in phase, for r = 1, p.
The geometric ratio is less than one for both norms. All supports are in a common ball. Integral
and phase claims follow by absolute convergence. The absolute fourth moment of each summand
is bounded by CAj Rj6 ; this series converges since βq 6 < 1. Radial symmetry gives B4 (W ) = 0,
so termwise integration proves the whole-moment assertion.

3.1

An isolated nonlinear active stage for a fixed smooth profile

This section proves the nonlinear activation estimate for one isolated annulus with a fixed smooth
radial profile. The smoothing length is independent of viscosity and of the small parameter below.
Theorem 3.2 (Isolated smooth activation). There exist a real, radial, compactly supported C ∞
vorticity Wϵ , with zero circulation, a number λϵ = σϵ + iτϵ with σϵ > 0, and a nonzero complex
number bϵ , with the following property. For every 2 < p < ∞ and K∗ > 0 there is q0 > 0 such
that the conclusions below hold for 0 < q < q0 . Put
Lq = log(1/q),

αq = q K∗ ,

κq = q 3/2 ,

δq = L−4
q ,

Hq =

1
δq
log .
σϵ
αq

There is a 2π-periodic family of smooth real seeds Φϑ , supported in two fixed annular collars, such
that
Z
Φϑ = 0,
B4 (Φϑ ) = 0,
Φϑ (ix) = Φϑ (x).
Let ω q,ϑ be the unforced Navier–Stokes solution with viscosity κq and initial vorticity Wϵ + αq Φϑ .
Then, uniformly in ϑ ∈ R,
sup ∥ω q,ϑ (t) − eκq t∆ Wϵ ∥L1 ∩Lp ≤ Cδq ,

(9)

∥K ∗ (ω q,ϑ (t) − eκq t∆ Wϵ )∥∞ dt ≤ Cδq ,

(10)

0≤t≤Hq

Z Hq
0

B4 (ω q,ϑ (Hq )) = δq ei(ϑ+τϵ Hq ) bϵ + o(δq ).

(11)

Here ∥f ∥L1 ∩Lp = ∥f ∥1 + ∥f ∥p ; all fourth absolute moments in (11) are finite. In particular,
(11) is a statement about the whole moment, rather than only a compactly tested moment. The
constants depend on the fixed profile, p and K∗ , but not on q or ϑ.
p

The logarithmic cap satisfies Hq δq → 0. In the fixed-data construction, q and hence this
positive cap are subsequently fixed.
22

3.1.1

The fixed profile and its positive angular spectrum

Fix ρ2 = 3/5 and choose a nonnegative, radial, radially decreasing χ ∈ Cc∞ (B1 ) with integral
one. Define
χϵ (x) = ϵ−2 χ(x/ϵ),
Wϵ = χϵ ∗ (−1Bρ + ρ2 1B1 ).
The convolution is smooth, radial, supported in B1+ϵ and has zero integral. It is constant on
Bρ−ϵ . Choose ϵ∗ < (1 − ρ)/4 and subsequently 0 < ϵ < ϵ∗ . The limits below refer first to the
mollified family as ϵ ↓ 0. After the spectral and moment limits here and the transfer-coefficient
limits in Lemma 4.8 and the interior-field calculation following it have been checked, choose one
sufficiently small ϵ satisfying all their nonvanishing and spectral conclusions, and retain that
choice for every later finite expansion and derivative order. Its radial derivative has the form
Wϵ′ (r) = Dρ,ϵ (r)−ρ2 D1,ϵ (r),

Da,ϵ ≥ 0,

supp Da,ϵ ⊂ [a−ϵ, a+ϵ],

Z ∞

Da,ϵ (r) dr = 1. (12)
0

To verify the sign, express the radial decreasing mollifier by its superlevel disks. The area of
intersection of two disks is nonincreasing in their center separation: along each line parallel to their
displacement, the overlap of the two interval sections is nonincreasing. Layer-cake integration
gives radial monotonicity of the disk convolution. Its value is one at the origin and zero at infinity,
giving the displayed derivative mass. The support assertion follows because the convolution is
exactly constant at points whose distance to the disk boundary exceeds ϵ.
Set
Z r
sWϵ (s) ds.
Ωϵ (r) = r−2
0

On the two derivative collars, respectively,
Ωϵ (r) = Ω0 (ri ) + O(ϵ),

(r1 , r2 ) = (ρ, 1),

(Ω0 (r1 ), Ω0 (r2 )) = (−1/5, 0).

(13)

Indeed disk translation changes its L1 norm by at most a constant times the translation length.
Thus ∥Wϵ − W0 ∥1 ≤ Cϵ, where W0 = −1Bρ + ρ2 1B1 . The radial circulation formula, on radii
bounded away from zero, gives |Ωϵ − Ω0 | ≤ Cϵ. The piecewise smooth Ω0 is Lipschitz there.
Direct integration gives the interface values.
For a positive angular mode m ≥ 1 use the convention f (r)eimθ . Put


hm (r, s) =

min(r, s)
max(r, s)

m

,

1
ψf (r) = −
2m

Z ∞

hm (r, s)f (s)s ds.
0

The function ψf eimθ solves ∆ψ = f eimθ , is regular at the origin and decays at infinity. This
follows either by solving the radial second-order equation on each side of r = s, or by checking
the derivative jump 1/s of the displayed Green kernel. Since ur = −imψf /r, the frozen Euler
linearization at Wϵ is
Lm f = −imΩϵ f + imψf Wϵ′ /r.
(14)
For m = 0 the linearization is zero: two radial vorticities have tangential velocities and do not
advect one another. Negative sectors are the complex conjugates of the positive sectors. From this
point onward all angular sectors belong to 4Z. No spectral assertion is made for other symmetry
classes.
Lemma 3.3 (The fixed smooth unstable pair). There are ϵ0 > 0 and η > 0 such that, for every
0 < ϵ < ϵ0 , 0 < η < σϵ /4 and the following modal conclusions hold. For each m ∈ 4N and each
23

integer k ≥ 0, realize (14) as a bounded operator on H k (I), using the Green potential of the zero
extension from a fixed interval I ⋐ (0, ∞) containing both derivative collars strictly in its interior.
The only spectral value of any of these individual operators with real part greater than η is the
algebraically simple m = 4 eigenvalue λϵ = σϵ + iτϵ . Its radial eigenfunction fϵ is smooth and
supported in the two derivative collars. For negative sectors use the conjugate operators; their
only spectral value with real part greater than η is λϵ in sector m = −4. The radial generator is
zero. Moreover
λϵ −→ ρ5 /2 + 3i/10 as ϵ ↓ 0.
For each fixed finite set S ⊂ 4Z and integer k ≥ 0, let Xk,S = m∈S H k (I) and let LS be its
bounded direct-sum generator. Its propagator has the decomposition
L

etLS = eλϵ t P+ + eλϵ t P− + Rk,S (t),

∥Rk,S (t)∥Xk,S →Xk,S ≤ Ck,S e2ηt ,

t ≥ 0.

(15)

The growing terms are included only if their sectors are present. The constants may depend on
ϵ, k, S. Smooth inputs supported in the fixed derivative collars remain supported in the union of
those two collars. No assertion is made about the spectrum or propagator of an infinite angular
direct sum.
Proof. Fix 0 < η < ρ5 /8 before the angular cutoff is chosen. First consider an eigenvalue λ in
an individual sector with ℜλ > 0. The eigenvalue equation forces f = 0 off the derivative collars
and gives
imψf (r)
g(r) =
f (r)r dr = g(r)Wϵ′ (r) dr,
.
λ + imΩϵ (r)
There is no division by Wϵ′ here. In the Hilbert space L2 (|Wϵ′ | dr) the equation for g is
λg = −imΩϵ g −

i
2

Z

hm (r, s)g(s)Wϵ′ (s) ds.

Each same-collar block is skew-adjoint, since the sign of Wϵ′ is constant there and hm (r, s) is real
and symmetric. Angular multiplication is also skew-adjoint. The two opposite-collar blocks have
operator norms at most their Hilbert–Schmidt norms, bounded by
Cam
∗ ,

a∗ = (ρ + ϵ∗ )/(1 − ϵ∗ ) < 1,

because their two weighted masses are 1 and ρ2 and their kernels are bounded by am
∗ . Consequently
ℜλ ≤ Cam
(16)
∗ .
This fixes a finite angular cutoff in any half-plane ℜλ ≥ η > 0, independently of ϵ.
For one of the finitely many positive sectors, eliminate f to obtain the equation on the two
collars
Z
i
W ′ (s)ψ(s)
ψ(r) = −
hm (r, s) ϵ
ds =: Tϵ (λ)ψ(r).
(17)
2
λ + imΩϵ (s)
Work first in the sup-norm space of continuous functions on the two collars. Let P evaluate at
r1 , r2 and let E extend a vector to constants on the two collars, so P E = I. The receiving-radius
derivative of the kernel is bounded there for fixed m. Hence
∥(I − EP )Tϵ (λ)∥ ≤ Cϵ

24

uniformly on compact subsets of ℜλ ≥ η. For ψ = Ep + w, P w = 0, solve the w equation by an
analytic Neumann series. The remaining two-dimensional equation has a determinant converging
uniformly on these compact sets to the determinant of
pi =

i X cj hm (ri , rj )
pj ,
2 j λ + imΩ0 (rj )

(c1 , c2 ) = (−1, ρ2 ).

The sign follows from W0′ = − cj δrj . Using interface variables f (r)r dr =
equation is equivalent to the matrix eigenvalue problem
P

P

ci ri2 qi δri , this

!

Mm q = λq,

i 2m/5 − 1 ρm
Mm =
.
−ρm+2 3/5
2

For m = 4 its roots are ±ρ5 /2 + 3i/10, both simple. For m = 8, 12, . . . they are imaginary, since
the real-matrix discriminant is


2(m − 4)
5

2

− 4ρ2m+2 > 0.

At m = 8 the first term is 64/25 and the second is 4(3/5)9 < 1; thereafter the first increases
and the second decreases. Large |λ| gives no omitted roots: once |λ| > Cm , the operator in (17)
has norm less than one, uniformly in ϵ, because the denominator then has size comparable with
|λ|. Uniform determinant convergence and Rouché’s theorem therefore give one simple root near
the growing m = 4 root and no other roots in ℜλ ≥ η. Combined with (16), this proves the
claimed positive spectrum for all sufficiently small ϵ. By the growing-root limit, decrease ϵ0 so
that 4η < σϵ throughout 0 < ϵ < ϵ0 . The chosen η is unchanged in this step.
We give the operator-domain and multiplicity details needed for the propagator claim. Choose
a closed interval I ⋐ (0, ∞) containing both collars strictly in its interior. On H k (I) define ψf by
the Green integral against the zero extension of f from I. This extension need not be globally H k :
multiplication of the potential by Wϵ′ restricts output strictly inside I. The radial elliptic equation,
or direct differentiation of its Green kernel, gives two local derivatives. Thus the coupling in (14)
is compact on H k (I) and the angular multiplier is bounded there. Its spectrum is imaginary:
off the multiplier’s value range the inverse is multiplication by (λ + imΩϵ )−1 . The compactoperator Fredholm alternative consequently reduces all spectrum in ℜλ > 0 to eigenfunctions.
The equation bootstraps such eigenfunctions to smoothness and forces their support into the
collars, so the roots already found are exactly the positive spectrum on every H k (I).
To check algebraic simplicity independently of a sup-norm determinant, start on X0 = L2 (I).
Write Af = −imΩϵ f , Cf = ψf |collars and Bψ = imWϵ′ ψ/r. Then Lm = A + BC and Tϵ =
C(λ − A)−1 B. Off the imaginary multiplier range, the resolvent identity is
(λ − Lm )−1 = (λ − A)−1 + (λ − A)−1 B(I − Tϵ )−1 C(λ − A)−1 .

(18)

Here C : X0 → C(collars) is bounded by its elliptic gain, and B : C(collars) → X0 is bounded.
The invertible complementary Schur factor and the simple two-by-two determinant zero imply a
rank-one simple pole in (18). The residue is nonzero, since an eigenfunction exists. On Xk =
H k (I) the resolvent equation and its radial elliptic bootstrap give, locally uniformly for λ off the
imaginary multiplier range,
∥v∥H k (I) ≤ Ck (∥f ∥H k (I) + ∥v∥L2 (I) ),
25

(λ − Lm )v = f.

The Xk resolvent is the restriction of the X0 resolvent; the simple pole has no higher-order
singularity by this estimate and has the same nonzero smooth one-dimensional range. This proves
algebraic simplicity on every space used below. The m = −4 conclusion follows by conjugation;
m = 0 is the identity propagator.
After removing the growing Riesz projections, the remaining bounded generator has spectral
bound at most η. Its exponential is bounded by Ce2ηt . For clarity this last fact does not assume a
spectral growth theorem for unbounded generators. For a bounded operator, integrate etz (z−L)−1
on a finite rectangular contour enclosing its remaining spectrum and having right edge ℜz = 2η.
Expansion of the resolvent on a large circle and termwise integration gives the exponential series;
contour deformation gives the rectangle formula. The resolvent is bounded on this fixed contour,
proving the estimate. The finitely many sectors and derivative orders give (15). Coefficients
initially smooth and supported in the collars remain so under this propagator: multiplication
preserves that support and the coupling has smooth output flat at the collar endpoints. This
proves the lemma.
3.1.2

An exactly neutral smooth seed

Normalize the growing radial eigenfunction fϵ so that its first collar mass tends to c1 r12 = −ρ2 .
The finite-dimensional reduction then implies
fϵ (r)r dr ⇀ −ρ2 δρ − iρ3 δ1 .
Indeed its interface vector tends to (1, −iρ), the growing eigenvector of M4 . Define
Z ∞

bϵ = 2π

r5 fϵ (r) dr.

(19)

0

The weak convergence on the two compact collars gives
bϵ −→ 2π(−ρ6 − iρ3 ) ̸= 0.
Consequently bϵ ̸= 0 for every sufficiently small ϵ. This condition is included in the final fixed
choice of the smoothing length. Since L4 is i times a real operator, fϵ is an eigenfunction of L4
with eigenvalue −λϵ = −σϵ + iτϵ . Define the real seed
Φϑ (r, θ) = e

iϑ



bϵ
fϵ (r) − fϵ (r) e4iθ + complex conjugate.
bϵ


(20)

Its circulation is zero and its only initial
sectors are ±4. For the convention B4 = z̄ 4 ω, angular
R 5
4iθ
integration gives B4 (f (r)e ) = 2π r f (r) dr. Therefore B4 (Φϑ ) = 0 exactly, while the growing
coefficient is eiϑ . The stable coefficient has modulus one.
R

3.1.3

The finite seed–viscosity expansion

For the rest of the proof abbreviate the fixed profile to W , and write σ = σϵ , τ = τϵ , Wκ (t) =
eκt∆ W and Uκ = K ∗ Wκ . The latter is an exact radial heat solution. Put X = L1 ∩ Lp with the
sum norm. Splitting the Biot–Savart convolution at radius one gives
∥K ∗ g∥∞ ≤ Cp ∥g∥X .
′

(21)

On the near part use Hölder and |x|−1 ∈ Lp (B1 ), which holds because p′ < 2. On the far part
use its bounded kernel and the L1 norm. This estimate requires no zero-circulation assumption.
26

The frozen full linear operator is Lf = −(K ∗ W ) · ∇f − (K ∗ f ) · ∇W . Its sectors are those
in (14). Choose integers M, N ≥ 1, to be fixed below, and construct
fapp =

M X
N
X

αj κℓ fj,ℓ ,

ωapp = Wκ + fapp .

(22)

j=1 ℓ=0

Set f1,0 (0) = Φϑ and all other initial coefficients zero. With Wr (t) = tr ∆r W/r!, solve the
triangular recursion
(∂t − L)fj,ℓ = 1ℓ≥1 ∆fj,ℓ−1 −

ℓ
X


(K ∗ Wr ) · ∇fj,ℓ−r + (K ∗ fj,ℓ−r ) · ∇Wr



r=1

X

−

(K ∗ fj1 ,ℓ1 ) · ∇fj2 ,ℓ2 .

(23)

j1 +j2 =j, j1 ,j2 ≥1
ℓ1 +ℓ2 =ℓ

Every right-hand side uses earlier coefficients in the order j + ℓ. They are smooth and supported
in the fixed collars: Laplacians preserve the support of smooth flat functions; ∆r W , r ≥ 1, has
that support; and every advective factor differentiates such a supported function. The linear
source is supported where W ′ is supported. Each coefficient has zero circulation, because each
forcing is a Laplacian or divergence of an integrable vector field. Its frequencies belong to 4Z and
satisfy |m| ≤ 4j.
Lemma 3.4 (Coefficient bounds and seeded residual). For every finite derivative order k, the
finitely many coefficients in (23) satisfy
∥fj,0 (t)∥H k ≤ Cj,k ejσt ,
∥fj,ℓ (t)∥H k ≤ Cj,ℓ,k (1 + t)D ejσt ,

(24)
ℓ > 0.

(25)

There is a finite D, enlarged as needed, such that if a(t) = αeσt ≤ δ0 ≤ 1 on [0, H] and κ(1 +
H)D ≤ c, the residual
Rapp = ∂t ωapp + (K ∗ ωapp ) · ∇ωapp − κ∆ωapp
satisfies
∥Rapp (t)∥X ≤ C a(t)M +1 + κN +1 (1 + t)D a(t) .




(26)

All constants are uniform in ϑ.
Proof. The first coefficient is the exact growing-plus-decaying solution from (20). For j ≥ 2,
ℓ = 0, induction and Sobolev multiplication bound its forcing by Cejσt . Convolution against the
growing exponent σ, or the remaining exponent 2η < σ, is bounded by a constant times ejσt ,
because jσ−σ ≥ σ. This proves (24) with no polynomial factor. For positive ℓ the same induction
applies, with finite polynomial losses from Wr (t) and possible resonance with the first growing
mode. This proves (25). Derivative loss causes no infinite iteration: the coefficient set is finite
and the profile is smooth in every order. At each induction step apply the preceding coefficient
estimates in a sufficiently higher fixed order and the propagator lemma in the requested order.
Compact support and zero circulation also control K ∗ f and its needed derivatives, including low
frequencies. More explicitly |fb(ξ)| ≤ C|ξ| near zero, from its zero mass and finite first absolute
moment, while the high-frequency multiplier has order minus one.
27

Expand the radial heat base only in its couplings to fapp . The integral Taylor formula and
heat contraction in L1 , Lp give
GN (t) := Wκ (t) −

N
X

κr Wr (t),

∥GN (t)∥X + ∥∇GN (t)∥X ≤ CκN +1 tN +1 .

r=0

Its residual couplings obey
∥(K ∗ GN ) · ∇fapp + (K ∗ fapp ) · ∇GN ∥X ≤ CκN +1 (1 + t)D a(t)
by (21) and the coefficient bounds. These are the only noncompact factors. Every other coefficient
is supported in one fixed bounded set, where its high Sobolev norms control the required L1 , Lp
product norms. The recursion cancels exactly all monomials with 1 ≤ j ≤ M , 0 ≤ ℓ ≤ N . Terms
of seed order j > M are bounded by CaM +1 : the pure-seed bounds contain no time polynomial,
and each positive-viscosity polynomial is absorbed after enlarging D and using κ(1 + H)D ≤ c.
Terms of viscosity order ℓ > N , including the diffusion of fj,N , have the second bound in (26);
aj ≤ a for a ≤ 1 absorbs their higher seed powers. Crucially the exact radial heat base is part of
(22), so no residual term without a seed factor occurs. Finally rotations of (20) change only unit
complex phases; all bounds are uniform in ϑ.
3.1.4

Nonlinear closure in the actual vorticity norm

Let ω κ be the NS solution from W + αΦϑ and put ζ = ω κ − ωapp . Its exact equation, with zero
initial data, is
∂t ζ + (K ∗ ω κ ) · ∇ζ = κ∆ζ − (K ∗ ζ) · ∇ωapp − Rapp .
(27)
Kato’s inequality gives its L1 estimate, and multiplication by |ζ|p−2 ζ gives its Lp estimate; transport disappears by divergence freedom and diffusion is nonpositive in both inequalities. Using
(21) and adding gives
d+
∥ζ∥X ≤ Cp ∥∇ωapp ∥X ∥ζ∥X + ∥Rapp ∥X .
(28)
dt
These inequalities follow first with smooth convex regularizations of the norm and compact cutoffs,
then by their limits. All solutions here are smooth at positive time, have finite X norm, and are
continuous in that norm at time zero, so the integrated inequality starts at zero. No derivative
bound on the actual NS vorticity enters its coefficient.
We choose parameters without circular dependence. Fix
C1 = Cp (∥∇W ∥X + 1),

r1 = (C1 /σ − 1)+ ,

before choosing expansion orders. Choose M with (M + 1)σ > C1 and N with
3
(N + 1) > K∗ r1 .
2

(29)

Construct the finite coefficients, then fix a cap threshold δ0 using the pure-seed coefficients alone
and reduce the positive-viscosity threshold so that ∥∇fapp ∥X ≤ 1 whenever a ≤ δ0 and κ(1+H)D
is sufficiently small. This is possible by the finite coefficient bounds: make the pure-seed gradient
at most 1/2 by the cap choice, and the finitely many positive-viscosity contributions at most 1/2
by the later viscosity threshold. Thus a later increase of N need not reduce a previously fixed
pure-seed cap. The exact heat gradient obeys ∥∇Wκ ∥X ≤ ∥∇W ∥X , so C1 is the already fixed
28

exponent in (28). The coefficient constants affect only the later cap threshold; they do not change
the previous choice of M, N .
Convolving (26) against eC1 (t−s) gives the all-time estimate
∥ζ(t)∥X ≤ Ca(t)M +1 + CκN +1 (1 + t)D+1 a(t)e(C1 −σ)+ t .

(30)

The first integral has denominator (M + 1)σ − C1 > 0. For the second use a bound by emax(C1 ,σ)t
and an additional polynomial factor if the two exponents coincide. At H = σ −1 log(δ/α) this
yields
∥ζ(H)∥X ≤ Cδ M +1 + CκN +1 (1 + H)D+1 δ(δ/α)r1 .
(31)
For the theorem’s parameters, (29) implies that the second bound is o(δq ): a strict positive power
of q beats every fixed power of log(1/q). Also κq (1 + Hq )D → 0, δq → 0, and αq /δq → 0. The
coefficient estimates and (30) therefore prove (9), including uniformity through the entire interval.
They also give the actual nonradial density bound
∥(I − Prad )ω κ (t)∥1 ≤ Ca(t) + Ca(t)M +1 + CκN +1 (1 + t)D+1 a(t)e(C1 −σ)+ t .

(32)

Indeed radial averaging contracts L1 , and the heat base is radial. This estimate concerns the
actual density; it is not inferred from a contour coordinate.
Similarly (21), the coefficient bounds and (30) imply
Z H

∥K ∗ (ω κ − Wκ )∥∞ dt ≤ Cδ + Cδ M +1 + CκN +1 (1 + H)D+1 δ(δ/α)r1 .

(33)

0

To integrate the approximate terms use 0H a(t)j dt ≤ δ j /(jσ). The error terms grow at a fixed
positive exponential rate; their nondecreasing polynomial factor can be evaluated at H, so there
is no additional factor H. This proves (10).
R

3.1.5

Whole-moment control and the cap

κ
κ
We verify the
tails needed to remove the moment cutoff. Write
p u = Uκ + v , with Uκ tangential,
RH κ
2
and BH = 0 ∥v ∥∞ dt. By (33), BHq = O(δq ). Let d(x) = 1 + |x| . Then |∇d| ≤ 1, ∆d ≤ 2,
and Uκ · ∇d = 0. Kato’s inequality with eλd gives, for every λ > 0,

Z

eλd(x) |ω κ (t, x)| dx ≤ m exp{λd0 + λBH + 2κλt + κλ2 t},

(34)

where m = ∥ω κ (0)∥1 and d0 = 1 + (1 + ϵ)2 . No initial exponential tail is presupposed beyond
compact support. To justify the unbounded test use
p

FL (d) =

eλd
.
1 + eλd /L

It has FL′ ≤ λFL and FL′′ ≤ λ2 FL ; therefore ∆FL (d) ≤ (λ2 + 2λ)FL (d). For fixed L add a compact
cutoff and let its radius tend to infinity. Finite mass and bounded drift make the cutoff errors
vanish. Gronwall then gives (34) uniformly in L, and monotone convergence removes L. The
drift is bounded here by (21) and the scalar L1 , Lp contractions.
Choose a fixed R > 1 + ϵ with dR − d0 > 2c∗ > 0. For small q, BHq < c∗ /2 and κq Hq → 0.
Exponential Markov and optimization of (34) give
(

(ds − d0 − BH − 2κt)2
|ω κ (t, x)| dx ≤ m exp −
4κt
|x|>s

Z

29

)

,

s ≥ R,

0 < t ≤ H.

At t = 0 the exterior tail vanishes by the initial support. Integrating this mass bound in s gives
Z

|x|4 |ω q,ϑ (t, x)| dx ≤ Ce−c/(κq Hq ) .

sup
0≤t≤Hq

(35)

|x|>R

For example the weighted tail equals R4 times the mass tail plus 4 R∞ s3 times the lower-radius
mass tail. The displayed Gaussian estimate bounds both terms after reducing c. This also proves
finiteness of the required absolute moments.
Let ξ be a smooth radial cutoff equal to one on BR and supported in BR+1 . Every formal
nonradial coefficient is supported in BR ; the exact heat base pairs to zero with both z̄ 4 and ξ z̄ 4 ,
since it is radial. At the cap the first coefficient has growing moment
R

αeλϵ H eiϑ bϵ = δei(ϑ+τϵ H) bϵ
and stable moment of size Cα2 /δ. The remaining pure-seed coefficients have compactly tested
moment O(δ 2 ); positive-viscosity coefficients have moment O(κ(1+H)D δ). The actual error pairs
against ξ z̄ 4 with bound ∥ξ z̄ 4 ∥∞ ∥ζ(H)∥1 . Using (31) and (35) proves
|B4 (ω κ (H)) − δei(ϑ+τϵ H) bϵ | ≤ Cδ 2 + Cα2 /δ + Cκ(1 + H)D δ + Cδ M +1
+ CκN +1 (1 + H)D+1 δ(δ/α)r1 + Ce−c/(κH) .
For δ = δq , every bound divided by δq tends to zero. This proves (11) and completes the theorem.
Remark 3.5 (Scaling and scope). Let A, R > 0 and set
ωA,R (t, x) = Aω q,ϑ (At, x/R),

ν
WA,R
(t, x) = A(eκq At∆ Wϵ )(x/R),

ν = κq AR2 .

Homogeneity of Biot–Savart gives K ∗ ωA,R (t, x) = AR(K ∗ ω q,ϑ )(At, x/R), so the physical cap
time is Hq /A. For 1 ≤ r < ∞, change of variables gives
ν
∥ωA,R (t) − WA,R
(t)∥r = AR2/r ∥ω q,ϑ (At) − eκq At∆ Wϵ ∥r .

Thus the separate density bounds are CAR2 δq in L1 and CAR2/p δq in Lp , and their sum is
sup
0≤t≤Hq /A

ν
∥ωA,R (t) − WA,R
(t)∥L1 ∩Lp ≤ CA(R2 + R2/p )δq .

The cap moment and integrated perturbation velocity satisfy
B4 (ωA,R (Hq /A)) = AR6 δq ei(ϑ+τϵ Hq ) bϵ + o(AR6 δq ),
Z Hq /A
0

ν
∥K ∗ (ωA,R (t) − WA,R
(t))∥∞ dt = R

Z Hq

∥K ∗ (ω q,ϑ (s) − eκq s∆ Wϵ )∥∞ ds ≤ CRδq .

0

All constants are independent of A, R. These are isolated-colour estimates. Coupled use also
requires bounds for the foreign-colour terms in the coefficient and error equations.

30

4

The canonical ancient orbit and signed coupling

4.1

The canonical ancient profile of the isolated smooth activation

We retain the fixed profile, eigenvalue and seeds constructed in Section 3.1. Abbreviate them
to W = Wϵ , λ = σ + iτ = λϵ , b = bϵ ̸= 0, and Φϑ . In this section the cap δ > 0 is fixed and
sufficiently small. It does not decrease with the vanishing parameter. An additional Sobolev
estimate permits identification of a unique isolated ancient profile. This identification requires no
convergence of an infinite formal series. Here δ specifies the ancient growing amplitude and the
linear preparation level. It does not assert that s = 0 is the exact first hitting time of |A(Ωδ )| = δ;
at that time A(Ωδ (0)) = δ + O(δ 2 ).
Theorem 4.1 (The isolated ancient limit). Fix an integer k ≥ 3 and 2 < p < ∞. There exists
δ∗ > 0 such that, for 0 < δ < δ∗ , there is a real quarter-turn invariant Euler solution Ωδ (s) on
−∞ < s < ∞, with zero circulation, having the following properties. For every fixed K∗ > 0, put
αq = q K∗ ,

κq = q 3/2 ,

Hq = σ −1 log(δ/αq ),

ϑq = −τ Hq .

Let ω q be the isolated Navier–Stokes solution with viscosity κq and initial vorticity W + αq Φϑq .
For every J < ∞ and every 0 < ε < k,
ω q (Hq + ·) −→ Ωδ

in C([−J, 0]; L1 ∩ Lp ) and in C([−J, 0]; H k−ε )

as q ↓ 0.

(36)

The parameter δ∗ and the limiting profile can be chosen independently of K∗ ; the threshold for
small q may depend on K∗ . On s ≤ 0, Ωδ is supported in one fixed ball and has a uniform H k
bound. Its whole fourth harmonic moment satisfies
B4 (Ωδ (s)) = δbeλs + O(δ 2 e2σs )

(s ≤ 0).

(37)

For s ≥ 0 it satisfies
|B4 (Ωδ (s))| ≤ Cδ (1 + s)4 .

(38)

These assertions concern the isolated annulus.
4.1.1

A high Sobolev error estimate without derivative loss
p

Fix once and for all a radius R > 1 + ϵ with a positive gap in d(x) = 1 + |x|2 , for example
R = 3+ϵ. This radius and the corresponding compact-support norm constants are fixed before the
energy exponent or expansion orders. The later cap smallness condition will make the integrated
perturbation drift smaller than this fixed gap. For this proof let
∥g∥Zk = ∥g∥1 + ∥g∥H k .
The finite expansion from (22) and (23) can be constructed in every finite derivative order. In
particular, the residual estimate holds in Zk . A useful version separates its pure-seed constant
from constants depending on the viscosity order:
∥Rapp (t)∥Zk ≤ [CM +CM,N κ(1+t)D ]a(t)M +1 +CM,N κN +1 (1+t)D a(t),

a(t) = αeσt ≤ 1. (39)

Here CM is independent of N . To verify the additional norm and this separation, the pureseed products are estimated by their compactly supported coefficient Sobolev bounds. Every
31

other omitted seed-order product contains at least one positive power of κ, and its finite time
polynomial is incorporated in D. The noncompact heat remainder products are
(K ∗ GN ) · ∇fapp + (K ∗ fapp ) · ∇GN .
The heat Taylor formula gives
∥K ∗ GN ∥W k,∞ + ∥∇GN ∥1 + ∥∇GN ∥H k ≤ Ck,N κN +1 tN +1 .
For the first term use heat Taylor on the smooth initial velocity K ∗ W and contraction of each
bounded derivative. For the other terms use heat contraction on ∇∆N +1 W in L1 and H k . The
coefficient bounds give ∥K ∗ fapp ∥W k,∞ + ∥∇fapp ∥1 + ∥∇fapp ∥H k ≤ Ca(t) after the positiveviscosity threshold is made small. Leibniz’s rule proves the claimed seeded remainder in both
norms.
Let ζ = ω κ − ωapp and uapp = K ∗ ωapp . Rewriting (27) gives
∂t ζ + uapp · ∇ζ + (K ∗ ζ) · ∇ζ = κ∆ζ − (K ∗ ζ) · ∇ωapp − Rapp .

(40)

We record the estimates that avoid a quasilinear derivative loss. Sobolev embedding and the
near–far kernel split give, for k ≥ 3,
∥K ∗ g∥∞ ≤ Ck (∥g∥1 + ∥g∥H k ),
∥∇K ∗ g∥H j ≤ Cj ∥g∥H j ,

(41)

j ≥ 0.

(42)

The second is Plancherel’s theorem for a bounded order-zero Fourier multiplier. In particular
∥∇K ∗ g∥∞ ≤ Ck ∥g∥H k .
For each integer derivative order up to k, the undifferentiated transport term has zero energy
pairing. The commutator with (K ∗ ζ) · ∇ is bounded by
Ck (∥∇K ∗ ζ∥∞ ∥ζ∥H k + ∥∇ζ∥∞ ∥∇K ∗ ζ∥H k−1 ) ≤ Ck ∥ζ∥2H k .
This commutator inequality follows from Leibniz’s rule: the endpoint terms put the first derivative
in L∞ and the last derivative in L2 , and the intermediate terms follow by Sobolev interpolation
in dimension two. The embeddings for both displayed endpoint terms hold because k > 2. No
derivative beyond order k of ζ occurs. Similarly,


∥(K ∗ ζ) · ∇ωapp ∥H k ≤ C ∥∇ωapp ∥H k +

k
X

j



∥∇ ωapp ∥∞ ∥ζ∥Zk .

j=1

For the term where no derivative lands on K ∗ζ, use (41); for every term with a positive number of
such derivatives use (42) in L2 and put the corresponding derivative of the smooth approximation
in L∞ . The H k transport estimate with uapp uses only its fixed W k,∞ bounds. Its coefficients
are bounded, independently of α, κ, H, by fixed norms of W plus one, once the approximate
perturbation is made small in the finitely many required norms.
Together with Kato’s L1 inequality this gives, for Y = ∥ζ∥Zk ,
d+ Y
≤ Ck Y + Ck Y 2 + ∥Rapp ∥Zk ,
dt

Y (0) = 0.

(43)

Diffusion has a nonpositive energy contribution. The constant Ck can be fixed before M, N : the
pure-seed approximation is made small by choosing its cap after M , and each positive-viscosity
coefficient is made small by choosing its viscosity threshold after N .
32

Set Ek = 2Ck , increasing it if necessary to dominate Cp (∥∇W ∥L1 ∩Lp + 1) and the compactsupport L1 ∩ Lp stability constant used below. Choose one finite M with
(M + 1)σ > Ek .

(44)

After its pure-seed coefficients are known, choose δ∗ > 0 small enough for all their required norms
and their pure-seed remainder. For each subsequently prescribed K∗ choose N so that, with
rk = (Ek /σ − 1)+ ,
3
(45)
2 (N + 1) > K∗ rk .
The N -dependent coefficients are controlled by decreasing the threshold for q, not by changing
δ∗ . In particular arrange CM,N κ(1 + H)D ≤ 1 in (39). On a bootstrap interval Y ≤ 1, (43) now
gives
Y (t) ≤ CM a(t)M +1 + CM,N κN +1 (1 + t)D+1 a(t)e(Ek −σ)+ t .
(46)
At the cap, the second term is
CM,N κN +1 (1 + H)D+1 δ(δ/α)rk = o(1)
in the prescribed polynomial window. The first is smaller than 1/2 after reducing δ∗ . Continuity
contradicts a first bootstrap exit, so the estimate holds through the full cap interval. It also
yields a uniform H k bound for the actual vorticity. This bound is obtained from the actual error
equation, rather than assumed from the formal approximation.
4.1.2

Finite homological coefficients and compactness

For a real angle θ define the rotation
Lrot = L − (τ /4)∂φ .

(Tθ f )(r, φ) = f (r, φ + θ/4),

It acts isometrically on every radial or full-plane Sobolev norm, commutes with the radial linearization, and respects Biot–Savart advection. The pure-seed coefficient fj,0 with seed phase
zero has a limit
e−jσt T−τ t fj,0 (t) −→ Fj in every finite H m as t → ∞.
(47)
For j = 1 the decaying part is O(e−2σt ) after this normalization, so
F1 = fϵ (r)e4iφ + complex conjugate.
For j ≥ 2, put gj = e−jσt T−τ t fj,0 . Its equation is
X

∂t gj = (Lrot − jσ)gj −

(K ∗ gj1 ) · ∇gj2 .

j1 +j2 =j

On its finite set of sectors, the homogeneous propagator decays at least as Ce−(j−1)σt by (15).
Induction, the coefficient bounds, and dominated convergence in Duhamel’s integral prove (47)
and the finite homological relations
Fj = −(jσ − Lrot )−1

X

(K ∗ Fj1 ) · ∇Fj2 ,

2 ≤ j ≤ M.

(48)

j1 +j2 =j

The inverse is the integral of the exponentially decaying propagator. All these functions are
smooth and supported in the fixed collars. Only the finitely many sectors in 4Z with |m| ≤ 4M
are used. No all-mode Sobolev spectral-growth assertion is needed.
33

Define the fixed polynomial ancient profile
PM (s) = W +

M
X

δ j ejσs Tτ s Fj ,

s ≤ 0.

(49)

j=1

For the phase ϑq = −τ Hq , (47) shows that every pure-seed term of the cap-shifted approximation
converges to the corresponding term in (49). The positive-viscosity terms tend to zero because
κq (1 + Hq )D → 0, while eκq (Hq +s)∆ W → W . For each fixed s ≤ 0, (46) gives the asymptotic error
CM (δeσs )M +1 , since its viscosity remainder tends to zero by (45).
We justify actual compactness separately. The kernel bound and (46), integrated as in (33),
give
Z
Hq

0

∥K ∗ (ω q (t) − Wκq (t))∥∞ dt ≤ CM δ + o(1).

Choose δ∗ so this is less than the positive radial gap fixed before the expansion orders. The
exponential Kato proof of (35), with this fixed cap and the already fixed radius R, gives
Z

sup
0≤t≤Hq

|x|4 |ω q (t, x)| dx ≤ Ce−c/(κq Hq ) .

(50)

|x|>R

Initial velocities have uniformly bounded finite energy, since the initial vorticities are compactly
supported and have zero circulation. The NS velocity energy inequality and the uniform vorticity
H k bound therefore bound the velocities in H k+1 . The equation bounds ∂t ω q uniformly in H k−2
on the cap interval.
On each shifted interval [−J, 0], Rellich compactness and interpolation give local compactness
in C(H k−ε ). The tail bound and the uniform H k , hence L∞ , bound give global L2 tightness.
Interpolation applied to a smooth exterior cutoff then gives global H k−ε tightness; the cutoff is
chosen outside the same fixed BR . Thus compactness holds globally in that norm. The mass
tail gives global L1 tightness, and local convergence plus uniform L∞ bounds gives compactness
in C(L1 ∩ Lp ) for every finite p. A diagonal subsequence over J gives an ancient Euler solution:
the transport term passes by the stated strong convergence, and κq ∆ω q → 0 on every compact
shifted interval. The solution is supported in BR on s ≤ 0, since the right side of (50) tends to
zero.
By lower semicontinuity of the H k norm and the strong L1 convergence, every such limit
satisfies, for all s ≤ 0,
∥Ω(s) − PM (s)∥Zk ≤ CM (δeσs )M +1 .
(51)
One may first take rational times; continuity in the weaker compactness norm and weak H k
continuity extend the bound to every time.
4.1.3

Uniqueness of the ancient subsequential limit

We use a norm in which no extra derivative of a comparison orbit is required. If Ω1 , Ω2 are two
ancient subsequential limits, their compact supports and uniform H k bounds imply a uniform
bound for ∥∇Ωj ∥L1 ∩Lp . Indeed compact support controls the L1 norm by the L2 derivative norm,
and H k , k ≥ 3, controls every finite Lp derivative norm. Their perturbations from W are small
in these norms after reducing δ∗ , so the stability coefficient is bounded by the fixed constant
dominated by Ek above.
For D = Ω1 − Ω2 the exact Euler equation is
∂s D + (K ∗ Ω1 ) · ∇D = −(K ∗ D) · ∇Ω2 .
34

The same Kato and Lp argument as (28) gives, for s0 ≤ s1 ≤ 0,
∥D(s1 )∥L1 ∩Lp ≤ eEk (s1 −s0 ) ∥D(s0 )∥L1 ∩Lp .
Both limits have the same polynomial (49). Using (51) at s0 = −T yields
∥D(0)∥L1 ∩Lp ≤ CM δ M +1 e−[(M +1)σ−Ek ]T −→ 0.
The same comparison with any fixed endpoint s1 ≤ 0 proves equality throughout the ancient
interval. The strict inequality (44) uses one finite M only. Thus every subsequence has the same
limit, proving the full convergence in (36). Repeating the argument for any different K∗ gives
the same limit; only its permissible viscosity expansion order and small-q threshold change.
For clarity, a direct H k difference stability estimate would require an H k+1 bound on the
comparison vorticity because it differentiates (K ∗D)·∇Ω2 k times. The L1 ∩Lp comparison above
deliberately avoids this extra hypothesis. The high Sobolev estimate is used for compactness and
for the available first derivatives.
4.1.4

Forward extension and the whole-moment asymptotics

The ancient construction initially gives compactly supported H k vorticity at zero and finiteenergy velocity in H k+1 , with k ≥ 3, and preserves zero circulation by strong L1 convergence.
Lemma 2.3 therefore extends this same solution uniquely to every s ≥ 0, with H k vorticity and
H k+1 velocity on each fixed interval. This first extension uses only the finite order k; the all-order
regularity of the fixed orbit is proved separately below. Its conserved L1 and Lp norms and (21)
give a time-independent velocity bound V .
The bounded velocity transports its support into BR+V s . Consequently
Z

|x|4 |Ωδ (s, x)| dx ≤ ∥Ωδ (0)∥1 (R + V s)4

(s ≥ 0),

which proves (38). On s ≤ 0, its support is in the fixed BR . Pairing (51) with z̄ 4 is therefore
bounded by its L1 error times R4 . The first homological coefficient has moment b, rotation
multiplies it by eiτ s , and every higher term is O(δ 2 e2σs ). This proves (37) and completes the
theorem.
Corollary 4.2 (Every fixed source window, including after the cap). For the isolated family in
Theorem 4.1, every fixed J < ∞ satisfies
ω q (Hq + ·) −→ Ωδ

in C([−J, J]; L1 ∩ Lp ),

sup |B4 (ω q (Hq + s)) − B4 (Ωδ (s))| −→ 0.
|s|≤J

Write Bq (s) = B4 (ω q (Hq + s)) and Bδ (s) = B4 (Ωδ (s)). For fixed 0 ≤ S, U < ∞ and a ∈ R, its
source moment therefore satisfies
Z U
−S

e−as |Bq (s) − Bδ (s)| ds −→ 0.

Proof. The negative window follows from (36) and (50). On 0 ≤ s ≤ J, let ζq (s) = ω q (Hq + s) −
Ωδ (s). Its exact equation is
∂s ζq + (K ∗ ω q (Hq + s)) · ∇ζq = κq ∆ζq − (K ∗ ζq ) · ∇Ωδ + κq ∆Ωδ .
35

The global Euler extension has bounded H k norm and compact support on this fixed interval.
Since k ≥ 3, its second derivatives belong to H 1 , which embeds in every finite Lp in dimension
two; compact support also bounds their L1 norms. Thus ∥∇Ωδ ∥L1 ∩Lp + ∥∆Ωδ ∥L1 ∩Lp ≤ CJ .
Kato’s inequality, the Lp inequality and (21) give
sup ∥ζq (s)∥L1 ∩Lp ≤ eCJ J ∥ζq (0)∥L1 ∩Lp + CJ κq −→ 0.


0≤s≤J

No actual NS derivative bound is used in this comparison.
We remove the moment cutoff separately. Scalar norm contraction gives one uniform post-cap
velocity bound V . Choose a fixed radius R′ > R + V J + d, also enclosing the Euler support on
[0, J], with d > 0. The positive post-cap propagator has a forward diffusion with |Xs − y| ≤
p
V J + 2κq |Bs |. For |y| ≤ R, its terminal fourth moment outside BR′ is at most CJ e−cJ /κq by
Gaussian endpoint moments. For unrestricted y its full fourth moment is at most CJ (1 + |y|4 ).
The positive domination of the signed solution by the propagation of |ω q (Hq )| therefore gives
Z

sup
0≤s≤J

|x|>R′

|x|4 |ω q (Hq + s, x)| dx ≤ CJ e−cJ /κq + CJ

Z
|y|>R

(1 + |y|4 )|ω q (Hq , y)| dy −→ 0

by (50). A compact radial cutoff of z̄ 4 , the just-proved L1 convergence, and this tail prove uniform
whole-moment convergence. On a fixed weighted time window the weight is bounded, so its final
assertion follows by integration.
Corollary 4.3 (A nonzero bilateral discounted moment). Define
Fδ (z) =

Z ∞
−∞

e−zs B4 (Ωδ (s)) ds.

It is analytic and is not identically zero on 0 < ℜz < σ. More precisely it extends meromorphically
to 0 < ℜz < 2σ as
δb
Fδ (z) =
+ Gδ (z),
(52)
λ−z
where Gδ is analytic on that larger strip. For every fixed real c, the exceptional set




β > 1 : Fδ (σ − ic)/β = 0

is discrete in (1, ∞). In particular one can select arbitrarily large β outside it, including for
c = 1/2.
Proof. The negative-time estimate (37) and the positive-time polynomial bound give absolute,
locally uniform convergence on 0 < ℜz < σ, including after any fixed number of z derivatives.
Subtract the leading negative-time exponential. Its integral is
Z 0

δbe(λ−z)s ds =

−∞

δb
.
λ−z

The remaining negative-time integral is analytic for ℜz < 2σ, and the positive-time integral
is analytic for ℜz > 0. This proves (52). Its pole at z = λ has nonzero coefficient, so the
transform cannot be identically zero on the smaller strip. Its zeros there are isolated. The curve
β 7→ (σ − ic)/β is nonconstant, lies in that strip for β > 1, and has no interior accumulation of
its zeros. Any terminal interval of large β contains values outside the zero set, proving the last
assertion. Accumulation as β → ∞ is possible at the boundary z = 0 and is not excluded or
needed.
36

Corollary 4.4 (Changing the canonical cap). If 0 < δ ′ < δ < δ∗ and s0 = σ −1 log(δ ′ /δ), then
(−∞ < s < ∞),

Ωδ′ (s) = T−τ s0 Ωδ (s + s0 )

Fδ′ (z) = e(z−iτ )s0 Fδ (z).

Thus the zero set of the transform is independent of the canonical cap. At z = (σ − ic)/β its
modulus scales by (δ ′ /δ)1/β .
Proof. The time-translated and rotated orbit has the polynomial (49) with δ replaced by δ ′ , since
δeσs0 = δ ′ . It has the same order-M + 1 remainder. The ancient uniqueness argument therefore
identifies it with Ωδ′ on negative times, and forward Euler uniqueness extends the identity to every
time. Rotation Tθ multiplies B4 by eiθ . Substituting v = s + s0 into the absolutely convergent
bilateral transform gives the stated exponential factor, which never vanishes. Its modulus at the
displayed point is eσs0 /β .
Corollary 4.5 (All finite regularity orders and finite jets of one orbit). The already fixed orbit
Ωδ belongs to every finite spatial Sobolev order at every time. For each integer ℓ ≥ k,
sup ∥Ωδ (s)∥L1 ∩H ℓ < ∞,
s≤0

and its H ℓ norm is bounded on each fixed positive-time interval. In particular its vorticity is
spatially C ∞ . For each fixed J ≥ 1 and integer r ≥ 0, the homological coefficients F1 , . . . , FJ
satisfy, for some sJ,r < 0 and CJ,r < ∞,
Ωδ (s) − W −

J
X
j=1

δ j ejσs Tτ s Fj

≤ CJ,r (δeσs )J+1 ,

s ≤ sJ,r .

(53)

L1 ∩H r

The coefficients are given by (48) for each finite target degree, and are independent of the larger
cutoff used to construct them. All intersection norms here are sum norms.
Proof. Fix an integer ℓ ≥ k. Repeat the finite-coefficient and actual-error construction above
in order ℓ, with amplitude δℓ smaller than δ and both the original and the order-ℓ permissible
thresholds. The profile and smoothing length remain fixed. Choose any fixed K∗ > 0. The exact
Navier–Stokes family used in this construction, with cap shift Hq = σ −1 log(δℓ /αq ) and seed phase
−τ Hq , is precisely the isolated family already identified by Theorem 4.1 at amplitude δℓ . Its full
L1 ∩ Lp limit is unique. The order-ℓ construction therefore gives the same Ωδℓ and a uniform H ℓ
bound on negative times. Lemma 2.3 yields its H ℓ bounds on every fixed positive-time interval.
Let sℓ = σ −1 log(δℓ /δ) < 0. Corollary 4.4 and forward uniqueness give
Ωδℓ (v) = T−τ sℓ Ωδ (v + sℓ )

(−∞ < v < ∞).

For s ≤ sℓ use the negative-time bound on Ωδℓ (s − sℓ ). For sℓ ≤ s ≤ 0 use its positive-time
bound on 0 ≤ v ≤ −sℓ . Rotation is an isometry, so these two bounds give the asserted uniform
order-ℓ bound for the same Ωδ on s ≤ 0. Translating fixed positive intervals gives its remaining
bounds. Their constants may depend on ℓ, δ. Since ℓ is arbitrary, Sobolev embedding proves
spatial smoothness. No intersection of infinitely many small-amplitude thresholds is used.
For the finite jet, fix J, r and choose ℓ ≥ max{k, r + 1, 3}. Choose a finite expansion degree
M ≥ J large enough that (M + 1)σ > Eℓ , and choose a sufficiently small δℓ < δ for this finite
construction. The same isolated family and uniqueness again identify its limit with Ωδℓ . The
37

pure-seed triangular recursion gives the same F1 , . . . , FJ independently of M ; passing its finite
expansion and actual-error estimate to the limit gives
Ωδℓ (v) − W −

J
X
j jσv

δℓ e

j=1

≤ CJ,r (δℓ eσv )J+1 ,

Tτ v Fj

v ≤ 0.

L1 ∩H r

Indeed the error has degree M + 1, and the finitely many discarded degrees J + 1, . . . , M have
the displayed degree-J + 1 bound. Translate by sℓ and rotate by τ sℓ . The amplitude identity
δℓ eσ(s−sℓ ) = δeσs and addition of the rotation phases give (53) for s ≤ sℓ . The constant and onset
time may depend on the finite target orders. This argument proves each finite jet and does not
assume convergence of an infinite formal series.
Remark 4.6 (Scope). The profile and source transform in this section belong to the isolated
annulus. Their use in a coupled receiver requires the foreign-colour and weighted source-history
estimates in addition to this isolated limit.

4.2

The smooth receiver’s protected coordinate and transfer coefficient

Throughout this section W = Wϵ is the one fixed profile of Section 3.1, λ = σ + iτ its growing
fourth-sector eigenvalue, and f+ = fϵ its normalized radial eigenfunction. All angular conventions
are unchanged.
Lemma 4.7 (A global growing-coordinate test). There is a smooth bounded complex test
Tadj (x) = a(r)e−4iθ
whose derivatives of every fixed order are bounded, such that
A(g) =

Z
R2

Tadj (x)g(x) dx,

A(f+ (r)e4iθ ) = 1,

A(f+ (r)e4iθ ) = 0,

and A(Lg) = λA(g) for smooth integrable perturbations for which the pairings are defined. Near
the origin Tadj is a constant multiple of z̄ 4 ; outside a fixed ball it is a constant multiple of z −4 .
In particular, ∆Tadj is compactly supported away from the origin.
Proof. On a fixed radial interval containing the derivative collars, the rank-one Riesz projection in
the positive fourth sector has form P+ g = f+ ℓ(g) with ℓ(f+ ) = 1 and ℓ(L4 g) = λℓ(g). Represent
this continuous L2 functional by a bilinear density 2πa(r)r dr; complex conjugation is included
in the representing density rather than in the pairing. Substitution of (14) gives
i ∞
(λ + 4iΩϵ (s))a(s) = −
a(r)Wϵ′ (r)h4 (r, s) dr.
(54)
2 0
On the collars, the radial Green equation and this nonvanishing denominator bootstrap a to
smoothness. Extend it to every s > 0 by the right side of (54) divided by its denominator. This
agrees with the original density where W ′ is supported, so it still satisfies the equation everywhere.
Integration of the Green kernel and Fubini then verify the global eigenfunctional identity.
For s smaller than every derivative collar, the numerator is a constant times s4 and Ωϵ is the
constant core angular velocity −1/5. Thus a(s) = Cs4 there and Tadj = C z̄ 4 . Beyond the profile
support the numerator is a constant times s−4 and Ωϵ = 0 by zero circulation. Thus a(s) = C ′ s−4
and Tadj = C ′ z −4 . These two extensions are harmonic; on the remaining fixed annulus
the test is
R
smooth. This proves all the regularity and Laplacian assertions. Normalize by 2π af+ r dr = 1.
Since the stable eigenvalue is −λ̄ ̸= λ, the eigenfunctional identity annihilates its eigenfunction.
The angular test kills every sector other than +4.
Z

38

Lemma 4.8 (Exact inverse-quintic forcing coefficient). Let a source w be real, integrable, supported in BRs , quarter-turn symmetric and of zero circulation. For each integer m ≥ 0 used
below, write
Z
Z
Bm (w) =

ȳ m w(y) dy,

|y|m |w(y)| dy < ∞.

R2

R2

These absolute moments are finite for this compactly supported source. At points in the exterior
region |z| > Rs its complex velocity is
∞
i X
B4ℓ (w)
U (z) =
.
2π ℓ=1 z̄ 4ℓ+1

(55)

For every finite L ≥ 1, its remainder after degree 4L satisfies
U (z) −

L
i X
1
B4ℓ (w)
≤
4ℓ+1
4L+5
2π ℓ=1 z̄
2π|z|
(1 − (Rs /|z|)4 )

Z

|y|4L+4 |w(y)| dy.

(56)

The first term forces the receiver’s positive fourth vorticity sector by B4 Fϵ (r)e4iθ , where
Fϵ (r) = −

iWϵ′ (r)
.
4πr5

Its growing projection is γϵ B4 , with γϵ = A(Fϵ e4iθ ). For the mollified family before the smoothing
length is fixed,
iρ−6 − ρ−1
γϵ −→ γ0 =
̸= 0 (ϵ ↓ 0).
(57)
8π
i
Proof. In complex notation Biot–Savart is U (z) = 2π
(z̄ − ȳ)−1 w(y) dy. For |z| > Rs , the
geometric series converges uniformly in y on the source support and absolutely after integration.
Quarter-turn symmetry gives Bm = (−i)m Bm , eliminating every degree not divisible by four;
zero circulation eliminates B0 . This proves (55). Summing the absolute values of the remaining
degrees and using |y| ≤ Rs proves (56). The exterior expansion is not used in an inner hole of an
annular support. For its first term,

R

ur = ℜ(U e−iθ ) =

iB4 4iθ
e + complex conjugate.
4πr5

The linearized forcing is −ur W ′ , giving Fϵ .
We check nonvanishing without assuming an adjoint convergence theorem. The inhomogeneous resolvent Schur reduction in the proof of Lemma 3.3 applies to this forcing. Its natural
radial measure has limit
2
X
ici
δr .
Fϵ (r)r dr ⇀
4πri4 i
i=1
Thus in interface variables its forcing vector tends to h = (i/(4πρ6 ), i/(4π)). The growing dual
of M4 , normalized on (1, −iρ), is ℓ0 (q) = (q1 + iq2 /ρ)/2. Its pairing with h is γ0 in (57). More
explicitly, on a small fixed contour around the growing root the inhomogeneous Schur equations
converge uniformly to (z − M4 )q = h. Integrating their resolvent in z gives convergence of
the rank-one projected forcing coefficient. The eigenfunction normalization agrees with that in
Section 3.1. This proves the limit and nonvanishing for all sufficiently small ϵ. This condition is
included in the final fixed choice of the smoothing length.
39

The direction of this interaction matters. A smaller source acts on the larger receiver by the
inverse-quintic exterior field above. A larger annulus acts in its interior through a cubic harmonic
field. They are different coefficients and cannot be interchanged.
For comparison, an exterior annulus’s interior harmonic field U (z) = c3 z̄ 3 has positive fourth
radial coefficient c̄3 r3 /2. Its frozen-base forcing is −c̄3 r3 W ′ /2, with growing projection γϵout c̄3 .
The same resolvent limit gives
γϵout −→ (ρ2 + i/ρ)/4 ̸= 0.
Indeed its interface forcing vector is (ρ2 /2, 1/2), paired against the same growing dual. Thus
older cubic fields can genuinely seed the growing coordinate; their size must be bounded, rather
than discarded by symmetry. All the required spectral and nonvanishing conditions now hold
for every sufficiently small smoothing length. Fix once one ϵ for which Lemma 3.3 holds and
bϵ , γϵ , γϵout are nonzero, and retain it in every later finite expansion and Sobolev order.
Lemma 4.9 (Duhamel coefficient for a child source). Let Ac = βAr , Rc = qRr , and use receiver
time s = Ar t. Suppose, in a frame removing all common solid rotation, that the source’s normalized fourth moment is B(u), with child time u = βs − s∗ . The receiver core has angular velocity
−Ar /5; this rotates the child’s moment by e4is/5 in the receiver frame. The ideal linear growing
receiver coefficient obeys
a′ (s) = λa(s) + βq 6 γϵ e4is/5 B(βs − s∗ ).
For the response with no independent growing coefficient at s = −∞,
e−λs a(s) = q 6 γϵ e−µβ s∗

Z βs−s∗

e−µβ u B(u) du,

−∞

µβ =

σ + i(τ − 4/5)
.
β

(58)

This integral exists whenever B(u) = O(eσu ) at −∞ and has polynomial growth at +∞, provided
β > 1.
Proof. The dimensional fourth moment is Ac Rc6 B. Dividing the child velocity by Ar Rr leaves
the factor βq 6 , so Lemma 4.8 gives the forcing magnitude. Rotation through angle Ωt multiplies
B4 by e−4iΩt . The parent constant core has vorticity −2Ar /5, hence angular velocity −Ar /5,
which gives the factor e4is/5 . Multiplication by the integrating factor e−λs and the substitution
u = βs − s∗ give (58). Its negative tail has exponential exponent σ − σ/β > 0; its positive tail
has discount σ/β > 0.
Lemma 4.10 (A nonzero bilateral source transform from an ancient orbit). Suppose a global
Euler source trajectory has a fourth moment B(u) with
B(u) = cbϵ eλu + O(e2σu )

(u → −∞),

c ̸= 0,

and |B(u)| ≤ C(1 + u)4 for u ≥ 0. Define
F(z) =

Z

e−zu B(u) du,

0 < ℜz < σ.

R

Then F is analytic and not identically zero. For every finite interval [B∗ , B∗ + 1] ⊂ (1, ∞) there
is a choice of β in that interval such that F(µβ ) ̸= 0.
Proof. The two explicit bounds give absolute locally uniform convergence of the integral and of
all complex derivatives, proving analyticity. Split at zero and subtract the leading negative-time
asymptotic. It gives
cbϵ
F(z) =
+ G(z),
0 < ℜz < σ,
λ−z
40

where G is analytic in 0 < ℜz < 2σ: the remainder on the negative half-line has rate 2σ, and the
positive half-line has polynomial growth. Thus its meromorphic extension has a nonremovable
simple pole at z = λ. In particular it cannot vanish identically on the original strip. The analytic
nonconstant curve β 7→ µβ lies inside that strip for real β > 1. If the transform vanished along any
real interval, its zeros would have an interior accumulation point and the identity theorem would
force identical vanishing. Its zeros along the displayed compact interval are therefore discrete,
and a nonzero choice exists.
The canonical ancient Euler orbit of Theorem 4.1 has precisely this negative-time estimate. It
comes from the high-Sobolev construction and not merely from the isolated X-norm cap theorem.
Its positive-time moment bound is elementary once a compactly supported bounded-vorticity orbit exists: L1 , Lp conservation bounds its velocity by (21), its support radius grows at most
linearly, and its fourth absolute moment is therefore polynomial of degree at most four. Lemma 4.10 isolates the precise ancient-orbit input needed for the noncancellation assertion.
Lemma 4.11 (Quantitative window and source errors). Let β > 1 be fixed, µ = µβ , and S, U ≥ 0.
Suppose F(µ) ̸= 0 and the source satisfies the bounds of Lemma 4.10. Define the truncation budget
Etr = F(µ) −

Z U

e−µu B(u) du .

−S

It satisfies
Etr ≤ Ce−(σ−ℜµ)S + Cβ (1 + U )4 e−ℜµU .
e has budget
Replacing the moment on that window by B
Z U

Erep =

e
e−ℜµu |B(u)
− B(u)| du.

−S

Set Q = q 6 |γϵ |e−σs∗ /β . If the three budgets satisfy
Etr < 14 |F(µ)|,

Erep < 14 |F(µ)|,

|a0 | < 14 Q|F(µ)|,

then adding the independent old growing coefficient a0 to the modified source response leaves a
growing coefficient of modulus greater than Q|F(µ)|/4. If s∗ = σ −1 log(δ/α) and both old and
child seed amplitudes have size α = q K∗ , the old-seed condition is algebraically feasible when
K∗ (1 − 1/β) > 6.
Here 0 < δ ≤ 1 is fixed or decreases logarithmically in 1/q; the remaining source and transfer
constants are fixed.
Proof. Put a = ℜµ = σ/β ∈ (0, σ). The negative tail is bounded by Ce−(σ−a)S /(σ − a). For
the positive tail, substitute u = U + v and use (1 + U + v)4 ≤ (1 + U )4 (1 + v)4 for U, v ≥ 0.
The integral of e−av (1 + v)4 is finite, giving the asserted positive-tail bound. The negative and
positive tails together constitute the single budget Etr . By the triangle inequality,
Z U
−S

e
e−µu B(u)
du ≥ |F(µ)| − Etr − Erep > 21 |F(µ)|.

41

Equation (58) multiplies this transform by a factor of modulus Q. A further triangle inequality
subtracting |a0 | leaves a modulus greater than Q|F(µ)|/4. Finally e−σs∗ /β = (α/δ)1/β , so
α
= q K∗ (1−1/β)−6 δ 1/β −→ 0
6
q (α/δ)1/β
under the stated power inequality. Fixed logarithmic cap factors do not alter this strict inequality.

4.2.1

Exact heat shielding and the remaining coupled terms

The neutral seed (20) has only sectors ±4 and B4 = 0. Its velocity is zero outside B1+ϵ : in
the exterior angular Green formula its coefficient is exactly its fourth moment. Thus its velocity
Vϑ = K ∗ Φϑ is smooth and compactly supported. Heat commutes with Biot–Savart, so a
passive physical seed of amplitude Aα and radius r has exterior heat velocity bounded, for
|x| ≥ R > 2(1 + ϵ)r, by
|eνt∆ [AαrVϑ (·/r)](x)| ≤ CAαr exp[−cR2 /(νt)].

(59)

Indeed convolution integrates a bounded function supported in that ball, and the Gaussian mass
at distance at least R/2 has the displayed bound. No damping estimate of a conserved harmonic
moment is invoked. The bound concerns a passive heat evolution; its nonlinear moment generation
is an additional term.
For precision, the actual foreign-colour equation yields an exact protected-coordinate identity.
Suppose a receiver colour w solves
wt + (K ∗ w + b) · ∇w = κ∆w,

div b = 0,

and put f = w − Wκ . Remove any common rigid rotation first. Testing with Lemma 4.7 gives
Z
d
A(f ) = λA(f ) − Tadj b · ∇W dx + E(t),
(60)
dt
where the exact error is the sum of
Z

κ
Z

∆Tadj f dx,

−

Z

f (K ∗ f ) · ∇Tadj dx,





Tadj (K ∗ (Wκ − W )) · ∇f + (K ∗ f ) · ∇(Wκ − W ) dx,
Z

f b · ∇Tadj dx,

−

Z

Tadj b · ∇(Wκ − W ) dx.

There is no derivative of the actual f after integrating its transport terms by parts. Smooth heat
Taylor and (21) give
|E(t)| ≤ C[κ(1 + t)∥f ∥X + ∥f ∥2X ] +

Z

|f ||b||∇Tadj | dx +

Z

Tadj b · ∇(Wκ − W ) dx .

(61)

For the term with ∇f , integration by parts bounds it by ∥∇Tadj ∥∞ ∥K ∗ (Wκ − W )∥∞ ∥f ∥1 ; the
other heat term uses ∥∇(Wκ −W )∥1 = O(κt). The quadratic self term uses (21). If b = Ω(t)Jx is a
rigid rotation, its base forcing vanishes and its perturbation contribution is exactly −4iΩ(t)A(f );
it can be removed by its integrating phase.
Equation (60) identifies the estimates required for coupled use. On the receiver collars one
must approximate b by the inverse-quintic child field of Lemma 4.8; its projected error is bounded
by the collar sup error times ∥Tadj ∇W ∥1 . One must also control the two weighted foreign terms in
(61) and prove a nonlinear density bound for this actual colour. The canonical source transform
must be approximated in the weighted norm of Lemma 4.11, including its post-cap history. The
actual foreign-colour estimates and weighted source-history control are additional inputs to a
coupled receiver argument.
42

4.2.2

A nonlinear receiver theorem for the concrete canonical input

The following theorem uses the moment of the canonical isolated Euler orbit and realizes its
exterior harmonic field through a prescribed smooth divergence-free extension. Its equation has
this auxiliary drift; the source and receiver are not asserted here to solve a jointly unforced system.
Fix an ancient orbit Ωδc from Theorem 4.1, with fixed small δc > 0, and put B(u) =
B4 (Ωδc (u)). Choose β > 1 for which
C∗ := γϵ F(µβ ) ̸= 0.
Such a choice is possible in every sufficiently large fixed interval by Lemma 4.10. Take a smooth
radial cutoff ζ0 equal to zero near the origin and equal to one on and outside the two receiver
derivative collars. For c ∈ C set
Ψc (r, θ) = −

ζ0 (r)
(ce4iθ + c̄e−4iθ ),
16πr4

Vc = ∇⊥ Ψc .

This is a smooth real divergence-free field with every fixed spatial derivative bounded by a constant times |c|. On the receiver collars it agrees exactly with the inverse-quintic field of a source
moment c.
Theorem 4.12 (Transmission for a prescribed canonical harmonic input). Fix K∗ > 0 such that
K∗ (1 − 1/β) > 6. For small q put
α = q K∗ ,

εq = q 6 ,

δq = [log(1/q)]−4 ,

t0 =

1
δc
log ,
βσ
α

Sq =

1
δq
log
,
σ
εq |C∗ |

Hq = t0 + Sq .

For 0 < κ ≤ q 3/2 define the prescribed foreign drift
bq (t, x) = βεq V e4i(t−t0 )/5 B(β(t−t0 )) (x),

0 ≤ t ≤ Hq .

Let wq solve the coloured equation
∂t wq + (K ∗ wq + bq ) · ∇wq = κ∆wq ,

wq (0) = W + αΦ0 .

Then uniformly in 0 < κ ≤ q 3/2 ,
sup ∥wq (t) − eκt∆ W ∥X ≤ Cδq ,
0≤t≤Hq

A(wq (Hq )) = δq ei(τ Sq +arg C∗ ) + o(δq ),
B4 (wq (Hq )) = bϵ δq ei(τ Sq +arg C∗ ) + o(δq ).
The whole fourth moment is finite. Multiplying the source moment in bq by a fixed unit complex
phase multiplies the leading cap phase by the same phase; all bounds are uniform in that phase.
Proof. For completeness, the positive-viscosity solution with this prescribed drift exists on the
whole displayed window. On a short interval its mild equation is a contraction in Ct (L1 ∩ Lp ):
the heat-gradient estimate is ∥∇eκt∆ ∥Lr →Lr ≤ C(κt)−1/2 , the Biot–Savart map sends X to L∞ ,
and bq is bounded there. Scalar convex-energy estimates give the L1 , Lp contractions and hence
a time-uniform velocity bound. Iterating the local construction prevents finite-time loss of the
mild solution. Smoothing follows by the same mild equation from the smooth spatial coefficients.
43

This argument uses only bounded drift on the finite window, not the conclusion of the theorem.
Use s = t − t0 and write bq = εq b1 (s). The ancient and future moment bounds give
(

∥b1 (s)∥C j ≤

Cj eβσs ,
s ≤ 0,
4
Cj (1 + s) , s ≥ 0.

In particular this is bounded by Cj eσs for all s, and its growing-mode weighted time integral is
finite, uniformly in the initial time −t0 . The first seed-order coefficient solves
(∂t − L)g1,0 = −b1 (t − t0 ) · ∇W,

g1,0 (0) = (α/εq )Φ0 .

Its growing coordinate is
e

−λs

α
A(g1,0 (t)) = eλt0 + γϵ
εq

Z βs

e−µβ u B(u) du.

−βt0

The first term tends to zero in modulus because K∗ (1 − 1/β) > 6. The integral tends to F(µβ )
as t0 , s → ∞. Thus its growing coefficient tends to C∗ . The complementary propagator has
exponent 2η < σ; integrating its ancient exponential forcing and its future polynomial forcing
gives a complementary bound Ce2ηs for s ≥ 0. For s ≤ 0 the whole first coefficient is bounded
by Ceσs . Indeed its forcing has ancient rate βσ > σ and (α/εq )eσt0 is bounded. All bounds are
uniform in q.
Expand in εq and κ exactly as in (22), replacing α there by εq . In the recursion add −b1 ·∇gj−1,ℓ
for j ≥ 2 and add −b1 · ∇Wℓ for j = 1. The initial first coefficient is the one above and the
other initial coefficients vanish. The support and finite angular-sector properties are unchanged.
Induction gives pure-seed bounds Cejσs and positive-viscosity bounds C(1 + t)D ejσs , uniformly
in t0 : the first coefficient has just been estimated, and for j ≥ 2 the external forcing is at most
Ceσs e(j−1)σs , leaving the same nonresonant gap jσ − σ. Positive-viscosity resonances produce
only the finite absolute-time polynomial. Consequently the residual estimate (26) holds with
a(t) = εq eσ(t−t0 ) ,

a(0) = q 6+K∗ /β δc−1/β ,

a(Hq ) = δq /|C∗ |.

The residual contains no forcing-free term: the exact heat base is retained, and b1 · ∇ of its heat
Taylor remainder has the seed factor εq . Its bound is CκN +1 (1 + t)D a(t) by the displayed input
estimate.
The prescribed drift is identical in the actual and approximate equations, so it disappears
from the transport part of their Kato comparison. Choose M using the fixed exponent C1 as
before, and choose N with
3
(N + 1) > (6 + K∗ /β)r1 .
2
The exact same X error calculation yields CaM +1 plus a viscosity remainder o(δq ) at the cap. It
is uniform for κ ≤ q 3/2 . The higher pure-seed terms contribute O(δq2 ) and the positive-viscosity
correctors contribute O(κ(1 + Hq )D δq ). The complementary first-coefficient response at Sq is
εq e2ηSq = o(δq ). Its growing response gives exactly the asserted protected-coordinate cap. These
estimates also give the density bound.
For the whole moment, the integrated self-perturbation velocity is O(δq ) as in (33). The
additional prescribed drift has integral at most Cq 6 (1 + Hq )5 = o(δq ), by its ancient and future
bounds. The radial exponential Kato tail argument therefore applies unchanged. All formal
coefficients are supported in the fixed collars, and the leading growing coefficient has moment bϵ .
The complementary, nonlinear and viscosity errors estimated above, followed by the Gaussian
tail bound, prove the whole-moment cap. This completes the proof.
44

For coupled use of Theorem 4.12, the actual Biot–Savart field must satisfy bounds for the
harmonic remainder, core leakage and feedback. Over a logarithmic stage, a finite harmonic
expansion must be carried to an order whose omitted remainder beats the comparison exponent.
The exterior remainder in (56) records the spatial estimate; the time-dependent coupled estimates
are additional requirements.

5

Actual cores, localization and the cold packet

5.1

Actual viscous outer cores and the local harmonic field

This section proves a conditional estimate for the actual coloured Navier–Stokes evolution. Its
nonradial-density hypothesis is explicit; it is not deduced from material contours or from named
unstable coordinates. We identify x = (x1 , x2 ) with z = x1 + ix2 and identify a velocity with
u1 + iu2 . Thus
i
K(z) =
,
Jz = iz.
2πz̄
Let P denote circular averaging. For f ∈ L1 ∩ L∞ , splitting the Biot–Savart integral at distance
d and optimizing d gives
q
(62)
∥K ∗ f ∥∞ ≤ C ∥f ∥1 ∥f ∥∞ .
Indeed the near integral is at most Cd ∥f ∥∞ and the far integral is at most C ∥f ∥1 /d.
Lemma 5.1 (Fourfold growth at the origin). If f ∈ L1 (R2 ) ∩ L∞ (R2 ) is invariant under z 7→ iz,
then
|K ∗ f (z)| ≤ C ∥f ∥∞ |z|.
Neither zero circulation nor regularity of f is required.
Proof. For r = |z| > 0, the integral over |y| ≤ 2r is at most C ∥f ∥∞ r, by integrating |z − y|−1
over B3r (z). For |y| > 2r use the absolutely convergent expansion
X z̄ n
1
=−
.
z̄ − ȳ
ȳ n+1
n≥0

The inverse moment vanishes unless n + 1 is a multiple of four, by changing variables under a
quarter turn. The surviving degrees are 3, 7, 11, . . .. Their absolute sum is at most
Cr

3

|f (y)|
dy ≤ C ∥f ∥∞ r3
4
|y|>2r |y|

Z

Z ∞
2r

s−3 ds ≤ C ∥f ∥∞ r.

Termwise integration is justified by the integrable majorant C|f (y)|/|y| on the far region. At
z = 0 the velocity vanishes by rotational covariance and continuity.
Lemma 5.2 (Exit controlled only by radial drift). Let a diffusion satisfy
√
dXs = v(s, Xs ) ds + 2ν dBs ,
X0 = x,
and let σ be its first exit from Bb , where |x| < b. Suppose, before exit,
|v(s, y) · y/|y|| ≤ V (s)

(0 < |y| < b),

V ≥ 0,

Z T

D :=
0

45

V (s) ds < b − |x|.

Then

(b − |x| − D)2
P(σ ≤ T ) ≤ 2 exp −
8νT

!

.

(63)

There is no bound on the tangential component of v.
Proof. For λ > 0 set ϕ(y) = cosh(λ|y|). Its even power series shows that it is smooth at zero. In
dimension two,
∆ϕ = λ2 cosh(λr) +

λ
sinh(λr) ≤ 2λ2 ϕ,
r

v · ∇ϕ ≤ λV (s)ϕ,

using sinh a ≤ a cosh a for a ≥ 0. Therefore Itô’s formula shows that
2

R s∧σ

e−2νλ (s∧σ)−λ 0

V (τ ) dτ

ϕ(Xs∧σ )

is a nonnegative supermartingale. Stopping at T gives
P(σ ≤ T ) ≤

cosh(λ|x|) 2νλ2 T +λD
2
e
≤ 2e−λ(b−|x|−D)+2νλ T .
cosh(λb)

Choose λ = (b − |x| − D)/(4νT ). The stopped calculation remains valid for bounded drifts
smooth for positive times: stop first before their time endpoint and then pass to that endpoint.
No approximation of a radial drift bound is used.
Consider real, fourfold symmetric colours solving
∂t wl + uν · ∇wl = ν∆wl ,

uν = K ∗

X

wl ,

(64)

l

with geometric scales
Al = A1 β l−1 ,

Rl = R1 q l−1 ,

1 < βmin ≤ β ≤ βmax ,

q ≤ 14 ,

βq ≤ 21 .

Assume
∥wl (0)∥1 ≤ C0 Al Rl2 ,

∥wl (0)∥∞ ≤ C0 Al .

(65)

In the countable case assume that the initial series converges absolutely in L1 ∩ Lp , p > 2, and
use its mild Navier–Stokes solution and the associated linear coloured evolutions. The common
velocity is bounded and smooth for positive times. Norm contraction identifies its vorticity
with the sum of the colours, and (62) implies uniform convergence of the velocity series because
P
l Al Rl < ∞.
Fix an index i, a window T = H/Ai , and
κi =

ν
.
Ai Ri2

Suppose that for some fixed a0 > 0 every l ≤ i has an initial constant core,
wl (0) = cl

on Ba0 Rl ,

|cl | ≤ C0 Al .

(66)

Write hl = (I − P )wl and impose the actual density hypothesis
sup ∥hl (t)∥1 ≤ δAl Rl2

(l ≤ i),

0≤t≤T

46

0 < δ ≤ 1.

(67)

No nonradial hypothesis is imposed on l > i.
The scalar stochastic representation used below is
√

dXs = −uν (t − s, Xs ) ds +

wl (t, x) = E[wl (0, Xt )],

2ν dBs ,

X0 = x.

(68)

Its sign is verified by applying Itô’s formula to wl (t−s, Xs ): the drift is −∂t wl −uν ·∇wl +ν∆wl = 0.
For completeness, first stop at t − ε, where the coefficients are smooth. Boundedness of the drift
gives a continuous limiting process at t. Its transition densities away from starting time are bounded by C/[νs]. This bound follows by L1 contraction, the Nash inequality ∥f ∥42 ≤ C ∥f ∥21 ∥∇f ∥22 ,
the L2 energy identity, and the corresponding adjoint estimate, composed at the midpoint. Thus
wl (ε) → wl (0) in L1 suffices to pass to the endpoint in expectation. Approximation of the bounded initial colour in L1 by continuous functions also justifies evaluation at the limiting endpoint.
This verifies (68) for the actual colours without smoothing an asserted radial bound.
Proposition 5.3 (Uniform persistence of actual constant cores). Consider the finite or countable
coloured evolution and geometric scales specified above, with ν, H > 0. Assume (65), (66), and
(67) on [0, T ], where T = H/Ai . There is C ≥ 1, depending only on C0 , a0 , βmin , βmax , with the
following property. Set
D=

√
δ+

βq
δq 2
+
,
β − q 2 1 − βq

c∗ :=

9
,
512

ε∗ :=

a0 (β − 1)
.
16CH

If

a0
ε∗
2
,
4C0 e−c∗ a0 /(κi H) ≤ ,
16
2
then, with a = a0 /2, for l ≤ i and 0 < t ≤ T ,
CHD ≤

2

(69)

2

∥wl (t) − cl ∥L∞ (BaR ) ≤ 4C0 Al e−c∗ a0 Rl /(νt) .

(70)

l

All constants are independent of i, viscosity, and the number of colours.
Proof. The maximum principle gives ∥hl ∥∞ ≤ 2C0 Al . The radial average induces a purely tangential velocity, so (62) and (67) imply
√
|(K ∗ wl ) · er | ≤ ∥K ∗ hl ∥∞ ≤ C δAl Rl
(l ≤ i).
Every unrestricted inner colour has ∥K ∗ wk ∥∞ ≤ CAk Rk , whence
X

∥K ∗ wk ∥∞ ≤ CAl Rl

k>l

βq
.
1 − βq

(71)

Temporarily assume that all probe-core defects satisfy
∥wk − ck ∥L∞ (BaR ) ≤ εAk .
k

For k < l and |z| ≤ a0 Rl , one has |z| ≤ aRk /2. Decompose the older colour into
wk = ck 1BaRk + (wk − ck )1BaRk + wk 1R2 \BaR .
k

The first term gives exactly (ck /2)Jz. The second gives a velocity bounded by CεAk |z| by
Lemma 5.1. The radial part of the third has zero velocity inside the disk. Its remaining density
is hk 1R2 \BaR , whose L1 norm is at most δAk Rk2 . Its exterior series starts at degree three, so
k

|K ∗ (wk 1R2 \BaR )(z)| ≤ CδAk |z|3 /Rk2 .
k

47

(72)

Here and below the density is evaluated at its actual viscous time. Summing the self, inner and
older bounds gives


ν

|u (t, z) · er | ≤ CAl Rl

ε
D+
β−1



(|z| ≤ a0 Rl , l ≤ i).

(73)

We used the geometric sums
X Ak

2 ≤

k<l

Rk

Al q 2
,
Rl2 β − q 2

X

Ak ≤

k<l

Al
.
β−1

Fix C large enough for (73).
On a bootstrap interval with ε ≤ ε∗ , the integrated radial drift in the backward representation
is at most a0 Rl /8, because Al T ≤ H and (69) controls both terms. Start at |x| ≤ a0 Rl /2 and
stop on exiting Ba0 Rl . If the backward path does not exit, its endpoint belongs to the initial
constant core. The possible discrepancy outside it is at most 2C0 Al . Lemma 5.2 therefore gives
(3a0 Rl /8)2
|wl (t, x) − cl | ≤ 4C0 Al exp −
8νt

!

,

which is (70). Subtracting cl is legitimate: a constant is itself a bounded scalar solution, and
bounded stochastic representation does not require its integrability.
The maximum normalized probe defect over the finite set l ≤ i is continuous for positive time.
It tends to zero at time zero, since the probe has a fixed margin inside the initial constant disk and
the common drift is bounded. If it first attained ε∗ , the preceding estimate, Rl ≥ Ri , t ≤ H/Ai ,
and (69) would instead bound it by ε∗ /2. This closes the bootstrap. The finite maximum adds
no dependence on i; every infinite inner sum was already controlled by (71).
Proposition 5.4 (Harmonic exterior field and actual density leak). Under Proposition 5.3, set
bi =

X
l≤i

K ∗ wl ,

Ωi =

1X
cl ,
2 l≤i

Hi =

X

K ∗ (wl 1R2 \BaR ),
l

l≤i

Li =

X

K ∗ ((wl − cl )1BaRl ).

l≤i

For |z| ≤ aRi /2 one has the exact decomposition
bi (t, z) = Ωi Jz + Hi (t, z) + Li (t, z).

(74)

The field Hi is divergence free and curl free on BaRi , and
|Hi (t, z)| ≤ CδAi |z|3 /Ri2 ,
−c∗ a20 /(κi H)

|Li (t, z)| ≤ CAi e

|DHi (t, z)| ≤ CδAi |z|2 /Ri2 ,
|z|.

(75)
(76)

The cubic coefficient of Hi is exactly
i X
wl (t, y)
c3,i (t) = −
dy.
2π l≤i |y|≥aRl ȳ 4
Z

(77)

With ηl (t) = ∥hl (t)∥1 /(Al Rl2 ),
|c3,i (t)| ≤ C

i−1
Ai X
(q 2 /β)m ηi−m (t) ≤ CδAi /Ri2 .
Ri2 m=0

48

(78)

Furthermore,
|Hi (t, z) − c3,i (t)z̄ 3 | ≤ CδAi |z|7 /Ri6 ,

(79)

3

(80)

6

|D(Hi (t, z) − c3,i (t)z̄ )| ≤ CδAi |z| /Ri6 .
The derivative denotes the real spatial Jacobian. No derivative bound on Li is asserted.

Proof. The constant disks induce (cl /2)Jz in the specified region, proving (74). Exterior densities
induce curl free velocities away from their supports, and all Biot–Savart velocities are divergence
free. Radial averaging commutes with radial truncation, and an exterior radial density induces
zero velocity inside its truncation radius. Consequently, on this region,
Hi (t, z) = −

i X X 4m+3
z̄
2π l≤i m≥0

hl (t, y)
dy.
4m+4
|y|≥aRl ȳ

Z

The inverse radial moment is zero, so this gives exactly (77). The coefficient integral is at most
∥hl ∥1
.
(aRl )4
This proves (78).
The series and its first spatial derivatives converge uniformly on |z| ≤ aRi /2. Indeed each
series ratio is at most 1/16, and the derivative majorant adds only the factor 4m + 3. Thus,
for each colour, its harmonic field and derivative are bounded by CδAl |z|3 /Rl2 and CδAl |z|2 /Rl2 ,
respectively. Removing m = 0 gives CδAl |z|7 /Rl6 and CδAl |z|6 /Rl6 . Summing uses
X Al

≤
d

l≤i

Rl

Ai /Rid
1 − q d /β

(d = 2, 6).

These prove (75), (79), and (80). Finally Lemma 5.1, (70), and
(76).

l≤i Al ≤ Ai β/(β − 1) prove

P

1/4

Scope for the localized tail problem. If R∗ = κi Ri ≤ aRi /2, then throughout BR∗ ,
2

|bi − Ωi Jz| ≤ CδAi |z|3 /Ri2 + CAi e−c∗ a0 /(κi H) |z|.
2

The integrated linear-leak coefficient is at most CHe−c∗ a0 /(κi H) . The actual density leak is
therefore quantified; no material hole has been retained by assumption. The harmonic derivative
estimate (75) is available for a coupled transfer or energy calculation, while the core leak has
only the growth
√ estimate (76). The parameter margins are compatible with H ≍ log(1/q) and
κi = q 3/2 if δH is sufficiently small. This compatibility does not prove (67); obtaining that
hypothesis and controlling the coupled inner aggregate over its whole receiver window remain
separate nonlinear tasks. In particular no fixed-data nonunique vanishing-viscosity theorem is
concluded here.

5.2

A quantitative parabolic localization estimate

The next estimate concerns an actual convection–diffusion equation, and gives a quantitative
substitute for compact material support. Its local cubic drift hypothesis must still be verified for
a coupled Navier–Stokes cascade.
49

Lemma 5.5 (Localized tail exit). Let f solve
U (t, x) = Ω(t)x⊥ + V (t, x) + b(t, x),

∂t f + U · ∇f = ν∆f,

div U = 0,

on 0 ≤ t ≤ T , with ∥f (0)∥1 ≤ m and supp f (0) ⊂ Br . Suppose
∥V (t)∥∞ ≤ C0 m(νt)−1/2 ,
|b(t, x)| ≤ L|x|

for all x,
3

|b(t, x)| ≤ C3 |x| + ℓ(t)|x|

for |x| ≤ R∗ .

Assume ℓ ≥ 0, Ω is integrable, and
C3 R∗2 T +

r ≤ R∗ /4,

Z T
0

ℓ(t) dt ≤ 41 ,

C0

m√
νT ≤ R∗ /8.
ν

There are absolute c, C > 0 such that
cR2
sup
|f (t, x)| dx ≤ Cm exp − ∗
νT
t≤T |x|≥R∗
Z

!

.

For each fixed integer k ≥ 0 there are constants c, Ck > 0 such that
Z

k

|x|≥R∗

|x| |f (T, x)| dx ≤ Ck me

Ck LT

R∗k exp

cR2
− ∗
νT

!

,

√
provided additionally νT ≤ cR∗ . The conclusion holds for smooth drifts and for limits for
which the parabolic solution and the corresponding diffusion representation are obtained by smooth
approximation.
Proof. Kato’s inequality bounds |f | by the positive scalar solution from |f (0)|. Normalize this
initial mass and represent the positive solution by the diffusion
√
dXt = U (t, Xt ) dt + 2ν dBt .
A time-dependent rotation removes Ω(t)x⊥ ; the rotated Brownian motion remains standard and
all norm assumptions are unchanged. Let σ be the first exit from BR∗ . Before that exit, deterministic Gronwall applied to the integral stochastic equation gives
#

Rt "

√
m√
0
sup |Xs | ≤ e
r + 2C0
νt + 2ν sup |Bs | .
ν
s≤t∧σ
s≤t
√
The stated margins imply that σ ≤ T requires sups≤T |Bs | ≥ cR∗ / ν. The coordinate reflection
bound for planar Brownian motion yields
C3 R∗2 t+

ℓ

cR2
P(σ ≤ T ) ≤ C exp − ∗
νT

!

.

An endpoint outside BR∗ requires this exit, proving the mass bound.
For the moment assertion, condition at σ. The remaining drift has linear growth
bounded
√
by L|x| plus ∥V (t)∥∞ . Its integrated bounded contribution is at most 2C0 (m/ν) νT ≤ R∗ /4.
Gronwall and Brownian moments therefore give, uniformly in the exit time,
E |XT |k | Fσ ≤ Ck eCk LT R∗k




on {σ ≤ T }.

Multiply by the indicator of this event and use its probability bound. Finally restore the initial
mass, which is at most m.
50

1/4

For a receiver with κi = ν/(Ai Ri2 ) and stage duration T = H/Ai , the choice R∗ = κi Ri
gives
R∗2
1
1/2
C3 R∗2 T ≤ Cxi κi H if C3 ≤ CAi xi /Ri2 .
= 1/2 ,
νT
κ H
i

A global bound L ≤ CAi costs only eCk H , independent of depth. The exponential exit factor
dominates that cost as κi ↓ 0 for logarithmic stage lengths. This parameter calculation does not
verify the required local cubic hypothesis for the actual foreign field.

5.3

Nonlinear inner-tail suppression through a receiver window

The estimate in this section is conditional on quantitative information about the actual foreign
velocity. It closes a longer-time nonlinear subproblem than the isolated-tail and first-diffusiontime estimates. It does not establish the required nonradial-density bound for the active and
older outer colours.
Write J(x1 , x2 ) = (−x2 , x1 ), z = x1 + ix2 , and
Jx
K(x) =
,
2π|x|2

Z

B4 (f ) =

z̄ 4 f (z) dz.

R2

A scalar is fourfold symmetric if f (Jx) = f (x); a vector field has the corresponding covariance if
b(t, Jx) = Jb(t, x). Constants denoted by C can change from line to line.
5.3.1

A Gaussian-energy estimate for a globally weak foreign field

Proposition 5.6 (Weak foreign field through a finite window). There are universal constants
c, λ0 , C > 0 with the following property. Let p > 2, and let f0 ∈ L1 ∩ Lp be real, fourfold
symmetric, supported in B1 , and satisfy
Z

f0 = 0,

B4 (f0 ) = 0,

m = ∥f0 ∥1 .

Suppose b is divergence free, has fourfold covariance, and, after removal of a time-dependent solid
rotation, satisfies
|b(s, x)| ≤ ℓ(s)|x|,

|b(s, x)| ≤ d|x|3 ,

(1 + s)ℓ(s) ≤ λ0 .

Let f solve
∂s f + (K ∗ f + b) · ∇f = ∆f.
For S ≥ 2, assume the displayed hypotheses on 0 ≤ s ≤ S − 1 and
mS 1/25 1 + log S ≤ c.
p

Then
sup

h

(81)
i

|B4 (f (s))| ≤ C m2 S 7/50 (1 + log S) + mdS 102/25 .

(82)

0≤s≤S−1

The constants are independent of a decomposition of the datum into smaller scales. The assertion
holds for smooth drifts and for the parabolic limits obtained by approximation with the stated
bounds.

51

Proof. Scalar contraction and the divergence-free Nash estimate give
∥f (s)∥1 ≤ m,

√
∥K ∗ f (s)∥∞ ≤ Cm/ s.

∥f (s)∥∞ ≤ Cm/s,

The last estimate follows by splitting Biot–Savart at a radius and optimizing. Take s0 = 1/100.
The positive scalar solution from |f0 | dominates |f |. Its diffusion representation, after removing
solid rotation, obeys
"
λ0 s

|Xs | ≤ e

#

√
√
1 + Cm s + 2 sup |Ba |

(0 ≤ s ≤ s0 ).

a≤s

The Gaussian exponential moment of the Brownian supremum is finite at this fixed small time.
With β = 3/10 it follows that
Z

2

eβ|x| /(1+s0 ) |f (s0 , x)| dx ≤ Cm.

Multiplication by the Nash L∞ bound gives a weighted L2 norm O(m). Fixed polynomial absolute
moments are O(m) on [0, s0 ]. The exact harmonic-moment equation and symmetrization of selfadvection therefore yield
|B4 (f (s0 ))| ≤ C(m2 + md).
(83)
For clarity, the self contribution is bounded by Cm
|x|2 |f |, because |z̄ 3 − ȳ 3 |/|z − y| ≤ C(|z|2 +
R
2
|y| ); the foreign contribution is bounded by d |x|6 |f |. These bounds also prove the result
directly if the final time is smaller than s0 .
For s ≥ s0 put
√
√
√
τ = log(1 + s), y = x/ 1 + s, w(τ, y) = (1 + s)f (s, x), E(τ, y) = 1 + s b(s, 1 + s y).
R

Then
L = ∆ + 12 y · ∇ + 1,

wτ = Lw − (K ∗ w + E) · ∇w,

with |E| ≤ λ(τ )|y|, λ ≤ λ0 , and |E| ≤ a(τ )|y|3 , where a(τ ) = de2τ . Let
2

Fβ = eβ|y| /2 w

2

2

Dβ = eβ|y| /2 ∇w

,

2
2

,

β = 3/10.

Before using the full Gaussian weight, we establish membership in its energy space. Put τ0 =
log(1 + s0 ) and, for R ≥ 1, set
ΦR (t) =

R2 t
,
R2 + t

θR (t) = Φ′R (t) =

R4
,
(R2 + t)2

2

1/2

ρR (y) = eβΦR (|y| ) ,

gR = ρR w.

2

These weights are bounded and smooth, 0 ≤ θRR≤ 1, and ρR ↑ eβ|y| . The short-time exponential
moment and Nash bound proved above give ρR w(τ0 )2 ≤ Cm2 , uniformly in R. For each
bounded weight, spatial cutoffs and integration by parts give the exact identity
⟨w, Lw⟩ρR = − ∥∇gR ∥22 + 21 ∥gR ∥22 − β

Z

2
|y|2 θR (|y|2 ) 12 − βθR (|y|2 ) gR
dy.



−1/2

Here the second derivatives of log ρRRcancel after substituting w = ρR gR . With cβ = β(1/2 −
2 . The divergence-free advection contributes
β) > 0, the last term is at most −cβ |y|2 θR gR
Z

β

2
(K ∗ w + E) · y θR gR
dy.

52

For τ ≥ τ0 , the Nash estimate
and the similarity change give ∥K ∗ w∥∞ ≤ Cs0 m. The foreign
R
2 , while Young’s inequality gives
contribution is at most βλ0 |y|2 θR gR
β ∥K ∗ w∥∞

Z

cβ
2
|y|θR gR
≤

Z

2
|y|2 θR gR
+ Cβ ∥K ∗ w∥2∞ ∥gR ∥22 .

4

Choose λ0 so that βλ0 ≤ cβ /4. It follows that, with constants independent of R,
cβ
2
2
1 d
2 dτ ∥gR ∥2 ≤ − ∥∇gR ∥2 − 2

Z

2
|y|2 θR gR
+ C(1 + m2 ) ∥gR ∥22 .

Gronwall gives a uniform bound on every fixed finite time interval. Monotone convergence and
2
2
weak lower semicontinuity show that Fβ is finite there, and that ∇(eβ|y| /2 w) and |y|eβ|y| /2 w are
square integrable in space-time. Thus Dβ is locally integrable in time. The transformed equation
has its time derivative in the dual of the space with norm ∥g∥2 + ∥∇g∥2 + ∥|y|g∥2 : the linear
drift and resulting quadratic potential define bounded bilinear forms on that space. The resulting
energy identity supplies the continuous weighted-L2 representative. Cutoffs in the now integrable
full weight justify the identities and energy inequalities below. This preliminary argument uses
no oscillator gap for the bounded weights.
2
Writing g = eβ|y| /2 w gives the exact identities
⟨w, Lw⟩β = 12 ∥g∥22 − ∥∇g∥22 − β(1/2 − β) ∥|y|g∥22 ,
Dβ = ∥∇g∥22 + β 2 ∥|y|g∥22 + 2β ∥g∥22 .
For aβ =

p

β(1/2 − β), completing squares in

P

2
j ∥∂j g + aβ yj g∥2 proves

Hβ := ∥∇g∥22 + β(1/2 − β) ∥|y|g∥22 ≥ 2aβ ∥g∥22 =

√
6
∥g∥22 .
5

Moreover Dβ ≤ 3Hβ : one can first bound it by 23 Hβ + 53 ∥g∥22 and then use the last inequality.
3
Hβ + 47
Splitting Hβ into 50
50 Hβ yields
1
1 2
⟨w, Lw⟩β ≤ − 50
Dβ + 25
Fβ .

(84)

√

6
1
Here 12 − 47
50 5 < 25 .
The Hardy–Littlewood–Sobolev estimate and Gaussian weighted Hölder give

∥K ∗ w∥4 ≤ C ∥w∥4/3 ≤ Cβ Fβ .
The two-dimensional Gagliardo–Nirenberg inequality gives
1/2

1/2

∥g∥4 ≤ C ∥g∥2 ∥∇g∥2 .
p

Since ∥|y|g∥2 + ∥∇g∥2 ≤ Cβ Dβ , self-advection contributes at most
3/2

3/4

CFβ Dβ

≤ Cβ Fβ Dβ
2

to the weighted energy. The foreign contribution is β E · y eβ|y| w2 , bounded by Cβ λDβ . Thus,
while Fβ is sufficiently small,
2 ′
1
1
1 2
2 (Fβ ) ≤ − 100 Dβ + 25 Fβ .
R

53

The initial weighted norm and a continuity bootstrap using (81) prove
Z log S

Fβ (τ ) ≤ Cmeτ /25 ,

Dβ (τ ) dτ ≤ Cm2 S 2/25 .

(85)

log(1+s0 )

Let P be angular averaging, G = P w, and h = (I − P )w. Every ordinary moment of h of
degree at most three vanishes: angular averaging removes radial moments, and fourfold symmetry
removes all other moments in these degrees. In
2

X = L2 (e|y| /4 dy),

Dh = ∥∇h∥2X ,

H = ∥h∥X ,

one consequently has
Dh ≥ 3H 2 .

(86)
2

Indeed L is self-adjoint in X, and its Hermite basis is ∂ α G, with G = (4π)−1 e−|y| /4 and eigenvalue
−|α|/2. Orthogonality to degrees at most three is equivalent to the stated ordinary-moment cancellations, because the corresponding dual Hermite polynomials span these polynomial degrees.
Expansion in this basis gives ⟨h, Lh⟩X ≤ −2H 2 ; since the left side is −Dh +H 2 , (86) follows. This
elementary Hermite expansion also follows from the tensor one-dimensional harmonic-oscillator
factorization.
Angular averaging is orthogonal for all the radial weights used here, and commutes with L.
Writing Va = K ∗ a, the angular equation is
hτ = Lh − (I − P ) div[VG h + Vh G + Vh h + EG + Eh].
The radial field VG is tangential, so its weighted energy contribution is zero. The h–h and E–h
contributions are bounded by CHDh and CλDh . Put DG = ∥∇G∥2X ≤ Dβ . Weighted integration
by parts, unweighted HLS, and GN give
1/2

1/4

1/2

|⟨h, − div(Vh G)⟩X | ≤ CHFβ DG Dh

1/2

≤ ηDh + Cη H 2 Fβ DG .

The strict Gaussian margin β − 1/4 = 1/20 absorbs cubic multiplication:
2

e|y| /8 EG

2

≤ Ca(τ )Fβ .

Consequently the E–G contribution is bounded by ηDh + Cη a(τ )2 Fβ2 . Take η and the bootstrap
constants sufficiently small. Using (86) yields
1/2

2
2
2 2
(H 2 )′ ≤ − 19
5 H + CFβ DG H + Ca(τ ) Fβ .

By (85) and Cauchy–Schwarz,
Z

1/2

Fβ DG dτ ≤ Cm2 S 2/25 1 + log S ≤ Cc2 .
p

Variation of constants therefore gives
h

i

H(τ ) ≤ Cm e−19τ /10 + de51τ /25 .
Factors depending on the fixed initial time are absorbed in C.

54

(87)

Finally B4 (f (s)) = e2τ B4 (w(τ )). Harmonicity cancels diffusion, and differentiating e2τ cancels
the dilation term. Radial self-advection vanishes. HLS and Gaussian weighted Hölder show that
d
B4 (f (eτ − 1)) ≤ Ce2τ [Fβ H + a(τ )Fβ ].
dτ
Substitution of (85) and (87), followed by integration, gives a first term Cm2 S 1/25+1/10 (1 + log S)
and remaining terms at most CmdS 102/25 , using m ≤ 1. Together with (83), this proves the
proposition.
The bounded-weight argument and the subsequent uniform estimates also justify passage from
smooth data and drifts to the stated parabolic limits. No initial L∞ norm enters a constant.
5.3.2

Comparison with the actual local foreign field

Proposition 5.7 (Local harmonic field and a rough small leak). Let ν, r, T, R∗ > 0, RR∗ ≥ 4r,
and p > 2. Let f0 ∈ L1 ∩ Lp be real, fourfold symmetric, supported in Br , and satisfy f0 = 0
and B4 (f0 ) = 0. Put M = ∥f0 ∥1 , and let f solve on [0, T ]
ft + (K ∗ f + b) · ∇f = ν∆f.
After removal of a solid rotation, suppose b is divergence free, has fourfold covariance, and admits,
on B2R∗ , the decomposition
b = bh + e,

|b(t, x)| ≤ L|x|

3

globally,
2

|bh (t, x)| ≤ C3 |x| , |Dbh (t, x)| ≤ C3 |x| ,

(88)

|e(t, x)| ≤ ϵL|x|.
where bh is divergence free with fourfold covariance and 0 ≤ ϵ ≤ 1. All bounds hold on 0 ≤ t ≤ T .
Its streamfunction is defined on B2R∗ . Assume
√
√
C3 R∗2 T + ϵLT ≤ 14 ,
C(M/ν) νT ≤ R∗ /8,
νT ≤ cR∗ .
(89)
Take a fixed radial cutoff of that streamfunction, equal to one in BR∗ and zero outside B2R∗ , and
denote its velocity by eb. Let g solve the same self-consistent equation as f , with foreign field eb
and the same initial datum. Set ℓ = CC3 R∗2 . If M/ν and ℓ(r2 /ν + T ) are sufficiently small, then
h

q

sup |B4 (f (t)) − B4 (g(t))| ≤ CM R∗4 eCLT ϵL T + r T /ν



t≤T

q

2

i

+ (1 + LR∗ T /ν)e−cR∗ /(νT ) .

(90)

No derivative bound on the leak e is required. The assertion holds for smooth drifts and for
parabolic limits obtained by approximation with the stated bounds.
Proof. Choose the streamfunction ψh zero at the origin. Integration along radial segments gives
|ψh (x)| ≤ CC3 |x|4 . A streamfunction cutoff gives
|eb| ≤ CC3 |x|3 ,

|eb| ≤ ℓ|x|,

Deb

∞

≤ ℓ.

It is divergence free, has fourfold covariance, and equals bh in BR∗ . A second useful bound is
|eb(x)| ≤ CL|x|: from ϵ ≤ 1 and (88), |bh | ≤ 2L|x| on B2R∗ , so |ψh | ≤ CL|x|2 there, and cutoff
derivatives preserve this linear growth. Thus this bound does not require C3 R∗2 ≤ L.
55

We first verify the velocity-gradient estimate needed for comparison. Normalize r = ν = 1
and write m = M/ν; the cutoff growth and Lipschitz constant are normalized in the same way.
Nash gives
√
∥g(t)∥∞ ≤ Cm/t,
∥ug (t)∥∞ ≤ Cm/ t,
ug = K ∗ g.
For Mk (t) =

R

|x|k |g(t, x)| dx, Kato’s inequality gives
M2′ ≤ 4m + Cmt−1/2 mM2 + 2ℓM2 ,
p

3/4

M4′ ≤ 16M2 + Cmt−1/2 m1/4 M4

+ 4ℓM4 .

Since M2 (0), M4 (0) ≤ m, a continuity bootstrap gives
M4 (t) ≤ Cm(1 + t)2

M2 (t) ≤ Cm(1 + t),

when m and ℓ(1 + T ) are small. On these trial bounds the integrated self terms are respectively
O(m2 (1 + t)) and O(m2 (1 + t)2 ); ℓT ≪ 1 absorbs the foreign terms. Hence
√
√
∥xg(t)∥4 ≤ Cmt−3/4 1 + t.
∥xg(t)∥1 ≤ Cm 1 + t,
The second follows from ∥xg∥44 ≤ ∥g∥3∞ M4 .
With Ht the heat kernel, scaling and the smooth kernels’ exterior decay give
∥∇K ∗ Ht ∥∞ ≤ Ct−1 ,
∥∇K ∗ div Ht ∥∞ ≤ Ct−3/2 ,

∥∇K ∗ div Ht ∥1 ≤ Ct−1/2 ,

∥∇K ∗ div Ht ∥4/3 ≤ Ct−3/4 .

For the nonlinear differentiated kernel the tail is O(|x|−3 ), so the last three norms are finite before
scaling. The self flux ug g has L1 norm Cm2 t−1/2 and L∞ norm Cm2 t−3/2 . Split the heat-mild
time integral at t/2, using the kernel’s L∞ norm first√and its L1 norm second; its contribution
√
to ∇ug is Cm2 /t. The foreign flux has L1 norm Cℓm 1 + t and L4 norm Cℓmt−3/4 1 + t. Use
the kernel’s L∞ norm first and its L4/3 norm second. This contribution is
√
Cℓmt−1/2 1 + t ≤ Cm/t,
because ℓ(1 + T ) is small. The initial heat contribution is Cm/t. Undoing normalization proves
∥∇ug (t)∥∞ ≤ C(M/ν)/t.

(91)

Put δ = f − g, v = K ∗ δ, and F = (b − eb)f . Then
δt − ν∆δ + (uf + eb) · ∇δ + v · ∇g = − div F.
Let ψ be a streamfunction of δ. Testing in H −1 gives
2 ′
2
1
2 (∥v∥2 ) + ν ∥δ∥2 ≤ (∥∇ug ∥∞ +

Deb

∞

) ∥v∥22 + ∥v∥2 ∥F ∥2 .

(92)

Here v · ∇ψ = 0. Thus the term v · ∇g vanishes when paired with ψ, and the same identity
replaces uf = ug + v by ug in the other self transport term. Integration by parts bounds that
term by ∥∇ug ∥∞ ∥v∥22 . The identical calculation applies to eb. The circulation of δ is zero, and its
polynomial moments are finite, so v ∈ L2 and these tests are justified by approximation.

56

Both solutions converge to their common datum in L1 ∩ Lp as t ↓ 0. Interpolation, using
p > 2, therefore gives ∥δ(t)∥2 → 0. Their first absolute moments are uniformly
bounded near
R
zero by the Kato moment estimates and the linear-growth bounds. Since δ = 0,
b ξ)| ≤ |ξ|
|δ(t,

Z

|x| |δ(t, x)| dx.

b ξ)|2 dξ, the part with |ξ| < a is O(a2 ) uniformly near
In the Fourier identity ∥v∥22 = C |ξ|−2 |δ(t,
zero. The part with |ξ| ≥ a is at most Ca−2 ∥δ(t)∥22 . First taking t ↓ 0 and then a ↓ 0 proves
∥v(t)∥2 → 0.
The true coefficient in (92) is integrable at time zero, although the upper bound (91) is
singular there. Indeed g0 ∈ Lp , p > 2, gives from the same mild formula
R

∥∇ug (t)∥∞ ≤ C0 (t−1/p + t1/2−1/p ),

0 < t ≤ 1.

The constant may depend on the actual initial Lp norm and on the bounded cutoff drift. The
′
kernel norms are Ct−1/p and Ct−1/2−1/p in Lp ; the latter time singularity is integrable, and
(ug + eb)g has bounded Lp norm near zero by contraction and the initial L1 –Lp Biot–Savart
bound. Together with the initial trace just proved, this starts the integrating factor in (92) from
v(0) = 0. Only after writing its source integral do we use (91) on [s, t], obtaining
∥v(t)∥2 ≤

Z t
0

(t/s)CM/ν eℓ(t−s) ∥F (s)∥2 ds.

(93)

Common smooth approximations have bounded initial Lp norms, so no arbitrary initial term at
a positive time is discarded in this argument.
The localization estimate, applied at each time s ≤ T , gives
Z

2

|x|≥R∗

|x|2 |f (s, x)| dx ≤ CM eCLs R∗2 e−cR∗ /(νs) .

The cutoff field and the actual field have global growth CL|x|. Combining this tail estimate with
Nash yields
R∗
2
(b − eb)f 1|x|≥R∗ ≤ CLM √ eCLs e−cR∗ /(νs) .
2
νs
Inside BR∗ the difference of drifts is exactly e. A weighted Kato estimate gives
CM (r2 + νs)eCLs . Consequently
s

ef 1|x|<R∗

2

≤ CϵLM

R

|x|2 |f (s)| ≤

r2 + νs CLs
e
.
νs

Substitute these bounds into (93). Since CM/ν < 1/4, integration of s−1/2−CM/ν is legitimate,
and yields, uniformly for t ≤ T ,


q

q

2



∥v(t)∥2 ≤ CM eCLT ϵL(T + r T /ν) + LR∗ T /ν e−cR∗ /(νT ) .

(94)

The bounded factor eℓT is absorbed into C.
For a radial cutoff χ∗ equal to one in BR∗ and zero outside B2R∗ ,
4

∇(χ∗ z̄ )

2

Z

≤ CR∗4 ,

χ∗ z̄ 4 δ ≤ CR∗4 ∥v∥2 .

Both f and g satisfy the localized exit estimate, with the margins (89). Their exterior absolute
2
fourth moments are at most CM R∗4 eCLT e−cR∗ /(νT ) . Combining this with (94) proves (90).
57

5.3.3

Physical normalization, the available power margin, and scope

Set
m = M/ν,

d = CC3 r4 /ν,

S = 2 + νT /r2 .

To supply the full time interval required by Proposition 5.6, extend the cutoff comparison field eb
by zero on (T, T + r2 /ν] and continue its solution g there. This extension preserves divergence,
fourfold covariance, and the cubic and linear growth bounds. It is a bounded measurable-intime, spatially Lipschitz drift and is covered by the parabolic approximation formulation. On the
original interval the normalized weak-field coefficient is ℓnorm = (r2 /ν)ℓ and satisfies
(1 + s)ℓnorm (s) ≤ ℓ(r2 /ν + T ),
and it is zero afterwards. Consequently the same smallness hypothesis supplies the normalized
drift through s = S − 1; the comparison on [0, T ] is unchanged. The space-time change x = ry,
t = r2 s/ν, and f (t, x) = νr−2 F (s, y) gives these quantities exactly; moreover B4 (f ) = νr4 B4 (F ).
Applying Proposition 5.6 to the cutoff-field solution and Proposition 5.7 to the actual solution
gives
"

sup |B4 (f (t))| ≤ CM r

4

t≤T

M 7/50
C3 r4 102/25
S
(1 + log S) +
S
ν
ν
q

h

#

q

2

i

+ CM R∗4 eCLT ϵL(T + r T /ν) + (1 + LR∗ T /ν)e−cR∗ /(νT ) .

(95)

This requires (81) and the small normalized linear-growth bound of the cutoff field in addition
to the local hypotheses. No cutoff has been applied to the actual nonlinear solution.
For this geometric cascade budget fix β and CH > 0, assume 0 < δ ≤ 1 and 1 ≤ H ≤
CH log(1/q), and take a receiver (Ai , Ri ) with
r = qRi ,

M ≤ CβAi q 2 Ri2 ,

κ=

ν
= q 3/2 ,
Ai Ri2

T = H/Ai ,

R∗ = κ1/4 Ri .

Suppose the actual outer-colour estimates furnish
L ≤ CAi ,

C3 ≤ CδAi /Ri2 ,

ϵ ≤ Ce−c/(κH) .

Then m = O(βq 1/2 ), d = O(δq 5/2 ), and S = O(q −1/2 H). Dividing the first line of (95) by Ai Ri6
gives


O β 2 q 643/100 H 7/50 (1 + log S) + βδq 323/50 H 102/25 .
(96)
The powers are 6.43 and 6.46. The localization terms are smaller than every q-power for logarithmic H, since
R∗2
1
=√
,
νT
κH

LT = O(H),

√
C3 R∗2 T = O(δ κH).

The remaining tail margins follow from




ℓ(r2 /ν + T ) = O δ[q 5/4 + q 3/4 H] ,
√
√
√
√
(M/ν) νT
νT
7/8
= O(βq
H),
= q 3/8 H,
R∗
R∗




mS 1/25 1 + log S = O βq 12/25 H 1/25 1 + log S .
p

p

58

3/2

Also r/R∗ = q 5/8 and ϵLT = O(He−c/(q H) ). All these quantities tend to zero uniformly in the
stated horizon range. Writing Lq = log(1/q), the two terms of (96) are bounded by
Cβ 2 q 643/100 L57/50
+ Cβδq 323/50 Lq102/25 = o(q 6 ),
q
uniformly for 0 < δ ≤ 1 and 1 ≤ H ≤ CH Lq . A fixed prefactor in the support radius changes
only the constants.
The actual outer-core argument supplies the required harmonic-plus-leak hypothesis conditionally on the density bound
sup ∥(I − P )wl (t)∥1 ≤ δAl Rl2 ,

l ≤ i.

t≤T

Its harmonic exterior multipole series can be differentiated to give |Dbh (x)| ≤ CC3 |x|2 throughout
B2R∗ when this disk lies strictly inside the constant-core probe region; the exponentially small core
density defect contributes the rough leak. When these inputs are obtained from Proposition 5.3,
its core margins (69) must also hold. Thus the present section closes full-window nonlinear innertail suppression under this explicit outer-density input. It does not prove that input for the active
colour and older seeds, a signed outward transfer, or distinct vanishing-viscosity accumulation
paths. The zero initial fourth moment used here is the one of the prescribed initial inner stack:
P
termwise integration is justified by l Al Rl6 < ∞. A restart from a later state retains its actual
fourth moment. If only B4 (f0 ) = 0 is dropped while all other hypotheses of Proposition 5.6
remain in force, the same proof adds |B4 (f0 )| to (82). A noncompact restarted state additionally
requires a separate Gaussian-space entry estimate.
Remark 5.8 (Selected-cutoff scope). The displayed q 6.43 and q 6.46 budget uses κi = q 3/2 at the
selected cutoff level. At later, larger receivers the same physical viscosity has a smaller normalized
value; their aggregate of smaller colours includes the intended active child. The budget is not
an erasure estimate for that entire aggregate at every later stage. After a source leaves its small
cap neighbourhood, its constant-core probe on a canonical source window requires a separate
comparison; it is not a consequence of the small-density hypothesis over that post-cap window.

6

The actual inactive prefix

6.1

Fixed seeds: separate initial and forced prefix contributions

The following scalar comparison separates the common initial seeds from the forced response. Its
coupling inequality is a hypothesis.
Lemma 6.1 (Two prefix norms). Let β ≥ 4, 0 < q < 1, S ≥ 0, b ≥ 0, and Ai = β i−1 . Fix
a receiving index j ≥ 2. Suppose nonnegative continuous functions xi , 1 ≤ i < j, satisfy, on
0 ≤ θ ≤ S,
xi (θ) ≤ M beλ0 θ + M

Z θ
0



X
eλ0 (θ−s)  Kil xl (s) + fi (s) ds,
l<j

λ0 =

σ+η
,
β

where M, C, σ > 0, η, r∗ > 0, fi ≥ 0 are integrable, and the nonnegative matrix K obeys
(

Kii ≤ Cr∗ /β,

Kil ≤

Cβ l−j q 2(i−l) ,
Cβ l−j q 6(l−i) ,

59

l < i,
i < l < j.

(97)

Assume βq 6 < 1. Define
C
rB =
β

βq 6
q 2 /β
r∗ +
+
1 − βq 6 1 − q 2 /β

!

,

λB = λ0 + M rB .

Then xi ≤ hi + pi , where the initial contribution satisfies
max hi (θ) ≤ M beλB θ .

(98)

i<j

For D > β with Dq 8 < β, put wi = (Dq 6 )j−i and
C
rP =
β

β
Dq 8
r∗ +
+
D − β β − Dq 8

!

,

λP = λ0 + M rP .

The forced contribution satisfies
pi (θ)
≤M
i<j
wi

Z θ

P (θ) := max

fi (s)
ds.
i<j
wi

eλP (θ−s) max

0

(99)

All constants are independent of j. One can choose η, r∗ small, then D large and q small, so both
λB and λP are strictly smaller than σ/2.
Proof. The row sums of K are bounded by rB : for l = i + d the inner coefficient is at most
Cβ i−j (βq 6 )d , and for l = i − d the outer coefficient is at most Cβ i−j (q 2 /β)d . Use β i−j ≤ 1/β and
sum the geometric series.
Solve the positive comparison system
Y ′ = (λ0 I + M K)Y + M f,

Y (0) = M b1.

Iteration of (97), or its integral comparison principle, gives x ≤ Y . The variation-of-constants
formula splits Y = h + p, with
(λ0 I+M K)θ

h=e

Z θ

M b1,

p=M

e(λ0 I+M K)(θ−s) f (s) ds.

0

The unweighted maximum norm gives (98). For the weighted maximum norm, the inner row
acquires (β/D)d and the outer row acquires (Dq 8 /β)d . Their geometric sums give rP and hence
(99). Since λ0 is close to σ/β ≤ σ/4, first make the diagonal term small. Increasing D makes
the remaining q-independent term in rP small; subsequently decreasing q makes all other terms
small.
Corollary 6.2 (The fixed-seed baseline is small in its actual field). Assume the hypotheses of
Lemma 6.1, with λB ≤ λ∗ < σ/2, in the clock θ = Aj t. Suppose H ≥ 0 and the completed cap
clocks satisfy
Tj,N ≤ S/AN + H

N
−1
X

A−1
m ,

N ≥ j,

S ≥ Hβ/(β − 1).

m=j

Let 0 ≤ δ ≤ δ∗ , where δ∗ < ∞ is fixed, and put b = δe−σS . Then through Tj,N the initial
contribution is bounded uniformly by
B = M δe−(σ−λ∗ )S .
60

Its normalized inward feedback on the receiver j is at most
C

X

β i−j q 2(j−i) hi ≤ CB

i<j

q 2 /β
.
1 − q 2 /β

(100)

The forced contribution has feedback at most
CP

Dq 8 /β
.
1 − Dq 8 /β

In particular, for a fixed h0 ≥ 0 and H = h0 log(1/q) one can take S = C0 log(1/q) so that B =
O(q 6 ) and its feedback is O(q 8 ), with constants independent of depth and uniform for 0 ≤ δ ≤ δ∗ .
Proof. Writing a = β j−N ∈ (0, 1] gives
Aj Tj,N ≤ Sa +

Hβ
(1 − a) ≤ S.
β−1

Apply (98). The actual inward row in (100) is a geometric series with ratio q 2 /β. For pi ≤ wi P its
ratio is Dq 8 /β. Choose a fixed C0 ≥ h0 β/(β −1) with C0 (σ −λ∗ ) > 6. Then B ≤ M δ∗ q C0 (σ−λ∗ ) =
O(q 6 ), and its receiving-field row adds a factor O(q 2 ). If Dq 6 < 1, the local chart bound follows
from xi ≤ B + wi P ≤ B + Dq 6 P whenever the latter quantity is below r∗ .
For the coloured Navier–Stokes evolution, the spatial q 2 and q 6 rows and the forced term must
be estimated separately; see Proposition 6.4 and Section 6.3.
Remark 6.3 (A generic smooth-profile self rate). The same positive-matrix proof works with
λ0 = γ/β for any fixed γ > 0. Choose β > 4γ/σ, then choose the foreign-row parameters so
M rB and M rP are less than σ/4 − γ/β. For a differential inequality whose diagonal rate is
at most γβ i−j , use γ/β as its common upper bound. Thus a fixed-profile Kato rate can be
used independently of the unstable eigenvalue. The transform selection in Lemma 4.10 permits
choosing a good β in any fixed sufficiently large interval, so this parameter change causes no
conflict with the isolated source transform.

6.2

Actual inactive densities and a large amplitude ratio

Fix the smooth radial reference W and p > 2. The inactive comparison uses the actual coloured
Navier–Stokes equation without differentiating an inactive colour. Its fixed-profile transportstability rate is made small in the receiver clock by choosing the amplitude ratio sufficiently
large. The first two closure corollaries assume an explicit packet bound; Section 6.3 derives a
bound from the actual packet.
Let P denote angular averaging. Use
Al = A1 β l−1 ,
h

Rl = R1 q l−1 ,
−2

i

Gl (t, x) = Al eνtRl ∆ W (x/Rl ),
ζl = wl − Gl ,

hl = (I − P )wl = (I − P )ζl .

For any density v put
∥v∥Xl =

∥v∥p
∥v∥1
+
,
2
2/p
Al Rl
Al R
l

61

xl (t) = ∥ζl (t)∥Xl .

Assume the initial colours are real, fourfold symmetric, have zero circulation, are supported in
Bc0 Rl , equal a constant in Ba0 Rl , and satisfy
∥wl (0)∥1 ≤ C0 Al Rl2 ,

∥wl (0)∥∞ ≤ C0 Al .

Here a0 , c0 , C0 are fixed. For the smooth seeded family these facts follow from the fixed compact
derivative collars. The series of colours is absolutely convergent in L1 ∩ Lp , and βq < 1, so its
common velocity and coloured evolution are well defined as before.
Choose a fixed dimensionless annulus A and d∗ > 0 such that
dist(supp ∇W, R2 \ A) ≥ d∗ .
Set Al = Rl A. Choose q afterwards so that these annuli lie inside the older constant-core probes
and are separated from the small receiver truncation balls used below.
Proposition 6.4 (Stopped inactive row with explicit parabolic errors). There are constants
γ, C, c > 0, depending only on the fixed smooth profile, p, and the displayed initial prefactors,
with the following property. Fix j ≥ 2 and
T ≤ S/Aj ,

κj =

ν
≤ q 3/2 ,
Aj Rj2

1 ≤ S ≤ C1 log(1/q).

Stop at the first time any i < j reaches xi = δ. Assume β ≥ 4, 0 < q < 1, βq ≤ 1/2, 0 < δ ≤ 1,
xi (0) < δ for i < j, q sufficiently small, and
√
(101)
CS( δ + βq) ≤ c.
In the clock θ = Aj t, every inactive i < j obeys
X
X
d+ xi
≤ γβ i−j xi + C
β l−j q 2(i−l) xl + C
β l−j q 6(l−i) xl + fi (θ) + Ej−i ,
dθ
l<i
i<l<j

where



fi (θ) =

(102)



1
1Ai  K ∗ hl  · ∇Gi
Aj
l≥j
X

at t = θ/Aj
Xi

is the actual current/deep nonradial forcing. With
"

CS

em = Ce

#

cq −2m
exp −
,
κj S

m ≥ 1,

one can take
h

Ek = C β −k ek +

X
r>k

β −r er +

X

β −r q 6(k−r) er + ek

1≤r<k

X

i

β −r q k−r .

(103)

1≤r<k

The sums may be enlarged to infinite sums of nonnegative terms. No small nonradial-density
assumption is imposed on colours l ≥ j.

62

Proof. The exact radial heat profile is an isolated Navier–Stokes solution. For the actual common
velocity u, subtraction gives




(∂t + u · ∇ − ν∆)ζi = − K ∗ ζi +

X

K ∗ wl  · ∇Gi .

(104)

l̸=i

Every radial foreign velocity is tangential and annihilates ∇Gi , so the foreign fields on the right
can be replaced by K ∗ hl . In particular diffusion produces no separate angular source. If every
initial colour equals its radial reference Gl (0), then ζi = hi = 0 solves the comparison system
with the prescribed initial data. For arbitrary radial initial colours the common velocity remains
tangential, and

hi (t) = 0,
ζi (t) = eνt∆ wi (0) − Gi (0) .
This difference is radial and need not vanish. In the all-radial zero-seed member of the reference
family the colours equal their references.
The scaled Biot–Savart split gives ∥K ∗ ζi ∥∞ ≤ Cp Ai Ri xi . Heat contraction of the fixed
reference gradient in L1 and Lp , and the scalar Kato inequalities, bound the self term by γAi xi .
For example γ can be chosen as a constant multiple of Cp (∥∇W ∥1 + ∥∇W ∥p ). No derivative of ζi
enters this estimate; the actual nonlinear transport term disappears from the norm inequalities.
We verify the localized foreign-field estimates under the stop. For any fixed index l, the global
radial component obeys
|u(t, x) · er | ≤ CAl |x| + CAl Rl βq.
(105)
Indeed every radial average has tangential velocity. For k ≤ l, the fourfold origin-growth bound
P
gives |K ∗ hk (x)| ≤ C ∥hk ∥∞ |x| ≤ CAk |x|, and k≤l Ak ≤ CAl . For k > l use ∥K ∗ wk ∥∞ ≤
P
CAk Rk and k>l Ak Rk ≤ CAl Rl βq. The same estimate holds for only the foreign radial field.
The constants in these sums are uniform for β ≥ 4 and βq ≤ 1/2.
Apply the actual outer-core bootstrap to the stopped finite prefix l ≤ j − 1. Its nonradialdensity hypothesis follows from ∥hl ∥1 ≤ 2Al Rl2 xl ≤ 2δAl Rl2 . All colours k ≥ j enter that
P
bootstrap through the unrestricted, convergent velocity sum k>l Ak Rk . The parameter margin
(101) and small q therefore give the individual core estimate
2

∥wl (t) − cl ∥L∞ (BaR ) ≤ CAl e−cRl /(νt) ,

a = a0 /2.

l

(106)

The bootstrap error threshold is of order S −1 ; its improvement follows from κj S ≪ 1 just as
in the outer-core proof. Retaining the individual radius in that proof, rather than its maximum
over l, gives (106).
√
Fix c2 > c0 . On Bc2 Rl the own nonradial velocity is bounded by C δAl Rl . Older constant
cores contribute only rotation; their harmonic remainders contribute CδAl Rl q 2 /β and their core
leaks are exponentially small. All inner velocities contribute at most CAl Rl βq. Choose q so that
this ball is strictly inside every older core probe disk. By (101), the integrated radial drift is less
than a fixed fraction of (c2 − c0 )Rl . The radial exit estimate for the positive scalar evolution from
|wl (0)| gives
Z
2

|x|>c2 Rl

|wl (t, x)| dx ≤ CAl Rl2 e−cRl /(νt) .

(107)

For its post-exit fourth moment use ϕ(x) = (Rl2 + |x|2 )2 . Equation (105) gives u · ∇ϕ ≤ CAl ϕ,
while ν∆ϕ ≤ CνRl−2 ϕ. Conditioning at exit from Bc2 Rl consequently gives E[ϕ(Xt ) | Fexit ] ≤

63

2

CRl4 eC(Al +ν/Rl )T ≤ CRl4 eCS . Multiplication by the exit probability proves
Z

2

|x|>c2 Rl

|x|4 |wl (t, x)| dx ≤ CAl Rl6 eCS e−cRl /(νt) .

(108)

Angular averaging preserves radial integrals. Inside Bc2 Rl use ∥hl ∥1 ≤ 2Al Rl2 xl ; outside use (108).
It follows that
Z
|x|4 |hl (t, x)| dx ≤ CAl Rl6 [xl (t) + ej−l ].
(109)
For an inner colour i < l < j, take a fixed small η > 0 such that BηRi is inside the initial
constant disk of colour i and c0 q < η/4. In this ball the radial velocity
√ estimate based on index i
has the same small integrated bound: the own nonradial term is C δAi Ri , older remainders are
controlled by (106), and all inner colours enter by CAi Ri βq. Starting the lth positive colour in
Bc0 Rl , radial exit from BηRi therefore gives
Z
|x|>ηRi

2

|wl (t, x)| dx ≤ CAl Rl2 e−cRi /(νt) .

(110)

This receiver-radius estimate is essential.
On Ai , for l < i, split hl into its core and exterior. Equation (106) bounds the core density by
2
2
CAl e−cRl /(νt) ; the origin-growth estimate bounds its velocity by CAl |x|e−cRl /(νt) . The exterior
part has mass at most 2Al Rl2 xl and a fourfold inverse-moment expansion starting at degree three.
Thus
|K ∗ hl (x) · er |
≤ CAl [q 2(i−l) xl + ej−l ].
(111)
Ri
For i < l < j, truncate hl at radius ηRi . Every radial truncation has zero circulation because
P hl = 0, and fourfold symmetry
cancels harmonic moments of degrees one to three. Its exterior
−5 R
4
|y| |hl (y)| dy. For the remaining density use (110), ∥hl ∥∞ ≤ CAl ,
field is bounded by CRi
and Biot–Savart interpolation. This gives
h
i
|K ∗ hl (x) · er |
≤ CAl q 6(l−i) (xl + ej−l ) + q l−i ej−i .
Ri

(112)

The fixed exponent c in em has been reduced to absorb square roots and polynomial factors.
Away from Ai , ∇Gi is the Gaussian convolution of the compactly supported reference gradient.
The fixed margin d∗ Ri gives, with τi = νt/Ri2 , scaled L1 and Lp tails after multiplication by
2
|x| + Ri bounded by Ce−cd∗ /τi . The Gaussian kernel has this decay whenever its argument is at
least d∗ Ri ; integration and fixed polynomial weights are absorbed by decreasing c. The foreign
version of (105) consequently bounds the total omitted forcing by CAi ej−i . Combining this with
the self estimate, (111), (112), and the explicitly retained fi proves (102)–(103).
Corollary 6.5 (Inactive density closure under packet history). Let σ > 0 be the growth rate of the
fixed smooth reference. Choose β sufficiently large that γ/β < σ/4, then a fixed D > β sufficiently
large, then q sufficiently small. Set wi = (Dq 6 )j−i and b = δe−σS . Assume xi (0) ≤ M b, and the
packet-history bound
Z θ
fi (s)
sup
eλP (θ−s) max
ds ≤ C∗ δ,
(113)
i<j
wi
θ≤Aj T 0
where λP < σ/2 is the weighted comparison rate below. Take S = C2 log(1/q), with C2 sufficiently
large and fixed. Fix a finite Lδ > 0 and choose the cap so that
q Lδ ≤ δ(q) ≤ 1,

q

CS( δ(q) + βq) ≤ c.
64

(114)

All constants and exponents are fixed before q is decreased. Then the stopped solution remains
strictly below its stopping threshold:
h

i

xi (t) ≤ B + wi CC∗ δ + o(q K ) < δ/4,

B ≤ M δe−(σ−λB )S ,

for every i < j and t ≤ T . Here K is any fixed prescribed power, the error in the forced weighted
norm is also o(δq 6 ), and the constants are independent of depth. Alternatively, the lower bound
in (114) may be replaced by the direct relative-error condition
em
η(q)SeλP S = o(δ(q)q 6 ),
η(q) = sup
.
(115)
6 m
m≥1 (βDq )
The conclusion is conditional on (113).
Proof. Bound each self coefficient in (102) by λ0 = γ/β. This replaces, rather than identifies
with, the earlier sharp spectral rate (σ + ε)/β. The unweighted off-diagonal row sum is at most
"

#

C
q 2 /β
βq 6
rB =
+
.
β 1 − βq 6 1 − q 2 /β
The row sum in the forced weights wi is at most
#

"

Dq 8
β
C
+
.
rP =
β D − β β − Dq 8
Thus λB = λ0 + rB and λP = λ0 + rP are both below σ/2 in the prescribed parameter order. The
positive comparison equation splits exactly into initial, packet, and parabolic-error contributions.
Its initial part is bounded by M beλB θ , giving the stated B.
For the explicit error put
em
η(q) = sup
.
6 m
m≥1 (βDq )
This is smaller than every power of q, uniformly when κj ≤ q 3/2 and S = O(log(1/q)). Indeed
successive ratios in this supremum are less than 1/2 for small q, and its first term has Gaussian
exponent of order q −7/2 /S. A direct summation of the four terms of (103) gives
"

#

Ek
Dq 6
1
βq
sup
≤ Cη(q) 1 +
+
+
.
6
k
6
(Dq
)
1
−
Dq
D
−
1
1
−
βq
k≥1
For example the third sum becomes D−k r<k β −r q −6r er ≤ η(q)D−k r<k Dr . The fourth uses
P
−r k−r ≤ β −k βq/(1 − βq). Its weighted convolution is therefore at most
r<k β q
P

P

HG (q) := Cη(q)SeλP S = o(q L )

for every fixed L > 0.

Choose L > Lδ + 6. Then (114) gives
HG (q)
o(q L )
≤
−→ 0.
δ(q)q 6
q Lδ +6
Thus the error is o(δq 6 ) in the forced weighted norm. Under the alternative (115), this relative
bound holds directly.
The positive-matrix variation of constants and (113) yield the asserted bound. Choose C2
so C2 (σ − λB ) > 6; then B ≪ δq 6 . Finally maxi<j wi = Dq 6 , so small q makes the complete
right side less than δ/4, contradicting a first stopping time. The estimate bounds the comparison
error; radial data have no angular forcing by (104).
65

Corollary 6.6 (A static packet bound suffices for inactive density). For the inactive-density
conclusion alone, the packet-history hypothesis (113) may be replaced by
sup max

θ≤Aj T i<j

fi (θ)
≤ C∗ δ.
wi

Fix C2 > 6/σ. Choose β, D, and then q, so that
C2 (σ − λB ) > 6,

C2 λP < 1.

With S = C2 log(1/q) and the cap condition (114), or the alternative relative condition (115)
together with the radial margin, the actual inactive densities remain below δ/4. Their normalized
inward nonradial velocity on the current receiver’s fixed collars is




O δSq 8−C2 λP + o(δq 6 ) = o(δq 6 ).
The static packet bound is an explicit input. The packet estimate in Section 6.3 closes the inactive
stop directly, without requiring this static input to be O(δ).
Proof. The stated parameter choices are possible since λB , λP can be made arbitrarily small by
taking the fixed β, D large and then q small. The packet contribution to the forced weighted
norm is at most CC∗ δSeλP S . Thus
h

i

xi ≤ B + (Dq 6 )j−i CC∗ δSq −C2 λP + o(q K ) .
The baseline B = o(δq 6 ) and the nearest forced bound CδSq 6−C2 λP are both below the stopping
threshold. The parabolic errors are o(δq 6 ) in the forced weighted norm by the preceding relativeerror calculation, so they are below the stopping threshold as well. For the inward field use the
actual older-core multipole estimate:
X

β i−j q 2(j−i) xi ≤ B

i<j

q2
Dq 8
−C2 λP
+
CC
δSq
+ o(q K ).
∗
β − q2
β − Dq 8

The error field is at most O(q 8 )HG = o(δq 6 ), and C2 λP < 1 makes the principal packet term
o(δq 6 ).

6.3

An actual packet bound closes the inactive density stop

All colours in this section are the actual scalar colours of the common unforced Navier–Stokes
velocity. Use the notation and fixed profile hypotheses of Proposition 6.4. In addition choose q
so small that βq 2/p ≤ 1/2.
Let S(t, s) be the positive scalar transport–diffusion propagator for that common velocity and
put
X

X
Pj (t) = S(t, 0)
|wl (0)| ,
vj =
wl .
l≥j

l≥j

The symbol Pj here denotes a positive envelope, not angular averaging. The initial sum is in
L1 ∩ Lp and is supported in Bc0 Rj . Positivity and monotone convergence imply
X

|wl (t)| ≤ Pj (t),

∥Pj ∥1 ≤ CAj Rj2 ,

2/p

∥Pj ∥p ≤ CAj Rj .

l≥j

The positive aggregate is used only in L1 ∩ Lp . The actual packet velocity is controlled by the
individual colour contractions.
66

Lemma 6.7 (Actual packet moments). For every fixed integer n ≥ 2, with θ = Aj t and κj =
ν/(Aj Rj2 ), one has
Z

(Rj2 + |x|2 )n/2 Pj (t, x) dx ≤ Cn Aj Rjn+2 (1 + θ +

q

κj θ)n eCn θ/β .

(116)

Constants are independent of j, depth, viscosity and all seed phases. The moment bound holds
without the inactive stop.
Proof. Write u = u<j + u≥j . Individual colour contractions and Biot–Savart interpolation give
∥u≥j ∥∞ ≤ C

X

Al Rl ≤ CAj Rj .

l≥j

The fourfold origin-growth inequality applied to each older colour yields |u<j (x)| ≤ CAj |x|/β for
β ≥ 4. The constants in these two sums are uniform for β ≥ 4 and βq ≤ 1/2. Thus the actual
radial drift is bounded by L|x| + V , with L = CAj /β and V = CAj Rj . Tangential rotation does
not affect radial weights.
R
R
R
The positive scalar propagator preserves mass. Let M = Pj (0) = Pj (t) and Qn = (Rj2 +
|x|2 )n/2 Pj . Differentiation of the weight, positivity, and Hölder with respect to Pj dx give
Q′n ≤ Cn LQn + Cn V M 1/n Q(n−1)/n
+ Cn νM 2/n Qn(n−2)/n .
n
If M = 0 there is nothing to prove. Otherwise Y = (Qn /M )1/n satisfies Y ≥ Rj and
Y ′ ≤ Cn LY + Cn V + Cn ν/Y.
√
A supersolution is eCn Lt (Cn Rj + Cn V t + Cn νt), after enlarging Cn : the derivative of the squareroot term controls Cn ν/Y for t > 0. The initial support bounds Y (0) by Cn Rj , proving the
claim. To justify the moment calculation without assuming moments in advance, start with a
finite truncation of the positive initial aggregate. The common velocity is bounded on the physical
window, so its positive diffusion representation gives finite moments. Apply the displayed uniform
calculation and pass monotonically to the full positive datum. Equivalently one may use truncated
radial weights. The constants in the refined estimate use only L, V above, not the crude global
drift bound.
Proposition 6.8 (Actual force on a stopped inactive prefix). Under the stopped hypotheses and
parameter margins of Proposition 6.4, set
M(S) = (1 + S +

q

κj S)4 eC4 S/β .

For every i < j, k = j − i, and 0 ≤ θ ≤ Aj T ≤ S, the actual packet force in that proposition
obeys
"
#
cp q −2k
6k
k
fi (θ) ≤ Cq M(S) + Cq exp −
.
(117)
κj S
In particular, for the weights wi = (Dq 6 )j−i with fixed D > 1,
fi (θ)
≤ CM(S) + ϵ(q),
wi
θ≤Aj T i<j
sup max

where ϵ(q) is smaller than every fixed power of q, uniformly for κj ≤ q 3/2 and S = O(log(1/q)).
67

Proof. Let h≥j = (I −P )vj , where P again denotes angular averaging. Its fourth absolute moment
is bounded by twice that of Pj . Choose a fixed η > 0 so that BηRi is inside the constant core
√ probe
of colour i, and is separated from Ai . On this ball the own radial field is bounded by C δAi Ri ,
the older fields by their actual core/harmonic bounds, and all inner fields by CAi Ri βq. The stop
and its radial margin therefore give a displacement strictly smaller than ηRi /2 up to T . Every
positive packet constituent starts in Bc0 Rj ⊂ BηRi /4 . The radial exit estimate consequently gives
Z

2

|x|>ηRi

Pj (t, x) dx ≤ CAj Rj2 e−cRi /(νt) .

This estimate uses the unrestricted convergent inner velocity sum, not a small-density assumption
on the packet.
Radial truncation of h≥j has zero circulation and vanishing
harmonic moments of degrees one
R
to three. Its field on Ai is therefore bounded by CRi−5 |x|4 |h≥j |. For the exterior leakage use
the last exit estimate and the Lp contraction of Pj . The elementary kernel split gives, for p > 2,
p

p−2

∥K ∗ f ∥∞ ≤ Cp ∥f ∥12(p−1) ∥f ∥p2(p−1) .
Indeed the near integral is at most Cp d1−2/p ∥f ∥p , and the far integral at most C ∥f ∥1 /d; optimize
d. Angular averaging preserves radial tail masses and contracts Lp . Thus the leaked nonradial
2
packet has velocity at most CAj Rj e−cp Ri /(νt) . Combining this with the fourth moment and
the uniform scaled heat-gradient bound on Gi proves (117): the force normalization is velocity
divided by Aj Ri .
After division by wi , the first term is at most CD−k M(S). The second is at most
CD−k q −5k e−cp q

−2k /(κ S)
j

.

For small q successive such terms decrease, and the first is smaller than every q-power because
κj S ≤ q 3/2 C log(1/q). This proves the weighted assertion.
Corollary 6.9 (Unconditional stopped inactive closure). Fix K > 6 and any fixed C2 > 6/σ.
Suppose xi (0) ≤ M q K for every i < j. Choose β sufficiently large, then D sufficiently large, and
then q sufficiently small so that
C2 λB < K − 6,

C2 (λP + C4 /β) < 1,

βq 2/p ≤ 1/2.

Use a comparison-row horizon constant at least C2 . Put
δ = (log(1/q))−4 .

S = C2 log(1/q),

For every T ≤ S/Aj with κj ≤ q 3/2 the actual older densities satisfy
xi (t) ≤ M q K−C2 λB + C(Dq 6 )j−i S(1 + S)4 e(λP +C4 /β)S + o(q K ) < δ/4.

(118)

The core persistence conclusion and the harmonic older-field estimates hold on this window, with
the packet force supplied by Proposition 6.8. On the current colour’s fixed collars, the normalized
older inward nonradial field is




O(q K+2−C2 λB ) + O S(1 + S)4 q 8−C2 (λP +C4 /β) + o(q K ).
These statements control magnitudes; they give no nonzero signed impulse.
68

Proof. Use the positive comparison matrix from the proof of Corollary 6.5. Its initial unweighted
part is bounded by M q K eλB S . Its packet part is bounded in the forced weights by CSM(S)eλP S +
o(q K ), using Proposition 6.8. The parabolic error sum is the same sum as in that corollary. The
cap δ = (log(1/q))−4 satisfies q Lδ ≤ δ for every fixed Lδ > 0 once q is small, so its error is also
o(δq 6 ). Since κj S ≪ 1, this proves (118). Write a = C2 (λP + C4 /β) < 1. The nearest forced
density is bounded by CDq 6−a S(1 + S)4 , and the initial term has exponent K − C2 λB > 6.
Their sum and√the Gaussian error are o(δ), giving the strict δ/4 improvement. The radial margin
holds since S δ = O((log(1/q))−1 ) and Sβq → 0. A first stopping time is thus impossible.
P
Finally sum the actual older core harmonic row i<j β i−j q 2(j−i) xi ; the initial and packet parts
give the two displayed field powers. The choices of β, D are possible because λB , λP , C4 /β can be
made arbitrarily small in this parameter order. This closes the density stop and the older-field
magnitude estimates. A signed transfer or protected active cap requires additional information.

7

Actual source activation and discounted history

7.1

A global weighted foreign-field estimate for the adjoint error

The following estimate controls an exact foreign pairing in the scalar growing-coordinate equation.
It does not assume that the foreign velocity is uniformly small throughout the plane in the current
colour’s units.
Let Tadj be the smooth growing adjoint test from Section 4.2. Only its already proved bounds
|∇Tadj (y)| ≤ C min{|y|3 , |y|−5 }
are used here, with the usual bounded extension across |y| = 1. Normalize a current colour to
(Aj , Rj ) = (1, 1). Let b be the actual sum of all foreign fields, after subtracting the constant
rotation Ω<j Jy of the older cores.
Lemma 7.1 (Global weighted foreign field). Assume the actual older core/harmonic conclusions
of Propositions 5.3 and 5.4, with older nonradial density parameter δ ≤ 1 and normalized core
defect at most ε. Assume β ≥ 4, βq ≤ 1/2, and q is small enough that the current unit ball is
inside every older probe core. Then
!

δq 2 + q 4 + ε
sup |b(y)| |∇Tadj (y)| ≤ C βq +
.
β
y

(119)

Consequently, for every f ∈ L1 ,
Z

!

δq 2 + q 4 + ε
|f | |b| |∇Tadj | ≤ C βq +
∥f ∥1 .
β

All constants are independent of the current index and packet depth.
Proof. The younger fields have global norm at most C l>j (Al /Aj )(Rl /Rj ) ≤ Cβq by the individual colour contractions. Multiplication by the bounded ∇Tadj gives their term.
On |y| ≤ a/(2q), the actual older harmonic/core-leak decomposition, with the rigid rotation
removed, gives
δq 2 3
ε
|b<j (y)| ≤ C
|y| + C |y|.
β
β
P

69

The geometric sums are q 2 /(β − q 2 ) for the cubic coefficient and 1/(β − 1) for the core leak. For
|y| ≤ 1, multiplying this bound by C|y|3 gives at most C(δq 2 + ε)/β. For 1 ≤ |y| ≤ a/(2q) use
C|y|−5 , giving the same bound.
P
On the complementary region, the individual fourfold origin-growth bound gives | l<j K ∗
wl (y)| ≤ C|y|/β in current units. The subtracted constant rotation has the same bound, since
P
its coefficient is at most C l<j Al /Aj ≤ C/β. Thus |b<j ||∇Tadj | ≤ C|y|−4 /β ≤ Cq 4 /β there.
This uses the sum of older amplitudes, which is finite and uniformly bounded in current units;
the sum of older velocity maxima in current units would not give a depth-uniform estimate and
is not used. Combining the regions proves the result.
Corollary 7.2 (A harmless linear scalar loss). Suppose on an actual stage of length H ≤
C0 log(1/q) its perturbation satisfies ∥f (t)∥X ≤ Ca(t), where a(t) = a(0)eσt , and the preceding
lemma holds uniformly. Write
!

δq 2 + q 4 + ε
χq = C βq +
.
β
Then the exact foreign cross-advection pairing in the growing-coordinate identity contributes at
most Cχq H to the error divided by the growing envelope at any time in the stage. If β is fixed,
δ ≤ 1 and ε is superpolynomially small, this loss tends to zero uniformly in depth.
Proof. The scalar identity and variation of constants weight the pairing at time t by eσ(s−t) .
The lemma bounds it by Cχq a(t). Division by a(s) cancels the exponential envelope, leaving
Cχq s ≤ Cχq H.
This is a linear loss, not a quadratic error in f . The foreign radial-base forcing and its heatprofile correction must still be estimated separately on the derivative collars, with their actual
retained coefficients. The corollary proves neither a signed lower bound nor the density envelope
assumed in its statement. It removes the need for an unjustified global foreign-field bound of
order a(t) in this one scalar term.

7.2

Actual postcap source moments under a stopped-parent core

The following proposition is a conditional estimate for an actual scalar colour in the common
Navier–Stokes velocity. It requires the displayed actual parent-core and positive cap-envelope
hypotheses. It does not assume that the signed colour has already been identified with a canonical
Euler profile. In particular, it supplies a uniformly integrable future history, not convergence or
noncancellation of that history.
Use dimensionless child coordinates, and let f solve
∂u f + v(u, y) · ∇f = κ∆f,

0 ≤ u ≤ U,

where v is the actual common velocity expressed in these coordinates. Let P be its positive scalar
evolution with P (0) = P0 ≥ |f (0)|. Then positivity gives |f (u)| ≤ P (u). A rigid rotation of the
coordinates may be made at any time; all assumptions and conclusions below concern radial
quantities and are unaffected by it.

70

Proposition 7.3 (Discounted actual
future-tail estimate). Fix β, σ, C0 , Ca , Cd > 0. Suppose
R
R ≥ 8, H0 ≥ 1, 0 < κ ≤ 1, M = P0 > 0, and
Z

(1 + |y|)4 P0 (y) dy ≤ C0 M,

(120)
"

c0 R2
(1 + |y|) P0 (y) dy ≤ C0 M exp −
κH0
|y|>R/4

Z

#

4

(121)

for a fixed c0 > 0. Assume the actual radial drift satisfies
|v(u, y) · er | ≤ a|y| + D

(|y| < R),

(122)

2

|v(u, y) · er | ≤ a0 |y| + D

(y ∈ R ),

(123)

where
0 ≤ a ≤ Ca δ/β,

0 ≤ a0 ≤ Ca /β,

0 ≤ D ≤ Cd .

There is a fixed constant C∗ depending only on the displayed constants such that, if
eaU (R/4 + DU ) ≤ R/2,

C∗ δ ≤ σ/4,

(124)

κu)4 eC∗ δu/β + E,

(125)

then, for every 0 ≤ u ≤ U ,
Z

|y|4 P (u, y) dy ≤ CM (1 + u +

√

where one can take
E = CM eC∗ U/β (1 + R + U +

q

#

"

cR2
κ(H0 + U ))4 exp −
.
κ(H0 + U )

(126)

The constants C, c > 0 depend only on the fixed constants and c0 . For every 0 ≤ J ≤ U and
every real measurable phase ϑ(u),
Z U
J

e−σu/β eiϑ(u) B4 (f (u)) du ≤ Cβ,σ M (1 + J)4 e−σJ/(2β) +

β
E.
σ

(127)

In particular the same bound controls any complex moment integral with weight e−µu and Re µ =
σ/β.
Proof. All statements can first be proved for the actual smooth positive-time Navier–Stokes velocity and then by approximation. Its positive scalar representation is the diffusion with drift v
and covariance 2κI. For radial estimates a predictable
rotation eliminates the tangential compoR
nent of its drift: if Yu = Ou Xu , then the martingale Ou dBu is again a planar Brownian motion
and |Yu | = |Xu |. Equivalently one may use the radial exit estimate proved above. The local drift
bound and Gronwall give, up to the first exit time τR from BR ,
"

sup |Xs | ≤ e

au

|X0 | + Du +

√

#

2κ sup |Bs | .
s≤u

s≤u∧τR

The stopped process is kept at its exit point in this inequality. For |X0 | ≤ R/4, (124) and the
Brownian supremum tail therefore imply
"

#

cR2
Pr(τR ≤ U ) ≤ C exp −
.
X0
κU
71

(128)

For U = 0 this means zero exit probability. The predictable rotation can be regularized at the
origin; for a smooth fourfold Navier–Stokes drift v(0) = 0 and its tangential angular speed is
locally bounded, so removal of that regularization causes no difficulty. No angular drift bound
enters the constants.
Split P0 into its restrictions to BR/4 and its complement. On unexited trajectories of the first
part, the path bound and the fourth moment of the Brownian supremum give
i

h

EX0 [(1 + |Xu |)4 1{τR >u} ] ≤ Ce4au (1 + |X0 |)4 + (Du)4 + (κu)2 .
Integration against the initial positive density and (120) bound this contribution by the first term
of (125), after enlarging C∗ .
The global radial bound also gives, conditionally from any starting time s and point x,
i

h

E[(1 + |Xu |)4 | Xs = x] ≤ CeC∗ (u−s)/β (1 + |x|)4 + (D(u − s))4 + (κ(u − s))2 .
This follows from the same radial rotation and Gronwall argument, now using a0 . Conditional
application at τR , where |XτR | = R, and (128) bound all exited trajectories of the first initial
part by
√
2
CM eC∗ U/β (1 + R + U + κU )4 e−cR /(κU ) .
For the second initial part, the global conditional bound and (121) give at most
√
2
CM eC∗ U/β (1 + U + κU )4 e−c0 R /(κH0 ) .
Reducing c and combining these estimates gives (126), and hence (125). This proves in particular
that the entire fourth moment of f is well
defined.
R
Since |f | ≤ P , one has |B4 (f )| ≤ |y|4 P . The cap-smallness condition in (124) makes its
discounted first term bounded by
CM (1 + u)4 e−3σu/(4β) ,
because κ ≤ 1. Integrating from J to infinity yields Cβ,σ M (1 + J)4 e−σJ/(2β) . The constant-error
term integrates to at most (β/σ)E. Unit complex phases do not change any absolute value,
proving (127).
Corollary 7.4 (Uniform scale margin for an actual source). Fix all profile constants and a fixed
finite β. Let q ↓ 0 and assume
R = cR /q,

U ≤ CU β log(1/q),

H0 ≤ CH log(1/q),

κ ≤ q 3/2 ,

δ = [log(1/q)]−4 ,

with the preceding proposition’s explicit actual drift and cap-envelope hypotheses. For small q,
the margins in that proposition hold, and E/M is smaller than every fixed power of q, uniformly
in viscosity, depth and seed phases. The actual discounted source histories then have uniformly
integrable future tails:
Z U
J

e−σu/β |B4 (f (u))| du ≤ Cβ,σ M (1 + J)4 e−σJ/(2β) + M o(q K )

for any fixed prescribed K > 0.

72

Proof. Here aU = O(δ log(1/q)) → 0 and DU/R = O(q log(1/q)) → 0, so the displacement
margin holds. The exponential in (126) has an exponent bounded above by
−

cq −7/2
,
log(1/q)

whereas its remaining factors are bounded by a fixed power of q −1 times a fixed power of log(1/q).
This proves the superpolynomial error estimate, uniformly for every smaller κ. The discounted
estimate follows from the proposition.
Remark 7.5 (How the actual drift hypotheses arise). For a child colour at scale (Aj , Rj ), its own
field and all descendant fields have normalized velocity bounded by a fixed D, by individual
P
L1 , L∞ contractions and l≥j Al Rl ≤ CAj Rj . The global older fourfold fields obey a radial
bound C|y|/β. Inside a fixed fraction of the parent’s protected constant core, the stopped actual
older densities and their harmonic/core-leak estimate give instead a radial bound Cδ|y|/β, after
the rigid rotation has been removed and exponentially small leaks have been absorbed into that
coefficient. These facts supply (122)–(123) when that stopped core estimate is available. The
positive cap-envelope moment and parent-radius leak (120)–(121) remain explicit inputs here.
They must be obtained from the actual preparation-stage localization; they are not consequences
of a signed cap-coordinate estimate alone.

7.3

The actual deepest selected source

We retain the fixed smooth profile W , its unstable eigenvalue λ = σ + iτ , its normalized growing
coordinate A, and its neutral seeds Φϑ . All colours below solve the scalar equation with the same
actual unforced Navier–Stokes velocity.
Lemma 7.6 (Weighted off-diagonal Nash estimate). Let S(t, s) be the positive scalar propagator
for the actual divergence-free Navier–Stokes velocity u, or its rigidly rotating frame, and diffusion
coefficient κ > 0 in two dimensions. Let ϕ be a fixed bounded smooth function
satisfying |∇ϕ| ≤ 1,
R
|∆ϕ| ≤ Cϕ . Suppose |u · ∇ϕ| ≤ V (t), where V ≥ 0, and put D(t, s) = st V . For ℓ ≥ 0 one has
∥eℓϕ S(t, s)e−ℓϕ ∥L1 →L∞ ≤


C
exp ℓD(t, s) + κℓ2 (t − s) + Cϕ κℓ(t − s) .
κ(t − s)

(129)

If a positive datum of mass M is supported where ϕ = 0, while ϕ = d0 > 0 on a set E, and
D(t, 0) < d0 /4, then, provided κt is sufficiently small,
∥1E S(t, 0)f0 ∥∞ ≤

CM −c/(κt)
e
,
κt

∥1E S(t, 0)f0 ∥1 ≤ CM e−c/(κt) .

The constants do not depend on a bound for the tangential velocity or on derivatives of u.
Proof. For F = eℓϕ f , direct conjugation gives the drift u + 2κℓ∇ϕ and zeroth-order coefficient
ℓu · ∇ϕ + κℓ2 |∇ϕ|2 − κℓ∆ϕ.
Integration of its positive equation bounds its L1 growth by Λ1 = ℓV + κℓ2 + Cϕ κℓ. In the
L2 energy identity half the drift divergence cancels the Laplacian term,
R so its growth coefficient
t

is at most Λ2 = ℓV + κℓ2 . Apply the planar Nash inequality to e− s Λ1 F : its L1 norm does
not increase, and its L2 dissipation is at least κ∥∇F ∥22 after the same scalar integrating factor.
73

Rt

Solving the resulting scalar differential inequality gives the L1 → L2 bound C[κ(t − s)]−1/2 e s Λ1 .
The adjoint calculation has the same bounds, with the signs of u and ℓ reversed. Composition at
the midpoint proves (129). The L1 bound before Nash is M exp(ℓD + κℓ2 t + Cϕ κℓt). Multiply
both bounds by e−ℓd0 and optimize ℓ > 0. Since D < d0 /4 and κt is small, this proves the last
assertions. The energy calculations can first start at positive time and use smooth data. For the
actual initial L1 ∩Lp vorticity, p > 2, the physical velocity is bounded by the Biot–Savart split; its
rotating-frame addition is tangential to radial cutoff boundaries. Thus the integrations by parts
are justified by cutoff limits. Their estimates use only V , not that possibly large velocity bound.
Truncating the positive initial datum and passing monotonically gives the L1 version, with the
same constants. The vorticity generating u may have either sign: divergence freedom is the only
property of that velocity used in these energy identities. For any signed transported aggregate
g, scalar positivity gives |g| ≤ S(t, 0)|g(0)|, or domination by the sum of the absolute initial
colours. That latter positive datum need only be in L1 ; no summed initial L∞ norm appears in
any step.
Choose once and for all
6<K<

643
,
100

C2 > K/σ + 1.

Use the actual packet estimate to choose β, D so large that
eB = C2 λB < K − 6,

eP = C2 (λP + C4 /β) < min{1, 8 − K}.

(130)

Here the comparison rates are those of Corollary 6.9; they tend to zero in its stated parameter
order. Choose q sufficiently small after these fixed choices. In particular βq 2/p < 1/2 and
βq < 1/2.
Theorem 7.7 (Actual selected-source activation and protected first cap). Let Al = A1 β l−1 ,
Rl = R1 q l−1 , and
wl (0, x) = Al W (x/Rl ) + q K ηl Φϑl (x/Rl ) ,




0 ≤ ηl ≤ 1.

Fix any j ≥ 1, impose ηj = 1, and set ν = q 3/2 Aj Rj2 . The phases and all other amplitudes are
arbitrary. Let wl be the actual scalar colours of the common unforced Navier–Stokes solution.
Normalize the jth source to radius and amplitude one, and remove the rigid rotation
Ω<j =

W (0) X Al
.
2 l<j Aj

ej (t) and put fj = w
ej − eκt∆ W , κ = q 3/2 , α = q K ,
Denote its normalized, rotated vorticity by w
σt
a(t) = αe . Fix one sufficiently large finite expansion order M , with M > CW /σ if a referencestate reset is required below. There are constants B, δ∗ > 0, allowed to depend on this finite order
and independent of j and all phases, such that through H∗ = σ −1 log(δ∗ /α),

∥fj (t)∥X ≤ Ba(t),

Z H∗

∥K ∗ fj (t)∥∞ dt ≤ CBδ∗ .

(131)

0

The first hitting time tj of |A(fj )| = δq , δq = (log(1/q))−4 , is well defined and protected: in a
fixed neighbourhood of that crossing, almost everywhere,
d
|A(fj (t))| ≥ σδq /2.
dt
74

It obeys
tj = σ −1 log(δq /α) + O(δq + q γ (1 + log(1/q))D0 )
for some γ > 0 and finite D0 .
For that fixed finite M , let sq be the exact δq cap on one fixed canonical reference orbit. Choose
ϕj so that A(Tϕj Ωref (sq )) = A(fj (tj )). Then the full vorticity state satisfies
ej (tj ) − Tϕj Ωref (sq )∥X ≤ CM δqM +1 + CM q γ (1 + log(1/q))DM δq + Cκtj .
∥w

(132)

The last, radial term is the explicit difference between the exact Navier–Stokes heat base and the
Euler base. It is not a seeded error. One can choose M > CW /σ, where CW is the one-sided X
comparison constant of a fixed small ancient segment. All fixed positive-envelope moments at tj
are bounded uniformly; outside one fixed ball they are smaller than every q-power. This theorem
concerns the deepest source selected by diffusion. It does not assert inductive transmission to all
larger receiving annuli.
Proof. The initial series converges absolutely in L1 ∩ Lp . Individual scalar contractions give
P
∥wl ∥1 ≤ CAl Rl2 , ∥wl ∥∞ ≤ CAl and hence l>j ∥K ∗ wl ∥∞ ≤ CAj Rj βq. These are actual-colour
bounds and require no hypothesis on the deep colours’ nonradial densities.
Work in the normalized rotating frame. Let CΦ bound the fixed autonomous first coefficient
in X divided by eσt , and fix B > 4(CΦ + 1) now. All later field estimates use this fixed B; their
constants may depend on it. Stop provisionally when ∥fj (t)∥X = Ba(t). The terminal cap δ∗ is a
provisional small number, to be fixed after the finite expansion orders, before the final choice of
q. By Corollary 6.9, all larger colours already obey their unconditional inactive-density estimate
through S = C2 log(1/q); their cores and harmonic estimates hold there. Their nonrotational
field on any fixed source collar is, in every fixed spatial derivative norm,
O(q K+2−eB ) + O(q 8−eP (1 + S)5 ) + o(q P )

(133)

for every fixed P . Derivatives here are taken only on the harmonic part; the core leak has no
asserted derivative bound and is included in the last velocity remainder.
The stopped source core persists directly, without a source-core hypothesis in the bootstrap.
Write its own velocity exactly as K ∗ eκt∆ W + K ∗ fj ; the first term is tangential everywhere
and the second is bounded by CBa(t). Larger cores have already been proved independently by
the inactive-prefix result. Their harmonic and leaked nonrotational contributions on the fixed
source-core disk have integral tending to zero by (133). Every smaller velocity contributes at
most Cβq. Thus the integral of the absolute radial drift on that disk is at most
CBδ∗ /σ + CβqS + o(1).
Choose δ∗ and then q so this is less than a0 /8. In the exact backward representation the sign of
the drift is reversed, but the same absolute radial bound holds. Starting in Ba0 /2 and stopping
on exit from the initial constant core Ba0 gives
ej (t) − W (0)∥L∞ (Ba /2 ) ≤ Ce−c/(κt)
∥w
0

by the radial exit lemma and the individual colour maximum principle. Consequently, in the
P
el are harmonic plus a rough
source core all foreign fields acting on the aggregate g = l>j w
exponentially small leak; after rotation their cubic constant is at most CBδ∗ . Their global
growth is C|x|, uniformly in j.
75

The aggregate g has mass norm Mg ≤ Cβq 2 , initial support in BCq , zero circulation and zero
initial B4 . Apply the already proved full nonlinear cold-tail estimate with r = Cq, R∗ = κ1/4 and
√
window S. Its margins hold since Mg /κ = O(βq 1/2 ), κS → 0, and CBδ∗ κS → 0. It yields
sup |B4 (g(t))| ≤ Cq 643/100 (1 + S)D1 .

(134)

t≤H∗

Every constant in this stopped calculation is independent of j.
For completeness, higher moments require
√ no signed cancellation. Nash for the positive passive envelope gives ∥K ∗ g(t)∥∞ ≤ CMg / κt. Inside B2R∗ , foreign radial drift is bounded by
(CBδ∗ R∗2 + ϵ(t))|x|. Its integrated linear-growth coefficient tends to zero. The elementary radial
moment calculation, or the integrating-factor diffusion comparison, therefore gives for each fixed
integer n ≥ 2, before exit,
Z

|x|n |g(t, x)| dx ≤ Cn Mg q +

√

q

n

κS + Mg S/κ .

√
The last displacement is smaller than κS since Mg /κ → 0. Radial exit from BR∗ has probability
2
Ce−cR∗ /(κS) , because both integrated self displacement and the cubic displacement are small
compared with R∗ . After exit the global foreign linear growth and integrable self displacement
amplify a fixed moment by at most Cn eCn S , absorbed by that probability. Thus the wholemoment bound is
Z

sup

|x|n |g(t, x)| dx ≤ Cn q 2+3n/4 (1 + S)n/2 + o(q P ).

(135)

t≤H∗

In particular B8 (g) = O(q 8 (1 + S)4 ). Using only the absolute fourth moment instead of (134)
would fail, since its available power is merely q 5 .
There is also an exponentially small actual density leak on the source collars. Choose in
Lemma 7.6 a fixed radial ϕ which rises from zero to d0 entirely inside the protected source core,
and is constant thereafter. On the support of its derivative the integrated radial drift is at most
CBδ∗ + CβqS + o(1) < d0 /4. Apply that lemma to the actual positive envelope of g, not to a new
Navier–Stokes solution. It bounds its mass and L∞ norm beyond the rise by CMg e−c/(κt) and
CMg (κt)−1 e−c/(κt) . Kernel interpolation makes the corresponding leaked velocity smaller than
every q-power through S.
Take finitely many ACTUAL coefficients of the old interior harmonic series and of the cold
exterior multipole series, retaining degrees 3, 7, . . . and 4, 8, . . ., respectively. Extend their streamfunctions by fixed smooth radial cutoffs, vanishing near zero and outside one fixed ball, so the
fields agree on a broad annulus containing both derivative collars. The result bapp is real, smooth, divergence free and has a fixed finite set of sectors. For every finite derivative order
required below,
∥bapp (t)∥C k ≤ εq αeσt ,

εq ≤ Cq γ0 (1 + S)D2 ,

∥b − bapp ∥L∞ (collars) ≤ Cq P ,

(136)
(137)

where γ0 > 0. Indeed one may take any number smaller than
min{643/100 − K, 8 − K − eP , 2 − eB } > 0.
The fourth cold coefficient uses (134); the higher coefficients use (135); the old coefficients use
(133). Retaining enough moments makes the remainder exponent P arbitrarily large: the next
76

cold moment has power 2 + 3n/4, while old Taylor remainders gain arbitrarily many factors of
q. All density/core leaks are superalgebraic. This step is valid for each fixed finite P, k and does
not assert a derivative bound for a rough leak.
Expand the receiving/source colour in α, κ as in the isolated proof, adding the ACTUAL
coefficient functions b1 = bapp /α to the triangular recursion. Explicitly, in addition to the righthand side of (23), add
−1m=1 b1 (t) · ∇Wℓ (t) − 1m≥2 b1 (t) · ∇gm−1,ℓ (t),

Wℓ (t) = tℓ ∆ℓ W/ℓ!.

Here g1,0 (0) = Φϑj and all other initial coefficients vanish. Their dependence on this actual run
does not require analyticity in either parameter: b1 (t) is one fixed coefficient function for that
run, and the recursion cancels the actual equation algebraically.
We make the finite derivative hierarchy explicit. Fix an integer r ≥ 4 and estimate coefficient
(m, ℓ) at Sobolev order
dm,ℓ = r + 2(M − m) + 2(N − ℓ),

1 ≤ m ≤ M,

0 ≤ ℓ ≤ N.

The Laplacian input (m, ℓ − 1) has two more assigned derivatives. Every heat coupling (m, ℓ − s),
s ≥ 1, has at least two more; each nonlinear input has strictly smaller seed order and therefore
at least two more assigned derivatives, sufficient for its one differentiated factor. The external
differentiated input (m − 1, ℓ) also has two more. Use the already established C k bounds for b1
up to the finite maximum assigned order, and smoothness of W up to that order plus the finitely
many heat Laplacians. The finite-sector propagator acts at each of these assigned orders without
radial derivative loss: base transport is angular multiplication on each sector. In two dimensions
H d , d ≥ 4, is an algebra; on the fixed coefficient support, zero circulation and the Fourier lowfrequency estimate give ∥K ∗ g∥H d ≤ Cd ∥g∥H d . These facts prove all coefficient estimates by
induction in m + ℓ. There is no infinite derivative iteration and no regularity assumption on a
rough actual core leak. Coefficients remain smooth, supported in the fixed collars, and of zero
circulation, since every forcing is a Laplacian or the divergence of a compactly supported flux. If
the retained input stream frequencies are at most 4Nh , coefficient (m, ℓ) has frequencies at most
4mNh ; thus only one fixed finite frequency set is used throughout this hierarchy.
The first coefficient differs from the isolated coefficient by Cεq (1 + t)eσt , by the sharp finitesector semigroup. For orders m ≥ 2, the forcing has exponent mσ and the gap mσ − σ > 0;
induction bounds the coefficient differences by Cm εq (1 + S)emσt . The finite coefficient bounds are
uniform since εq S → 0. Positive-viscosity terms have only finite time polynomials. The source
factor is retained in the heat Taylor remainder.
Use the ACTUAL foreign drift in the transport part of the comparison between the actual
source and this expansion. It cancels from the Kato norm calculation. The additional residual is
(b − bapp ) · ∇ωapp . All nonheat coefficient gradients have compact collar support, so (137) controls
them. Away from a broad collar annulus the exact heat gradient is Gaussian; its product with
the foreign radial field is uniform because |b · er | ≤ C(|x| + βq) globally, by origin growth for old
colours and the unrestricted deep velocity sum. Thus the residual obeys
′

∥R∥X ≤ CaM +1 + CκN +1 (1 + t)D a + Cq P + o(q P ),
where P ′ can also be arbitrarily large. The Kato coefficient is the same fixed C1 = Cp (∥∇W ∥X +1)
as before, after a small terminal cap. For clarity the parameter choices and bootstrap improvement
can be made in the following order. Fix C1 before the expansion orders. Choose (M + 1)σ > C1 ,
then N with
3
P > K max{1, C1 /σ} + 1,
2 (N + 1) > K(C1 /σ − 1)+ + 1,
77

then choose the finite harmonic truncation and derivative hierarchy above. Use the B > 4(CΦ +1)
fixed at the start of the density bootstrap. The finite pure-coefficient bounds are uniform when
εq (1 + S) is small. Choose δ∗ sufficiently small that the summed higher pure coefficients are
at most Ba/32, the pure gradient sum is at most 1/2, the pure residual comparison is at most
Ba/32, and the stopped source core has its fixed radial margin. All remaining constants are
finite and fixed. Now choose q small so the forced first-coefficient correction, positive-viscosity
sums and comparisons, spatial mismatch comparison, and remaining gradients are each within
their respective Ba/32 or gradient-1/2 budgets. In particular ∥∇ωapp ∥X ≤ ∥∇W ∥X + 1, so the
coefficient is precisely the previously fixed C1 . The explicit comparison error is bounded by
′

CaM +1 + CκN +1 (1 + t)D+1 a(t)e(C1 −σ)+ t + Cq P (1 + t)eC1 t + o(q P eC1 t ),
3

uniformly through H∗ . Dividing by a(t), the second term is bounded by Cq 2 (N +1)−K(C1 /σ−1)+ (1+
S)D+1 ; the third by Cq P −K max{1,C1 /σ} (1 + S). Both tend to zero, with explicit positive powers.
Choose P ′ large enough for the last superalgebraic term. The approximate density and comparison error sum to less than Ba/2. Continuity of ∥fj ∥X /a therefore rules out its first stop and
proves (131). The core and cold-tail estimates held on that sole provisional density stop and were
obtained before its improvement. Therefore no conditional source-density input remains.
The autonomous pure coefficients approach their universal ancient homological coefficients
at an exponential rate. To check the rate, use the rotating generator Lrot of the ancient-profile
construction and normalize coefficient m by its factor emσt and the common angular rotation.
The first normalized coefficient differs from its growing limit by O(e−2σt ). For m ≥ 2 its equation
is
X
′
Fm
(t) = (Lrot − mσ)Fm (t) −
(K ∗ Fm1 (t)) · ∇Fm2 (t).
m1 +m2 =m

The finite-sector propagator of the first operator decays at least as Ce−(m−1)σt . Its limiting constant is the unique homological resolvent coefficient. Subtract that constant; the differentiatedinput hierarchy above controls the forcing difference. Induction gives a bound Cm e−ct , with any
fixed 0 < c < σ, in all the finitely required orders. Taking c = σ/2 turns its cap-time remainder
into a positive q-power times a finite logarithmic loss. The above actual-input coefficient difference is O(εq S), so at a cap with linear amplitude δq the actual source is the universal ancient
polynomial up to
CM δqM +1 + CM q γ (1 + S)DM δq .
This displayed bound concerns fj relative to the radial heat base. For the full vorticity state used
in (132), one must additionally use the exact heat identity
∥eκtj ∆ W − W ∥X ≤ κtj ∥∆W ∥X .
It gives the explicit unseeded radial term Cκtj in that equation. The radial heat term has zero
protected coordinate, so it does not alter the first-cap phase or the coordinate crossing estimate
below. The protected scalar identity has quadratic self error O(a2 ), heat error O(κ(1 + t)a) and
projected base forcing O(εq a). Its other foreign error is small: on a fixed ball the old field is
(133), the cold global velocity is O(βq), and outside that ball the source’s positive envelope has
Gaussian tails. It follows that near a = δq its radial crossing derivative is at least σδq /2 for small
q. More explicitly the same expansion and comparison, using that the protected test kills the
stable first coefficient, give on preparation
|A(fj (t)) − αeiϑj eλt | ≤ Ca(t) a(t) + εq (1 + S) + q γ (1 + S)D .


78



For a(t) ≤ 2δq this estimate excludes any cap before the indicated small neighbourhood of the
linear crossing. Within that neighbourhood the positive radial derivative gives one crossing. This
proves protection and the time estimate. Matching the canonical exact coordinate removes its
amplitude and phase bias; local invertibility of the canonical radial coordinate proves (132).
Finally, the integrated radial drift on any fixed ball surrounding the initial source support is at
most CBδ∗ + CβqS + o(1), small compared with a fixed boundary gap. Forward radial exit of its
actual positive passive envelope is therefore bounded by Ce−c/(κt) . After exit, the uniform global
radial growth amplifies each fixed moment by at most Cn eCn S , absorbed by that probability.
This proves the cap-time positive-envelope assertions and completes the theorem.
The universal finite polynomial above is a jet of the same canonical orbit for every finite order,
by Theorem 4.5. Its finite-order constants and its range of small amplitudes may depend on that
order; no infinite formal series is used.
Lemma 7.8 (Actual cap-to-reference comparison and two-scale envelope reset). Fix the profile, p > 2, and one sufficiently small canonical ancient segment before choosing the later finite
expansion orders. On this segment fix a reference orbit with A(Ωref (0)) = δref > 0 and write
A(Ωref (u)) = cref eλu + O(e2σu ),

aref = |cref |.

Fix C0 < C2 , the parameter margins of Corollary 6.9, and βq 2/p , βq ≤ 1/2. Put L = log(1/q),
δ = L−4 , and let sq < 0 be the reference orbit’s exact δ-cap time. Normalize an actual child
colour of index j ≥ 2 to unit amplitude and radius and remove the solid rotation of its older cores.
All actual colours here belong to the geometric stack of Lemma 3.1, with α = q K , Al = β l−1 ,
Rl = q l−1 , and the allowed seed phases and amplitude reductions. The clock and parameter
margins stipulated above are those of Corollary 6.9 in these child units. Suppose 0 < κ ≤ q 3/2 ,
its actual cap time is τ ≤ C0 L, and, in the exact-coordinate phase gauge,
e
∥w(t)
− eκt∆ W ∥X ≤ Cpast δeσ(t−τ )

(0 ≤ t ≤ τ ),

e ) − Ωref (sq )∥X ≤ Eq .
∥w(τ

(138)
(139)

Here Cpast is fixed independently of depth. Assume
δ −r (Eq + q) −→ 0,

r = CW /σ,

where CW > 0 is a fixed one-sided X comparison exponent on the chosen canonical segment. Set
tref = τ − sq .
The actual older-field and descendant hypotheses needed on the advance follow from Corollary 6.9, Proposition 5.4, and the individual scalar contractions. Consequently
sup
0≤v≤−sq

e + v) − Ωref (sq + v)∥X +
∥w(τ

Z −sq
0

e + v) − Ωref (sq + v)∥X dv ≤ Cδ −r (Eq + q). (140)
∥w(τ

There is no hypothesis comparing δref with a terminal finite-expansion amplitude. If
δ −r−1/β (Eq + q) −→ 0,

(141)

the actual signed fourth-moment history on [sq , 0] converges to the canonical history in the weighted absolute norm with weight e−σu/β .
For sufficiently small fixed δref and subsequently small q, the actual positive envelope of the
child, and that of its entire source/deep packet, have uniformly bounded moments of every fixed
79

order at tref . They have superalgebraic moment tails outside one fixed ball. More strongly, choose
a fixed cR > 0 so R = cR /q lies inR a fixed fraction of the parent’s protected core. If P (t) denotes
either positive envelope and M = P (0), then for every fixed integer n ≥ 2,
"

#

cn R2
(1 + |y|) P (tref , y) dy ≤ Cn M exp −
.
κC2 L
|y|>R/4

Z

n

(142)

All constants are independent of depth and of fixed trial past/reset constants; the threshold for q
may depend on those trial constants.
Proof. The canonical first jet and exact-cap monotonicity give
sq = σ −1 log(δ/aref ) + O(δ),

−sq = O(log L).

(143)

Thus tref + J < C2 L for every fixed J and small q. The canonical support and its H k bounds on
the fixed negative segment are supplied by Theorem 4.1. With k ≥ 3 they give fixed X bounds
for both ∇Ωref and ∆Ωref , as in the proof of Corollary 4.2.
We verify the actual foreign-field input independently of (138) after τ . Write
eB = C2 λB < K − 6,

eP = C2 (λP + C4 /β) < 1.

Corollary 6.9 protects every older core and its harmonic field through the full child-clock interval [0, C2 L], without an active-child density assumption. On any fixed child ball, its weighted
harmonic row and the core-leak decomposition give, after removing the solid rotation,
∥b<j ∥∞ ≤ C{q K+2−eB + q 8−eP (1 + C2 L)5 } + o(q P ).
The decomposition holds on the entire older probe core by (74)–(76); every fixed child ball lies
there for small q. Individual colour L1 , L∞ contractions give the unrestricted descendant bound
∥b>j ∥∞ ≤ Cβq, as in Lemma 6.7. Therefore on the canonical support
∥b∥∞ ≤ ρq := C{βq + q K+2−eB + q 8−eP (1 + C2 L)5 } + o(q P ) = O(q).

(144)

In particular neither an actual child-core hypothesis nor a continuation of the finite-expansion
envelope is used here.
e + v) − Ωref (sq + v) and E(v) = ∥ζ(v)∥X . Its exact equation is
Set ζ(v) = w(τ
e + b) · ∇ζ = κ∆ζ − (K ∗ ζ + b) · ∇Ωref + κ∆Ωref .
∂v ζ + (K ∗ w

The actual foreign field stays in divergence-free transport. Lemma 2.5 and (21) imply
E ′ (v) ≤ CW E(v) + C(ρq + κ).
Hence E(v) ≤ CeCW v (Eq + q), and integration of this exponential, using CW > 0 and (143),
proves (140). This proof never uses a derivative of the actual colour or of a rough density leak.
We next reset the positive envelopes at a fixed radius. Choose a fixed ball enclosing their
original support with a fixed radial margin. Before the cap, (138) and (21) bound the child’s
integrated nonradial speed by CCpast δ. On the advance, the radial component of its own velocity
is at most
∥K ∗ (Ωref (u) − W )∥∞ + Cp E(u − sq ).
80

The fixed canonical negative envelope, which follows from its first finite jet in (51), gives
Z 0

∥K ∗ (Ωref (u) − W )∥∞ du ≤ Caref = O(δref ).

sq

The comparison contribution tends to zero by (140). Older nonrotational fields contribute o(1)
and descendants contribute CβqC2 L = o(1). The resulting absolute radial displacement is
O(Cpast δ) + O(δref ) + o(1).
Fix δref sufficiently small for the prescribed margin, and then choose q for the fixed trial constants.
The radial exit estimate of Lemma 5.2 gives the fixed-radius Gaussian exit bound. Conditional
global moment growth, using (105), multiplies it by at most Cn eCn C2 L (1 + C2 L)n , which is
absorbed by its Gaussian. Thus all fixed envelope moments are bounded and their fixed-radius
moment tails are smaller than every prescribed q-power. The packet has bounded initial mass
and the same fixed initial support by the positive-envelope construction preceding Lemma 6.7; it
uses the same actual common drift, so the argument applies without a summed initial L∞ norm.
For completeness, the parent-radius estimate is proved directly, rather than deduced from the
fixed-radius tail. Choose cR so BR is inside a fixed fraction of the parent’s older probe core. The
actual older harmonic and leak estimates give there
|b<j · er | ≤ C

δq 2 3
ϵcore
|y| + C
|y| ≤ aq |y|,
β
β

aq ≤ Cδ/β + o(q P ).

The child’s own and descendant signed velocities have total speed at most a fixed D, by the
individual contractions and the convergent sum of Al Rl . The full actual radial drift on BR is
therefore bounded by aq |y| + D. Outside this ball its unrestricted radial bound is C|y|/β + D, as
in Lemma 6.7.
Let H = tref ≤ C2 L and enclose the original packet support in the fixed BRin . Since aq H =
o(1) and DH = O(L) ≪ R,
eaq H (Rin + DH) ≤ R/8
for small q. Run the radial diffusion calculation at radius R/4. After predictable removal of
√
tangential drift, its Brownian supremum must exceed a fixed multiple of R/ κ to exit. Thus
Pr{τR/4 ≤ H} ≤ C exp[−cR2 /(κH)].
Condition at that exit and use the unrestricted global radial bound. The same moment calculation
as in the proof of Proposition 7.3 gives
Z
√
2
(1 + |y|)n P (H, y) dy ≤ Cn M eCn H/β (1 + R + H + κH)n e−cR /(κH) .
|y|>R/4

Its prefactor is a fixed power of q −1 times a fixed logarithmic power, whereas R2 /(κC2 L) ≥
cq −7/2 /L. Reducing the Gaussian exponent constant proves (142). Truncation of the positive
initial packet and monotone passage justify the calculation when that initial packet has no finite
L∞ norm.
Finally, cut the fourth moment off at the fixed reset radius. Its difference is bounded by the
cutoff norm times E, plus superalgebraic tails. Since e−σu/β ≤ Cδ −1/β on [sq , 0], (140) gives
Z 0

e−σu/β |Bactual (u) − B4 (Ωref (u))| du ≤ Cδ −r−1/β (Eq + q) + o(q P ).

sq

This proves the weighted assertion under (141).
81

Remark 7.9 (The outer source). When j = 1, the older field is empty. The comparison and
fixed-radius reset in Lemma 7.8 remain valid with that term zero. No parent-radius handoff is
needed at the outermost scale; the R = cR /q estimate is asserted only for j ≥ 2.
Corollary 7.10 (Actual reference-state reset and fixed source windows). Choose the order in
Theorem 7.7 with M > CW /σ, and normalize one sufficiently small canonical reference state in
time and phase so that
A(Ωref (0)) = δref > 0,

A(Ωref (s)) = cref eλs + O(e2σs )

(s → −∞).

2 ), but the two parameters are not identified. Let s < 0
Write aref = |cref |; then aref = δref + O(δref
q
be the exact canonical hitting time of |A| = δq on Ωref , and remove the actual source’s constant
phase so that its first-cap coordinate equals A(Ωref (sq )). Put Lq = −sq and tref = tj + Lq . For
every fixed J < ∞, after the same phase and rigid-rotation removal,

ej (tref + ·) −→ Ωref
w

in C([−J, J]; L1 ∩ Lp ),

uniformly in cascade depth and all other seed phases. Whole fourth moments converge uniformly on each such window. At tref the actual source’s positive passive envelope, and the positive
envelope of its entire source/deep packet, have all fixed moments bounded uniformly and have
superalgebraic moment tails outside one fixed ball. In particular no power of log log(1/q) is lost
in this reference-time moment reset.
Proof. Theorem 7.7 is used only through its shrinking exact cap tj . Its full-state reset, including
the radial heat-base defect, gives
Eq ≤ CM δ M +1 + CM q γ (1 + C2 L)DM δ + Cκtj .
Its preparation envelope supplies (138) with a fixed Cpast . The choice M > CW /σ makes δ −r (Eq +
q) → 0, so Lemma 7.8 (with its empty-older-prefix variant when j = 1) proves convergence at the
fixed reference and on every fixed earlier reference window, together with both positive-envelope
resets. It is immaterial whether the selected source’s terminal finite-expansion amplitude exceeds
δref .
The clock lies inside the required inactive horizon directly. Indeed the selected-source cap
clock and (143) give
K
1
tref = tj − sq = L + log aref + o(1).
σ
σ
The choice C2 > K/σ + 1 therefore places every fixed reference window inside [0, C2 L] for small
q. On a fixed later window the canonical Euler support and its X gradient/Laplacian bounds
depend only on that window. The independently available actual older-field bound and descendant velocity sum still apply there. The same one-sided comparison extends convergence to
C([−J, J]; X).
To remove the fourth-moment cutoff on such a later window, the child and descendant total
velocity is uniformly bounded by scalar contractions. On a sufficiently large fixed ball the actual
older radial field is small; the initial reference envelope has the fixed-radius Gaussian tail just
proved. Radial exit and the unrestricted global moment bound give uniform fourth-moment
tightness on that fixed window. Strong L1 convergence on a cutoff ball then proves uniform
whole-fourth-moment convergence. This completes the corollary without extending the selectedsource expansion envelope beyond its proved endpoint.
82

Lemma 7.11 (Gaussian handoff at the parent radius). Fix any j ≥ 2 and any viscosity with
normalized value 0 < κ ≤ q 3/2 at this child. The child may have been activated by its incoming
packet; it need not be the deepest source selected by diffusion. Use only the common geometric
initial colour bounds and constant cores, and assume every actual larger nonradial density is at
most δq Al Rl2 through a normalized horizon 0 < H ≤ CH log(1/q). Let Fj (t) be the actual positive
scalar envelope transported from the absolute value of the normalized source datum. The same
conclusion holds for the positive envelope of the entire source/deep packet, transported from the
sum of its initial absolute colours. At every prepared cap or reference handoff time 0 < t ≤ H,
there is a fixed r0 > 0 such that, for every fixed integer n ≥ 0,
"

#

cn q −2
(1 + |x| )Fj (t, x) dx ≤ Cn exp −
.
κH
|x|>r0 /q

Z

n

(145)

This estimates positive-envelope leakage at the parent radius independently of any fixed-radius cap
reset.
Proof. Take r0 /q strictly inside the parent’s protected core, for example r0 = a0 /4. The actual
larger cores follow from the stipulated density stop by applying the outer-core proposition at
the parent index. Its horizon in parent units is H/β = O(log(1/q)),
its normalized viscosity
p
is βq 2 κ ≤ βq 7/2 , and its radial margin tends to zero because (H/β) δq = O(1/ log(1/q)) and
qH → 0. Thus no selected-source cold-tail theorem or child small-density assumption is being
invoked in this handoff lemma. The actual old core and harmonic decomposition gives on this
disk a radial bound Cδq |x|/β + ϵ(t)|x|, with superalgebraic leak integral. The own source and
all smaller velocities together are at most C, by the convergent individual scalar L1 , L∞ bounds.
Consequently the radial displacement bound for exit from this disk is at most


CH + Cδq H/β + superalgebraic error r0 /q.
Since H ≪ q −1 and δq H = o(1), this is less than a fixed fraction of its radius. The initial source support has fixed radius. The radial exit lemma bounds its exit probability by
C exp[−cq −2 /(κH)]. Globally all older colours satisfy the origin-growth bound |u<j (x)| ≤ C|x|/β,
while the source/deep packet velocity is bounded by C. Conditioning at exit therefore bounds
its later fixed moment by Cn q −n eCn H/β . Both this polynomial prefactor and the fixed q-power
from H = O(log(1/q)) are absorbed by reducing the constant in the exit exponential. Positivity
of the actual envelope and its fixed initial mass complete (145).
Proposition 7.12 (Actual positive source history before the parent cap). Start the source history
at the fixed reference reset of Corollary 7.10. The same statement applies to any other actual
source colour having the same scalar contractions and the following uniform reference-envelope
bounds: for every fixed integer n ≥ 4,
Z

n

(1 + |y|) P0 (y) dy ≤ Cn M,

Z

M=

P0 ,

with constants independent of q and depth. For the prepared geometric children these bounds are
supplied by Lemma 7.8. Their separate Gaussian parent-radius handoff is also available and is
used in the logarithmically shrinking-cap application below. The present general future-moment
bound uses all the displayed fixed moments; a fourth moment bound alone is not assumed to supply
them. On a following interval 0 ≤ u ≤ Uq ≤ C0 β log(1/q) assume that every larger colour is still
in its actual nonradial stop ∥(I − P )wl ∥1 ≤ δAl Rl2 , and that its actual constant core persists on
83

the appropriate probe disk with the Gaussian defect estimate. These are explicit hypotheses for
this following interval; the preceding selected-source theorem alone does not establish the whole
parent interval. Assume the normalized viscosity satisfies 0 < κ ≤ q 3/2 . For sufficiently small
fixed δ > 0 one has
ej (tref + u))| ≤ C(1 + u)4 eCδu/β + o(q P ),
|B4 (w
for every fixed P , uniformly in depth. Choose δ additionally so Cδ < σ/2. Then this is an
integrable uniform positive-history bound under the receiving discount e−σu/β .
Proof. On the disk |z| ≤ a0 /(4q) the actual old harmonic series gives |bold · er | ≤ Cδ|z|/β + ϵ(t)|z|.
P
Indeed the series is bounded by Cδ k≥1 β −k q 2k |z|3 , and q 2 |z|2 is bounded on this disk. The
core leak has superalgebraic time integral. At every time the source’s own velocity is bounded
by C, while the sum of all smaller velocities is bounded by Cβq, without any nonradial stop.
Thus actual radial drift there is at most C(1 + βq) + (Cδ/β + ϵ(t))|z|. For its actual positive
passive envelope use the weight (1 + |z|2 )2 . Its diffusion contribution is at most Cκ(1 + |z|2 ),
its additive-drift contribution at most C(1 + |z|2 )3/2 , and its linear-drift contribution at most
Cδ(1 + |z|2 )2 /β. Hölder and scalar Gronwall for the fourth root give the displayed polynomial
times weak exponential bound before exit.
Write R = cR /q, L = log(1/q), and let U ≤ CU βL. On BR use |v · er | ≤ a|y| + D, with
a = Ca δ/β after absorbing the Gaussian core leak, while globally use |v · er | ≤ a0 |y| + D with
a0 ≤ Cg /β and D ≤ Cd . Choose the fixed δ sufficiently small that
4Ca δ < σ2 .

θ := Ca CU δ < 14 ,

The first condition gives eaU ≤ q −θ . Put b = Re−aU and split the initial positive envelope at
rcut = b/8. Then rcut ≥ cq −1+θ → ∞, and DU ≤ b/8 for small q.
For paths beginning in Brcut , let Xu be the actual diffusion and set Yu √
= e−au Xu . Its radial
drift, until X exits BR , is at most De−au and its diffusion coefficient is 2κe−au . If X exits
BR by U , then Y has already exited Bb ; until this latter exit the local drift hypothesis applies
because |Xu | = eau |Yu | ≤ R. Applying the cosh test from Lemma 5.2, using only the upper radial
bound, gives
"
#
"
#
cb2
cR2 e−2aU
Pr{τR ≤ U } ≤ 2 exp −
= 2 exp −
.
κU
κU
Indeed the test generator is bounded by [2κe−2au λ2 + De−au λ] cosh(λ|Y |), and the exit gap is at
least b − rcut − DU ≥ 3b/4. No tangential drift bound is used. At U = 0 the exit probability is
zero.
The stopped fourth-moment calculation gives, for the unexited part,
Z

|y|4 P (u, y) dy ≤ CM (1 + u +

√

κu)4 e4au

before adding exited and initially exterior paths.

This follows by testing (1 + |y|2 )2 and taking the fourth root; the initial fourth moment is at most
C4 M . Conditional global moment growth after exit gives a remainder bounded by
Cg′ U/β

CM e

√
(1 + R + U +

"

#

cR2 e−2aU
κU ) exp −
,
κU
4

where Cg′ depends only on the global growth and additive speed. This is smaller than M q P for
every fixed P > 0: with κ ≤ q 3/2 , its Gaussian exponent has size at least cq −7/2+2θ /L, and all its
other factors are fixed powers of q −1 and L.
84

For the initially exterior part, the nth reset moment gives
Z
|y|>rcut

−(n−4)

(1 + |y|)4 P0 (y) dy ≤ Cn M rcut

.
′

Propagating this part with the global radial bound costs at most CeCg U/β (1 + U )4 . For any
prescribed P , choose one finite n such that (1 − θ)(n − 4) > P + Cg′ CU + 1. The resulting exterior
contribution is M o(q P ). Thus, uniformly on 0 ≤ u ≤ U ,
Z

|y|4 P (u, y) dy ≤ CM (1 + u)4 e4Ca δu/β + M o(q P ).

The second cap-smallness condition makes this bound integrable under e−σu/β , with uniformly
vanishing discounted future tails. This proves the stated generic future-history estimate. In the
actual application δ = δq , the strict displacement margin of Proposition 7.3 also holds, and its
sharper Gaussian-error estimate is used by Lemma 9.1.
Remark 7.13 (The core part of a stopped parent environment). If every larger colour is stopped at
the sharper density level δq = (log(1/q))−4 , the core hypothesis in Proposition 7.12 follows from
the already proved outer-core proposition, applied at the parent index. Its total time horizon
in parent units
is O(log(1/q)), its normalized viscosity is βq 2 κ ≤ βq 7/2 , and its radial margin
p
is O(log(1/q) δq + βq log(1/q)) = o(1). Thus only the actual parent/larger density stop is a
substantive input in this usual logarithmic window; a separate post-cap small-density assumption
on the source is unnecessary.
Corollary 7.14 (The actual whole-history source transform). Use the fixed reference gauge of
Corollary 7.10 and the actual post-reference environment of Proposition 7.12. Assume also Uq →
∞. Extend the actual source moment by zero before its physical initial time and after Uq . Then,
with µβ = [σ + i(τ − 4/5)]/β,
Z

e−µβ u Bactual,ref (u) du −→

Z

e−µβ u B4 (Ωref (u)) du,

R

R

uniformly in depth and all other seed phases. The same conclusion holds for the weighted absolute
difference of the two moment functions. The actual history is used only through Uq ; the zero
extension is a definition of the truncated history, not a claim about later actual dynamics. If the
canonical transform is nonzero at the chosen β, the actual one is nonzero for all sufficiently small
q, with a uniform lower bound in this fixed reference gauge.
Proof. On the preparation interval, the actual source norm envelope and fixed-radius Gaussian
ej (t))| ≤ Cαeσt . The exact reference shift satisfies αeσtref = aref (1 + o(1)). Thus
tails imply |B4 (w
the actual reference moment is bounded by Ceσu for u < 0, including the part between the
shrinking cap and the reference state. Its tail outside the fixed source radius is superalgebraic
and smaller than α, so it is included in that same envelope. The post-reference bound gives
C(1 + u)4 eCδu/β for u > 0 up to its endpoint, with superalgebraic errors uniformly on a logarithmic horizon. The canonical moment has the corresponding negative exponential and positive
polynomial bounds. Since β > 1 and Cδ < σ/2, both are integrable under e−σu/β . The extra
superalgebraic errors times the whole logarithmic horizon and its possible fixed q-power amplification still tend to zero. The fixed-window whole-moment convergence now proves the weighted
absolute-difference assertion by truncating at [−J, J], first choosing J and then q. This gives the
transform assertion and its nonzero conclusion.
85

Remark 7.15 (Why the fixed reference gauge matters). If the transform is instead centred at the
shrinking exact cap, the time and phase shift multiplies both actual and canonical transforms by
the same factor whose modulus is comparable to (δq /aref )1/β . The absolute lower bound then
shrinks by that factor, while the relative transform error remains small. Using the fixed reference
state as the receiving force origin avoids that artificially shrinking coefficient. Merely comparing
a fixed window around the shrinking cap would omit the later nonlinear source pulse.
Remark 7.16 (The reference inputs in the invariant class). For the inductive reset write
Eq ≤ Creset δ M +1 + q χ LD .
The fixed choice M + 1 > CW /σ + 2 implies (141), including the extra factor 1 + |sq | used
in Proposition 9.2. For each fixed trial reset constant the reference comparison error and its
integrated radial contribution tend to zero. The reference and its localization radius are fixed
before that later finite order.
The reference past before its small cap has weighted norm at most CCpast δ 1−1/β . In small-cap
time v ≤ 0, this follows from the envelope Cpast δeσv and the factor e−σsq /β = O(δ −1/β ). The
advance has fixed canonical history norm plus the vanishing error just proved.
At reference time (142), with n = 4, supplies (121) with H0 = C2 L, while the fixed-radius
moment reset supplies (120). Under the current parent’s protected-core stop, its actual older
harmonic/leak field satisfies (122); its unrestricted origin-growth bound gives (123); the child’s
own and descendant velocity sum gives the fixed D. Thus Proposition 7.3 supplies the prospective
future history without an assumed signed source waveform.
The upper history prefactor can be chosen independently of a fixed trial parent stop. In the
proof of that proposition the local stop constant occurs in eCCstop δu/β , while the prefactor uses
only the reset moment and D. On u ≤ C2 βL, this factor is at most two after choosing q for
the fixed trial stop; its discounted exponent is at most −σu/(2β). Conditional global moment
growth in the Gaussian error uses the unrestricted bound, independently of that stop. The
future weighted norm is consequently bounded by a fixed reference constant. Hence, for the finite
derivative orders selected before the class constants,
Minput ≤ Mref + CCpast δ 1−1/β + o(1),
where Mref may depend on those finite orders, fixed stream cutoffs, profile, reference, clocks, and
β, but not on trial past/reset/stop constants or depth. Orders can be chosen using the already
fixed comparison exponents, then their input constant computed, then the output class constants
fixed, and finally q decreased.
Fixed-window comparison and the two discounted tails also give uniform o(1) closeness of the
signed transform to the fixed canonical transform. This provides its nonzero lower bound once
the full prospective receiver interval is reached; the independent upper input bound is available
on every shorter stop. These are the reference and handoff inputs needed by Proposition 10.1.

8

Finite nonlinear expansions for retained actual forcing

8.1

Finite actual forcing: density closure and exact-cap reset

A finite expansion with the actual foreign-field coefficients gives an actual density envelope and
an exact-cap state reset. The recursion uses those coefficients as measurable functions of time.
A nonzero growing coefficient and pointwise bounds on the protected late window supply the
first-cap conclusion.
86

8.1.1

Spaces, operators and the derivative reserve

Let C be the union of the two closed derivative collars of the fixed smooth radial profile W . Choose
a closed radial interval I ⋐ (0, ∞) containing these collars in its interior. For an integer J ≥ 1,
let Ejr denote the real finite angular-sector space of functions supported in C whose sectors are
m ∈ 4Z, |m| ≤ 4Jj, with radial coefficients in H r (I). The norm is the sum of their radial Sobolev
norms. On these supported finite-sector functions it is equivalent to the full-plane H r norm,
with constants depending on I, J, j, r. Initially and in every recursion below the functions are
smooth, flat at the collar endpoints, and have zero integral. The coefficient equations therefore
use a common closed supported subspace on each finite Sobolev level; there is no assertion of an
unrestricted all-angular-mode spectral estimate.
Write
Lf = −(K ∗ W ) · ∇f − (K ∗ f ) · ∇W,
Q(f, g) = −(K ∗ f ) · ∇g.
The inherited finite-sector spectral lemma gives
etL = eλt P+ + eλ̄t P− + R(t),

∥R(t)∥H r →H r ≤ Cr e2ηt ,

2η < σ,

λ = σ + iτ.

(146)

This is used only in one of the finitely many spaces Ejr . The growing eigenfunction F+ lies in
sector +4 and is normalized by A(F+ ) = 1; its real conjugate lies in sector −4. The global test
defining A selects sector +4. For the pattern rotation set
(Tθ f )(r, φ) = f (r, φ + θ/4),

Lrot = L − (τ /4)∂φ .

For vector fields its definition is the pushforward Tθ V (x) = R−θ/4 V (Rθ/4 x), where R is the planar
rotation matrix. Equivalently its polar components are shifted by θ/4. This gives K ∗ (Tθ f ) =
Tθ (K ∗ f ) and Tθ [V · ∇f ] = (Tθ V ) · ∇(Tθ f ). This rotation is applied to the entire function,
separately in every angular sector. It is not replaced by the scalar factor e−ijτ s on an order-j
coefficient. The operators commute, and rotations are isometries in radial Sobolev norms. Thus
etLrot ≤ Ceσt and the growing real space has eigenvalue σ.
Fix truncation orders M, N ≥ 1 and a desired final Sobolev order k. Set
k∗ := max{4, k},

rjℓ = k∗ + 2(M − j) + 2(N − ℓ),

R = k∗ + 2M + 2N + 3.

(147)

The reserve is explicit. A diffusion source uses gj,ℓ−1 , whose order exceeds rjℓ by two. A quadratic
or foreign-advection source uses a coefficient with strictly smaller seed order, whose order exceeds
the requested order by at least two. A heat-interaction source uses gj,ℓ−r , again with at least two
additional derivatives. All profile derivatives used below are fixed derivatives of W ; W is smooth
in every finite order.
For zero-integral functions supported in one fixed ball,
∥K ∗ f ∥H r+1 (R2 ) ≤ Cr ∥f ∥H r ,

∥Q(f, g)∥H r ≤ Cr ∥f ∥H r ∥g∥H r+1 ,

r ≥ 2.

(148)

To check the first low-frequency estimate, compact support and zero mass give |fb(ξ)| ≤ C|ξ| ∥f ∥1
for |ξ| ≤ 1. The Biot–Savart multiplier has order −1, so it is bounded there by this cancellation
and is of order −1 at high frequency. Plancherel proves the first estimate. For the second,
H r (R2 ) is an algebra for r > 1, and ∇g ∈ H r ; multiplication and the first estimate prove the
assertion. On the supported spaces H k∗ also controls all the L1 , Lp product norms and the two
extra derivatives in residual Laplacians. These elementary bounds justify, rather than hide, every
transport derivative in the finite induction.
87

8.1.2

The finite recursion and its exact residual

Put X = L1 ∩ Lp , p > 2, with its sum norm. In a scale family write L = log(1/q), δ = L−4 , with
0 < q < 1; all error exponents χ > 0 are fixed before q is decreased. Splitting Biot–Savart at
radius one gives ∥K ∗ f ∥∞ ≤ Cp ∥f ∥X . Let 0 ≤ H < ∞, t0 ≥ 0, α ≥ 0, ε > 0, 0 < κ ≤ 1, and
a(t) = εeσ(t−t0 ) ,

s = t − t0 ,

Wκ (t) = eκt∆ W,

Wr (t) =

tr r
∆ W.
r!

Let B(t, x) be real and divergence free, spatially C ∞ for almost every time, and locally integrable
in time with values in every fixed finite C m norm. Only its C R norm is used in the quantitative
bounds. This qualitative regularity is automatic for the finite smooth stream basis with actual
moment coefficients. Its polar velocity components have finitely many angular sectors m ∈ 4Z,
|m| ≤ 4J. The retained field is bJ = εB. Assume
α σt0
e ∥Φ∥H r10 ≤ A0 ,
ε

D(t) := e

−σ(t−t0 )

∥B(t)∥C R ,

Z H

D(t) dt ≤ B0 ,

(149)

0

where Φ ∈ E1r10 is a real smooth zero-integral seed. The bounded constants A0 , B0 are the input
ball; no pointwise upper bound on D is imposed.
Set g10 (0) = (α/ε)Φ, with all other initial coefficients zero. For 1 ≤ j ≤ M , 0 ≤ ℓ ≤ N , solve
(∂t − L)gjℓ = 1ℓ≥1 ∆gj,ℓ−1 −

ℓ
X


(K ∗ Wr ) · ∇gj,ℓ−r + (K ∗ gj,ℓ−r ) · ∇Wr



r=1

X

+

Q(gj1 ℓ1 , gj2 ℓ2 ) − 1j=1 B · ∇Wℓ − 1j≥2 B · ∇gj−1,ℓ .

(150)

j1 +j2 =j, j1 ,j2 ≥1
ℓ1 +ℓ2 =ℓ

Every coefficient on the right precedes (j, ℓ) in the ordering by j + ℓ. The angular sectors are
at most 4Jj. All outputs are supported in C: Laplacians preserve the support of smooth flat
functions; ∆r W with r ≥ 1 has this support; and every advective term differentiates a supported
function. Each forcing has zero integral by divergence freedom or the Laplacian identity. The
bounded finite-sector generator and Duhamel therefore construct a unique smooth coefficient,
absolutely continuous in time in every required lower Sobolev order. Measurable time coefficients
are sufficient.
Define
F (t) =

M X
N
X

εj κℓ gjℓ (t),

wapp = Wκ + F.

j=1 ℓ=0

The recursion is an algebraic identity at the given run’s B, κ. Dependence of B on the actual
solution and on κ does not change that identity.
Lemma 8.1 (Coefficients and finite residual). Under (149), with the reserve (147),
∥gj0 (t)∥H rj0 ≤ Cejσs ,

∥gjℓ (t)∥H rjℓ ≤ C(1 + t)djℓ ejσs ,

djℓ = 2ℓ(j + ℓ),

ℓ ≥ 1.

(151)

The constants depend on the fixed finite orders and input ball, not on t0 , H, ε, κ or the particular
waveform. One may take dR = 4N (M + N ) + N + 3. If a ≤ 1 and κ(1 + H)dR ≤ c, the residual
in the retained field satisfies
∥∂t wapp + (K ∗ wapp + εB) · ∇wapp − κ∆wapp ∥X ≤ C aM +1 + κN +1 (1 + t)dR a (1 + D(t)). (152)


88



Proof. For g10 , (146) and the weighted initial bound show that its homogeneous contribution is
at most Ceσs . Its forcing convolution is at most
Z t

C
0

eσ(t−r) ∥B(r)∥C R dr = Ceσs

Z t

D(r) dr.
0

The reserve supplies its requested derivatives. Suppose preceding coefficients have the claimed
bounds. A quadratic pure-seed source has size Cejσ(r−t0 ) in the requested Sobolev order, by
(148). For j ≥ 2 its convolution has the gap (j − 1)σ, so its bound is Cejσs with no polynomial.
A Rpure external-advection source has size Cejσ(r−t0 ) D(r); dividing its convolution by ejσs leaves
C 0t e−(j−1)σ(t−r) D(r) dr ≤ CB0 . This proves the pure-seed estimate.
For positive viscosity order, the heat terms carry (1 + r)r from Wr , the diffusion term has the
preceding polynomial, and the quadratic terms add their two polynomial degrees. A convolution
at j = 1 can add at most one degree; for j ≥ 2 the strict gap adds none. The choice djℓ = 2ℓ(j +ℓ)
dominates all these degrees:
djℓ ≥ dj,ℓ−1 + 1,

djℓ ≥ r + dj,ℓ−r + 1,

djℓ ≥ dj−1,ℓ ,

djℓ ≥ dj1 ℓ1 + dj2 ℓ2 .

The second inequality is used for r ≥ 1 and ℓ ≥ r; the third for j ≥ 2; the last for the displayed
quadratic indices. For the direct B∇Wℓ term its degree ℓ is smaller. The weighted D convolution
adds no degree beyond the preceding factor. This proves (151) by the finite induction, using
precisely the reserve already specified.
P
r
For completeness we give the residual identity. Put GN = Wκ − N
r=0 κ Wr and Qr h =
−[(K ∗ Wr )∇h + (K ∗ h)∇Wr ]. Write QG h for the same expression with GN replacing Wr . After
cancellation of the retained monomials the residual is exactly
RJ = −

M
X

X

εj κN +1 ∆gjN −

j=1

εj κℓ+r Qr gjℓ

1≤j≤M, 0≤ℓ≤N, 1≤r≤N
ℓ+r>N

X

−

εj1 +j2 κℓ1 +ℓ2 Q(gj1 ℓ1 , gj2 ℓ2 )

1≤j1 ,j2 ≤M, 0≤ℓ1 ,ℓ2 ≤N
j1 +j2 >M or ℓ1 +ℓ2 >N

+ εM +1

N
X

κℓ B · ∇gM ℓ + εB · ∇GN − QG F.

(153)

ℓ=0

Here juxtaposition with a gradient means a dot product. The exact radial heat base has zero
residual on its own. The integral heat Taylor formula gives
∥GN ∥X + ∥∇GN ∥X ≤ CκN +1 tN +1 .
Use the Biot–Savart X bound and (151) to estimate QG F . Every remaining factor in (153),
other than the heat remainder, is supported on the fixed collars. Pure terms with seed order
exceeding M are CaM +1 ; their pure coefficient bounds have no polynomial. A positive viscosity
power multiplied by any of the finitely many polynomial factors is absorbed by increasing dR
and requiring κ(1 + H)dR ≤ c. Terms with viscosity order exceeding N are CκN +1 (1 + t)dR a,
since aj ≤ a. The external top-seed term uses ∥B∥ = eσs D and is CaM +1 D. The external heat
remainder is CκN +1 (1 + t)dR aD. This proves (152), including all viscosity terms and without
assuming a pointwise bound on D.

89

8.1.3

Comparison with the actual foreign velocity

Suppose the actual colour solves
∂t w + (K ∗ w + b) · ∇w = κ∆w,

w(0) = W + αΦ.

The actual b is divergence free; it may depend on the entire interacting solution. Assume the
usual positive-time smoothness and Ct X initial trace, sufficient for the scalar Kato inequalities.
A removed rigid rotation is allowed. Let U be a fixed annular neighbourhood of C, separated
from its complement by a positive radial heat-gradient gap. Assume
sup |b − bJ | ≤ mq ,

|b(t, x) · er | ≤ Cb (1 + |x|),

(154)

x∈U

with fixed Cb . The second is used only against the exact radial heat gradient outside U. No
derivative of b is assumed. The stream extensions giving B agree with the retained harmonics on
U. These are exactly the collar omission and actual global radial-growth inputs supplied by the
packet/core estimates.
Lemma 8.2 (Actual density comparison). Fix C1 = Cp (∥∇W ∥X + 1) before choosing M, N, J.
Choose (M + 1)σ > C1 . There is a later threshold a∗ > 0 such that if a(t) ≤ a∗ and the finite
viscosity smallness conditions hold, then ∥∇wapp ∥X gives the fixed exponent C1 and
∥w − wapp ∥X ≤ CaM +1 + CκN +1 (1 + t)dR +1 ae(C1 −σ)+ t
+ Cmq (1 + t)eC1 t + C(1 + H)eC1 H (1 + εeσH )e−c/(κH) .

(155)

The last Gaussian term is understood as zero at H = 0. Consequently in a scale family
ε = qr ,

σt0 ≤ θL,
r, γ, χ > 0,

H ≤ CH L,

κ ≤ qγ ,

θ ≥ 0,

0 < CH < ∞.

mq ≤ q P Ldm ,

if
γ(N + 1) > CH (C1 − σ)+ + χ,

P > r + θ + CH (C1 − σ)+ + χ,

(156)

then
∥w − wapp ∥X ≤ CaM +1 + Cq χ Ld∗ a,

∥w − Wκ ∥X ≤ Ca.

(157)

These estimates require no nonzero transmission coefficient.
Proof. The finite coefficient bounds make the pure perturbation gradient at most 1/2 after choosing a∗ sufficiently small. Choose the later viscosity threshold so the remaining finite gradients are
at most 1/2. Heat contracts ∥∇W ∥X . This yields the already fixed C1 ; the coefficient constants
change thresholds, not that exponent. For z = w − wapp the exact equation is
(∂t + (K ∗ w + b) · ∇ − κ∆)z = −(K ∗ z) · ∇wapp − RJ − (b − bJ ) · ∇wapp .
The actual foreign field is part of divergence-free transport and cancels from the L1 , Lp norm
inequalities. The Biot–Savart X bound gives
(∥z∥X )′ ≤ C1 ∥z∥X + RJ

X

+ ∥(b − bJ ) · ∇wapp ∥X .

The initial error is zero. Smooth norm regularizations and compact cutoffs justify the integrated
inequality from time zero.
90

On U the mismatch is at most Cmq (1 + a). Outside it, ∇F = 0 and the weighted radial heatgradient norm is Ce−c/(κt) , after decreasing c to absorb fixed polynomial heat factors. The actual
radial-growth bound and ∥bJ ∥∞ ≤ aD therefore bound the omitted force by C(1 + aD)e−c/(κt) .
Convolve with eC1 (t−v) . For the top pure residual, division by a(t)M +1 leaves
Z t

e−[(M +1)σ−C1 ](t−v) (1 + D(v)) dv ≤ [(M + 1)σ − C1 ]−1 + B0 .

0

For the viscous residual, division by a(t) leaves at most C(1 + t)dR +1 e(CR1 −σ)+ t (1 + B0 ). The
constant collar mismatch gives Cmq (1 + t)eC1 t . The Gaussian force uses D ≤ B0 and monotonicity of its exponential, proving (155) after enlarging its harmless prefactor. In the scale
family, a(0) ≥ q r+θ and (156) makes both non-Gaussian errors relative to a(t) positive powers
of q times fixed logarithms. The Gaussian dominates all these fixed power losses uniformly for
smaller viscosities. Finally the finite coefficient sum has norm Ca and the error has the same
bound, proving (157).
8.1.4

A normalized finite polynomial loses the forcing waveform

Extend B by zero after H, and put S = H −t0 . Suppose the decomposition holds on −t0 ≤ s ≤ S,
with the following future bound for s ≥ 0:
B(t0 + s) = B main (s) + E(s),
B main (s)
Z S
−t0

CR

≤ CF (1 + s)dF ehs ,

0 ≤ h ≤ h∗ < σ,

e−σs ∥E(s)∥C R ds ≤ ηq .

(158)

The main and remainder are finite-sector fields of the same class as B and are zero-extended after
S. Before s = 0 only the total weighted bound (149) is used; no past convergence hypothesis is
needed for the coefficient reset. Define the full first-order growing amplitude
α
Cq = eλt0 A(Φ) +
ε

Z H

e−λ(t−t0 ) A(−B(t) · ∇W ) dt.

(159)

0

It includes every retained input and the initial seed. It equals the infinite-future growing amplitude of the zero-extended first coefficient. It is not replaced by a canonical target coefficient.
Let F1 (C) = CF+ + C̄F+ , and, for j ≥ 2, define
Fj (C) = −(jσ − Lrot )−1

X

(K ∗ Fa (C)) · ∇Fb (C).

(160)

a+b=j

The inverse is the convergent integral of the finite-sector semigroup et(Lrot −jσ) . It is not a formal
inverse. All coefficients are smooth, supported in C, of zero mass, and belong to sectors 4n with
|n| ≤ j and n ≡ j modulo two. Induction proves that they are real polynomials of total degree j
in C, C̄, and that Fj (C) = |C|j Targ C Fj (1) for C ̸= 0.
Lemma 8.3 (Finite coefficient memory loss). Under (149) and (158), for any fixed 0 < d <
min(σ − 2η, σ − h∗ ), the pure coefficients obey
e−jσs T−τ s gj0 (t0 + s) − Fj (Cq )

H rj0

≤ Cj (1 + s)dj e−ds + Cj ηq ,

0 ≤ s ≤ S.

(161)

The constants depend only on the fixed finite orders, input ball, future envelope, and the strict rate
gap. They do not depend on the waveform, preparation length, or an incoming canonical-profile
error. No lower bound on |Cq | is required for this coefficient statement.
91

Proof. The coefficient bounds imply that every normalized coefficient at s = 0 is uniformly
bounded in its reserved order. Write hj (s) = e−jσs T−τ s gj0 (t0 + s). For j = 1 the growing
component is F1 (C(s)), where the partial integral in (159) defines C(s). Its missing main future
tail has norm at most C(1 + s)dF e−(σ−h)s ; its missing remainder has norm at most Cηq . Its
complementary component has propagator decay e−(σ−2η)s after normalization. Its main forcing
has bound C(1 + s)dF e−(σ−h)s in this frame. Duhamel convolution of these two decays is bounded
byR C(1 + s)d1 e−ds for the stated strictly smaller d. The remainder convolution is bounded by
C 0s e−σv ∥E(v)∥C R dv ≤ Cηq . This proves the first case.
For j ≥ 2 the normalized equation is exactly
h′j = (Lrot − jσ)hj −

(K ∗ ha ) · ∇hb − e−σs (T−τ s B(t0 + s)) · ∇hj−1 .

X
a+b=j

Its homogeneous propagator decays as Ce−(j−1)σs by (146). Subtract (160). Each quadratic
difference has at least one lower difference, and its other factor is uniformly bounded in the
needed reserved order. The reserve supplies one extra derivative in that lower factor, so (148)
applies without loss. By induction these terms are bounded by C(1 + s)d e−ds + Cηq in the
target order. The direct main external term is C(1 + s)dF e−(σ−h)s ; its remainder has weighted
time norm at most Cηq . Duhamel and (j − 1)σ > d yield the same bound for the normalized
difference, including the decaying bounded initial value at zero. Increasing a finite polynomial
degree handles a possible convolution polynomial. This proves the finite induction and (161).
8.1.5

Exact protected-coordinate matching

Define the real Banach-valued polynomial
PM (z) = W +

M
X

Fj (z),

z ∈ C.

j=1

Its protected coordinate is ΨM (z) = A(PM (z)). It is a real polynomial in z, z̄, is rotation
equivariant, ΨM (eiθ z) = eiθ ΨM (z), and DΨM (0) is the identity on R2 . In particular it has the
form z[1 + O(|z|)]; the parity actually eliminates its quadratic term, but that stronger fact is not
needed.
Lemma 8.4 (Exact polynomial and ancient cap gauge). There is a fixed ρ > 0 such that ΨM has a
unique inverse on |ζ| < ρ, with inverse derivative uniformly bounded. Suppose ∥v − PM (z)∥X ≤ e,
iθ
|z| < ρ, and A(v) = δeiθ with 0 < δ ≪ ρ. Then for z∗ = Ψ−1
M (δe ),
|z − z∗ | ≤ Ce,

∥v − PM (z∗ )∥X ≤ Ce.

(162)

For the inherited canonical ancient orbit, Corollary 4.5 supplies every fixed finite jet
Ωref (u) − PM (cref eλu )

X

≤ CM |cref eλu |M +1 ,

u ≪ 0,

cref ̸= 0.

If uδ is its small exact cap, the rotation ϕ matching its protected coordinate exactly to δeiθ satisfies
∥v − Tϕ Ωref (uδ )∥X ≤ C[e + δ M +1 ].

92

(163)

Proof. Choose 2ρ so small that sup|z|≤2ρ ∥DΨM (z) − I∥ ≤ 1/4. The integral mean-value formula
then gives the quantitative inverse estimate |ΨM (z) − ΨM (z ′ )| ≥ (3/4)|z − z ′ | in that ball. The
inverse function theorem, or its contraction proof applied to z = ζ − [ΨM (z) − z], supplies the
inverse on a smaller ball. Since A is bounded on X, the state error bounds the coordinate error
by Ce, and the inverse estimate proves the first bound in (162). The derivative of PM is bounded
on this fixed ball, so the second follows.
At the canonical exact cap, its jet remainder is O(δ M +1 ) because |cref eλuδ | ≍ δ. The same
inverse estimate changes its linear asymptotic parameter to the unique parameter matching its
full protected coordinate, with error O(δ M +1 ). Rotation equivariance makes that parameter,
after the exact coordinate-matching rotation, precisely z∗ . The polynomial and ancient-state
errors therefore add to the bound (163). Matching only the linear parameter would instead leave
a lower-order error; it is not used here.
8.1.6

State reset and the separate instantaneous cap condition

Combine the lemmas in a scale family with ηq ≤ q χE LdE , χE > 0, and 0 < c∗ ≤ |Cq | ≤ C ∗ .
Define
δ
Sq = σ −1 log
,
δ = L−4 ,
ε|Cq |
and assume the prospective interval contains t0 + [Sq − L0 , Sq + L0 ], where L0 is fixed. At a time
in that window put z(t) = εeλ(t−t0 ) Cq . The coefficient memory estimate, the positive-viscosity
correctors, the comparison bound, and ∥Wκ − W ∥X ≤ CκH give
∥w(t) − PM (z(t))∥X ≤ Cδ M +1 + q χ∗ Ld∗ ,

(164)

for some fixed χ∗ > 0. Indeed Sq = (r/σ)L + O(log L), so the intrinsic memory transient is
a positive power of q times fixed logarithms. The other errors have the strict powers already
stated. The exponent χ∗ is independent of an incoming canonical state error: that error may
change the waveform and Cq , but is not used in the memory estimate. Exact coordinate matching
by Lemma 8.4 then gives the same order of ancient-state reset at any exact cap in this window.
A weighted time norm alone does not supply a crossing slope. For a protected first-cap conclusion, impose the following separate pointwise actual-field bounds on the late window whenever
3δ/4 ≤ |A(w)| ≤ 5δ/4, writing f = w − Wκ :
Z

Tadj b · ∇W +

Z

|f | |b| |∇Tadj | +

Z

Tadj b · ∇(Wκ − W ) ≤ ρq δ,

ρq → 0.

(165)

These are instantaneous hypotheses, not consequences of (149). In the actual receiver application
they follow, respectively, from the absolute packet moments and older collar bounds, the global
weighted adjoint lemma, and the weighted smooth heat-profile estimate with the global radialgrowth bound. These pointwise criteria are checked separately in the receiver application.
Corollary 8.5 (Protected actual first cap). Assume the preceding scale-family bounds, the nonzero coefficient bound, (165), and a sufficiently wide prospective interval. Then for small q there is
a unique first crossing |A(w(tq ))| = δ. It lies in the fixed late window and satisfies
d
|A(w)| ≥ σδ/2 a.e. for 3δ/4 ≤ |A(w)| ≤ 5δ/4 in that window.
dt
The upper slope is Cδ almost everywhere, and its exact-cap state satisfies
tq = t0 + Sq + o(1),

w(tq ) − Tϕq Ωref (uδ )

X

93

≤ Cδ M +1 + q χ∗ Ld∗ ,

where the canonical coordinate and actual coordinate are matched exactly. Any continuous family
of actual solutions satisfying these uniform bounds has continuous cap times and phases.
Proof. The all-time density bound and boundedness of A give |A(w)| ≤ Ca(t). Choose L0 so
large that Ce−σL0 /c∗ < 1/4. Then no cap occurs for t − t0 ≤ Sq − L0 . In the fixed late window
(164) and the polynomial coordinate yield A(w) = z(t) + O(δ 2 ) + o(δ) uniformly. The moduli at
the window endpoints straddle δ, so continuity gives a crossing. The adjoint identity proved in
Section 4.2, with (157) and (165), gives
A′ = λA + EA ,

|EA | ≤ Cδ 2 + ρq δ + Cκ(1 + H)δ.

The last term tends to zero relative to δ. The coordinate is absolutely continuous. Its modulus
derivative exists almost everywhere and equals σ|A| + Re(AEA )/|A|. On the stated level interval
it is at least 3σδ/4 − Cδ 2 − o(δ) ≥ σδ/2. The upper bound follows from the same identity. Let
Cspeed be its fixed upper-speed constant. For all sufficiently small q, the crossing lies in the middle
half of the late window. Choose
dcap = min{L0 /4, (8Cspeed )−1 }.
On [tq − dcap , tq + dcap ] the radius stays in the level band: before a first exit its change is at
most Cspeed δdcap ≤ δ/8, whereas exit requires a change of δ/4. The almost-everywhere lower
slope integrates on this entire interval. At earlier times the envelope margin applies or the radius
is below 3δ/4; a previous return from a larger radius would require a downward band crossing.
Thus for each fixed d ∈ (0, dcap ] there is the explicit uniform precap margin
sup

|A(w(t))| ≤ δ − min{δ/4, σδd/2}.

0≤t≤tq −d

A second crossing within the protected window would require a downward passage through this
level interval, contradicted by the strictly positive slope. The endpoint estimate also gives tq −
(t0 + Sq ) = o(1). Apply the exact-gauge lemma to (164) at this crossing. Finally a uniform
early margin, common crossing slope, and uniform convergence of the continuous coordinate
under solution parameters prove continuity of the first hit; its nonzero coordinate makes its
circle-valued phase continuous.
The finite density and reset estimates use upper field-history bounds. The protected first-cap
corollary additionally uses a nonzero Cq and (165).

9

Transmission to an actual receiver

9.1

An actual receiver and the loss of source memory

The receiving step starts from the child preparation envelope and its phase-matched exact-cap
state. The finite expansion is applied to the actual harmonic coefficients of the coloured Navier–
Stokes evolution.
Write L = log(1/q), δ = L−4 , α = q K , where K > 6, and fix the smooth reference W , its
protected coordinate A, its growing eigenvalue λ = σ + iτ , and its canonical ancient orbit Ωref :=
Ωδref . The reference amplitude δref > 0 is fixed, sufficiently small, independently of q. After
one fixed time translation and rotation, normalize the canonical orbit so A(Ωδref (0)) = δref > 0.

94

The bilateral transform is multiplied by a nonzero factor by this normalization; its nonvanishing
selection is therefore unchanged. Choose a fixed β in a sufficiently large interval so that
Z

e−µβ u B4 (Ωδref (u)) du ̸= 0,

C∗ := γϵ

µβ = [σ + i(τ − 4/5)]/β.

R

All choices below are made before q.
9.1.1

Discounted histories of an arbitrary prepared actual child

The following conditional history estimate applies to a child prepared by an incoming packet. It
uses its preparation envelope and exact-cap reset independently of the receiving-stage conclusion.
Lemma 9.1 (Prepared-child reference history). Fix the smooth profile, its all-order canonical
reference orbit, β > 1, logarithmic clock constants and finite spatial derivative orders. Write
L = log(1/q), δ = L−4 and µβ = [σ + i(τ − 4/5)]/β. Assume the actual colours belong to the
geometric stack of Lemma 3.1, with the clock and parameter margins of Corollary 6.9 in child
units. Use child units, with all older constant rotations removed. Let the actual child wc have
normalized viscosity κc ≤ q 3/2 , an exact cap time τc satisfying c0 L ≤ τc ≤ C0 L, and the common
geometric initial colour bounds. Suppose, for fixed finite trial constants Cpast , Creset > 0,
∥wc (t) − eκc t∆ W ∥X ≤ Cpast δeσ(t−τc ) ,

0 ≤ t ≤ τc .

Let sq < 0 be the canonical exact δ-cap time, and match the constant phase ϕ exactly in the
protected coordinate. Suppose
Eq = Creset δ M +1 + q χ LD ,

∥T−ϕ wc (τc ) − Ωref (sq )∥X ≤ Eq ,

χ > 0,

and
δ −CW /σ−1/β (1 + |sq |) Eq + q(1 + |sq |) −→ 0.




(166)

Here CW is the fixed one-sided comparison constant on the small canonical ancient segment. The
sufficient fixed choice M + 1 > CW /σ + 2 has strict slack in this condition.
Put tref = τc − sq . On the interval ending at this reference time, assume the actual inactiveprefix estimates and the geometric preparation localization hypotheses of Lemma 7.8, with strict
clock slack. On the following prospective interval assume the actual parent/larger constant-core
and harmonic conclusions on the provisional parent density stop ∥wi − Wκi ∥X ≤ Cs δ, where
Cs > 0 is fixed and finite. In particular the child’s older radial drift in a fixed fraction of the
parent core has coefficient at most C(1 + Cs )δ/β. Use the Gaussian parent-radius handoff of
Lemma 7.11 at tref . Let the following child horizon satisfy
0 ≤ Uq ≤ CU βL.
The complete age and parent-unit horizon are assumed to lie in the fixed logarithmic windows of
the cited inactive/core estimates.
For every such shorter stopped interval the absolute differentiated main-input history is bounded by one fixed ball independent of the trial constants, after a later q threshold depending on those
constants. If the complete prospective interval has been reached and Uq → ∞, define
(

Bq (u) =

B4 (T−ϕ wc (tref + u)),
0,
95

−tref ≤ u ≤ Uq ,
otherwise.

Then

Z

e−σu/β |Bq (u) − B4 (Ωref (u))| du −→ 0.

(167)

R

Consequently its complex discounted transform converges to the full canonical transform at µβ .
For clarity, Bq is the actual physical history truncated at the controlled prospective endpoint
and zero-extended thereafter. The canonical history in (167) is defined on the entire real line.
No estimate for the genuine actual infinite-time future after parent density/core control has ended
is asserted or needed.
The quantifiers are as follows. The fixed profile, reference, β, clocks and finite spatial orders
determine the input ball first. For each fixed finite triple (Cpast , Creset , Cs ) satisfying the stated
preparation/reset assumptions, there is a later q threshold. Within that class every estimate and
convergence above is uniform in depth, source phase and allowed nondistinguished seed homotopies. The threshold need not be uniform over unbounded trial constants. Thus the constants
may be selected once from the finite-forcing output bounds and then one common q chosen, without
multiplying an inherited input constant at every descending stage.
Proof. Write Lq = −sq . The canonical exact-cap asymptotic gives
Lq = O(log L),

e−σsq /β ≤ Cδ −1/β .

All phases below have already been removed by T−ϕ . Rigid rotations preserve every radial
positive-envelope estimate.
Localization before the reference. On the preparation interval the child’s own nonradial radial
velocity has time integral at most CCpast δ. The descendant velocity contributes at most CβqC0 L.
The actual older nonrotational harmonic field and core leaks on a fixed child ball are smaller, by
Corollary 6.9. For any fixed trial Cpast these displacements tend to zero. The initial support and
a fixed radial margin therefore give, at every preparation time, positive-envelope Gaussian tails
outside one fixed ball. Conditional global origin-growth estimates control every fixed post-exit
moment; their fixed q-power amplification is absorbed by the Gaussian. The exact radial heat
base has the corresponding Gaussian moment tails. These conclusions use the actual positive
scalar envelope and the common drift, not a gradient of the actual child.
On the cap-to-reference segment, Lemma 7.8 gives the one-sided X comparison with the
phase-matched canonical ancient solution
sup ∥T−ϕ wc (τc + v) − Ωref (sq + v)∥X ≤ Cδ −CW /σ [Eq + q(1 + Lq )].

(168)

0≤v≤Lq

Indeed the canonical segment has fixed derivative bounds and fixed compact support. On that
support the descendant field is O(βq), the actual larger nonrotational field tends to zero with
a strict positive q power, and κc ≤ q 3/2 . The comparison forcing is therefore O(q) with fixed
parameters. Gronwall over Lq costs at most Cδ −CW /σ . No derivative of an actual moment
coefficient or actual colour occurs.
The canonical nonradial radial displacement on this segment is O(δref ); the comparison error
contributes o(1) by (166). Choose the fixed reference amplitude inside the fixed radial margin. Together with the descendant and older-field contributions, this gives the same fixed-radius
positive-envelope moment reset at the reference. In particular no factor log log(1/q) is introduced
into its moment constants. The separate parent-radius handoff gives the Gaussian exterior-tail
input required by Proposition 7.3; fixed-radius tightness alone is not used as a substitute for that
input.
96

The negative history. Let u = t − tref . For u ≤ sq , angular symmetry gives B4 (eκc t∆ W ) = 0.
Inside the fixed preparation ball the whole-moment pairing is bounded by a fixed multiple of the
perturbation L1 norm. Outside it use the preceding Gaussian positive-envelope and heat tails.
Thus
|Bq (u)| ≤ CCpast δeσ(u−sq ) + Gq (u),
−tref ≤ u ≤ sq ,
where the Gaussian remainder is superpolynomial on the logarithmic preparation horizon. Its
weighted integral is still superpolynomial: the earliest negative endpoint has size O(L), so the
discount costs only a fixed power of q −1 , whereas κc ≤ q 3/2 . The main term has the explicit
bound
Z sq
CCpast δe−σsq /β
e−σu/β CCpast δeσ(u−sq ) du ≤
≤ CCpast δ 1−1/β −→ 0.
σ(1 − 1/β)
−tref
The canonical moment on (−∞, sq ] has the same vanishing discounted bound, since it is O(eσu ).
Therefore the weighted absolute difference on the entire pre-small-cap interval tends to zero. This
calculation replaces the selected-source identity involving αeσtref .
On [sq , 0], the fixed-radius localization converts (168) into a whole fourth-moment difference
with the same bound plus Gaussian errors. Since
Z 0

e−σu/β du ≤ Cδ −1/β (1 + Lq ),

sq

its weighted absolute difference tends to zero by (166). The negative canonical history on this
segment has one fixed discounted norm. This proves a negative-history input bound consisting
of a fixed reference majorant, CCpast δ 1−1/β , and a vanishing reset contribution.
The controlled future. At reference the positive-envelope mass and fourth moment have fixed
majorants. The own and descendant velocities have a fixed bounded sum by the individual scalar
contractions and the convergent geometric velocity series. The older global radial drift is at most
C|y|/β; inside the parent probe core its coefficient is C(1 + Cs )δ/β, with Gaussian leaks. The
parent-radius handoff supplies its initial exterior Gaussian tail. Keep the local radial coefficient
a = C(1 + Cs )δ/β explicit in the proof of Proposition 7.3. Its unexited fourth moment has
a prefactor determined only by the initial envelope moments and fixed source and descendant
velocity bound, and a factor e4au . All stop-dependent constants in the exited part are absorbed
by its Gaussian after a later q threshold. The proposition and Corollary 7.4 thus yield on every
shorter stopped interval
|Bq (u)| ≤ C(1 + u)4 eC(1+Cs )δu/β + o(q P ),

0 ≤ u ≤ Uq ,

for every fixed P . The prefactor is fixed by the reference-envelope majorants and bounded source
and descendant speed. It does not multiply Cs . For each fixed trial Cs , a later threshold makes
C(1 + Cs )CU δL ≤ log 2,

C(1 + Cs )δ < σ/2.

Consequently the future absolute discounted input is bounded by a fixed multiple of 0∞ e−σu/β (1+
u)4 du, and its tail from J is bounded by
R

Cβ,σ (1 + J)4 e−σJ/(2β) + o(q P ).
The errors remain negligible under all logarithmic-window and fixed q-power losses. Spatial
differentiation of a retained fourth-harmonic stream field differentiates only its fixed smooth
97

spatial basis, so the same argument supplies every fixed spatial derivative input norm. Together
with the negative-history estimates, this gives a fixed input ball independent of the trial constants
after their later thresholds. It is an upper bound only and requires neither a nonzero signed
transform nor completion of a shorter density stop.
Once the complete prospective interval is available, every fixed window [0, J] lies within it for
small q. From (168), the actual reference state converges to the canonical reference state in X.
On each such following window the canonical derivatives are bounded, viscosity tends to zero, and
the actual older nonrotational and descendant fields on fixed support tend to zero. The same onesided comparison therefore gives convergence in C([0, J]; X). Positive-envelope radial localization
on this fixed window, followed by conditional moment growth, gives uniformly vanishing fourthmoment tails outside a sufficiently large fixed ball. Whole fourth moments consequently converge
on the window. All constants are uniform over the fixed prepared class.
The canonical positive history has its polynomial discounted tail. Truncate at J, apply fixedwindow whole-moment convergence, and then let J → ∞, using the uniform future tail just
established. Combining this with the two negative-history pieces proves (167). The complex
transform conclusion follows because |e−µβ u | = e−σu/β .
Every estimate used geometric normalized mass, amplitude or velocity sums, fixed spatial
orders and the common clocks. Its thresholds depend on the fixed trial constants, not on depth
or phase. Selecting those constants once and then taking the minimum of the finitely many
thresholds yields the stated invariant-class quantifiers. If the force origin is instead chosen at the
actual protected crossing of the fixed reference level, the same comparison and canonical positive
slope change its time and constant phase only by o(1); this preserves the history and transform
conclusions.
Proposition 9.2 (Actual receiving-stage implication). Consider actual colours wl at scales Al =
A1 β l−1 , Rl = R1 q l−1 , in the common unforced Navier–Stokes velocity. Their initial conditions
are the fixed smooth seeded profiles constructed in Section 3.1, with all seed amplitudes at most
α; every colour is real, fourfold symmetric and has zero circulation. Fix a receiving index i and
its immediate child c = i + 1. Work in receiver units and remove the common solid rotation of
all older cores. Assume κc ≤ q 3/2 and hence κi = βq 2 κc ≤ βq 7/2 .
The inductive child hypothesis is as follows. In child units it has an exact first cap τc , with
|A(wc (τc ))| = δ,
c0 L ≤ τc ≤ C0 L,

wc (t) − eκc t∆ W

X

≤ Cpast δeσ(t−τc )

(0 ≤ t ≤ τc ),

inf ∥wc (τc ) − Tϕ Ωδref (sq )∥X ≤ Creset δ M +1 + q χ LD ,

(169)
(170)

ϕ

where Cpast , Creset are fixed finite trial constants, and sq < 0 is the canonical exact-cap time
|A(Ωδref (sq ))| = δ; the phase in (170) is chosen in the same exact protected-coordinate gauge. Here
χ > 0 is fixed. Choose the fixed finite order M so that M + 1 > CW /σ + 2, where CW is the onesided cap-to-reference comparison constant. Then (166) holds and controls the discounted history
through the advance. The positive scalar envelope transported from |wc (0)| has at τc uniformly
bounded moments of each fixed order and Gaussian tails outside a fixed ball, as supplied by the
preparation localization lemma. All these constants are uniform in depth and in the allowed seed
homotopies.
Choose fixed C0 , C2 with C2 > C0 > max(K/σ, 6/σ), then β, D sufficiently large so that
C0 > 6β/[σ(β − 1)], the unconditional inactive closure holds and
ϵ0 := C2 σ/β < K − 6,

C2 λB < K − 6,
98

C2 (λP + C4 /β) < 1,

ϵ0 < 1/2.

(171)

For small q, the actual receiver has a unique protected first cap |A(wi (ti ))| = δ, with
ti = t0 + σ −1 log

δ
q 6 |C∗ |

+ o(1),

where t0 is the child’s fixed-reference time in receiver units. The reference phase ϕc used below
is measured in the receiver’s frame at t0 . If θc is its phase in the child’s frame with all older
rotations removed, then ϕc = θc + 4t0 /5 modulo 2π: the parent core rotation is −1/5 in receiver
time. Thus the source field has factor e4i(t−t0 )/5 eiϕc , rather than counting the reference rotation
a second time. It obeys an exponential preparation envelope
wi (t) − eκi t∆ W

X

≤ Cq 6 eσ(t−t0 )

(0 ≤ t ≤ ti ).

(172)

In its exact-cap phase gauge its state satisfies
′

′

∥wi (ti ) − Tϕi Ωδref (sq )∥X ≤ Cδ M +1 + q χ LD ,

(173)

for some fixed χ′ > 0, D′ < ∞. The exponent χ′ depends only on the fixed stage parameters,
and may be chosen independently of the incoming reset exponent χ. Advancing this cap to the
fixed reference preserves an upper clock bound C0 L, for small q. At the cap, and whenever the
coordinate lies in [3δ/4, 5δ/4] within its protected late window,
d
|A(wi (t))| ≥ σδ/2.
dt
The corresponding upper derivative bound is Cδ. The positive receiver envelope has uniformly
bounded fixed moments and Gaussian tails at the cap. The cap time and phase are continuous
under all allowed seed homotopies. If the child’s reference phase is ϕc , then the receiver cap phase
is ϕc + arg C∗ + τ Sq + o(1), uniformly, where Sq = σ −1 log[δ/(q 6 |C∗ |)].
Proof. Let Cs ≥ 1 be a fixed finite trial density-stop constant and put
Ts = inf{t ≥ 0 : ∥wi (t) − Wκi (t)∥X = Cs δ},

inf ∅ = +∞.

For small q the initial density is O(q K ) < Cs δ. While the child is current, Corollary 6.9 bounds
the receiver independently of a signed child transfer. Its logarithmic horizon contains the child
cap and its short reference advance by the strict clock slack. Lemma 7.8 therefore supplies the
fixed-reference state, phase and positive-envelope moment reset before Ts . In receiver units,
t0 ≤ C0 L/β + O(log L).
A shift to the child’s actual protected crossing of the fixed reference level changes this time and
phase by o(1) and preserves these inputs. Fix a prospective half-width L0 > 0, to be enlarged
below, and define
Sq∗ = σ −1 log

δ
,
q 6 |C∗ |

H+ = t0 + Sq∗ + L0 ,

b = min(H+ , Ts ).
H

b The receiver’s nonraFor small q, H+ ≤ C2 L. Corollary 6.9 controls all older colours on [0, H].
dial mass is at√most 2Cs δ in normalized units, so its actual constant core follows from Proposition 5.3: C2 L Cs δ → 0 and C2 Lβq → 0 for every fixed trial Cs . The associated harmonic and

99

Gaussian-leak estimates hold on this entire density-stopped interval, including any portion after
a hypothetical coordinate crossing.
b − t0 ) ≤ C2 βL.
Upper child-history bounds on shorter stops. Apply Lemma 9.1 with Us = β(H
The parent-radius Gaussian handoff supplies its initial exterior tail. In child units the own and
descendant velocities have a fixed bounded sum, the global older radial drift is at most C|y|/β,
and inside a fixed fraction of the parent core it is at most C(1 + Cs )δ|y|/β plus Gaussian leaks.
b because they require the density stop rather than a coordinate-cap
These bounds persist to H
restriction. The lemma gives
|Bc (u)| ≤ C(1 + u)4 eC(1+Cs )δu/β + o(q P ),

0 ≤ u ≤ Us ,

for every fixed P , with the prefactor determined by the fixed reference moments and bounded
source and descendant velocities, independently of Cs . For each fixed trial Cs , a later q threshold
makes C(1 + Cs )C2 δL ≤ log 2 and C(1 + Cs )δ < σ/2. Hence the future discounted input is in a
fixed ball independently of the trial constant. The pre-small-cap contribution is O(Cpast δ 1−1/β ),
and the weighted reset contribution tends to zero by (166). These upper bounds hold on every shorter density-stopped interval; no signed-transform lower bound or completion of the full
interval has yet been used.
All other actual harmonic coefficients. In receiver units the inner packet has total mass
O(βq 2 ), Lp norm O(βq 2/p ), and global velocity at most Cβq. Inside the protected receiving
core the older/current radial remainder has bound C(1 + Cs )δr3 plus Gaussian leakage. Radial
moment comparison and exit at a fixed core radius give, for every fixed n,
Z

|x|4n |inner packet(t, x)| dx ≤ Cn βq 4n+2 (1 + H)4n + o(q P )

b
(0 ≤ t ≤ H ≤ H).

(174)

√
Indeed its radial moment root is bounded, before exit, by CeCn (1+Cs )δt [q + βqt + κi t]; κi ≤ βq 7/2
makes the square-root term smaller than q up to logarithms. At exit the probability is e−c/(κi H) ,
and the global radial growth amplifies each fixed moment by only eCn H , which is absorbed by this
Gaussian. This calculation uses the positive aggregate, not its nonexistent summed L∞ bound.
The descendants below the immediate child have mass O(βq 2 ) in child units. The child’s
preparation envelope traps their positive aggregate in a fixed child ball up to its reference time;
afterwards their fourth moment is bounded by
Cβq 2 (1 + u)4 eC(1+Cs )δu/β + o(q P ).
Thus their fourth moment adds only
O(q 2−ϵ0 LD )
to the weighted child-reference transform. The factor q −ϵ0 comes from the finite negative-history
endpoint, whose source clock is at most C2 L + O(log L); its receiving discount is σ/β. This
argument controls the actual entire descendant packet and does not assert a separate smalldensity bound for its colours.
Retain exterior harmonics 4, 8, . . . , 4J of the inner packet and interior harmonics 3, 7, . . . , 4J −
1 of the older colours. Use their actual time coefficients, and fixed smooth stream cutoffs agreeing
on the derivative collars; call the resulting global field bJ . The leading child fourth harmonic gives
the first-order coordinate forcing
βq 6 γϵ e4i(t−t0 )/5 Bc (β(t − t0 )).
100

For each finite k, all retained higher inner harmonics have weighted C k norm, divided by q 6 , at
most CJ q 4−ϵ0 LD by (174). The unconditional inactive theorem bounds the actual older harmonics
on these collars by
O(q K+2−C2 λB ) + O(q 8−C2 (λP +C4 /β) LD ).
After source recentering, their weighted norm divided by q 6 is o(1) by (171). The own initial
seed contributes only O(q K−6−ϵ0 LD ) to the first growing coefficient. All these strict powers are
uniform in depth.
For the collar density leak, the inner positive aggregate has Gaussian L1 tail outside a fixed
protected radius and conserved finite Lp norm. Biot–Savart interpolation therefore gives a Gaussian velocity leak, despite its infinite sum of initial L∞ amplitudes. Older leaks have the individual
core estimates. The omitted harmonic remainder is O(q 4J+6 LD ) for the inner packet, and gains
arbitrarily many q-powers for the older Taylor series. Choose J so large that, writing P = 4J + 6,
P > 6 + ϵ0 + C2 (C1 − σ)+ + χ1
for a fixed χ1 > 0; older omission orders are chosen to meet the same inequality. This is a
relative envelope requirement, not only an absolute endpoint error bound: the first envelope
is a(0) = q 6 e−σt0 ≳ q 6+ϵ0 L−D , and a q P forcing error divided by a(t) amplifies by at most
e(C1 −σ)+ C2 L . The fixed-profile Kato exponent C1 is selected before J.
The actual nonlinear density bootstrap. Set ε = q 6 , s = t − t0 and a(t) = εeσs . Choose the
finite orders first. The main child history and fixed stream basis determine a fixed derivative
input ball. For every fixed triple (Cpast , Creset , Cs ) the preceding estimates place the actual input
in this same ball after a later q threshold. Thus for every spatial derivative order needed in the
finite recursion,
Z
H

0

ε−1 e−σ(t−t0 ) ∥bJ (t)∥C k dt ≤ Ck .

(175)

b and follows from Lemma 9.1 for the main harmonic and the
This bound holds for every H ≤ H
strict power estimates for all other harmonics. Spatial differentiation acts on the fixed stream
basis and leaves the actual time coefficients undifferentiated.
Apply Lemmas 8.1 and 8.2 to this actual run. In their notation take ε = q 6 , r = 6, θ = ϵ0 ,
CH = C2 and γ = 3: κi ≤ βq 7/2 ≤ q 3 for small q. For each retained coefficient use the explicit
reserve rjℓ = k∗ + 2(M − j) + 2(N − ℓ) and R = k∗ + 2M + 2N + 3, with k∗ = 4 for the X
conclusion. The finite stream field has polar sectors at most 4J; its order-j coefficients therefore
have sectors at most 4Jj. The seed bound in (149) follows from q K−6 eσt0 ∥Φ∥H r10 = o(1). The
other input bound is exactly (175) for this finite derivative order R. Actual harmonics agree on
the fixed broad annulus U, with omission mq ≤ q P LD ; Gaussian density leaks can be absorbed
in that bound. The actual global radial bound is |b · er | ≤ C(|x| + βq), uniformly in depth.
Choose (M + 1)σ > C1 and a finite N so that 3(N + 1) > C2 (C1 − σ)+ + χ2 . The harmonic
order was chosen to satisfy P > 6 + ϵ0 + C2 (C1 − σ)+ + χ2 . These are precisely the relative-error
inequalities (156). The coefficient recursion including every viscosity term is (150); its exact
cancelled residual is (153), not a pointwise assumed forcing remainder. The actual foreign field
stays in the divergence-free error transport. Consequently (157) gives

∥wi − wapp ∥X ≤ Ca(t)M +1 + q χ2 LD a(t),

∥wi − Wκi ∥X ≤ Ca a(t).

No signed growing-coefficient lower bound is used in this application.

101

(176)

The constant Ca is computed in the fixed input ball, before choosing Cs . Enlarge the fixed L0
so that its half-width meets the early-margin requirement of Corollary 8.5, using the coefficient
target lower bound |C∗ |/2. Then choose
Cs > max{1, 2Ca eσL0 /|C∗ |}.
Only afterwards decrease q to validate the input estimates and core margins for this fixed choice,
as well as the fixed child preparation and reset constants. Since a(H+ ) = δeσL0 /|C∗ |, (176)
b
gives ∥wi − Wκi ∥X < Cs δ/2 throughout [0, H].
Continuity excludes Ts ≤ H+ . Thus the full
deterministic prospective interval has been reached using upper input bounds alone.
The complete actual child history and growing amplitude. Now βt0 → ∞ and U+ = β(H+ −
t0 ) → ∞. Extend the actual phase-measured child moment Bc (u) by zero outside [−βt0 , U+ ].
Lemma 9.1 proves weighted absolute convergence on this complete interval and therefore
Z

e−µβ u Bc (u) du = eiϕc Fref (µβ ) + o(1).

(177)

R

The zero extension is only of the controlled finite history; no estimate of the actual infinite future
after H+ is used. Extend the retained field bJ by zero after H+ , and define its full actual first
growing amplitude by
Cq =

α λt0
1
e A(Φ) −
ε
ε

Z H+

e−λ(t−t0 )

Z



Tadj bJ (t) · ∇W dx dt.

(178)

0

This includes the own seed and every retained actual older, descendant and higher-sector input.
Equivalently it is lims→∞ e−λs A(g10 (s + t0 )) for this zero-extended first coefficient. Retained
sectors other than ±4 have zero direct growing projection, but remain in its full spatial coefficient
and in the nonlinear recursion. Its difference from the actual main-child discounted transform is
O(q χ4 LD ) for some fixed χ4 > 0, by the preceding power estimates. The main-child transform
converges to the canonical value by (177); that convergence is only o(1) here, since the finite
ancient-truncation error may be logarithmic. Hence
Cq = eiϕc C∗ + o(1),

|Cq | ≥ |C∗ |/2.

Universal coefficient reset. Use Lemma 8.3 with main field equal to the actual immediate-child
fourth harmonic. Its post-reference C R norm is C(1 + s)4 eC(1+Cs )δs , so h∗ = σ/2 is admissible
for the final small q. Every other retained field has weighted remainder norm ηq ≤ q χ3 LD by the
preceding strict power estimates. The pattern rotation is the vector pushforward/scalar rotation
of Section 8.1; all angular sectors are rotated separately. The full coefficient in (178) equals
(159), with the field zero-extended at H+ . Choose a fixed 0 < d < min(σ − 2η, σ/2). Lemma 8.3
proves, with the explicit derivative reserve, convergence to Fj (Cq ) for each pure coefficient. At
s = Sq +O(1) ≍ L its transient is a positive q-power. The actual main waveform need not be close
to a canonical waveform for this statement: only its finite input ball and future envelope are used.
Its canonical closeness was used for the lower bound on Cq . Formula (164), with the positive′
′
viscosity correctors and ∥Wκi − W ∥X ≤ Cκi H+ , gives a state error Cδ M +1 + q χ LD against the
universal polynomial. Lemma 8.4 matches its full protected coordinate by the two-real-variable
inverse, and identifies the matching all-finite-jet ancient state. This proves (173) once the exact
first cap is found. Neither the incoming waveform error nor an incoming reset exponent enters
this intrinsic transient exponent.

102

Protected first crossing. The global density envelope excludes a cap before H+ − 2L0 for
the fixed L0 chosen above. In the ensuing fixed window the settled first growing coordinate is
εeλs Cq + O(a2 ) + o(a). The polynomial future tail and the small weighted remainders make the
error uniform there. Its moduli at the two ends straddle δ. The actual adjoint-coordinate identity
gives, when the coordinate is in [3δ/4, 5δ/4] within this protected late window,
A′ = λA + O(δ 2 ) + o(δ).
For the instantaneous radial-base forcing, use (174), not merely the weighted history: the collar
field is O(q 6 LD ) plus the small older field. For the exact foreign cross-advection pairing use
Lemma 7.1:
Z

|f | |b| |∇Tadj | ≤ Cχq ∥f ∥1 = o(δ),

δq 2 + q 4 + εcore
χq = C βq +
β

!

.

This uses the global test weight and the actual older cores; it requires no claim that the receiver’s
own heat-base defect is Gaussian in its core. The heat-profile foreign correction is O(κi H),
using the global radial bound on b and the weighted smooth heat Taylor bound. It is o(δ); the
remaining heat errors and the quadratic self error have the stated orders too. Thus the three
separate pointwise criteria of (165) hold with ρq → 0. Corollary 8.5, applied with half-width L0 /2,
proves the protected first crossing and its lower and upper slopes. Its center uses |Cq |; replacing
that by |C∗ | changes the center by o(1). Consequently this narrower window is contained in the
controlled prospective interval for small q; no forcing beyond H+ is used. This gives the asserted
time and phase asymptotics. Uniqueness of recrossing is asserted in the protected window; no
later Euler or Navier–Stokes return to the level is excluded.
For the universal polynomial, let a = |z| denote the radius of its parameter in the growing
phase gauge. Its coordinate modulus is R(a) = a[1+O(a)]. dR/da = 1+O(a) > 0. Since a′ = σa,
its time derivative is dR/dt = σa[1 + O(a)]. Matching the actual first coordinate therefore selects
its time/amplitude and phase gauges uniquely; the matching error has the same orders as (173).
This step uses the exact cap, rather than identifying it with the linear preparation time.
Finally, (172) gives integrated receiver nonradial drift O(δ); the younger global velocity gives
O(βqH), and the older field is smaller. The positive receiver envelope has Gaussian tails outside
a fixed ball and bounded fixed moments at the cap by the radial exit lemma and global moment
growth. The same uniform preparation and crossing estimates hold under the stated homotopies.
Positive-viscosity continuous dependence, followed by the strictly positive crossing slope, proves
cap-time and cap-phase continuity. Its own cap clock is [6/σ + C0 /β]L + O(log L) < C0 L; the
additional reference advance is only O(log L), so the same strict clock margin remains. The lower
clock bound persists since ti ≥ (6/σ)L − O(log L). This completes the actual receiving-stage
implication.

10

Uniform descent and fixed-datum phase selection

10.1

Uniform constants, the descending induction, and phase selection

We record the constant selection explicitly. An assertion that every finite stage has some finite
constant would not suffice for the construction below. In this section the selected seed has
amplitude exactly α; other seed amplitudes may lie in [0, α].

103

Proposition 10.1 (A common invariant class of actual stages). Fix p > 2, the smooth profile
and a sufficiently small fixed reference orbit, normalized once by a time shift and rotation so that
A(Ωref (0)) = δref > 0. There are choices of K ∈ (6, 643/100), fixed β > 1, q > 0, and constants
Cpast , Creset , R0 , c0 , C0 , C2 > 0, all independent of the selected index N , such that the following
2 . For arbitrary
holds. Put α = q K , δ = (log(1/q))−4 , Aj = β j−1 , Rj = q j−1 , and νN = q 3/2 AN RN
seed phases and amplitudes in [0, α], with the N th amplitude exactly α, every actual colour j =
N, N − 1, . . . , 1 has a protected first cap. For the selected viscosity, κj = q 3/2 (βq 2 )N −j ≤ q 3/2 for
every j ≤ N . In its normalized older-core rotating frame, its cap time τj from the physical initial
time obeys
c0 log(1/q) ≤ τj ≤ C0 log(1/q) < C2 log(1/q),
and its preparation satisfies
wj (t) − eκj t∆ W

X

≤ Cpast δeσ(t−τj ) ,

0 ≤ t ≤ τj .

Its state in its exact-cap gauge lies within Creset δ M +1 + q χ (log(1/q))D of the corresponding state
on Ωref , for fixed finite M, χ > 0, D < ∞. The positive envelope is localized uniformly throughout
preparation and at its fixed-reference time. Its fixed moments and fixed-radius Gaussian tails
have common bounds; for j ≥ 2 the separate parent-radius Gaussian tails also have common
bounds. The first caps have common protection constants at each fixed physical scale and depend
continuously on the seed parameters.
Proof. Choose K, for example K = 31/5. The generic profile constants, including a one-sided
X comparison exponent CW uniform on a fixed small ancient segment and the finite residual
exponent C1 , are fixed first. Fix a small reference on that segment; the finite expansion orders
are selected below after the clock and amplitude ratios, and before the invariant class constants.
The later advance from the shrinking actual cap to this reference uses one-sided comparison;
it does not require the selected source’s terminal expansion amplitude to exceed the reference
amplitude. Choose fixed clocks C0 > K/σ and C2 > max{C0 , K/σ + 1} with strict slack. Fix
0 < c0 < min{K/σ, 6/σ}.
Choose a sufficiently large interval for β so that throughout it
6β
< C0 ,
σ(β − 1)

K >6+

K
,
β−1

ϵ0 := C2 σ/β < min{K − 6, 1/2}.

The unweighted inactive rate is made small by this large amplitude ratio. After selecting β as
below, choose the fixed finite weight D so that the weighted inactive rate and its remaining
geometric rows satisfy
C2 λB < K − 6,
C2 (λP + C4 /β) < 1.
Their residual off-diagonal smallness is supplied by the later choice of q, as in Corollary 6.9. The
bilateral canonical transform is analytic and not identically zero, so a β in this interval can be
chosen with C∗ ̸= 0. The preceding finite weight choice gives the required weighted-row smallness;
the large β controls the unweighted rate. All these parameters are fixed before q.
Fix a broad localization ball BR0 containing the initial profile and small ancient segment with
a fixed radial margin. We verify that a common input class, rather than an inherited arbitrary
constant, feeds the finite receiving recursion. Suppose provisionally that a child is in a trial
class with past-envelope constant Cpast and cap-reset constant Creset . Its own nonradial velocity
104

integrates to at most CCpast δ before its cap. The younger velocity contributes CβqC2 log(1/q),
and the older harmonic remainder is smaller. For any fixed trial constant these vanish as q ↓
0. The radial exit argument therefore gives the same localization ball and moment constants,
independent of this trial constant, once q is small. This holds at every preparation time, since
the integrated radial drift up to that time has the same bound. For the stronger parent-radius
tail, run a separate exit argument at radius c/q in source units. The older stopped-core radial
coefficient there is Cδ/β, and the source and descendants have bounded total speed. Their age
is O(log(1/q)), so the deterministic radial displacement is much smaller than q −1 . The exit
probability is C exp[−cq −2 /(κj C2 log(1/q))]. Conditional global moment growth adds only fixed
powers of q −1 ; the same Gaussian, with reduced exponent constant, controls the parent-radius
weighted tails.
Let sq be the reference orbit’s exact δ-cap time. Advancing to the fixed reference state takes
−sq = O(log log(1/q)). On this segment canonical nonradial velocity integrates to O(δref ), and
the actual comparison error has integral tending to zero if M is large enough. Choose δref small
within the fixed radial margin. A second positive-envelope localization at the reference state
hence gives the same fixed radius and universal moment constants, without a factor log log(1/q).
In the reference source clock, the pre-small-cap contribution to the weighted forcing norm is
at most
CCpast δ 1−1/β .
(179)
Indeed the past envelope is Cpast δeσv in small-cap coordinates v ≤ 0, while the reference discount
contributes e−σsq /β = O(δ −1/β ). The cap-to-reference segment is bounded by the fixed ancient
envelope and the vanishing forward-comparison error. The positive reference history is bounded
by Lemma 9.1 on the entire prospective interval through min(H+ , Ts ), including the part after
an attained coordinate cap. Here Ts is the trial density stop of Proposition 9.2. Its moment
prefactor is universal: the trial-stop factor eCCs δu/β is at most two on u ≤ C2 βL after a later
decrease of q for each fixed Cs . The controlled history is zero extended at its prospective endpoint;
no uncontrolled future is used. Consequently every needed spatially differentiated forcing norm
is bounded by
Minput ≤ Mref + CCpast δ 1−1/β + o(1).
The constant Mref depends only on the fixed reference/profile and fixed radius, not on either
trial constant or depth. The small error depends on the trial reset constant, but its coefficient
tends to zero. Once that full prospective interval has been reached, the main signed transform
is uniformly within o(1) of eiϕc C∗ ; fixed-window comparison followed by the two discounted tails
proves this uniformly on the trial class. Thus |Cq | ≥ |C∗ |/2 once a sufficiently long prospective
receiving interval has been reached. The upper input bound, which is valid on any shorter stop,
is used first to reach that interval.
Select all finite expansion orders before computing the class constants. Choose M + 1 >
CW /σ + 2 and (M + 1)σ > C1 ; choose enough actual harmonic terms and viscosity orders that
their generic residual losses are smaller than the entire growing envelope on 0 ≤ t ≤ C2 log(1/q).
In particular the omitted harmonic power P satisfies P > 6 + C2 σ/β + C2 (C1 − σ)+ + 1 for the
receiving stage, and the analogous selected-source inequality with a(0) = q K . There are only
finitely many required derivatives and sectors.
Compute finite-expansion bounds using the fixed input ball Minput ≤ Mref + 1. Choose Cpast
larger than twice the resulting cap-relative envelope constant, divided by the fixed lower growing
coefficient, and larger than twice the selected-source past constant. Now choose q so that the term
(179) and all other input errors are at most one. The output past envelope improves this same
105

Cpast ball. The fixed comparison and coefficient bounds give a finite output cap-reset constant
independent of the incoming reset error: the source waveform changes Cq , which exact-cap time
and phase normalization remove; the universal coefficient reset has only intrinsic positive-q-power
transients. Choose Creset larger than twice this output constant and the selected-source constant.
Decrease q again so that its inherited reset error advances to the fixed reference by o(1). This
constructs a genuinely invariant class of constants.
Choose a common positive exponent χ smaller than the intrinsic base and receiving exponents
and take a common finite logarithmic power. No input reset exponent is successively lost.
Theorem 7.7 gives the actual base j = N in this invariant class. Its normalized cap clock is
(K/σ) log(1/q) + O(log log(1/q)). Assume the class holds for j + 1. Proposition 9.2, with the
common input ball just established, supplies it for j. The receiving normalized clock from the
physical initial time obeys
τj+1
6
τj ≤
+ log(1/q) + O(log log(1/q)).
β
σ
The geometric sum and strict clock slack keep it below C0 log(1/q) after a final decrease of
q. The receiving growth interval gives a common positive lower clock c0 log(1/q). All core,
tail, moment and inactive estimates therefore stay inside their C2 windows, including the short
reference advance. Descending induction proves the assertions for every finite N with the same
constants. Only finite older prefixes are stopped; all infinite younger packets were controlled by
their convergent mass, Lp and velocity sums.
Finally decrease q so that βq 2/p ≤ 1/2 and βq ≤ 1/2. Then the datum series and every
required geometric sum converge, and βq 2 < 1 gives the selected viscosities decreasing to zero.
The estimates hold throughout a fixed late interval on both sides of each prospective cap
and give a strict early margin and an integrated positive radial slope on that late interval. The
quantitative first-cap lemma yields common protection constants. At the outer physical scale
they give fixed common numbers e, h, c, Lobs > 0 in Lemma 2.8: choose a sufficiently small
two-sided late half-width e, use the slope c = σδ/2 (after a harmless reduction), and take the
early margin h > 0 furnished by the settled-coordinate and prewindow estimates. The upper
scalar derivative gives a common Lipschitz bound Lobs . At any other fixed scale, the normalized
constants are converted by its fixed clock factor Aj . Positive-viscosity continuous dependence
and the protected crossing imply continuous cap times and phases under seed homotopies. All
estimates were uniform in those phases and in all nondistinguished amplitude reductions.
Proposition 10.2 (Surjective outer phase for each selected viscosity). With the preceding fixed
parameters, fix N ≥ 2, all phases except the N th, and put all seed amplitudes equal to α. The
map taking ϑN ∈ R/(2πZ) to the unit phase of the outer colour at its first cap for viscosity νN is
continuous and surjective.
Proof. Multiply every seed except the distinguished N th seed by r ∈ [0, 1]. The invariant actualstage proposition applies throughout this homotopy. At r = 0 the initial datum is the radial stack
plus just that seed. Rotational covariance of the unforced equation and its colour equations shows
that changing its phase rotates the entire solution, leaves the cap time unchanged and rotates the
outer cap coordinate once. Thus the outer phase map has degree one (with an immaterial sign if
the opposite rotation convention is used). Common protected caps give a continuous circle-valued
homotopy as r goes to one. Its degree stays nonzero, and a continuous circle map of nonzero
degree is surjective. For completeness, a map omitting a point factors through the circle minus
that point, an interval, and is homotopic to a constant; it has degree zero. This proves the last
assertion without a differentiability assumption.
106

10.1.1

Proof of the non-unique-selection theorem

Proof of Theorem 1.1. Use the fixed parameters in Proposition 10.1. All actual seeds will have
the same fixed amplitude α. Set the outer seed phase to zero and provisionally every other phase
to zero. The compact admissible stack lemma gives all required norm, support and circulation
properties for every phase choice. Choose Tf > C2 log(1/q) plus the common outer protection
window. Let
Z
ν
a (t) = Tadj (x)w1ν (t, x) dx
be the bounded smooth adjoint observable of the outer passive colour. There is no older-core
rotation at this outer scale. This is one fixed bounded
C 2 test with angular-four covariance,
R
allowed in (4). Pairing convergence follows from | Tadj (fn − f )| ≤ ∥Tadj ∥∞ ∥fn − f ∥1 . The scalar
identity for this noncompact test is obtained using expanding cutoffs; its bounded derivatives and
the bounded physical velocity make all cutoff errors tend to zero. The invariant stage theorem
gives cap times in one compact subinterval of (0, Tf ), a common precap margin and common
positive radial slope near the first crossing of radius δ. The observable is uniformly Lipschitz,
since its scalar equation gives
|(aν )′ | ≤ ∥w1 (0)∥1 [∥uν ∥∞ ∥∇Tadj ∥∞ + ν ∥∆Tadj ∥∞ ].
The velocity bound is uniform by the fixed-data L1 , Lp contractions and the kernel split, and
νN ≤ 1 after choosing q small.
We choose sparse indices Nk → ∞ and their phases successively. At the kth step keep all
earlier assigned phases and vary only ϑNk . Proposition 10.2 permits its outer cap phase to be
exactly +1 for even k, and −1 for odd k. Changing a phase at depth N changes the initial L1 ∩Lp
2 + R2/p ) → 0. Choose each next index so far out that all subsequent
norm by at most CαAN (RN
N
tail phase changes alter every earlier selected-viscosity outer observable by less than the desired
summable tolerance. This is possible by continuous dependence at each fixed positive viscosity
and absolute convergence of the remaining initial tail. In detail, after step k there are finitely
many positive viscosities and continuity radii; choose Nk+1 so that the norm of the entire freely
variable tail is below all those radii. The fixed smooth outer colour is the same in every datum.
Its scalar continuity follows from the fixed-viscosity mild equation. For two velocities, put h for
their outer-colour difference and v for their velocity difference. The heat-kernel gradient estimate
gives, for r = 1, 2,
∥h(t)∥r ≤ Cν −1/2

Z t
0

(t − s)−1/2 [∥u(s)∥∞ ∥h(s)∥r + ∥v(s)∥∞ ∥w1 (s)∥r ] ds.

Volterra comparison gives uniform colour continuity at fixed ν > 0 from uniform velocity continuity; no actual colour gradient is assumed. At step k choose a continuity radius rk > 0 at that
step’s assigned datum, for its selected viscosity and target accuracy 2−k . Require the worst-case
norm of every variable phase tail starting at Nk+1 to be below mini≤k ri . Protected first-cap
continuity then gives the stated phase tolerances. Tolerances may be chosen so the kth final cap
phase differs from its target by at most 2−k .
The resulting infinite phase assignment defines one fixed ω0 . No viscosity-dependent initial
datum is used. The selected viscosities νNk = q 3/2 (βq 2 )Nk −1 tend to zero. The fixed-data compactness theorem yields subsequences of the even and odd selected solutions converging strongly
in C([0, Tf ]; L1 ∩ Lp ) to Euler solutions ω + and ω − . Their velocities converge strongly, and the
passive-colour theorem identifies the limits of w1ν as the unique bounded transport colour with the
107

fixed initial outer density, determined by the total Euler path. Thus their observables converge
uniformly on [0, Tf ].
Extract also convergent cap times. Common precap protection and radial slope ensure each
limiting time is the first crossing of radius δ for its limiting observable: before the common cap
window there is a strict margin, and inside it the integrated radial lower slope excludes an earlier
crossing. The limiting cap coordinates are respectively +δ and −δ. If ω + = ω − , uniqueness
of the bounded passive colour in that total velocity gives the same observable. It would then
have the same first cap time and coordinate, contradicting these two values. Therefore the
total Euler paths differ. The compactness theorem already gives their strong initial trace and
renormalization, completing the proof.

References
[1] T. Gallay and C. E. Wayne, Invariant manifolds and the long-time asymptotics of the Navier–
Stokes and vorticity equations on R2 , Arch. Rational Mech. Anal. 163 (2002), 209–258.
https://arxiv.org/abs/math/0102197.
[2] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev
spaces, Invent. Math. 98 (1989), 511–547. https://doi.org/10.1007/BF01393835.
[3] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002. https://www.cambridge.org/core/
books/vorticity-and-incompressible-flow/393C35E544EDD0711CAA7F7AB05D7432.
[4] G. Ciampa, G. Crippa and S. Spirito, Strong convergence of the vorticity for the 2D Euler
equations in the inviscid limit, Arch. Rational Mech. Anal. 240 (2021), 295–326. https:
//doi.org/10.1007/s00205-021-01612-z. https://arxiv.org/abs/2008.12133.

108

