Vortex-sheet existence: source audit

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\newcommand{\R}{\mathbb R}

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\newcommand{\M}{\mathcal M}

Macro expands to the finite signed Radon measure space specified in the exact target.

Macro expands to the finite-measure space named in the target; the later norm consistently denotes total variation.

The macro expands to the finite signed-measure space specified in the target.

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\newcommand{\curl}{\operatorname{curl}}

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\newcommand{\Div}{\operatorname{div}}

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\newcommand{\norm}[2]{\left\|#1\right\|_{#2}}

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\title{Global weak Euler flows for finite signed vorticity on the plane}

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\date{8 October 2026}

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\noindent Research proof manuscript.

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\paragraph{Exact target.}

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Let $K(x)=(-x_2,x_1)/(2\pi|x|^2)$.

For x nonzero this is the positively oriented two-dimensional Biot-Savart kernel; its locally integrable extension as a distribution is unambiguous.

The displayed vector is perpendicular to x, has size 1/(2 pi |x|), and has positive unit circulation.

The kernel equals nabla-perp of (2pi)^-1 log|x|, with the required orientation and normalization.

The factor 1/(2pi), orientation (-x_2,x_1), and punctured-plane kernel match the target exactly; the value at zero is immaterial.

 Given any finite signed Radon measure
$\mu_0\in\M(\R^2)\cap H^{-1}(\R^2)$, with the \emph{inhomogeneous} Sobolev
space and no sign restriction, set $u_0=K*\mu_0$.

The hypotheses exactly match target_problem.txt: arbitrary finite signed measure and inhomogeneous H^-1. The convolution is well-defined locally almost everywhere by the later Tonelli estimate.

These are the exact signed finite-measure and inhomogeneous H^{-1} hypotheses; the convolution is legitimized by the decomposition proved below, without requiring zero circulation.

The datum class exactly matches the supplied target, including inhomogeneous H^-1 and signed measures; convolution is justified in S0035-S0052.

The admissible datum is exactly a finite signed measure in inhomogeneous H^-1; S0035-S0052 establish that its canonical convolution is meaningful without sign or circulation restrictions.

 The requested conclusion is a
global weak Euler solution
\[
 u\in C_w([0,\infty);L^2_{\rm loc}(\R^2;\R^2)),\qquad u(0)=u_0,
 \qquad \Div u=0,
\]
such that, at \emph{every} $t\geq0$,
\[
 \mu(t)=\curl u(t)\in\M(\R^2),\qquad u(t)=K*\mu(t),\qquad
 \sup_{0\leq t\leq T}\norm{\mu(t)}{\M}<\infty\quad(T<\infty).
\]

This is a statement of the requested target, not an inference of existence. Its weak local L2 continuity, initial value, divergence, every-time canonical curl, and finite-horizon variation agree with the supplied problem.

Read as the requested target, these quantifiers and function spaces match the supplied problem, including every time and weak local L2 continuity; they are not assumed as a premise.

This is the requested conclusion, not an additional proved assertion here; comparison with target_problem.txt verifies every time quantifier, local weak topology, canonical identity, and horizon TV bound.

This is an exact restatement of the requested conclusion, including every-time measure curl, canonical velocity, finite-horizon uniform variation, and weak local L2 continuity; it does not yet assert proof.

The velocity identity is precisely
\[
 \int_0^T\!\!\int_{\R^2}
 \bigl(u\cdot\partial_t\varphi+(u\otimes u):\nabla\varphi\bigr)\dd\,dt
 +\int_{\R^2}u_0\cdot\varphi(0)\dd=0
\]
for all divergence-free $\varphi\in C_c^\infty([0,T)\times\R^2;\R^2)$.

Integration by parts in time and space gives the displayed plus signs and initial term; compact support before T removes the terminal term, and divergence-free tests remove pressure. All products are locally integrable under the target regularity.

Integration by parts in time and row-wise divergence gives the displayed plus initial term; divergence-free compact tests remove pressure and agree with the target.

The displayed identity matches integration by parts in the target, with positive initial term and tests vanishing near T; pressure drops against divergence-free tests.

The test class, sign of the initial term, tensor contraction, and time endpoint agree with the exact target; all integrals are defined by locally bounded kinetic energy.

No global kinetic-energy assumption, compact support, zero circulation, or
energy inequality is added.

The source target uses inhomogeneous H^-1 and local energy only, with arbitrary circulation/support/sign; it does not add an energy inequality.

Inspection of the subsequent estimates uses finite measure and inhomogeneous H^{-1}, with a bounded background carrying nonzero circulation; no listed additional assumption is inserted.

Direct comparison with the target confirms none of these four additional restrictions is imposed; later estimates use a relative L2 perturbation of a possibly infinite-energy background.

Comparison with target and final construction shows no compact support, zero mass, global energy, or energy inequality is imposed on the input.

\section{Initial normalization and a bounded Euler background}

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We use $\nabla^\perp=(-\partial_2,\partial_1)$,
$(\nabla v)_{ij}=\partial_jv_i$, $(\Div S)_i=\sum_j\partial_jS_{ij}$, and
$\widehat f(\xi)=\int e^{-ix\cdot\xi}f(x)\dd$.

The rotated gradient, row-wise tensor divergence and gradient indices are compatible: div(grad v)=Delta v, div((grad v)^T)=grad div v. Fourier differentiation is multiplication by i xi under the declared convention.

The four conventions give curl(grad-perp psi)=Delta psi and Div(v tensor w)_i=partial_j(v_i w_j); the Fourier derivative multiplier is i xi_j.

All four declarations are well typed; row divergence of v tensor v is (v dot grad)v for div v=0, and Fourier differentiation is i xi.

All four declared conventions are compatible: div of a tensor acts on its second index, curl(nabla-perp f)=Delta f, and differentiation has Fourier multiplier i xi.

The multiplier of $K$ is $m(\xi)=-i\xi^\perp/|\xi|^2$.

Since K=grad-perp((2pi)^-1 log|x|) and Delta G=delta_0, its multiplier is i xi-perp times (-|xi|^-2), namely the stated negative imaginary multiplier, locally integrable at zero.

K=grad-perp G and Delta G=delta with G=log|x|/(2 pi), so multiplying -|xi|^{-2} by i xi-perp gives exactly -i xi-perp/|xi|^2.

K is the rotated gradient of G=(2pi)^-1 log|x|. Fourier differentiation of the distributional Green function gives m=-i xi-perp/|xi|^2, now locally integrable at zero. The Green transform itself requires a finite-part interpretation; the claimed K multiplier does not. Oddness and homogeneity of the explicit K fix the zero-frequency normalization.

K=nabla-perp[(2pi)^-1 log|x|], Delta G=delta_0, and Fourier differentiation give -i xi-perp/|xi|^2 with this convention.

\begin{lemma}[Canonical velocity and the physical-space decomposition]\label{lem:decomp}
Let $\mu\in\M\cap H^{-1}$ and $M=\norm\mu\M$.

The hypotheses and M=total variation are well-typed; no sign or zero-mass condition is introduced.

The lemma's hypotheses and M definition repeat the finite signed-measure and inhomogeneous H^{-1} assumptions; no positivity is used.

The assumptions and M are precisely the finite TV and inhomogeneous Sobolev hypotheses, including the degenerate zero datum.

The datum and total variation M are well typed; abbreviating the spatial domain does not strengthen the target hypothesis.

 The convolution $K*\mu$
is defined almost everywhere, belongs to $L^2_{\rm loc}$, and has divergence
zero and curl $\mu$.

Tonelli yields local L1 and almost-everywhere absolute convolution; div K=0 and curl K=delta_0 pass through finite-measure convolution. The decomposition and multiplier estimate at S0044-S0052 give local L2 using H^-1.

Tonelli gives local L1 and a.e. definition; the physical decomposition yields local L2, and differentiating K distributionally gives div zero and curl mu.

Uniform integrability of |K| over translated bounded balls gives a.e. convolution; the radial decomposition and bounded multiplier at S0047 give L2loc, while div K=0 and curl K=delta give the distributional identities.

Tonelli gives local L1 and a.e. convolution, the kernel distribution identities give div=0/curl=mu, and the low/high-frequency split in S0044-S0052 upgrades to local L2.

 Fix a nonnegative radial $a\in C_c^\infty(B_A)$ of
integral one.

For any positive radius A a normalized nonnegative radial compact smooth bump exists. Its first moments vanish by evenness; these are precisely the properties used later.

A nonnegative radial smooth probability function supported in an arbitrary sufficiently large ball exists by normalizing a radial bump; its first moments vanish.

A standard radial nonnegative compact bump can be normalized to mass one for any A>0; its parity and finite moments are available.

A radial nonnegative compact smooth bump normalized by its positive integral exists for any A>0; evenness and unit mass are exactly the later cancellation hypotheses.

 Set
\[
 \nu_0=a*\mu,\quad \sigma_0=\mu-\nu_0,\quad
 V_0=K*\nu_0,\quad v_0=(K-K*a)*\mu.
\]

Finite-measure convolution gives smooth integrable nu_0, finite signed sigma_0, and distributional canonical velocities. Associativity against compact tests makes all four definitions compatible.

Finite-measure convolution defines nu0, sigma0, V0 and v0 as distributions and locally integrable functions; subtraction is linear and each type agrees with later estimates.

Finite-measure convolution makes nu0 smooth integrable and sigma0 finite; the two velocity convolutions are distributions and locally integrable fields by the uniform ball bound.

Finite-measure convolution defines nu_0 and sigma_0; K*a is bounded, L_a is integrable, and the four definitions have the stated distributional meanings.

Then $K*\mu=V_0+v_0$, $v_0\in L^1\cap L^2$, $V_0\in C_0\cap C^\infty$
is bounded, and
\[
 \norm{v_0}1\leq C_aM,\quad
 \norm{v_0}2\leq C_a\norm\mu{H^{-1}},\quad
 \norm{\nu_0}1\leq M,\quad \norm{\sigma_0}\M\leq2M.
\]

L_a is integrable and compactly supported, giving L1 bound. Weighted multiplier supremum gives H^-1 to L2. Bounded Fourier transforms of finite measures and integrability of m times a-hat give bounded smooth C0 V_0; Young and triangle inequalities give M and 2M.

Radial circulation makes K-K*a integrable and compactly supported; its high-pass multiplier is bounded from H^{-1} to L2. Fourier inversion makes V0 smooth and vanishing at infinity. Convolution with a probability gives M and 2M bounds.

Radial circulation makes K-K*a compactly supported and L1, giving Young's TV bound. Its multiplier times sqrt(1+|xi|^2) is bounded, giving the H^-1 to L2 bound. Smooth Fourier factors give V0 in C0 and all derivatives; triangle/Young give M and 2M.

Splitting K=L_a+K*a yields the identity. Young gives L1 v_0 and measure bounds M,2M; the multiplier bound gives L2 v_0. Integrable differentiated low-frequency multipliers give bounded smooth C0 V_0.

All derivatives of $\nu_0$ belong to $L^1\cap L^\infty$, and every
positive-order derivative of $V_0$ belongs to $L^2\cap L^\infty$.
\end{lemma}

Differentiating a*mu gives L1/Linfinity bounds by the corresponding norms of derivatives of a times M. Positive-order differentiated velocity multipliers are integrable and square-integrable, yielding L2 and Linfinity derivatives.

For nu0 use D^alpha a in L1 and Linfinity against finite mu. For positive derivatives of V0 the multiplier xi^alpha m a-hat is both L1 and L2, multiplied by bounded mu-hat.

Derivatives of a are L1 and Linfty, so their measure convolutions are both. For V0, derivative multipliers are bounded near zero, rapidly decreasing at infinity, and in L1 and L2; multiplication by bounded mu-hat preserves both.

D^alpha a*mu has L1 and Linfinity bounds by total variation. For positive alpha, xi^alpha m a-hat is both L1 and L2; bounded mu-hat and inversion/Plancherel give the asserted V_0 derivatives.

\begin{proof}
For a bounded ball $B$, $\sup_y\int_B|K(x-y)|\dd<\infty$; Tonelli gives the
almost-everywhere definition and local integrability.

The integral of |x-y|^-1 over a radius-R ball is maximized at its center and equals O(R); Tonelli against |mu| bounds local L1 by C_R M and therefore makes convolution finite almost everywhere.

Splitting any translated ball into distance less than twice its radius and its complement bounds integral |K| by C radius. Tonelli applies to |K(x-y)| d|mu|(y) dx and proves finite local integrals.

For a radius-R ball, integrating |x-y|^-1 gives a bound CR uniform in its relative center. Tonelli against total variation gives the claimed a.e. definition and local L1, even for noncompact signed measures.

For a ball of radius R, integrating |x-y|^-1 is uniformly at most C R; integration against |mu| and Tonelli proves a.e. finite convolution and local L1.

 The kernel's divergence,
curl, and Fourier identities give the distributional assertions.

Distributional integration of the angular kernel gives zero divergence and unit curl mass; Fourier differentiation then gives m. Convolution commutes with derivatives on compact tests by the local integral estimate.

Integration on circles gives div K=0 and curl K=delta; the Fourier multiplier in S0033 then carries both distributional identities through convolution.

Testing the kernel against compact smooth functions yields div K=0 and curl K=delta; the Fourier sign was checked independently at S0033.

Testing K against compact derivatives gives div K=0 and curl K=delta_0; convolution then gives the two distributional identities, with Fourier sign fixed by S0033.

 These identities
extend from compactly supported to arbitrary finite measures by truncation in
total variation.

For mu_n=1_{B_n} mu, total-variation difference tends to zero. The uniform local convolution bound implies K*mu_n to K*mu in local L1, allowing every distributional derivative identity to pass to the limit.

The variation of mu restricted to |x|>R tends to zero; the uniform translated-ball estimate controls the convolution locally, permitting distributional passage of curl and divergence.

Restricting mu to expanding balls converges in TV. The same uniform ball kernel bound controls local L1 differences, permitting all distributional derivative identities to pass.

mu restricted to B_n converges in total variation; the uniform translated-ball kernel bound gives local L1 convergence of the velocities and hence passage of distributional div/curl identities.

 Equivalently, $m$ is locally integrable at frequency zero and
integrable against Schwartz functions, so uniform convergence of the Fourier
transforms of the truncated measures passes the formula to the limit.

The bound |m|=|xi|^-1 is locally integrable in two dimensions and has at most polynomial growth. Also sup|mu_n-hat-mu-hat| is bounded by total-variation error; pairing m times this difference with a Schwartz test tends to zero.

The multiplier is O(|xi|^{-1}) at zero, integrable in two dimensions, and has at most polynomial growth at infinity. The uniform Fourier error is bounded by the tail variation, so dominated testing against Schwartz functions applies.

In dimension two |m|=|xi|^-1 is integrable near zero and polynomially bounded at infinity. TV convergence gives uniform convergence of Fourier transforms, dominated after multiplication by any Schwartz test.

In dimension two |m|=|xi|^-1 is integrable near zero and against Schwartz tails. Total-variation convergence bounds the Fourier difference uniformly, so dominated integration proves the Fourier limit.

For radial $a$, circulation gives
\[
 (K*a)(x)=K(x)\int_{|y|<|x|}a(y)\,dy.
\]

Radial convolution is tangential and rotationally symmetric. Stokes gives circulation equal to enclosed a-mass, hence tangential velocity mass/(2pi radius), with no extra harmonic term because the convolution is regular at the origin.

Radial symmetry forces a tangential velocity; Stokes circulation around radius r is the mass of a inside that radius, yielding exactly K(x) times that mass.

Rotational symmetry makes K*a tangential, and circulation around radius R is enclosed mass by curl(K*a)=a; division by 2pi R gives the formula.

Radial symmetry makes K*a tangential; Stokes circulation on radius rho is the a-mass inside rho. Dividing by 2pi rho yields the displayed formula, also continuously interpreted at zero.

Consequently $L_a=K-K*a$ vanishes outside $B_A$ and has only the integrable
$|x|^{-1}$ singularity at zero.

The enclosed mass equals one outside the support radius, so L_a vanishes there. Near zero K*a is bounded and K has magnitude (2pi|x|)^-1, whose two-dimensional integral is finite.

Outside the support radius the enclosed mass is one; inside, K*a is bounded and K has integrable inverse-radius singularity. Thus the difference has the stated support and local behavior.

Outside radius A the enclosed mass is one; inside, K*a is bounded while K has only the locally integrable 1/r singularity.

For |x|>=A the enclosed a-mass is one, so L_a=0; near zero the only possible singularity is 1/|x|, which is integrable in two dimensions.

 Thus $L_a\in L^1$ and Young's inequality gives
the $L^1$ estimate for $v_0$.

The preceding pointwise kernel description gives finite L1 norm; integrating |L_a*mu| by Tonelli gives its norm at most ||L_a||1 M, the stated measure version of Young's inequality.

Integrating |L_a(x-y)| against d|mu|(y) gives ||L_a*mu||1 <= ||L_a||1 M, proving the needed measure version of Young directly.

The compact 1/r kernel is L1 in two dimensions; Fubini/triangle gives ||L_a*mu||1 <= ||L_a||1 TV(mu).

Compact support and the integrable singularity give ||L_a||_1<infinity; integrating |L_a(x-y)| against |mu|(dy) proves ||v_0||_1<=||L_a||_1 M.

 Evenness gives
$|1-\widehat a(\xi)|\leq C_a\min(|\xi|^2,1)$; hence
\[
 (1+|\xi|^2)|(1-\widehat a(\xi))m(\xi)|^2\leq C_a.
\]

Evenness cancels the linear Taylor term and integral one cancels the constant, so |1-a-hat|<=C|xi|^2 near zero and <=2 globally. Multiplying |m|^2=|xi|^-2 by 1+|xi|^2 gives a bounded product in both frequency regimes.

Use a-hat=integral cos(x dot xi)a(x)dx and |1-cos z|<=z^2/2 near zero; elsewhere use 2. Multiplication by |m|=|xi|^{-1} gives the displayed uniformly bounded weighted square.

Evenness cancels the linear Taylor term of exp(-ix dot xi); the finite second moment gives O(|xi|^2), while 1+||a||1 bounds high frequencies. Splitting |xi|<=1 and >=1 proves the displayed multiplier bound.

Unit mass and evenness cancel constant and linear terms in 1-a-hat; its bound is C min(|xi|^2,1). Multiplication by |m|^2=|xi|^-2 and 1+|xi|^2 is uniformly bounded in both frequency regimes.

Plancherel proves the $L^2$ estimate, using exactly the inhomogeneous hypothesis.

Integrating the pointwise multiplier inequality against |mu-hat|^2/(1+|xi|^2) and using Plancherel gives exactly the inhomogeneous H^-1 bound, including low frequencies.

The Fourier expression for v0 and the preceding multiplier bound yield integral |v0-hat|^2 <= C integral |mu-hat|^2/(1+|xi|^2); this is exactly the stated inhomogeneous norm.

Multiply the H^-1 Fourier integrand by the bounded factor in S0047, then apply Plancherel. This does not demand homogeneous H^-1 or zero mass.

Integrate the S0047 multiplier bound against |mu-hat|^2/(1+|xi|^2); Plancherel proves exactly the inhomogeneous H^-1-to-L2 bound, including nonzero circulation.

The multipliers $m\widehat a$ and $\xi^\alpha m\widehat a$ are integrable;
for $|\alpha|\geq1$ the latter multipliers $\xi^\alpha m\widehat a$ are also square integrable.

The current sentence explicitly restricts square integrability to the latter, positive-order multipliers. At zero m*a-hat has size |xi|^-1 and is L1 but generally not L2; xi^alpha*m*a-hat has size |xi|^(|alpha|-1), which is L1 and L2 for |alpha|>=1. Schwartz decay handles infinity.

Near zero, xi^alpha m has magnitude O(|xi|^{|alpha|-1}); at infinity a-hat is Schwartz. L1 holds also for alpha=0; L2 holds for |alpha|>=1 and is not asserted for the zeroth multiplier.

Near zero, m*a-hat is O(|xi|^-1), hence L1 in two dimensions but generally not L2. The current text expressly restricts square integrability to the latter multipliers with |alpha|>=1: their extra factor makes them bounded or smaller near zero. Rapid decay at infinity proves both asserted integrabilities.

Polar integration near zero gives L1 for m a-hat, behaving like |xi|^-1, and for xi^alpha m a-hat. For positive |alpha| the latter behave like |xi|^(|alpha|-1) and are L2; Schwartz decay handles infinity. The current sentence explicitly restricts its L2 claim to the latter multipliers, so nonzero circulation is allowed.

 Fourier inversion and the
Riemann--Lebesgue lemma give the assertions about $V_0$.

Multiplication by bounded mu-hat preserves the integrability estimates; Fourier inversion and Riemann-Lebesgue give bounded continuous derivatives and V_0 vanishing at infinity. The positive-order L2 assertions follow from the square-integrable multipliers.

The bounded mu-hat times each integrable derivative multiplier permits differentiation and Fourier inversion, yielding continuous derivatives vanishing at infinity; square-integrable positive derivative multipliers give L2 derivatives.

The multiplier times bounded mu-hat lies in L1 for every derivative, so Fourier inversion produces smooth bounded derivatives vanishing at infinity. Positive derivatives additionally have L2 by Plancherel.

Bounded mu-hat times each L1 multiplier permits classical Fourier inversion and vanishing at infinity by Riemann-Lebesgue; all derivative orders are covered by S0049.

 The convolution
identities and remaining bounds follow by finite-measure convolution.

Associativity can be tested against compact smooth functions using the uniform integral bound for K; Young gives ||nu_0||1<=M and ||mu-nu_0||M<=2M. Derivatives of nu_0 use derivative kernels of a.

Fubini is justified by the bounded translated local kernel integral and finite measure mass. Probability convolution gives ||nu0||1<=M, and the variation triangle inequality gives ||sigma0||M<=2M.

Associativity against compact tests is dominated by the uniform translated-ball bound; ||a*mu||1<=M and ||mu-a*mu||M<=2M follow by triangle inequalities.

Associativity follows against compact tests by absolute Fubini and the uniform translated-ball bound. Probability convolution contracts total variation, and sigma_0=mu-nu_0 has variation at most 2M.

 In
particular $K*\mu=V_0+v_0\in L^2_{\rm loc}$.
\end{proof}

Combining the distributional convolution splitting with bounded V_0 and global L2 v_0 yields the equality almost everywhere and membership in L2 on every bounded subdomain.

The distributional decomposition has locally L2 right side, because V0 is bounded and v0 globally L2; it therefore identifies the a.e. convolution with that local L2 function.

The distributional sum identity and V0 bounded plus v0 globally L2 establish the asserted local L2 canonical convolution.

The identity was justified distributionally and in local L1; V_0 is bounded and v_0 is global L2, giving the asserted local L2 equality.

\paragraph{External input E1, used only here.}

Heading identifying an external input.

This is document formatting, an environment declaration, a title/heading, or an audit-status notice; it asserts no mathematical fact and all source bytes remain represented.

Current exact source object inspected: document structure, heading, typography or administrative prose, with no substantive mathematical assertion or change to meaningful macro semantics.

Section or subsection heading only; mathematical content is separately checked in the subsequent permanent containers.

Crippa--Stefani, \emph{An elementary proof of existence and uniqueness for the
Euler flow in localized Yudovich spaces}, Calc. Var. PDE 63, 168 (2024),
Theorem 3.3, with Assumptions 2.1 and 3.1 and Definition 3.2,
\url{https://link.springer.com/article/10.1007/s00526-024-02750-4}.

The original Crippa-Stefani article was independently reopened for v3. The citation metadata, Theorem 3.3, Assumptions 2.1/3.1 and Definition 3.2 match. The displayed domain/range in Assumption 3.1 was retrieved as its original image and visually inspected; the additional Theorem 2.2 citation in S0058 is to this same article.

The named primary publisher article and PDF were inspected anew, including Theorem 2.2, the exact domain/codomain in Assumption 3.1, Definition 3.2, Theorem 3.3 and the uses of its divergence hypothesis in its proof. The article identity and all cited theorem numbers match.

The published primary article, exact numbering, Theorem 2.2, Assumptions 2.1/3.1, Definition 3.2, Theorem 3.3 and the relevant existence-proof steps were freshly inspected. The current citation identifies the required finite-p bounded-vorticity input; no unavailable source remains.

The primary publisher article was retrieved directly for v3. The bibliographic data, Theorem 3.3, Assumptions 2.1 and 3.1 and Definition 3.2 match the citation. Theorem 2.2 was additionally inspected, including its image-only range and Holder estimate, to verify the now explicit finite-p interface in S0058.

The specialization used here supplies a global Lagrangian vorticity solution for
arbitrary signed $L^1\cap L^\infty$ data on $\R^2$, with canonical Biot--Savart
velocity almost everywhere in time, bounded vorticity and velocity on finite
horizons, a log-Lipschitz velocity modulus, and nonincreasing $L^1$ norm.

With the verified finite-exponent kernel hypotheses at S0058, Theorem 3.3 applies to arbitrary signed nu_0 in L1 intersection Linfinity. Its canonical Lagrangian solution has finite-horizon vorticity/velocity bounds and log-Lipschitz modulus. Its proof uses the divergence hypothesis only on bounded integrable time-step vorticities, all of which lie in L1 intersection L3_ul. The manuscript subsequently derives the every-time representative and norm conservation.

Primary Theorem 3.3 supplies a global Lagrangian weak solution for signed L1 and Linfinity data, canonical velocity and the stated bounds and log-Lipschitz modulus. In this divergence-free specialization the flow is measure preserving, so the L1 norm is in fact constant.

Fresh primary-source matching gives the signed L1 intersect Linfty data class, global Lagrangian solution, a.e.-time canonical velocity, L1 nonincrease, horizon Linfty bounds and log-Lipschitz modulus. The full Assumption 3.1 domain is verified at p=3 using Theorem 2.2 with q=1; the bounded densities in the existence proof belong to it. No compact support, positive sign or finite energy is required.

Primary Theorem 3.3 supplies a global signed L1/Linfinity Lagrangian solution, canonical bounded velocity, finite-horizon vorticity bounds, log-Lipschitz modulus and the stated L1 upper bound. Current S0058 verifies its assumptions using p=3, whose domain contains every bounded-vorticity approximant in the theorem proof. Every-time representatives and incompressible norm preservation are subsequently derived locally.

Definition 3.2 includes the weak initial-value vorticity equation and the
Lagrangian pushforward representation almost everywhere in time.

Definition 3.2 explicitly specifies the weak initial-value transport identity and the almost-every-time pushforward. No stronger all-time representative is taken directly from that citation.

Primary Definition 3.2 contains the initial-value weak vorticity identity and the pushforward representation for almost every time; the manuscript correctly separates that from its every-time representative construction.

Definition 3.2 includes the weak vorticity identity with initial term and the a.e.-time pushforward representation; no stronger every-time conclusion is imported here.

Primary Definition 3.2 includes the weak initial-value vorticity equation, v=K omega, and pushforward for a.e. time; the manuscript preserves its a.e.-time scope here.

 It does not require compact support or finite global energy.

The inspected definition/theorem require only L1 intersection Linfinity and the kernel conditions; neither compact support nor global velocity L2 occurs among their hypotheses.

The primary hypotheses on the whole plane are L1 and Linfinity vorticity and kernel conditions; neither compact support nor global L2 velocity appears.

The published hypotheses are L1 and Linfty and the kernel/divergence conditions; compact support and finite global kinetic energy are absent.

The primary hypothesis is L1-intersection-Linfinity on the plane, without compact support or velocity L2. No stronger source assumption is silently imported.

The kernel hypotheses hold because $|K(z)|=(2\pi|z|)^{-1}$,
$|K(z)-K(w)|=|z-w|/(2\pi|z||w|)$, and $\Div K=0$; take $p=3$ in Assumption~3.1, whose bounded
$1/3$-H\"older velocity mapping is supplied by Theorem~2.2, and note that
bounded vorticity has uniformly bounded local $L^3$ norms.

Direct expansion gives |K(z)|=1/(2pi|z|), the exact inversion-distance identity |K(z)-K(w)|=|z-w|/(2pi|z||w|), and distributional div K=0. The original Assumption 3.1 display is K:L1 intersection Lp_ul -> C_b^(0,1-2/p). Taking p=3 and Theorem 2.2 with q=1 gives the bounded 1/3-Holder map on its whole stated domain. A bounded vorticity satisfies sup_x||omega||L3(B1(x))<=pi^(1/3)||omega||infinity. Thus every bounded-vorticity input used by Theorem 3.3 lies in this finite-p domain; its separate log-Lipschitz estimates do not require a Lipschitz endpoint mapping.

The kernel size is 1/(2pi|z|); squaring the inversion-distance expression proves |K(z)-K(w)|=|z-w|/(2pi|z||w|), and distributional Div K=0. In the original source, Assumption 3.1 concerns K:L1 intersect L^p_ul -> C_b^{0,1-2/p}, with a divergence bound and no-flow boundary condition. Take p=3; Theorem 2.2 with q=1 proves the bounded C_b^{0,1/3} map on the entire required L1 intersect L3_ul domain, and the divergence bound holds with any positive C3 because the divergence is zero. The plane has no boundary. For bounded omega, sup_x ||omega||L3(B1(x))<=pi^{1/3}||omega||infinity, so every bounded datum encountered in the proof of Theorem 3.3 belongs to this domain. Its log-Lipschitz bound comes separately from Corollary 2.4, as its proof explicitly uses.

Direct norm calculation gives |K(z)|=(2pi|z|)^-1, and inversion in the plane gives the exact stated difference identity. Distributionally Div K=0, so Fubini against compact test gradients gives Div(K*omega)=0 for every omega in the entire L1 intersect L3_ul domain. The original Assumption 3.1 map was inspected, including its displayed formula: at p=3 its domain is L1 intersect L3_ul and its range C_b^(0,1/3). Theorem 2.2 supplies precisely that full-domain bounded map with q=1, p=3. The boundary condition is empty on R2. Theorem 3.3 uses this divergence hypothesis on bounded densities, all lying in that domain since ||omega||L3_ul<=pi^(1/3)||omega||infty. Its log-Lipschitz conclusion is provided separately by the bounded-vorticity estimate, not by asserting that a fixed 1/3-Holder modulus is Osgood.

Direct inversion-vector algebra gives both kernel identities and distributional div K=0. I visually inspected the primary Assumption 3.1 map L1 intersect L_ul^p -> C_b^(0,1-2/p), and Theorem 2.2 with q=1,p=3 gives its bounded 1/3-Holder specialization under the checked kernel assumptions. The domain is the full L1 intersect L_ul^3 class, not only globally bounded inputs; its divergence is zero for every member and the plane boundary condition is empty. For bounded omega, sup_z||omega||_(L3(B1(z)))<=pi^(1/3)||omega||infinity. Theorem 3.3 applies its divergence bound to bounded-vorticity approximants, all in this domain; its log-Lipschitz conclusion comes from its bounded-vorticity estimate, not a false p=infinity Lipschitz endpoint.

The source is not being used as an unquoted smoothness theorem.

Only bounded vorticity, canonical velocity, transport and Lagrangian flow are used from E1; subsequent derivative assertions are re-derived by multipliers rather than by a smoothness conclusion from that theorem.

The cited input is used for existence, bounds and Lagrangian representation only; the subsequent proof explicitly derives every additional regularity property it needs.

All smoothness used later comes from explicit convolutions and L2 derivative multipliers; the source is used only for bounded-vorticity Lagrangian existence.

The subsequent proof only uses bounded vorticity and a flow; all extra time and L2/L3 properties are derived explicitly, so no unquoted smooth-solution interface is needed.

\begin{lemma}[Bounded background and its trace]\label{lem:background}
The datum $\nu_0$ in Lemma~\ref{lem:decomp} has a global bounded-vorticity Euler evolution
$\nu$, with velocity $V=K*\nu$ and a pressure $p_V$, such that on each finite
horizon $[0,T]$,
\[
 \sup_t(\norm{\nu(t)}1+\norm{\nu(t)}\infty+
 \norm{V(t)}\infty+\norm{V(t)}3)<\infty,
 \quad V(t)-V_0\in C([0,T];L^2).
\]

The current finite-p hypothesis match at S0058 applies to nu_0. The all-time volume-preserving flow then gives conserved L1/L2/Linfinity vorticity norms; near/far and weak-L2 layer cake give bounded and L3 canonical velocity. The integrated transport equation with the norm-one order-zero multiplier gives C_t L2 differences from V0. Curl-free L2 acceleration gives pressure. Each asserted conclusion follows without a global L2 norm of V.

The norm statements follow from the measure-preserving transported representative, near/far velocity estimate, weak-L2 layer cake, and the bounded Fourier operator applied to the time integral of V nu. Pressure follows from the curl-free acceleration argument below.

The current full-domain finite-p verification of E1 supplies its bounded-vorticity background without an endpoint Lipschitz assertion. The flow representative, near/far estimate, weak-L2 layer-cake bound and zero-order relative-velocity multiplier in the following proof establish the stated every-time horizon bounds and C_t L2 difference. All these downstream steps were reread and retain their hypotheses.

The datum properties S0039 and the fully matched finite-p external interface S0054-S0058 provide a global bounded-vorticity evolution. S0065-S0076 select every-time representatives, S0079-S0088 establish global L3 and relative-L2 continuity, and S0089 supplies the velocity pressure. The finite p=3 operator-domain check adds no restriction to nu0 or to the exact original datum.

Moreover $\norm{V(t)-V_0}2\leq
t\norm V{L^\infty_{t,x}}\norm{\nu_0}2$.
\end{lemma}

F_t is the time integral of V nu, with L2 norm at most t sup||V||infinity ||nu_0||2. The operator -K div has L2 norm at most one, producing the exact coefficient one and the stated estimate.

The operator -K*Div has Fourier operator norm one; integrating ||V nu||2 <= ||V||infinity ||nu0||2 over [0,t] proves the coefficient-one estimate.

Transport integration gives V(t)-V0=-K div integral(V nu); its Fourier operator norm is one and ||Vnu||2<=||V||infty||nu0||2, proving the exact coefficient t.

The L2 multiplier norm of -K div is at most one; ||integral_0^t V nu||_2<=t sup||V||_infinity ||nu_0||_2 by volume preservation, which proves the displayed estimate.

\begin{proof}
Apply E1, using its $L^1$ and $L^\infty$ bounds and canonical velocity.

nu_0=a*mu lies in L1 intersection Linfinity, hence also L1 intersection L3_ul. S0058 verifies the primary source Assumption 3.1 domain/range at p=3, divergence estimate and empty plane boundary condition, as well as Assumption 2.1. Inspection of Theorem 3.3 and its time-stepping proof confirms these hypotheses supply exactly the canonical Lagrangian solution, bounds and modulus used here.

nu0=a*mu lies in L1 intersect Linfinity by the decomposition. S0058 now explicitly verifies Assumption 3.1 at p=3: the full L1 intersect L3_ul domain maps to bounded C^{0,1/3} by Theorem 2.2 with q=1, its divergence is zero and the plane has no boundary. Bounded vorticity belongs to that domain, so every application of the divergence bound in Theorem 3.3 is covered. The source theorem supplies existence and log-Lipschitz bounds using its Corollary 2.4; neither a Lipschitz mapping at p=infinity nor smooth Euler existence is assumed.

nu0=a*mu is signed L1 intersect Linfty and therefore belongs to L1 intersect L3_ul. The whole-plane kernel satisfies Assumption 2.1 and, by Theorem 2.2 with q=1,p=3, the entire domain/range and zero-divergence requirements of Assumption 3.1. Theorem 3.3 yields exactly the canonical a.e.-time bounded-vorticity Lagrangian background used here.

nu0=a*mu is signed L1/Linfinity by S0039 and therefore belongs to L1 intersect L_ul^3. The primary existence theorem allows signed noncompact data; the full p=3 mapping, divergence and vacuous whole-plane boundary hypotheses were checked at current S0058. Its bounded-vorticity conclusion supplies exactly the source solution used here.

No smoothness upgrade is needed in this construction.

The remaining construction only requires bounded/log-Lipschitz V, transport, L2 gradient bounds and strong time continuity; all are obtained explicitly below without a smooth-background assertion.

All subsequent requirements are obtained from bounded vorticity, norm-preserving transport, and Fourier estimates; no differentiable-background existence claim is used.

The following reconstruction uses bounded log-Lipschitz velocity and elementary convolution/Fourier estimates only, so does not require a classical smooth Euler solution.

The estimates that follow use only the bounded/Lagrangian source conclusion, not derivatives of the evolving vorticity.

We first select an every-time representative.

Selection of an all-time representative is a permissible definition step; the following volume-preserving flow supplies it and agreement almost everywhere preserves the equation.

Choosing the canonical continuous representative is a permissible modification at a null set of times, with the actual construction specified in S0065-S0076.

The flow-based representative is defined at every time and subsequently shown equivalent to the original a.e.-time solution.

The planned representative choice is implemented by the unique Osgood flow at S0070 and is checked against the source's a.e. representative at S0071.

The log-Lipschitz modulus is Osgood, so the flow and
its inverse are unique, continuous, and have bounded displacement on finite
horizons.

The integral of 1/(r log(1/r)) diverges at zero. Osgood comparison gives uniqueness and continuous dependence for the bounded time-measurable field, also backward in time; displacement is bounded by T sup||V||infinity, and backward flow gives the inverse.

The reciprocal integral of r(1-log r) diverges at zero. The Osgood differential inequality gives uniqueness and stability, applied also to the reversed-time field for the inverse flow. Bounded velocity gives displacement at most T times its bound.

Integral dr/[r(1+|log r|)] diverges at zero, yielding Osgood uniqueness and continuous dependence forward/backward; bounded velocity gives displacement <=T||V||infty and completeness on R2.

The reciprocal integral of r(1+|log r|) diverges at zero; the Osgood difference inequality gives forward/backward uniqueness and continuous dependence. A finite-horizon velocity bound gives displacement <=T sup|V| and global flows.

 Mollify the velocity only in space by a probability kernel.

A nonnegative compact smooth unit-mass spatial kernel preserves divergence zero, boundedness and the same Osgood modulus, and converges uniformly in space for almost every time with a bound uniform over a finite horizon.

Convolving with a smooth probability kernel preserves divergence zero and approximates the bounded uniformly log-Lipschitz field uniformly in space, essentially uniformly in time on a finite horizon.

A fixed positive approximate identity preserves divergence and the uniform Osgood modulus; mollified fields are smooth in space, measurable in time, and uniformly bounded.

Spatial probability mollification is defined for bounded V, preserves divergence, converges uniformly in x uniformly for a.e. time by the uniform modulus, and does not require time smoothness.

The resulting smooth divergence-free flows preserve Lebesgue measure by
the Jacobian equation.

The spatially smooth velocity has a flow differentiable in initial position; its Jacobian solves J'=div V_epsilon J=0 with J(0)=1. Thus each flow preserves Lebesgue measure, even when time derivatives are only almost everywhere.

The spatially smooth field has bounded spatial derivatives after mollification, so its flow is differentiable in initial position. The determinant solves J'=div V_l(X_l)J=0, J(0)=1, proving preservation of Lebesgue measure.

The spatial Jacobian solves J'=div(V_moll)J=0 a.e. in time with J(0)=1. Thus its globally defined invertible flow preserves Lebesgue measure; time smoothness is unnecessary.

Spatially smooth bounded time-measurable divergence-free velocities have spatially differentiable Caratheodory flows. The determinant satisfies J'=div(V_epsilon) J=0, hence preserves Lebesgue measure.

 Their uniform Osgood modulus and uniform convergence
of the velocities give convergence of flows and inverse flows on compact
time-space sets.

For two approximate trajectories their difference is bounded by the integrated uniform velocity error plus the common Osgood modulus of the difference. Osgood stability makes it vanish uniformly on compact time-space sets; the backward equation supplies the same estimate for inverses.

For the difference of two flows use displacement bound e(t)<=T||V_l-V||infinity+integral omega(e(s))ds; Osgood stability makes e tend uniformly to zero. The backward ODE gives the identical inverse-flow conclusion.

Uniform spatial modulus gives uniform mollification convergence (after harmless assignment at exceptional times). Osgood stability on finite horizons applies in both time directions and uniformly for initial points in compact sets.

The uniform spatial modulus and uniform mollification error yield convergence of forward and backward flows by the same Osgood integral inequality; bounded displacement extends the argument on compact time-space sets.

 For a compact continuous test, bounded displacement confines
its transported support to a fixed bounded set; dominated convergence passes
volume preservation to the original flow $X_t$.

Bounded speed confines inverse images of a fixed compact test support to a common bounded enlargement. Uniform flow convergence gives pointwise convergence of compositions, bounded by the test supremum on that enlargement; dominated convergence passes preservation of integrals to the limiting flow.

A compact test transported by a flow of displacement at most C_T remains within a fixed compact set; its integrand is bounded there. Dominated convergence transfers the mollified change-of-variables identity.

Bounded displacement puts preimages of a compact support in one bounded enlargement. Uniform convergence of flows and bounded tests permit dominated convergence of the volume-preservation identity.

A compact test can only be reached by points within T sup|V| of its support. This is a common finite-measure domain, permitting dominated convergence in the measure-preserving change-of-variable identities.

Define $\nu(t)=\nu_0\circ X_t^{-1}$ at every time.

The unique volume-preserving invertible flow defines the composition at every time, up to spatial null sets. Its initial identity gives nu(0)=nu_0.

The unique homeomorphic flow has an inverse for every time; composing the measurable nu0 with it defines the representative without changing its a.e. spatial class.

The unique invertible flow and measurable nu0 define nu0 composed with its inverse at every time; volume preservation makes the definition independent of null-set representatives.

The homeomorphic measure-preserving flow has an inverse at every time, so composition defines nu(t), including t=0, without changing its Lp equivalence class.

 It agrees with the source
solution almost everywhere in time and preserves the $L^1$, $L^2$, and
$L^\infty$ norms.

The source pushforward and the unique flow agree almost everywhere in time; volume preservation converts pushforward into inverse composition. Change of variables preserves each Lp norm for p=1,2,infinity and either sign of nu_0.

The primary pushforward and measure preservation identify this composition with the original solution a.e. in time. Changing variables preserves each of its L1, L2 and Linfinity norms.

The a.e. pushforward in Definition 3.2 becomes this composition since the flow preserves volume. Change of variables preserves signed L1, L2, and Linfty norms at each time.

The source pushforward and unique inverse flow identify the representatives a.e. in time; measure preservation gives exact L1, L2 and Linfinity norm conservation for signed data.

 Approximating $\nu_0$ in $L^1$ or $L^2$ by compact smooth
functions and using volume preservation and continuity of the inverse flow
gives $\nu\in C_tL^1\cap C_tL^2$.

Approximate nu_0 in Lp for p=1 or 2 by compact smooth functions; inverse-flow composition preserves approximation errors. Such smooth compositions vary continuously by bounded displacement and dominated convergence, establishing C_t L1 and C_t L2.

Approximate nu0 by compact smooth functions in L1 or L2. Composition is an isometry, and bounded displacement plus continuous inverse flow gives convergence for the smooth approximants by dominated convergence; the three-term approximation proves norm continuity.

For compact smooth approximants, inverse-flow convergence and bounded displacement give global Lp continuity; the isometry of transport controls approximation error uniformly for p=1,2.

Approximate nu_0 in Lp, p=1,2, by compact smooth f. Transport is an Lp isometry; continuity and bounded displacement give dominated convergence for f composed with inverse flows. The triangle inequality then proves continuity for nu_0.

 Set $V(t)=K*\nu(t)$ at every time.

For every all-time representative nu(t), L1 intersection Linfinity permits absolutely convergent canonical Biot-Savart convolution; this defines V(t) without a time exceptional set.

The near/far kernel estimate makes K*nu defined and bounded for every time; this definition fixes the canonical normalization, including t=0.

nu(t) has finite L1 and Linfty at every t, so near/far splitting defines bounded canonical convolution for all times.

Each nu(t) is in L1 and Linfinity, so the near-far convolution converges boundedly at every spatial point and defines the stated every-time velocity.

It agrees with the source velocity almost everywhere, so its equation is
unchanged.

The redefined nu agrees with the original almost everywhere in time and convolution is linear, so the same is true of V. Products used in the distributional equation therefore agree almost everywhere as well.

Changing a time-dependent function on a time-null set leaves its spacetime distribution unchanged, so the source equation survives the representative choice.

A.e.-time equality of nu and of its canonical velocity preserves their distributional transport equation; altering a null time set cannot create a new distribution.

The revised nu equals the external source a.e. in time and K is linear, so V agrees a.e.; time-space distributional equations are unchanged by this null-time replacement.

 For $f=\nu(t)-\nu(s)$ the near--far bound
$\norm{K*f}\infty\leq r\norm f\infty+(2\pi r)^{-1}\norm f1$
shows continuity of $V$ in $L^\infty$ by choosing $r$ small and then $t-s$
small.

The near kernel has integral r and the far kernel is bounded by (2pi r)^-1. Here ||f||infinity<=2||nu_0||infinity; first make the near term small by choosing r, then use C_t L1 to make the far term small. This includes continuity at zero.

The near kernel integrates to r and the far kernel is at most 1/(2 pi r). The difference has uniformly bounded Linfinity norm and L1 norm tending to zero; choosing r and then the time difference proves Linfinity continuity.

The near kernel has L1 norm r and the far kernel supremum 1/(2pi r). Since ||f||infty<=2||nu0||infty and ||f||1 tends to zero, first choosing r then |t-s| proves Linfty continuity.

The integral of |K| over B_r is r and its exterior supremum is (2pi r)^-1. The first contribution is uniformly small with r and the second tends to zero by C_tL1, proving C_tLinfinity.

 Thus this representative has the canonical initial velocity and
continuous positive-time local velocity pairings.

The inverse-flow identity at zero gives the canonical V_0, and Linfinity time continuity implies all compact velocity pairings and local L2 continuity.

At zero the composition is nu0, giving V0; Linfinity continuity implies continuous pairings with compactly supported velocity tests and local L2 continuity.

At zero the inverse flow is identity, and Linfty continuity on finite horizons implies continuity against every local integrable velocity test.

At zero the flow is identity, hence V(0)=K*nu_0=V_0. Uniform velocity continuity implies continuity of every compact local pairing, including the initial one.

The Biot--Savart near--far split also bounds the velocity:
\[
 \norm V\infty\leq C\bigl(r\norm\nu\infty+r^{-1}\norm\nu1\bigr)
 \quad(r>0).
\]

Integrating the absolute kernel over |z|<r gives r, and on the complement its supremum is 1/(2pi r); triangle inequality gives the bound with a universal constant for every r>0.

Split the convolution into |z|<r, whose kernel integral equals r, and |z|>=r, whose supremum is 1/(2 pi r); applying this to |nu| yields the stated bound.

The near/far computation is uniform in x for signed nu after replacing by absolute values; both norm terms are finite for every r>0.

The same polar near-far estimate holds for every r>0; conserved vorticity L1/Linfinity norms give a finite uniform velocity bound by taking any fixed r.

Thus $V\nu$ belongs to both $L^1$ and $L^2$ uniformly on finite horizons.

Bounded V times nu in L1 and L2 belongs to those spaces, with norms bounded by sup||V||infinity times the conserved corresponding vorticity norms. The bounds hold uniformly on each finite horizon.

Bounded V multiplies nu in L1 and L2, both uniformly preserved on finite horizons; hence the products satisfy both stated integrability bounds.

Bounded V times nu in L1 and L2 gives both product norms with constants uniform on the fixed horizon by conserved vorticity norms.

||V nu||_p<=||V||_infinity ||nu||_p for p=1,2; the L2 vorticity bound follows from L1-intersection-Linfinity and then conservation, uniformly on each finite horizon.

For completeness, $K*\mu$ for any finite measure satisfies
\[
 |\{|K*\mu|>\lambda\}|\leq C\norm\mu\M^2\lambda^{-2}.
\]

For positive lambda and M>0, splitting at r=M/(pi lambda) leaves far convolution <=lambda/2; near convolution has L1 <=rM. Chebyshev then gives area <=2rM/lambda=C M^2 lambda^-2. For M=0 the convolution vanishes and the assertion is immediate.

For positive lambda and nonzero mass the near/far split and Chebyshev give measure <=C M^2/lambda^2. If M=0 the convolution is zero and the assertion holds directly.

For lambda>0 and nonzero M split at M/(pi lambda), use the far bound lambda/2 and near L1 norm Mr. Chebyshev yields CM^2/lambda^2; M=0 is immediate separately.

For lambda>0, splitting into bounded far field and L1 near field gives distribution function <=C M^2/lambda^2. The zero measure case is identically zero and needs no positive splitting radius.

Indeed split $K$ at radius $r=\norm\mu\M/(\pi\lambda)$; the far part has
supremum at most $\lambda/2$, and the near part has $L^1$ norm at most
$Cr\norm\mu\M$.

For M>0 and lambda>0 the radius is positive, the far estimate equals M/(2pi r)=lambda/2, and Tonelli gives near L1 <=rM. The zero-measure endpoint is trivial separately and requires no division by a radius.

At r=M/(pi lambda), the far supremum is M/(2 pi r)=lambda/2. The near convolution L1 norm is at most rM, giving the two estimates; the zero-mass case is handled directly before choosing r.

At the stated positive radius the far convolution is bounded by M/(2pi r)=lambda/2; the near kernel has norm r, hence its convolution norm <=rM. The zero-measure case is identically zero.

For M>0 the chosen radius gives far convolution <=M/(2pi r)=lambda/2; near convolution has L1 norm <=r M. For M=0 the claimed near/far bounds are trivial because the convolution is zero.

 Chebyshev proves the bound.

If |near+far|>lambda while |far|<=lambda/2, then |near|>lambda/2; Chebyshev gives 2||near||1/lambda and hence the weak-L2 bound with the correct exponent.

Since a level set of the sum at lambda lies in the near-part level set at lambda/2, Markov's inequality bounds it by 2||near||1/lambda <=C M^2/lambda^2.

The superlevel set for the sum lies in the near-part superlevel lambda/2, of size <=2||near||1/lambda=CM^2/lambda^2.

Chebyshev at lambda/2 gives measure <=2rM/lambda=C M^2/lambda^2; this uses no cancellation or sign restriction.

 Combining it with boundedness
of $V$ and layer cake gives $\norm V3^3\leq C\norm\nu1^2\norm V\infty$.

Layer cake gives ||V||3^3=3 integral_0^B lambda^2 area{|V|>lambda} d lambda <=C ||nu||1^2 B. The distribution vanishes above B=||V||infinity, and the lambda^-2 singularity cancels exactly.

Layer cake gives ||V||3^3=3 integral_0^B lambda^2 measure{|V|>lambda}d lambda <= C||nu||1^2 B; the upper integration endpoint is the Linfinity bound.

Layer cake gives ||V||3^3=3 integral_0^||V||inf lambda^2 measure{|V|>lambda} d lambda <=CM^2||V||inf, including zero M.

The layer-cake integral 3 integral_0^B lambda^2 meas{|V|>lambda} d lambda is <=C ||nu||_1^2 B; boundedness cuts off the integral at B=||V||_infinity.

Integrate the vorticity equation:
\[
 \nu(t)-\nu_0=-\Div F_t,\qquad F_t=\int_0^t V(s)\nu(s)\,ds\in L^1\cap L^2.
\]

The transport equation integrated in time, using its initial datum and C_t L1 vorticity, yields nu(t)-nu_0=-div integral_0^t Vnu. The bounded products are Bochner integrable into L1 and L2; hence the identity holds at every time.

The weak vorticity equation integrates to nu(t)-nu0=-Div F_t for all times by continuity; Bochner integrability follows from V bounded, nu in continuous L1 and L2, and finite horizon.

The weak initial transport equation and C_t L1 representative integrate to nu(t)-nu0=-div F_t at every t. Vnu is Bochner integrable in L1 and L2 by established continuity/bounds.

The source weak transport equation and C_tL1 representative integrate to the displayed distribution identity. V nu is strongly measurable and uniformly L1/L2, so its Bochner integral belongs to both spaces.

The Fourier multiplier of $-K*\Div$ has operator norm at most one, yielding
$V(t)-V_0=-K*\Div F_t$ and the stated estimate and continuity.

The multiplier of -K div is -xi-perp tensor xi/|xi|^2, with operator norm one away from zero. V*nu is strongly continuous into L2: use C_t Linfinity of V, C_t L2 of nu, and bounded norms. Its Bochner primitive is C_t L2; applying the bounded multiplier gives the canonical relative velocity identity and strong L2 continuity, including zero.

The Fourier matrix of -K*Div is a signed rank-one matrix xi-perp tensor xi/|xi|^2, of norm one. Applying it to F_t gives V(t)-V0 distributionally and in L2, with the claimed bound and continuity.

The multiplier of -K div is -xi-perp tensor xi/|xi|^2, of Euclidean operator norm one. The canonical Fourier formula identifies V(t)-V0 without a harmonic addition. Moreover V is continuous into Linfty and nu into L2, so Vnu is Bochner-continuous into L2; F_t=integral_0^t Vnu ds is strongly continuous there. Applying the bounded multiplier proves the claimed C_t L2 difference, even when V and V0 individually have infinite energy.

The Fourier symbol -m(xi) i xi-dot has operator norm one. Applying it to the integrated vorticity identity gives V(t)-V0=-K div F_t in L2, without a harmonic addition because both velocities are canonical. The Bochner integral F_t is Lipschitz in L2, so the same bounded operator proves strong relative-L2 continuity at every time including zero.

 Likewise
$\partial_tV,\nabla V\in L^\infty_tL^2$.

The time derivative is the order-zero multiplier applied to Vnu in L-infinity_t L2. Each spatial derivative has multiplier xi_j xi-perp/|xi|^2 applied to nu in L-infinity_t L2. Both are uniformly bounded on finite horizons.

The same bounded multiplier applied instantaneously to V nu controls partial_t V in L2; multipliers i xi_j m control grad V in L2 using nu in L2.

The same degree-zero multiplier acts on Vnu for partial_t V, and xi tensor xi-perp/|xi|^2 acts on nu for grad V; Plancherel gives both horizon L2 bounds.

partial_t V=-K div(V nu) has a bounded order-zero L2 multiplier; nabla V has multiplier i xi tensor m acting on nu with norm one. Conserved ||nu||_2 and bounded ||V||_infinity give both assertions.

 The same argument between
any two times gives $\norm{V(t)-V(s)}2\leq C_T|t-s|$.

Replacing the lower time endpoint zero by s in the transport integral gives an L2 difference bounded by integral_s^t ||Vnu||2, at most C_T |t-s|.

Replacing [0,t] by [s,t] in the Bochner integral yields ||V(t)-V(s)||2 <= ||V||infinity ||nu0||2 |t-s| on each finite horizon.

Integrate Vnu between s and t instead of zero and t; its uniform L2 bound yields the stated Lipschitz relative-velocity estimate for all times.

Repeating S0083 over [s,t] gives ||V(t)-V(s)||_2<=sup||V nu||_2 |t-s| at every time by continuity of the representatives.

 With
$B_T=\sup_t\norm{V(t)}\infty$, interpolation gives
$\norm{V(t)-V(s)}3^3\leq2B_T\norm{V(t)-V(s)}2^2$.

B_T is finite by the near/far estimate. Pointwise |V(t)-V(s)|<=2B_T; multiplying its squared magnitude and integrating gives exactly ||difference||3^3<=2B_T ||difference||2^2.

For f=V(t)-V(s), ||f||infinity<=2B_T and integral |f|^3<=||f||infinity integral |f|^2, giving the exact stated inequality.

Pointwise |V(t)-V(s)|<=2B_T, so integration of |difference|^3 <=2B_T|difference|^2 gives exactly the displayed inequality.

For f=V(t)-V(s), ||f||_infinity<=2B_T and integral|f|^3<=||f||_infinity integral|f|^2, proving the coefficient and exponent exactly.

Therefore $V\in C([0,T];L^3)$.

Each V(t) belongs to L3 by weak-L2 plus boundedness; the L2 difference estimate and interpolation inequality make its L3 differences tend to zero, including the initial time.

Each V(t) is in L3 by layer cake, and the preceding difference bound tends to zero with t-s; hence the map is continuous into global L3.

Each V(t) is L3 by S0082, and S0087 plus the L2 difference estimate gives strong L3 continuity including t=0.

Every V(t) is L3 by S0082 and the difference estimate S0087 with S0086 tends to zero; this proves strong C([0,T];L3), not merely local continuity.

 The vector
$\partial_tV+(V\cdot\nabla)V$ is an $L^2$ curl-free field by transport;
it is a pressure gradient in distributions.

V is bounded with gradient in L2, so convection lies in L2, as does partial_t V. The distributional curl of convection is div(Vnu), obtained by spatial smoothing/product convergence; transport cancels it. A curl-free distribution on the plane is a gradient, with its negative used as p_V.

The two acceleration terms are L2 because V is bounded and grad V,partial_t V are L2. The identity curl((V dot grad)V)=Div(V nu) follows by smoothing and local Sobolev convergence; transport cancels this with partial_t nu. A curl-free distribution on the simply connected plane is a gradient.

V bounded and grad V, partial_t V in L2 make the vector L2; spatial mollification justifies curl((V dot grad)V)=div(V nu), canceled by transport. The distributional Poincare lemma on R2 then yields a gradient with pressure chosen with the opposite sign.

V is bounded with nabla V and partial_t V in L2. The distribution identity curl((V dot nabla)V)=div(V nu) follows by local Sobolev approximation; transport cancels it with partial_t nu. A curl-free distribution on the plane is a gradient, whose negative defines p_V.

 Thus the velocity Euler equation
used below is indeed available without a global-energy assumption on $V$.
\end{proof}

Choosing p_V as the negative potential of the curl-free acceleration gives the velocity Euler equation in distributions. Its derivation needed only differences in L2, never global energy of V.

Taking p_V to be the negative potential of the preceding acceleration gives the velocity Euler equation without ever requiring V itself to be in global L2.

The curl-free reconstruction supplies pressure, and all terms are locally integrable distributions; global L2 of V itself was never used.

The resulting divergence-free velocity satisfies partial_t V+div(V tensor V)+nabla p_V=0; only relative L2, not global velocity energy, entered S0083-S0089.

\section{An initial Euler--Reynolds subsolution}

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The convention for Euler--Reynolds is
\begin{equation}\label{eq:ER}
 \partial_tu+\Div(u\otimes u)+\nabla p=\Div R,
 \qquad \Div u=0,
\end{equation}
where $R$ is symmetric.

This explicitly defines the Euler-Reynolds sign convention and symmetric stress class. The divergence and tensor indices agree with S0032 and with all later integrations by parts.

This fixes the sign convention, row-wise divergence and symmetric stress; all later packet and cross-stress signs were checked against this equality.

The declared sign convention has symmetric R and row divergence. Direct integration against divergence-free tests yields the usual negative stress contribution used in the anchor lemma.

This fixes the positive div R sign and symmetric-stress convention used consistently in the seed, packet correction, and weak stress testing.

 Let $b=\mathcal F^{-1}(e^{-|\xi|^4})$ and
$b_t(x)=t^{-2}b(x/t)$ for $t>0$.

Inverse Fourier transform of the stated Schwartz multiplier defines b. The positive-time two-dimensional dilation preserves integral and has multiplier b-hat(t xi); no zero-time kernel value is required.

Inverse Fourier transform of exp(-|xi|^4) is defined in the Schwartz class, and the dilation t^-2 b(x/t) has Fourier transform exp(-t^4|xi|^4) for positive t.

The inverse Fourier transform of exp(-|xi|^4) is well defined in Schwartz space; dilation t^-2 gives unit mass and Fourier multiplier exp(-t^4|xi|^4) for t>0.

With the declared inverse Fourier normalization, the rapidly decaying multiplier defines b; t^-2 b(x/t) has multiplier exp(-t^4|xi|^4), for strictly positive t.

 It is a real radial Schwartz kernel of
integral one; it need not be positive.

The even real radial Schwartz multiplier has a real radial Schwartz inverse and integral b=b-hat(0)=1. Positivity is not used or guaranteed; all measure/Young bounds later use ||b||1 instead.

The frequency function is real, radial, even and Schwartz, so its inverse is real radial Schwartz. Integral b equals its Fourier transform at zero, namely one; none of the bounds use positivity.

Real even radial Fourier data give a real radial Schwartz inverse; Fourier value at zero is one. Positivity is neither required nor used; L2 contraction uses its nonnegative multiplier, while L1 estimates retain C_b.

A real even radial Schwartz multiplier has a real radial Schwartz inverse; integral b=b-hat(0)=1. No positivity is used, and later L1 estimates retain the factor C_b.

 Write $C_b=\norm b1$.

C_b is finite because b is Schwartz; it is at least one by its integral and is the correct constant for L1 convolution estimates without positivity.

C_b denotes the finite L1 norm of this Schwartz function, retaining the needed total-variation constant for a possibly signed kernel.

C_b is a finite fixed positive constant because b is Schwartz and has integral one.

C_b is finite by the Schwartz property and is the correct total-variation bound for signed convolution kernels.

\begin{proposition}[Initial Euler--Reynolds seed]\label{prop:seed}
For the preceding decomposition and background, put
\[
 W(t)=b_t*v_0,\qquad u^{(0)}=V+W,\qquad p^{(0)}=p_V,
\]
\[
 R^{(0)}=W\otimes W+V\otimes W+W\otimes V
 -4t^3\bigl[\nabla\Delta W+(\nabla\Delta W)^T\bigr].
\]

The definitions use well-typed smooth positive-time W and the established V,p_V. Every stress term is globally L1; the symmetric derivative tensor carries the negative 4t^3 sign needed for partial_t W=-4t^3 Delta^2 W.

All four definitions are well typed: W is global L1 and L2 for t>0, V bounded, and the symmetric derivative tensor has finite L1 norm. Direct substitution gives the seed stress with the negative time-derivative coefficient.

All three velocity/pressure definitions are well formed. The symmetric derivative stress has divergence -4t^3 Delta^2 W, precisely the derivative of the biharmonic multiplier; the quadratic/cross terms expand the nonlinear increment.

All fields are defined by the preceding datum/background. The products are global L1 because W is L1/L2 and V bounded; the derivative term is global L1 by S0107. The tensor is symmetric.

This is a positive-time distributional solution of \eqref{eq:ER}.

Adding the background Euler equation to the W equation and expanding the quadratic flux leaves the derivative stress with divergence partial_t W and the three W-dependent product stresses. Divergence remains zero.

partial_t W=-4t^3 Delta^2 W, and the divergence of the symmetrized gradient-Delta tensor is Delta^2 W; expansion of the square cancels all seed terms in the stated Euler-Reynolds equation.

Insert partial_t W=-4t^3 Delta^2 W and div W=0 into the displayed R; the old background equation cancels and every remaining term matches with the stated sign.

Subtract the V Euler equation: partial_t W=-4t^3 Delta^2 W is div of the derivative stress and the remaining three products are precisely the quadratic difference, proving the stated ER equation.

 On $0<t\leq T$,
\begin{align*}
 \norm{u^{(0)}-V}2&\leq\norm{v_0}2,\\
 \norm{\curl u^{(0)}}1&\leq\norm{\nu_0}1+C_b\norm{\sigma_0}\M,\\
 \norm{R^{(0)}}1&\leq\norm{v_0}2^2
 +2C_b\norm V\infty\norm{v_0}1+Ct\norm{\sigma_0}\M.
\end{align*}

The multiplier has modulus <=1, giving the L2 contraction. Curl equals nu+b_t*sigma_0, giving conserved ||nu_0||1 plus C_b variation. Product stress uses ||W||2^2 and 2||V||infinity||W||1; the derivative bound is 4t^3 O(t^-2) variation=O(t) variation.

Plancherel contraction gives the first estimate; measure convolution and norm-preserving nu give the second; quadratic and cross terms cost ||v0||2^2 and 2 C_b||V||infinity||v0||1, while second-kernel-derivative scaling gives Ct||sigma0||M.

Multiplier modulus <=1 gives the L2 bound; TV convolution gives the curl bound. W tensor W costs ||v0||2^2, the two V-W terms cost 2C_b||V||infty||v0||1, and t^3||D^2b_t||1=O(t) yields the last stress term.

Multiplier contraction gives ||W||_2<=||v_0||_2; Young gives curl bound ||nu_0||_1+C_b||sigma_0||; the two cross products cost 2||V||_infinity C_b||v_0||_1 and t^3 D^2 b_t costs C t||sigma_0||.

It has strong $L^2_{\rm loc}$ initial trace $u_0$ and canonical velocity
$u^{(0)}(t)=K*(\nu(t)+b_t*\sigma_0)$.

V(t)-V_0 tends to zero globally in L2, while W(t) tends to v_0 globally in L2; together they give the strong local initial trace. Associating K with b_t*sigma_0 in distributions proves the canonical convolution identity.

V(t)-V0 tends to zero globally in L2 and W(t) tends to v0 in L2, so their sum has strong local L2 initial trace. Fourier convolution gives curl W=b_t*sigma0 and W=K*(b_t*sigma0), with the canonical background added.

V(t)-V0 tends to zero in global L2 and b_t*v0 tends to v0 in L2. Associativity of finite-measure convolution gives W=K*(b_t*sigma0), so the canonical identity holds at every positive time.

V(t)-V_0 tends to zero in L2 and W(t)-v_0 does too, proving strong local trace. curl W=b_t*sigma_0 and the canonical convolutions commute against compact tests, giving the stated identity.

On compact positive-time intervals $R^{(0)}\in C_tL^1$,
$u^{(0)}\in C_tL^3$, and the velocity is strongly continuous in local $L^2$.
\end{proposition}

On positive-time compact intervals W is continuous in L1,L2,Linfinity and differentiated convolution kernels in L1. Product continuity with V in L3 and W in L3/2 gives C_t L1 stress; interpolation gives C_t L3 velocity, and the established L2 differences give local strong continuity.

On any compact interval away from zero, kernel dilation and derivatives depend continuously in all required norms. Product estimates then give continuous L1 stress and L3 velocity, as well as strong local L2 velocity continuity.

On a compact interval separated from zero, b_t and its needed derivatives vary continuously in their Lp norms. Products use L2-L2 or L3-L3/2 Holder; background continuity was verified, giving all three stated topologies.

Positive-time kernel continuity yields W continuity in L1/L2/Linfinity; S0088 handles V. Products in conjugate spaces and D^2 b_t L1 continuity give C_tL1 stress and C_tL3/local-L2 velocity.

\begin{proof}
The multiplier of $b_t$ is $e^{-t^4|\xi|^4}$, so
$\partial_tW=-4t^3\Delta^2W$.

Dilation gives multiplier exp(-t^4 |xi|^4); differentiating at t>0 gives -4t^3 |xi|^4 exp(-t^4 |xi|^4). Delta^2 has positive multiplier |xi|^4, hence the displayed minus sign.

Fourier dilation yields exp(-t^4|xi|^4); differentiating it gives -4t^3|xi|^4 exp(-t^4|xi|^4), the multiplier of the asserted fourth-order heat equation.

Dilation yields Fourier factor exp(-t^4|xi|^4); differentiation gives -4t^3|xi|^4, exactly -4t^3 Delta^2.

Fourier scaling gives exp(-t^4|xi|^4); differentiation produces -4t^3|xi|^4, precisely -4t^3 Delta^2 in physical space with the declared Fourier sign.

 Since $\Div W=0$,
$\Div[\nabla\Delta W+(\nabla\Delta W)^T]=\Delta^2W$.

Convolution preserves div v_0=0. In components, summing j derivatives of partial_j Delta W_i gives Delta^2 W_i; the transposed term is partial_i Delta div W=0.

Convolution preserves Div W=0. Component expansion gives Div(grad Delta W)=Delta^2 W and Div((grad Delta W)^T)=grad Delta Div W=0.

Row divergence of grad Delta W is Delta^2 W; row divergence of its transpose is grad Delta div W=0. Divergence commutes with convolution.

Index expansion gives div(nabla Delta W)=Delta^2 W and div((nabla Delta W)^T)=nabla Delta div W=0; mollification preserves div W=0.

Expanding $(V+W)\otimes(V+W)$ proves \eqref{eq:ER} with the stated signs.

The quadratic expansion has W tensor W and both mixed terms, all with plus sign. Their divergence together with the negative derivative stress equals exactly the residual left after subtracting the background Euler equation.

Subtract the background equation from the proposed seed equation; the remaining three quadratic tensors occur once on each side and partial_t W equals the divergence of the displayed derivative stress.

Expand V+W into V tensor V, W tensor W, and the two cross tensors. The seed derivative equals the divergence of the negative derivative stress, while the V terms satisfy Euler.

The quadratic expansion has W tensor W plus the two V/W cross tensors; adding the derivative stress with negative sign reproduces partial_t W and the stated positive-div-R convention.

Plancherel gives the $L^2$ bound and $W(t)\to v_0$ strongly in $L^2$.

The Fourier multiplier lies between zero and one and converges pointwise to one as t decreases to zero. Plancherel gives contraction and dominated convergence of (multiplier-1) times v_0-hat gives strong L2 convergence.

The Fourier multiplier lies in [0,1] and tends pointwise to one. Plancherel gives contraction and dominated convergence against |v0-hat|^2 gives strong L2 convergence.

The multiplier lies in [0,1] and converges pointwise to one. Plancherel and dominated convergence prove contraction and strong convergence without assuming positivity of b.

The multiplier is bounded by one and tends pointwise to one; dominated convergence against |v_0-hat|^2 gives contraction and strong convergence, including nonzero v_0.

Finite-measure convolution gives $\curl W=b_t*\sigma_0$ and the curl bound.

Distributional curl commutes with smooth convolution and curl v_0=sigma_0. Young's measure estimate bounds ||b_t*sigma_0||1 by C_b||sigma_0||M; addition of nu supplies the stated seed curl bound.

Spatial distributional derivatives commute with Schwartz convolution, giving curl W=b_t*sigma0. Integrating against variation bounds its L1 norm by C_b||sigma0||M.

curl v0=sigma0 distributionally, so curl W=b_t*sigma0; its L1 norm is at most ||b_t||1 TV(sigma0)=C_b TV(sigma0).

curl v_0=sigma_0 follows from the decomposition; curl commutes with smooth convolution. Young for a finite signed measure gives ||b_t*sigma_0||_1<=C_b||sigma_0||.

Furthermore $\norm W1\leq C_b\norm{v_0}1$.

Dilation preserves ||b_t||1=C_b, so Tonelli/Young gives ||W||1<=C_b||v_0||1 with no positivity assumption.

Integrating the absolute convolution of b_t and v0 gives ||W||1 <=||b_t||1||v0||1=C_b||v0||1.

Young's inequality gives ||b_t*v0||1<=||b_t||1||v0||1=C_b||v0||1.

Direct Tonelli gives ||b_t*v_0||_1<=||b_t||_1||v_0||_1=C_b||v_0||_1 for all positive t.

 The identity
$\Delta W=\nabla^\perp\curl W$ yields
\[
 \norm{\nabla\Delta W}1\leq C\norm{D^2b_t}1\norm{\sigma_0}\M
 =Ct^{-2}\norm{\sigma_0}\M.
\]

For divergence-free W, direct component expansion gives Delta W=grad-perp curl W. Differentiating b_t twice has L1 norm t^-2 times the fixed kernel norm, so convolution against sigma_0 gives the displayed bound and exponent.

For divergence-free W the vector identity Delta W=grad-perp curl W has the displayed positive sign. Differentiating twice the kernel and scaling D^2 b_t gives t^-2 in L1, proving both transitions.

For divergence-free W, component expansion gives Delta W=nabla-perp curl W. Two derivatives of b_t scale with L1 norm t^-2, yielding both displayed transitions.

The two-dimensional identity Delta W=nabla-perp curl W uses div W=0. Two kernel derivatives scale in L1 as t^-2; convolution with sigma_0 proves both transitions.

Thus the derivative stress costs $Ct\norm{\sigma_0}\M$, and the other
terms are bounded as displayed.

The symmetric derivative term contributes at most 8t^3||grad Delta W||1=O(t)||sigma_0||M. The remaining terms use L2 contraction for W squared and Linfinity V times L1 W for both mixed products.

Multiplying the previous derivative bound by 4t^3 leaves Ct. The other terms are controlled by the L2 product and bounded-background times L1-W estimates, with transpose contributing the factor two.

Multiply the derivative bound by 4t^3 and account for the transpose with a fixed constant; the remaining stress estimates are the L2 square and Linfty-L1 products already derived.

Multiplication by 4t^3 and adding the transpose turns the t^-2 derivative bound into C t. The quadratic and cross products have the L2-square and two Linfinity-times-L1 bounds in S0098.

 Lemma~\ref{lem:background} proves the trace.

The background lemma gives ||V(t)-V_0||2 to zero, while S0104 gives W(t) to v_0 in L2. Since V_0+v_0=u_0, the seed has strong local L2 initial trace.

The background lemma gives V(t)-V0->0 in L2; combining with W(t)->v0 and V0+v0=u0 proves the seed trace.

The background lemma gives V(t)-V0 ->0 in L2; adding the independently verified W trace gives the exact canonical u0 locally strongly in L2.

This invokes S0061 together with S0104: both V-V_0 and W-v_0 vanish in L2, so u^(0) tends to V_0+v_0=u_0 strongly locally.

Fubini against compact tests, or the Fourier identity, gives the canonical
convolution.

In Fourier distributions, v_0-hat=m sigma_0-hat and multiplication by b_t-hat commutes with m. Equivalently absolute testing permits Fubini. Thus W=K*(b_t*sigma_0), and adding V gives the canonical identity.

K convolved with b_t*sigma0 has Fourier expression m exp(-t^4|xi|^4)(1-a-hat)mu-hat, identical to W. Distributional testing is justified by Schwartz decay and the finite-measure local kernel bound.

K*(b_t*sigma0) and b_t*(K*sigma0) have the same tempered Fourier distribution; local kernel integrability also justifies their compact-test Fubini identity. No harmonic term is introduced.

K has uniformly bounded local L1 mass under translations and b_t has finite L1 mass; absolute Fubini against compact tests justifies K*(b_t*sigma_0)=b_t*(K*sigma_0)=W.

 On positive-time compact intervals, convolution bounds give
$W\in C_t(L^1\cap L^2\cap L^\infty)$ and hence
$W\in C_tL^{3/2}\cap C_tL^3$.

The kernels b_t vary continuously in L1 and L2 on compact positive times. Young with v_0 in L1 or L2 yields the three stated norms, including Linfinity via L2 kernels; interpolation yields L3/2 and L3 continuity.

Away from zero, b_t varies continuously in L1 and Linfinity, and convolution with v0 in L1 and L2 gives the stated three norm continuities. Interpolation supplies L^{3/2} and L3 continuity.

Continuity of the scaled Schwartz kernels in L1 and Linfty on t>=a>0 combines with v0 in L1 intersect L2; interpolation gives both 3/2 and 3.

On t in [a,b] with a>0 the kernels vary continuously in L1 and Linfinity. Young with v_0 in L1/L2 gives all endpoint continuities; interpolation supplies 3/2 and 3.

 The derivative stress is continuous
in $L^1$ by continuity of $D^2b_t$ in $L^1$.

Positive-time dilation is continuous in L1 for D2 b_t and multiplication by continuous t^3 preserves this; convolution against the fixed finite measure is bounded into L1.

The derivative stress is a scalar t^3 times a fixed second-derivative convolution against sigma0; its L1 continuity follows from L1 continuity of the dilated kernel derivatives.

The derivative stress is t^3 times D^2b_t*sigma0, up to fixed rotations/transposes; continuity of D^2b_t in L1 and of t^3 gives C_t L1.

D^2 b_t varies continuously in L1 for t bounded away from zero. Convolution with a fixed finite measure and multiplication by smooth t^3 preserve L1 continuity.

 Products $W\otimes W$
are $L^1$-continuous by $C_tL^2$, while $V\otimes W$ and its transpose
are $L^1$-continuous by $V\in C_tL^3$ and $W\in C_tL^{3/2}$.

For W squared, difference factorization and Cauchy-Schwarz give L1 continuity from C_t L2. For each mixed product, Holder with exponents 3 and 3/2 gives continuity from the established two factors, including the transpose.

For W squares use ||f tensor f-g tensor g||1 <=(||f||2+||g||2)||f-g||2. For each mixed product use the L3 and L^{3/2} Holder inequality; the transpose has the identical norm.

The product difference splits as a difference times a bounded factor. Holder with (2,2) or (3,3/2) proves every indicated symmetric pair continuous in L1.

The product map L2 times L2 to L1 and the product map L3 times L3/2 to L1 are continuous by Holder; all factors have these strong time continuities.

This proves all asserted continuity without differentiating the old background.
\end{proof}

The preceding checks cover all stress terms and velocity norms; derivatives of the old velocity were never needed for these time-continuity conclusions.

S0111-S0113 prove all three claimed continuous norms using norm continuity, with no time derivative of V required.

The preceding product estimates use strong norm continuity only; no time or spatial derivative of the old background is taken.

S0111-S0113 account for every seed continuity claim and use only already established background norms, preserving the rough-background interface of LER.

\begin{remark}[Why a subcritical global increment norm is required]\label{rem:cross}
Every repair below must have an \emph{arbitrarily small} $L^{3/2}$ velocity
increment, in addition to its $L^2$ and curl-variation estimates.

The stated extra increment requirement is sufficient for the subsequent whole-plane mixed products because the background is globally L3. The following explicit swirl shows that L2 and curl variation alone give no such product guarantee.

This states a construction requirement fulfilled later. With V only globally L3, small L^{3/2} increments provide the Holder-dual control for global L1 stress.

The construction requires this extra subcritical control to make whole-plane cross stresses integrable and arbitrarily small; finite L2 and curl TV alone do not supply it, as the next explicit example verifies.

This specifies the norm needed for the global Holder closure, rather than adding an input condition. S0249 and S0270 supply an arbitrarily small L^(3/2) correction, while S0247 proves its pairing with the current global L3 velocity is integrable.

 Indeed,
$\norm{V\otimes w}1\leq\norm V3\norm w{3/2}$ closes the whole-plane
cross-stress bound.

The pointwise tensor norm is |V||w|, and 1/3+2/3=1. Holder therefore supplies the exact global L1 product bound without support restrictions.

Pointwise |V tensor w|=|V||w|, and 1/3+2/3=1 gives the displayed global Holder bound.

Holder uses conjugate exponents 3 and 3/2; ||V||3 is finite on each horizon and the second factor is freely small.

Holder with conjugate exponents 3 and 3/2 yields the global tensor L1 estimate; spatial localization would not suffice for the unbounded-plane background.

 Finite energy and finite curl variation do not suffice:
a smooth radial swirl equal to $e_\theta/(r\log r)$ for $r\geq e^2$ belongs
to $L^2$, has integrable curl, but is not in $L^{3/2}$.

Smoothly joining the swirl through a finite annulus causes no integrability issue. In the tail, energy is 2pi integral dr/(r log^2 r)<infinity and curl=-1/(r^2 log^2 r) has the same finite radial integral; L3/2 requires integral r^-1/2(log r)^-3/2 dr, which diverges.

For the tail swirl f(r)=1/(r log r), L2 reduces to integral dr/(r log^2 r)<infinity; curl=(rf)'/r=-1/(r^2 log^2 r), with finite L1 tail. The L^{3/2} integral is integral r^{-1/2}(log r)^{-3/2}dr and diverges. A smooth core extension adds only finite norms.

For the exterior swirl, energy is integral dr/(r log^2 r)<infinity and curl=-1/(r^2 log^2 r) has finite L1. Its 3/2 power integral is integral r^-1/2(log r)^-3/2 dr, which diverges. A smooth core patch changes none of these tails.

For a smooth cutoff of f(r)=1/(r log r), L2 tail is integral dr/(r log^2 r)<infinity; curl=(r f)'/r=-1/(r^2 log^2 r) is L1; the 3/2 tail integral r^-1/2(log r)^-3/2 diverges.

 Against a smooth
unit-circulation background equal to $e_\theta/(2\pi r)$ at infinity,
$\int|V\otimes w|=\infty$ because $\int_R^\infty dr/(r\log r)=\infty$.

A smooth radial compact-vorticity unit-circulation background has the specified tail. There |V tensor w|=(2pi r^2 log r)^-1; the area element 2pi r dr leaves integral dr/(r log r), divergent as log log r.

Multiplying the two tail magnitudes and the polar Jacobian leaves a positive constant times integral dr/(r log r), which diverges. A smooth radial unit-circulation vorticity produces the specified background outside its support.

The absolute tensor norm is |V||w|, whose area integral equals integral dr/(r log r) up to the stated constant. A smooth unit-mass radial vortex gives exactly the stated exterior background.

A smooth radial unit-circulation vorticity produces the stated far-field background. The tensor norm equals product of lengths, and polar integration gives a constant times integral dr/(r log r)=infinity.

Multiplying $w$ by a small constant does not repair this divergence.
\end{remark}

For any nonzero scalar multiplier, the tail integral is multiplied by its positive absolute value and remains infinite. The assertion concerns arbitrarily small nonzero multiples; the zero multiple is trivially excluded by its purpose.

Every nonzero scalar multiple multiplies the nonnegative divergent integral by its absolute value; taking that scalar small cannot make it finite.

Any nonzero scalar rescales the divergent nonnegative cross-stress integral without making it finite; the statement concerns small nonzero perturbations.

Any nonzero constant multiple leaves that positive divergent cross integral infinite; zero multiplication is immaterial to the intended small nonzero increment example.

\section{Endpoint repair: elementary building blocks}

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\subsection{An exact translating dipole}

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The Bessel functions used below may be defined by their convergent power
series, whose termwise derivatives give the Bessel equation and
$J_1'(z)=J_0(z)-J_1(z)/z$.

The power series J_n(z)=sum_m (-1)^m(z/2)^(2m+n)/(m!(m+n)!) converges locally uniformly with all derivatives. Coefficient comparison gives the Bessel ODE and J1'=J0-J1/z for z nonzero, with a removable continuation at zero.

For integer n, J_n(z)=sum_k (-1)^k(z/2)^{2k+n}/(k!(k+n)!) converges with all derivatives on compact sets. Coefficient substitution gives the Bessel equation and differentiating the n=1 series gives the stated recurrence.

The entire power series has infinite convergence radius, so derivatives can be taken on compact sets. Comparing coefficients yields z^2J1''+zJ1'+(z^2-1)J1=0 and J1'=J0-J1/z (with the removable zero interpreted by series).

J_0=sum(-z^2/4)^k/(k!)^2 and J_1=(z/2)sum(-z^2/4)^k/(k!(k+1)!) converge locally uniformly with derivatives; coefficient comparison gives the Bessel ODE and the displayed recurrence.

There is a positive zero $\beta\in(3,4)$ of $J_1$.

The exact alternating-series evaluations at 3 and 4 have opposite signs; continuity gives a root strictly between them, hence positive.

The exact rational lower and upper partial-sum bounds in S0124 have opposite signs, and the power series is continuous, proving a zero in the open interval (3,4).

The exact alternating-series bounds at 3 and 4 have opposite signs, and J1 is continuous, so a positive zero lies strictly between them.

The rational endpoint estimates in S0124 have opposite strict signs and J_1 is continuous, giving beta strictly between 3 and 4.

For completeness, the alternating series gives
$J_1(3)\geq(3/2)(223/1024)>0$ by truncating after its cubic term in
$(3/2)^2$, and $J_1(4)\leq-2/45<0$ by truncating after its fourth such term.

Exact rational arithmetic: summing J1(3) terms m=0..3 gives 669/2048=(3/2)(223/1024)>0; summing J1(4) terms m=0..4 gives -2/45<0. Their next terms are respectively positive and negative, giving the asserted bound directions.

Exact rational summation independently gives J1's k=0..3 partial sum at 3 as 669/2048=(3/2)(223/1024), and its k=0..4 sum at 4 as -2/45. Alternating remainder signs produce the displayed inequalities and strict signs.

Exact rational arithmetic gives the n=3 partial sum at 3 as 669/2048=(3/2)(223/1024) and the n=4 partial sum at 4 as -2/45. Alternating remainders give the claimed lower and upper bounds respectively.

Exact arithmetic: the cubic sum at z=3 is 223/1024, multiplied by 3/2; the quartic sum at z=4 is -1/45, multiplied by 2. Alternating remainder signs give the stated lower/upper bounds.

The omitted terms alternate with decreasing magnitudes.

The magnitude ratio of successive J1 terms is (z/2)^2/((m+1)(m+2)). After the stated truncation indices it is strictly below one at z=3 and 4 and decreases with m; signs alternate.

The absolute successive term ratio is (z/2)^2/((k+1)(k+2)); after each displayed truncation this is below one and decreases, proving the alternating-tail bounds.

The successive term magnitude ratio is (z/2)^2/((k+1)(k+2)); beyond the specified truncations it is below one and decreasing for both z=3,4.

Successive absolute series terms have ratio (z^2/4)/((k+1)(k+2)); after each stated truncation it is below one and decreases, validating the alternating-tail estimates.

The intermediate value theorem gives the asserted zero.

J1 is continuous on [3,4] by its entire power series. Strict opposite endpoint signs invoke the intermediate value theorem and place the zero in the open interval.

Continuity on [3,4] follows from uniform series convergence there; opposite strict endpoint signs satisfy the intermediate value theorem and place the zero strictly inside.

The strict opposite-sign exact bounds and continuity on [3,4] meet every hypothesis of the intermediate value theorem.

Continuity of the entire power series on [3,4] and the strict endpoint signs satisfy every hypothesis of the intermediate value theorem.

If $J_0(\beta)=0$ also, the recurrence gives $J_1'(\beta)=0$.

At positive beta the recurrence has nonzero denominator. Substituting J1(beta)=0 and the hypothetical J0(beta)=0 makes J1'(beta)=0.

At a zero of J1, the identity J1'=J0-J1/z gives J1'(beta)=J0(beta), so simultaneous vanishing of J0 forces zero derivative as claimed.

Substitute J1(beta)=0 and hypothetical J0(beta)=0 into the coefficient-verified recurrence to obtain zero derivative.

The recurrence at the positive beta with J_1(beta)=0 gives J_1'(beta)=J_0(beta), so simultaneous vanishing yields both zero Cauchy data.

Uniqueness for the regular second-order Bessel ODE near $\beta>0$ would then
make $J_1$ identically zero, contrary to $J_1(z)=z/2+O(z^3)$.

On (0,infinity) the second-order Bessel equation has continuous regular coefficients. Zero value and derivative at beta force the zero solution throughout this connected interval, contradicting the first nonzero power-series term z/2.

On every interval bounded away from zero the Bessel ODE has continuous coefficients and the zero Cauchy data have a unique solution, by the linear integral equation and Gronwall. Propagation along (0,infinity) contradicts the nonzero leading term z/2.

At beta>0 the second-order Bessel ODE has continuous coefficients and unique solutions for value/derivative data. Zero Cauchy data force J1=0 on the connected positive axis, contradicting its nonzero first series coefficient.

At beta>0 the Bessel ODE coefficients are continuous and locally Lipschitz in the dependent variables; uniqueness gives zero on the connected positive axis, contradicting the series coefficient z/2 near zero.

Thus $J_0(\beta)\ne0$.

The contradiction in S0128 excludes a common positive zero, so division by beta J0(beta) in the profile is legitimate.

The contradiction rules out J0(beta)=0, so every denominator in the dipole formula is nonzero.

The contradiction excludes simultaneous roots; all later divisions by beta J0(beta) are legitimate.

The contradiction in S0127-S0128 proves the required nonzero denominator J_0(beta), making the later profile well defined.

In polar coordinates set
\[
 \psi(\rho,\theta)=
 \begin{cases}
 -\rho\sin\theta+\dfrac{2J_1(\beta\rho)}{\beta J_0(\beta)}\sin\theta,
 &\rho\leq1,\\
 -\sin\theta/\rho,&\rho\geq1,
 \end{cases}
 \qquad D=\nabla^\perp\psi.
\]

The denominator beta J0(beta) is nonzero by the preceding recurrence/ODE argument. At zero the interior profile equals x2[-1+1/J0(beta)-beta^2|x|^2/(8J0(beta))+O(|x|^4)], a convergent power series in Cartesian coordinates; this defines its removable value and all derivatives. The branches agree at rho=1, and the exterior formula is never used at zero.

Both branches give -sin(theta) at radius one. Near the origin J1(beta rho)sin(theta) is x2 times a convergent series in |x|^2, so the polar expression extends smoothly there. The stream function and its perpendicular gradient are therefore well-defined with the stated branch domains.

The polar expression has a removable origin singularity: J1(beta rho)sin(theta) is a convergent Cartesian series in x2(x1^2+x2^2)^k. Thus the interior stream function is analytic at zero, with value zero. At rho=1 the two branch traces agree by J1(beta)=0. D is the declared rotated gradient; no atom or puncture is inserted at the origin.

Both branches agree at rho=1 by J_1(beta)=0; the interior J_1(beta rho) sin(theta) has a smooth Cartesian extension at rho=0. The checked nonzero denominator and positive beta remove singular cases.

With $q=\psi+x_2$, $D-e_1=\nabla^\perp q$ and
$\curl D=-\beta^2q$ in $B_1$, while $\curl D=0$ outside.

grad-perp x2=-e1, giving D-e1=grad-perp q. In the disk q is a first angular-mode Bessel eigenfunction, so Delta q=-beta^2 q; Delta x2=0. Outside, -sin(theta)/rho is harmonic, hence curl D=Delta psi=0.

grad-perp x2=-e1 gives D-e1=grad-perp q. The angular-mode Bessel ODE gives Delta q=-beta^2 q in the disk, while Delta(-sin(theta)/rho)=0 outside; curl D=Delta psi.

nabla-perp x2=-e1 gives D-e1=nabla-perp q. The angular-mode Bessel ODE yields Delta q=-beta^2q in the disk, while the exterior -sin(theta)/rho is harmonic.

nabla-perp x_2=-e_1, so D-e_1=nabla-perp q. The Bessel radial eigenfunction has Delta q=-beta^2 q inside; the exterior -sin(theta)/rho is harmonic, giving zero exterior curl.

The relations $J_1(\beta)=0$ and $J_1'(\beta)=J_0(\beta)$ give the matching
values and first derivatives of the stream function.

Writing psi=f(rho) sin(theta), interior and exterior values at 1 are both -1; interior f'=-1+2J1'(beta)/J0(beta)=1 and exterior f'=1. Tangential first derivatives agree by differentiating the shared boundary trace.

At rho=1 the Bessel term vanishes. The interior radial derivative is -sin(theta)+2 J1'(beta)sin(theta)/J0(beta)=sin(theta), equal to the exterior derivative; tangential traces also agree.

At rho=1 the interior value is -sin(theta), and its radial derivative is (-1+2J1'(beta)/J0(beta))sin(theta)=sin(theta), equal to the exterior derivative. Tangential first derivatives follow by differentiating the common trace.

At rho=1 the streamfunction is -sin(theta) from both sides. Interior radial derivative -sin(theta)+2J_1'(beta)sin(theta)/J_0(beta)=sin(theta), matching the exterior derivative; angular derivatives follow from the common trace.

The Bessel equation also gives $J_1''(\beta)=-J_0(\beta)/\beta$,
so the second radial derivatives match.

At a root, beta^2 J1''+beta J1'=0, yielding J1''=-J0/beta. Interior f''=2beta J1''/J0=-2, exactly the exterior second derivative of -rho^-1 at rho=1.

The ODE at J1(beta)=0 gives beta^2 J1''+beta J1'=0. Thus the interior second radial derivative is -2 sin(theta), equal to differentiating -sin(theta)/rho twice at one.

At the root the ODE gives J1''(beta)=-J1'(beta)/beta=-J0(beta)/beta. The interior second derivative is therefore -2sin(theta), matching the exterior second radial derivative.

Bessel's equation at its zero gives beta^2 J_1''+beta J_1'=0. Thus the interior second radial derivative is -2 sin(theta), equal to the exterior second radial derivative at one.

Tangential differentiation of the matching traces gives the other second
derivatives; hence the velocity is continuously differentiable across the circle.

Differentiating the common value and normal-derivative traces tangentially yields the tangential-tangential and normal-tangential second derivatives. Polar-to-Cartesian coefficients are smooth near rho=1, so psi is C2 there and D is C1 across the circle.

Differentiating the common value and common radial-derivative traces tangentially gives the remaining Hessian traces. Since the interface is away from the polar origin, Cartesian Hessians agree, making D continuously differentiable across it.

Differentiating the equal trace and equal first radial derivative tangentially matches pure tangent and mixed Hessian components; together with radial second derivatives this makes psi C2 and D C1 across the circle.

Differentiating common value and radial-derivative traces tangentially matches the mixed and tangential second derivatives; the Cartesian coordinate transformation is smooth near rho=1, so psi is C2 and D is C1.

In particular $q=0$ on
the circle and $(D-e_1)\cdot\nabla\curl D=0$ distributionally.

q vanishes on the circle by J1(beta)=0. Inside, (grad-perp q) dot grad(-beta^2 q)=0; outside curl is zero. Since q=0 at the boundary, differentiating its zero extension produces no boundary measure, proving the distributional identity.

q has zero trace at rho=1 and curl D=-beta^2 q there, with zero exterior curl. Inside, grad-perp q dot grad q=0. Integration by parts across the circle has no jump of curl, proving the distributional transport assertion.

q vanishes at rho=1, and interior vorticity is -beta^2q, so (nabla-perp q) dot grad q=0. Its continuous zero boundary trace prevents an interface delta; exterior vorticity is zero.

q vanishes on the boundary. Inside, curl D=-beta^2 q and nabla-perp q dot nabla q=0; outside curl is zero. Its matching zero trace precludes a boundary delta, proving the distributional transport identity.

 The pressure
\[
 P=-\tfrac12|D-e_1|^2-\tfrac{\beta^2}{2}q^2\mathbf1_{B_1}
\]
gives $-\partial_1D+\Div(D\otimes D)+\nabla P=0$.

Set w=D-e1. Then w-perp=-grad q and curl w=-beta^2 q in the disk, so convection is grad(|w|^2/2+beta^2 q^2/2). Outside curl vanishes. The stated negative potential therefore cancels -partial_1 D+div(D tensor D), with no boundary jump because q=0 and D is C1.

Set w=D-e1=grad-perp q. Then w-perp=-grad q and curl w=-beta^2 q, so (w dot grad)w=grad(|w|^2/2+beta^2 q^2/2) inside. Outside the beta term vanishes. The stated negative pressure cancels this acceleration exactly.

With w=D-e1, curl w=-beta^2q in the disk and w-perp=-grad q. Thus w dot grad w=grad(|w|^2/2+beta^2 q^2/2) inside and grad(|w|^2/2) outside, canceled by the displayed P.

Set w=D-e_1. Then (w dot nabla)w=beta^2 q nabla q+grad(|w|^2/2) inside and only the latter outside; the displayed negative pressure cancels both. q=0 and C1 matching eliminate interface terms.

 This follows from
$(w\cdot\nabla)w=(\curl w)w^\perp+\nabla(|w|^2/2)$, and matching on the
circle leaves no boundary distribution.

Component expansion gives (w dot grad)w-(curl w)w-perp=grad(|w|^2/2), with the stated plus sign for the chosen perpendicular convention. The matched velocity and continuous q^2 extension ensure the distributional derivative contains no surface delta.

Component expansion gives (w dot grad)w-grad(|w|^2/2)=curl(w)(-w2,w1). The fields and relevant pressure traces match, and q vanishes at the interface, so distributional first derivatives add no boundary measure.

Component expansion verifies the stated vector identity with the manuscript's orientation. D is C1 and q=0 on the interface, so piecewise pressure and velocity terms introduce no boundary distribution.

Component expansion gives (w dot nabla)w=(curl w)w-perp+grad(|w|^2/2), with the stated rotation sign. C1 velocity and continuous pressure traces yield no spatial boundary distribution.

 Outside the unit disk,
$D=\nabla(-x_1/|x|^2)$; $D$ is even under $x\mapsto-x$.

Direct differentiation gives both exterior expressions as ((x1^2-x2^2),2x1x2)/|x|^4. The stream function is odd under inversion, so its rotated gradient is even in both core and exterior.

Differentiating -x1/|x|^2 gives ((x1^2-x2^2),2x1x2)/|x|^4, identical to grad-perp(-x2/|x|^2). The stream function is odd under inversion, so D is even in both regions.

Differentiating -x2/|x|^2 with nabla-perp gives grad(-x1/|x|^2). The stream function is odd under inversion on both branches, hence its rotated gradient is even.

Differentiating -x_1/|x|^2 gives ((x_1^2-x_2^2),2x_1x_2)/|x|^4, equal to nabla-perp(-x_2/|x|^2). This is even; the interior streamfunction is odd, so its rotated gradient is also even.

For a unit direction $\xi\in\mathbb S^1$, rotate the velocity and pressure profiles together, and write $D_r(x)=r^{-1}D(x/r)$ and $P_r(x)=r^{-2}P(x/r)$.

The direction is now explicitly unit length. Rotate velocity by the rotation matrix and pressure by composition with its inverse; this preserves Euler and the unit translation speed. For r>0 the explicitly defined D_r=r^-1 D(x/r) and P_r=r^-2 P(x/r) make convection, pressure gradient and r^-1 directional derivative all scale as r^-3.

Proper planar rotation sends e1 to any unit xi and preserves divergence and tensor equations. Scaling velocity by r^-1 and pressure by r^-2 makes every equation term scale as r^-3; translations later retain the rotated profiles consistently.

The direction is explicitly unit length, so an orthogonal rotation maps e1 to xi and transports velocity and scalar pressure together. The current definition P_r=r^-2 P(x/r), along with D_r=r^-1 D(x/r), makes the three terms in the traveling equation all scale r^-3. Rotations preserve the subsequent norms, parity, divergence and curl identities.

For a rotation O with O e1=xi, define the rotated velocity by O D(O^T x) and the rotated pressure by P(O^T x); unit length is explicitly stipulated. D_r=r^-1 D(x/r) and P_r=r^-2 P(x/r) then give the same r^-3 factor in the translating, quadratic-divergence and pressure-gradient terms. Orthogonal rotations preserve the norms.

The direction is suppressed where no confusion arises.

The suppressed direction is fixed in each of the formulas and restored by rotation; the later construction uses exactly four specified unit directions.

The suppressed direction refers to the fixed rotation just defined; each use occurs within a specified direction slot and introduces no change of profile.

The suppressed direction is the fixed unit scanning direction; rotation invariance leaves all profile constants unchanged.

The suppressed direction is always the active one of the four fixed unit directions; no later formula changes its value during a visit.

 Then
\begin{equation}\label{eq:dipole}
 -r^{-1}\partial_\xi D_r+\Div(D_r\otimes D_r)+\nabla P_r=0,
 \quad \Div D_r=0,
\end{equation}
and
\begin{equation}\label{eq:dipole-norms}
 \norm{D_r}2=C_2,\quad \norm{\curl D_r}1=C_\omega,\quad
 \norm{D_r}{3/2}=C_{3/2}r^{1/3},\quad \norm{\nabla D_r}1=C_\nabla.
\end{equation}

The explicitly defined pressure P_r=r^-2 P(x/r) has gradient scaling r^-3, as do convection and r^-1 partial_xi D_r. Change of variables gives ||D_r||p=r^(2/p-1)||D||p; a spatial derivative costs r^-1. Thus L2, curl L1 and gradient L1 are invariant, and the L3/2 exponent is 1/3.

The translating equation scales by r^-3, and divergence remains zero. Change of variables gives ||D_r||p=r^{2/p-1}||D||p and ||grad D_r||1=||grad D||1, hence all four displayed norm scalings and the curl scaling.

The scaled pressure is P_r(x)=r^-2P(x/r); all equation terms scale r^-3. For p, ||D_r||p=r^(2/p-1)||D||p, and first derivatives scale r^-2; these give exactly the four norm formulas.

The explicit jointly rotated definitions in S0139 give r^-3 times the unit-profile equation and divergence zero. In dimension two, ||r^-1 D(x/r)||_p=r^(2/p-1)||D||_p; curl and gradient scale as r^-2. Thus L2, curl-L1 and gradient-L1 are invariant and L^(3/2) scales as r^(1/3); finiteness follows from the matched core and r^-2 exterior decay.

The constants are finite by boundedness in the core and the exterior
$|x|^{-2}$ decay; the curl is supported in $B_r$.

The profile is bounded and C1 on its core; the tail velocity is O(|x|^-2), with gradient O(|x|^-3), giving the required integrals for p=2,3/2 and gradient L1. Curl vanishes outside the radius-r disk by harmonicity and the matched boundary.

D and grad D are bounded in the core; outside, D=O(rho^-2) and grad D=O(rho^-3). These are integrable in each claimed norm, and the harmonic exterior has zero curl.

The core is bounded, exterior D is O(|x|^-2), exterior grad D is O(|x|^-3), and curl vanishes outside the disk. Polar integration verifies every norm used at p=2,3/2 and derivative L1.

Core D and nabla D are bounded; the explicit exterior derivative of a degree -1 potential gives D=O(rho^-2), nabla D=O(rho^-3). These are in the claimed spaces, and exterior harmonicity confines curl to the closed radius-r disk with zero boundary trace.

\subsection{A symmetric globally integrable anti-divergence}

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For a vector field $f$ supported in $B_R$, with
\[
 \int f=0,\qquad \int x_jf_i(x)\dd=0\quad\hbox{for every }i,j,
\]
define
\[
 (\mathcal Af)_{ij}=\partial_i\Delta^{-1}f_j+
 \partial_j\Delta^{-1}f_i-\delta_{ij}\Div\Delta^{-1}f.
\]

The integral conditions are Lebesgue moments of a compactly supported integrable vector field. Delta^-1 is convolution with (2pi)^-1 log|x|; the first derivatives in the formula are locally integrable kernels. Every index ranges over 1,2 and the result is a two-by-two tensor.

For integrable compactly supported f with the stated finite moments, convolution with derivatives of log|x| defines the operator locally. The formula is a symmetric two-by-two tensor and its moment conditions are exactly those used for its global L1 estimate.

For integrable f of compact support the two moments are finite; convolution with grad G defines the displayed symmetric tensor as a locally integrable distribution. Its moment hypotheses are exactly those needed for global L1 below.

The Newtonian convolution formula is defined for the compact L1 zero-moment inputs actually used below. Both vector mean and each component first moment are explicitly required, exactly the cancellations needed for integrable far field.

It is symmetric and $\Div\mathcal Af=f$.

Swapping i,j leaves the formula invariant. Summing partial_j gives partial_i Delta^-1 div f+f_i-partial_i Delta^-1 div f=f_i in distributions; the first and third terms cancel exactly.

Interchanging i and j preserves the expression. Taking partial_j and summing produces partial_i Delta^{-1}Div f+f_i-partial_i Delta^{-1}Div f=f_i.

Swapping i,j leaves the definition unchanged. Summing partial_j gives partial_i div Delta^-1 f+f_i-partial_i div Delta^-1 f=f_i, including all signs.

Interchanging i,j leaves the tensor invariant. Summing partial_j gives partial_i div Delta^-1 f+f_i-partial_i div Delta^-1 f=f_i, proving the right inverse with the declared tensor convention.

 Its kernel has degree $-1$.

Each kernel entry is a linear combination of first derivatives of G, of degree -1 in dimension two; no second-order singular integral is needed here.

Each kernel term is a first derivative of the two-dimensional logarithmic fundamental solution and is homogeneous of degree -1.

Each tensor-kernel component is a linear combination of first derivatives of log|x|, homogeneous of degree -1 in two dimensions.

Each kernel component is a first derivative of the logarithmic Green function, hence homogeneous of degree -1 away from zero; its singularity is locally integrable.

On $B_{3R}$, integration of that kernel gives $CR\norm f1$.

For support y in B_R, the integral over x in B_3R of |x-y|^-1 is bounded by its integral over B_4R, namely O(R). Tonelli yields C R ||f||1 uniformly in y.

For y in B_R and x in B_3R, the difference x-y lies in B_4R; integral of |x-y|^-1 over that ball is C R. Fubini gives the local bound.

For y in B_R, the translated integration set lies in B_4R; integral of C/|z| there is CR. Tonelli yields the local L1 bound uniformly for signed f.

For y in B_R, integral over x in B_3R of |x-y|^-1 is <=C R. Tonelli then gives the local L1 bound C R||f||_1.

 Outside $B_{3R}$,
subtract its constant and linear Taylor terms using the two moment conditions;
the remainder is at most $CR^2|x|^{-3}\norm f1$.

For |x|>3R, Taylor-expand the degree-minus-one kernel at x through first order in -y. The mean and all first moments cancel; its second derivatives are O(|x|^-3) along the segment, leaving C R^2 |x|^-3 ||f||1.

Taylor expansion of the degree -1 kernel at x has second derivative O(|x|^-3) when |y|<=R and |x|>3R. Zeroth and first moment cancellation removes the first two terms, leaving C R^2 |x|^-3||f||1.

Taylor expansion T(x-y)=T(x)-DT(x)y+O(R^2|x|^-3) is uniform when |x|>3R; the two prescribed moments kill the first two terms componentwise.

For |x|>3R, expand the degree -1 kernel at x through linear order in y. The mean and component first moments cancel both terms; second derivatives are O(|x|^-3), giving C R^2 |x|^-3||f||_1.

 Integration gives
\begin{equation}\label{eq:anti}
 \norm{\mathcal Af}1\leq CR\norm f1.
\end{equation}

The far radial integral is R^2 integral_3R^infinity rho^-3 rho d rho=O(R); adding the near estimate gives ||Af||1<=C R ||f||1 with a universal kernel constant.

Integrating the far bound gives C R^2 integral_3R^infinity rho^-2 d rho=C R; adding the local estimate proves the full L1 inequality.

The outer integral is CR^2 integral_3R^infty rho^-2 d rho=CR; adding the core estimate proves the norm bound with a universal kernel constant.

Polar integration of R^2 rho^-3 over rho>3R is C R; adding S0147 proves the global estimate with a dimension/kernel constant independent of f and R.

The output need not be compactly supported.

Convolution with the noncompact degree-minus-one kernel generally produces a nonzero algebraic tail even for compact input; the moment conditions improve its decay, not its support.

The inverse-Laplacian derivative kernels have nonzero tails in general; the moment cancellation improves decay but does not make those tails vanish identically.

The nonlocal Newtonian inverse can retain a nonzero second-moment tail even when f is compact; compact support of the output was not used.

The construction is convolution with a noncompact Green kernel; the moment cancellations ensure integrability, not compact support, and later arguments consistently retain that tail.

 Cancellation of only the mean
would leave a generally nonintegrable $|x|^{-2}$ far field.

After canceling only the zeroth moment, the leading derivative-kernel term has degree -2 and is generically nonzero when a first moment is nonzero. Its absolute radial integral has logarithmic divergence in dimension two.

With only zeroth moments removed, the leading Taylor derivative has degree -2 and a nonzero first-moment coefficient in general. Its absolute radial integral is integral dr/r, which diverges.

With only mean cancellation the leading tail is DT(x) applied to first moments, of degree -2. A generic nonzero first-moment choice gives nonzero angular magnitude and logarithmically divergent planar L1.

If only mean vanishes, the nonzero first-moment term is a derivative of the degree -1 kernel, generally rho^-2; its absolute radial integral is logarithmically divergent in dimension two.

Differentiation of the profile gives
\[
 \partial_rD_r=r^{-1}D_r-\Div(D_r\otimes x/r).
\]

Differentiating r^-1 D(x/r) gives -r^-1 D_r-r^-1(x dot grad)D_r. Tensor divergence of D_r tensor (x/r) is r^-1[(x dot grad)D_r+2D_r]; subtracting it from r^-1 D_r gives precisely the same expression.

Direct differentiation gives partial_r D_r=-r^-2(D+y dot grad D), y=x/r. Expanding Div(D_r tensor x/r)=r^-2(2D+y dot grad D) proves the stated identity, including the extra r^-1 D_r term.

Differentiate r^-1D(x/r) to get -r^-2(D+y dot grad D). Meanwhile div(D_r tensor x/r)=r^-2(2D+y dot grad D), giving exactly the displayed identity.

Differentiate r^-1 D(x/r) to get -r^-2(D+y dot nabla D). Expanding div(D_r tensor x/r)=(x dot nabla D_r+2D_r)/r yields exactly the displayed relation.

The vector $f_r=\Div(D_r\otimes x/r)$ is supported in $B_r$, because the
exterior dipole is homogeneous of degree $-2$; $\norm{f_r}1\leq C$.

Outside B_r, homogeneity gives (x dot grad)D_r=-2D_r, so the tensor divergence vanishes. In the core f_r=r^-2 f_1(x/r), whose L1 norm is scale invariant and finite from C1 regularity.

Euler's homogeneous identity gives 2D_r+x dot grad D_r=0 outside B_r. Inside f_r scales as r^-2 times a bounded fixed profile, so its L1 norm is constant. C1 matching prevents an interface distribution.

Outside the core Euler homogeneity gives 2D+y dot grad D=0. Inside, f_r is r^-2 times a fixed integrable function of x/r, so its L1 norm is radius independent; matching eliminates interface distributions.

Outside B_r, homogeneity gives x dot nabla D_r=-2D_r and hence f_r=0, with no interface delta since D is C1. In the core f_r=r^-2 f_1(x/r), giving r-independent L1 norm.

Evenness gives all first moments zero.

D_r is even, D_r tensor x is odd, and its divergence f_r is even. Thus x_j f_{r,i} is odd on its centered compact support, giving all first moments zero.

D_r is even; differentiating its odd product with x gives even f_r. Each x_j f_{r,i} is therefore odd and compactly supported, with integral zero.

D is even, D tensor x is odd, and its divergence is even. Multiplication by x_j makes each first-moment integrand odd on symmetric compact support.

D_r is even and x is odd, so div(D_r tensor x/r) is even; multiplication by x_j is odd on its symmetric compact support, making every first moment zero.

 Its mean is zero by the boundary flux
on any circle outside $B_r$, since the angular mean of the exterior dipole
is zero.

Integrate the divergence on a radius-R circle with R>r: the flux is (R/r) integral D_r dS. The exterior angular field has components proportional to cos(2theta), sin(2theta), each with zero integral; hence the vector mean vanishes.

The total flux is integral D_r (x dot n)/r on a circle. The exterior dipole angular components are proportional to cos(2theta),sin(2theta), whose means vanish; hence the vector mean of f_r is zero.

The divergence integral equals the outer-circle flux proportional to the angular integral of exterior D=(cos2theta,sin2theta)/rho^2, which is exactly zero.

On radius R>r the flux is (R/r)D_r times arclength; the exterior components are r R^-2(cos2theta,sin2theta), whose angular integrals vanish. The divergence theorem therefore gives integral f_r=0.

 Therefore $F_r=-\mathcal Af_r$ satisfies
\begin{equation}\label{eq:radial}
 \partial_rD_r=r^{-1}D_r+\Div F_r,
 \qquad \norm{F_r}1\leq Cr.
\end{equation}

Both moments are available, so the anti-divergence satisfies div F_r=-f_r and ||F_r||1<=C r ||f_r||1<=Cr. Substitution into the differentiated profile identity gives the stated plus div F_r sign.

The anti-divergence gives Div(-A f_r)=-f_r and ||A f_r||1<=C r||f_r||1<=Cr. Substituting into S0152 proves both assertions.

Set F_r=-A f_r; divergence is -f_r and the radius-R estimate with R=r gives ||F_r||1<=Cr. Substitution into S0152 gives the claimed radial derivative.

All hypotheses of S0149 hold for f_r. Taking F_r=-A f_r changes -div(D_r tensor x/r) into +div F_r and gives ||F_r||_1<=C r||f_r||_1<=C r.

\subsection{Variable radius and a compact source}

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On each visit take $r,\eta,X$ to be $C^1$ in time, with $r(t)>0$, $\eta(t)\geq0$, and $X'=\eta\xi/r$, and write $T_X f(x)=f(x-X)$.

The current hypotheses explicitly give C1 time paths on every visit, r>0, eta>=0 and X'=eta xi/r. Thus division and the following chain/product rules are legitimate on visits. T_X is explicitly the spatial translation f(x-X); the later construction pastes visits by zero velocity traces.

On each visit the C1 functions with strictly positive radius make the scaling and translation differentiable; nonnegative amplitude and X'=eta xi/r match the translating-dipole speed. T_X consistently means f(x-X).

The current source explicitly assumes r, eta and X are C1 on each visit, with positive radius and X'=eta xi/r. On compact visit subintervals r is bounded below, so the chain rule for translated and rescaled C1 profiles is valid. The later constructed visits meet these hypotheses through smooth positive coefficients and flat ramps. Resets occur only between zero-amplitude visits; no continuity of a reset center is required for the physical field.

The current hypotheses explicitly require C1 time dependence of r, eta and X, r>0, eta>=0 and Xprime=eta xi/r. Together with the C1 spatial dipole these justify the packet chain rule on each visit. The declared translation T_X f(x)=f(x-X) fixes the sign of its derivative. Later visits have smooth parameters, and their zero endpoint traces justify separate pasting.

 By
\eqref{eq:dipole}--\eqref{eq:radial}, the packet
$v(t,x)=\eta(t)D_{r(t)}(x-X(t))$ satisfies, after adding gradients to pressure,
\begin{equation}\label{eq:packet}
 \partial_tv+\Div(v\otimes v)+\nabla p
 =(\eta r)'T_XD_r/r+\Div(\eta r'T_XF_r).
\end{equation}

Direct differentiation gives eta' T_XD_r+eta r' T_X partial_rD_r-eta^2/r T_X partial_xiD_r. The last term and self-convection cancel up to the scaled pressure by the dipole equation. Substitution of partial_rD_r=D_r/r+div F_r leaves (eta r)' T_XD_r/r+div(eta r' T_XF_r), with both translations now explicit.

Differentiate eta T_X D_r: amplitude and radial terms combine as (eta r)'T_X D_r/r+Div(eta r'T_X F_r). The motion contributes -eta^2 partial_xi D_r/r, which cancels the nonlinear and pressure terms by S0141.

Direct differentiation gives eta' T_X D_r + eta r' T_X partial_r D_r - eta X' dot T_X grad D_r. Using X'=eta xi/r cancels the nonlinear dipole translation term after adding eta^2 T_X P_r to pressure. The radius identity then leaves (eta r)' T_X D_r/r + Div(eta r' T_X F_r). T_X commutes with spatial divergence; the source now displays both translations explicitly. No time derivative of F_r is taken.

Differentiate eta T_X D_r: eta-prime T_X D_r+eta r-prime T_X partial_r D_r-eta Xprime dot T_X grad D_r. The translation term cancels the nonlinear dipole term because Xprime=eta xi/r. Substituting partial_r D_r=D_r/r+div F_r combines the first two scalar coefficients into (eta r)prime/r. Spatial divergence commutes with T_X, so both displayed right-hand profiles are translated as written; the scaled dipole gradient is absorbed into pressure.

Choose a radial cutoff $\chi$ equal to zero on $B_1$ and one outside $B_2$,
let $\alpha=1/5$, and put
\[
 \Phi_\xi(x)=-\xi\cdot x/|x|^2,\quad
 \Pi_r(x)=r\chi(x/r^\alpha)\Phi_\xi(x),\quad
 Z_r=D_r-\nabla\Pi_r.
\]

A smooth radial cutoff with the specified constant regions exists. Phi is defined away from zero, and its cutoff product extends smoothly by zero near zero. Since the exterior D_r equals r grad Phi, subtracting grad Pi removes exactly that tail beyond 2r^alpha.

A smooth radial cutoff with the two stated constant regions exists. Phi is odd and homogeneous -1 away from zero, but the zero cutoff removes its singularity. Pi_r is smooth near zero and Z_r is a well-defined localized remainder.

A smooth radial cutoff exists; the exterior dipole is r grad Phi_xi. The cutoff removes its singularity near the origin and defines an even compact remainder once r<r^(1/5).

A smooth radial cutoff exists. The odd potential Phi is multiplied by a radial cutoff vanishing near zero, so Pi is smooth there; for small r its exterior gradient equals the dipole tail, with alpha=1/5 fixed for later scale separation.

For small $r$, $Z_r$ is even and supported in $B_{2r^\alpha}$; direct radial
integration gives $\norm{Z_r}1\leq Cr(1+|\log r|)$.

For sufficiently small r, the core radius r is below r^alpha. Both D_r and grad Pi are even and agree outside 2r^alpha. The core integral is O(r), the annulus r<rho<2r^alpha contributes O(r integral d rho/rho), and the cutoff-derivative shell contributes O(r).

For r<1, r<=r^{1/5}; outside 2r^{1/5}, grad Pi_r= r grad Phi=D_r. Both vector terms are even. The core costs Cr, the annulus integral of r/rho^2 costs Cr|log r|, and the cutoff derivative contributes Cr.

For small r, outside 2r^(1/5) the two gradients coincide. The core contributes O(r), the annulus from r to r^(1/5) contributes Cr integral d rho/rho, and cutoff derivative annulus contributes O(r). Odd potentials have even gradients.

For r<1, r<r^alpha and outside 2r^alpha both tails coincide. Pi is odd, so its gradient and Z are even. Core L1 cost is O(r); integrating r rho^-2 from r to 2r^alpha costs O(r|log r|), including the cutoff annulus.

 Its integral is
$2\pi r\xi$: on a large circle the respective boundary integrals for $D_r$
and $\nabla\Pi_r$ are $\pi r\xi$ and $-\pi r\xi$.

For xi=e1, integrate D_r=grad-perp psi_r on a large radius-R circle: psi_r=-r sin(theta)/R gives integral D_r=(pi r,0). On that circle Pi=-r cos(theta)/R, and integral grad Pi=integral_boundary Pi n=(-pi r,0). Subtraction gives 2pi r e1; rotations give 2pi r xi. The negative gradient boundary integral is essential.

For xi=e1, boundary integration of grad-perp psi gives pi r e1, while the gradient of -r x1/|x|^2 gives -pi r e1 through its scalar boundary values. Their difference is 2pi r e1; rotation gives the stated xi vector.

For R beyond the cutoff, the outer-circle integral of D_r is pi r xi. On the same circle Pi_r(Rn)=-r xi dot n/R, so integral Pi_r n ds=-pi r xi. Subtracting gives 2pi r xi for the compactly supported difference Z_r. This uses boundary integrals, not nonexistent separate absolute integrals of the two long-range fields.

For xi=e1 and a large circle of radius R, the exterior streamfunction gives integral_B D_r=pi r e1. Since Pi_r=-r cos(theta)/R on that circle, integral_B grad Pi_r=integral n Pi_r R dtheta=-pi r e1. Their compact difference therefore has integral 2pi r e1. Simultaneous rotation gives the two fluxes and difference for every unit xi.

Thus $S_r=Z_r/r$ has integral $2\pi\xi$ and all first moments zero.

Division by positive r gives mean 2pi xi. The compact field Z_r is even, so its first moments, and those of S_r, vanish by oddness.

Division by r yields mean 2pi xi. Since Z_r is even and compactly supported, each first moment of Z_r/r vanishes by oddness.

Divide the compact remainder's integral by r; its parity makes all first moments vanish. Those moments are absolutely defined because the remainder is compactly supported.

Dividing S0162 by r gives mean 2pi xi. Z is even and compactly supported, so every first moment vanishes by odd integration; dividing by positive r preserves it.

Fix a nonnegative radial smooth probability bump $U\in C_c^\infty(B_1)$ and define $U_H(x)=H^{-2}U(x/H)$, supported in $B_H$.

The source now fixes U in C_c-infinity(B1), nonnegative, radial, with integral one, and defines U_H=H^-2 U(x/H) for positive H. Change of variables gives unit mass and support in B_H; radial symmetry cancels first moments. All later derivative/scaling constants are consequently norms of this one fixed profile.

Normalize a nonnegative radial smooth bump of compact support inside B1. The H^-2 dilation has mass one, retains radial symmetry and smoothness, and its support lies in B_H.

The source now fixes one U in C_c^infty(B1), radial, nonnegative and of unit integral, and expressly sets U_H=H^-2 U(x/H). Substitution proves support in B_H, mass one, evenness and zero first moments. The fixed profile gives ||grad U_H||1=H^-1||grad U||1 and all later scaling constants independently of H. A bump concentrated on H^2 violates this stated definition.

A nonnegative radial C-infinity function compactly supported in B1 can be normalized to unit integral and then fixed once. Its explicitly stated dilation H^-2 U(x/H) is nonnegative, radial, smooth, unit mass and supported in B_H for H>0. Change of variables gives ||grad U_H||1=H^-1||grad U||1 and first moment at most H; the fixed choice supplies uniform cross-scale constants in quadrature and projection estimates.

If $2r^\alpha\leq H$, the difference $S_r-2\pi U_H\xi$ has its zeroth and
first moments zero.

The radius condition puts both terms in B_H. The mean cancels because integral S_r=2pi xi and integral U_H=1; radiality makes the bump's first moment zero and evenness gives the same for S_r.

The support hypothesis puts both terms in B_H. Their means are both 2pi xi, and both even profiles have zero first moments, proving all cancellations.

Both terms are supported within B_H; their masses are 2pi xi and their first moments vanish by radial/even symmetry. Translating the zero-mean difference preserves its zero first moments.

The source difference is supported in B_H when 2r^alpha<=H. Its mean is 2pi xi-2pi xi=0; both summands are even and have zero first moments, so all hypotheses of S0149 are available.

 Equation~\eqref{eq:anti} gives
\[
 \norm{\mathcal A(S_r-2\pi U_H\xi)}1\leq CH(1+|\log r|).
\]

Applying the anti-divergence estimate with support radius H yields C H times the input L1 norm. The latter is <=C(1+|log r|)+2pi by S0161 and the unit-mass bump, giving the displayed bound.

The compact source has L1 norm <=C(1+|log r|) and support radius H; applying S0149 with the established zeroth and first moments gives exactly CH(1+|log r|).

Apply the verified anti-divergence bound with support radius H and L1 input bounded by C(1+|log r|)+2pi; this yields CH(1+|log r|).

Apply S0149 with support radius H and input norm <=||Z_r||_1/r+2pi||U_H||_1<=C(1+|log r|); this gives the stated H factor without invoking an L1 singular-integral bound.

Equation~\eqref{eq:packet} therefore becomes
\begin{equation}\label{eq:packet-source}
 \partial_tv+\Div(v\otimes v)+\nabla p=\mathcal U+\Div E,
 \qquad \mathcal U=2\pi(\eta r)'U_H(x-X)\xi,
\end{equation}
where
\begin{equation}\label{eq:E-bound}
 E=\eta r'T_XF_r+(\eta r)'T_X\mathcal A(S_r-2\pi U_H\xi),\quad
 \norm E1\leq C|\eta r'|r+C|(\eta r)'|H(1+|\log r|).
\end{equation}

Use T_XD_r/r=T_XS_r+grad(T_XPi_r/r). Absorb the latter gradient into pressure. The zero-moment decomposition S_r=2pi U_H xi+div A(S_r-2pi U_H xi), followed by translation, gives the displayed E=eta r' T_XF_r+(eta r)' T_X A(S_r-2pi U_H xi). Translation is an L1 isometry, so the two stated norm terms follow with exactly their coefficients.

Replace D_r/r by S_r+grad(Pi_r/r), absorbing the gradient in pressure. Split S_r into 2pi U_H xi plus a zero-moment remainder, represented by Div A. Translate both stresses by X; triangle inequality and S0156,S0166 give the stated bound with the correct coefficients.

Write T_X D_r/r=T_X S_r+grad(T_X Pi_r/r), absorb its scalar gradient into pressure, then write S_r=2pi U_H xi+Div A(S_r-2pi U_H xi). This yields exactly Ucal and the two translated symmetric terms in E. Translations preserve their L1 norms. The anti-divergence bound gives C|(eta r)'|H(1+|log r|), and ||F_r||1<=Cr gives C|eta r'|r. The signs and every translation agree with direct differentiation.

Translate the decomposition D_r/r=S_r+grad(Pi_r/r) and div A(S_r-2pi U_H xi)=S_r-2pi U_H xi. Its gradient goes into pressure. This yields source 2pi(eta r)prime U_H(x-X)xi and exactly the two translated stress terms shown. Translation preserves L1; ||F_r||1<=Cr and the zero-moment anti-divergence estimate <=CH(1+|log r|) prove both terms of the bound. No time derivative is applied to the stress.

The endpoint estimate $\norm{\curl v}1=|\eta|C_\omega$ is independent of $r$.

Curl of eta D_r(x-X) has L1 norm |eta| times the scale-invariant curl norm. Neither translation nor positive radius affects it.

Curl differentiates the dilated velocity once, and two-dimensional integration cancels r^-2. Translation changes no norm; multiplication by eta gives |eta|C_omega.

Translation preserves norms and curl D_r scales r^-2 on area r^2; multiplying by eta gives |eta|C_omega, independent of all concentration radii.

Translation is an L1 isometry and curl(eta D_r)=eta curl D_r. The critical derivative scaling in S0141 gives exactly |eta| C_omega, independent of radius.

\section{Two-scale raster and local stress repair}\label{sec:repair-interface}

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\subsection{Stress coefficients and a fixed cycle}

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For a prescribed symmetric stress piece $S$, use the directions
$e_1,e_2,(e_1+e_2)/\sqrt2,(e_1-e_2)/\sqrt2$.

All four directions are unit vectors. Their rank-one symmetric tensors span the three-dimensional symmetric matrix space, with the paired diagonal directions providing opposite off-diagonal entries.

The four listed vectors are unit vectors; their rank-one symmetric tensors span the symmetric two-by-two tensors modulo an isotropic scalar, as verified by the coefficient expansion.

The four listed unit directions have tensor products spanning all symmetric 2x2 matrices modulo an isotropic term; the explicit nonnegative decomposition below verifies the required cone representation.

All four vectors are unit vectors; their rank-one tensors span symmetric 2x2 tensors modulo an isotropic part. S0176 checks the sign of each off-diagonal contribution.

For a fixed smooth compactly supported tensor $S$, choose a nonnegative
compact smooth function $h$ equal to one on a neighborhood of its support.

The compact support of S has a compact neighborhood admitting a smooth nonnegative cutoff equal to one there. It can be chosen compactly supported, allowing small regularization mass without losing smoothness near zeros.

Compact support of S admits a larger smooth compact cutoff equal to one on a neighborhood. Nonnegativity is obtainable by a standard radial cutoff enclosing the compact set.

Compact support has a bounded open neighborhood with a smooth nonnegative cutoff equal to one there; this separates every possible zero of h from support of S.

A compactly supported smooth tensor has compact support; a smooth nonnegative cutoff equal to one on a larger neighborhood exists and can be fixed before gamma and all fine scales.

For $\gamma>0$ put
\[
 A=\sqrt{S_{12}^2+\gamma^2h^2},\qquad
 B=\sqrt{S_{11}^2+S_{22}^2+\gamma^2h^2}.
\]

Both radicands are nonnegative and gamma is strictly positive. On the support neighborhood of S they are bounded below by gamma^2; outside it S vanishes, making the square-root branches simply gamma h.

For gamma>0 the two nonnegative square roots are defined pointwise. They are strictly away from zero on the support of S because h=1 there; the second is the Euclidean norm of three components.

The radicands are nonnegative. Where S may be nonzero h=1 and gamma>0, so the principal square roots have strictly positive radicands; elsewhere their smoothness is checked next.

The nonnegative square roots are well defined. On the support of S, h=1 and gamma>0 prevents a zero radicand; away from that neighborhood S=0, giving the smooth extension gamma h.

These functions are smooth: near the zero set of $h$, the entries of $S$
already vanish, so both square roots reduce to $\gamma h$.

Near every zero of h the tensor S vanishes identically because h equals one near supp S. There the nonnegative square roots equal gamma h; elsewhere positive radicands have smooth square roots. These local descriptions cover the plane.

A neighborhood of the zero set of h is disjoint from supp S. There the roots equal gamma h, since h>=0; elsewhere their radicands are positive. These local formulas prove smoothness even at the zero set.

Every point where h=0 has a neighborhood outside support S because h=1 on a neighborhood of that compact support. There both radicals equal gamma h as h>=0; where h>0 the radicals are smooth by composition.

At every zero of h a neighborhood is disjoint from support S, so the radicands equal gamma^2 h^2 and the nonnegative square roots equal gamma h. Where h>0 the roots are smooth compositions with strictly positive radicands.

They satisfy $A\geq|S_{12}|$ and $B\geq\max(S_{11},S_{22})$.

The nonnegative square root of S12^2 plus a nonnegative term dominates |S12|. Likewise B dominates each absolute diagonal entry, hence in particular their maximum with the stated signs.

Each square root dominates the absolute value of every individual entry it contains. In particular B dominates both diagonal entries themselves, whether positive or negative.

Each square root dominates every absolute component occurring under it; hence A>=|S12| and B>=|S11|,|S22| imply the stated weaker maximum bound even for negative diagonal entries.

The first radicand dominates S_12^2; the second dominates both diagonal squares. Taking nonnegative roots yields A>=|S_12| and B>=|S_ii|>=S_ii, hence the stated weaker maximum bound.

Define
\[
 a_1=B-S_{11},\quad a_2=B-S_{22},\quad
 a_3=A-S_{12},\quad a_4=A+S_{12},\quad q=A+B
\]
Then $-S+qI=\sum_i a_i\xi_i\otimes\xi_i$.

The third and fourth tensors sum to diagonal A and off-diagonal (a3-a4)/2=-S12. Adding the first two gives diagonal A+B-S11 and A+B-S22. Thus the tensor equals -S+(A+B)I with no missing factor two.

Expanding the diagonal directions gives diagonal contributions (a3+a4)/2=A and off-diagonal contribution (a3-a4)/2=-S12. Adding a1,a2 gives q-S11,q-S22, exactly -S+qI.

Direct matrix expansion gives diagonal B-S11+A=q-S11 and B-S22+A=q-S22; the off-diagonal is (a3-a4)/2=-S12. All five definitions and the final identity match exactly.

The diagonal direction pair contributes A I with off-diagonal -S_12 because (a_3-a_4)/2=-S_12. Adding the axis contributions yields diagonal A+B-S_11 and A+B-S_22, exactly -S+(A+B)I.

All coefficients are smooth, compactly supported, and nonnegative, with
$\sum_i\norm{a_i}1\leq C\norm S1+C\gamma\norm h1$.

Smoothness/support follow from A,B,h and S; nonnegativity follows from S0175 and |S12| domination. Pointwise A<=|S12|+gamma h and B<=|S11|+|S22|+gamma h bound the sum of coefficients by C|S|+C gamma h, then integrate.

Smoothness and compact support follow from S,h, and positivity from the root inequalities. Pointwise sum a_i=2A+2B-S11-S22 <=C|S|+C gamma h; integrating proves the bound.

The previous domination proves nonnegativity, and common compact support/smoothness were established. A<=|S12|+gamma h and B<=|S11|+|S22|+gamma h bound the sum by a universal multiple of |S|+gamma h, then integrate.

Smoothness and support follow from S0174; the inequalities S0175 make all coefficients nonnegative. Pointwise sum 2B-S_11-S_22+2A is <=C|S|+C gamma h, so integration gives the stated universal L1 estimate.

The final term can be made arbitrarily small before choosing fine scales.

With h fixed and integrable, gamma||h||1 tends to zero as gamma decreases. It is fixed positively before any fine-scale limit, preserving smoothness and finite derivative constants.

With h already fixed and integrable, choosing gamma sufficiently small makes C gamma||h||1 as small as prescribed, independently of subsequent fine parameters.

With fixed integrable h, gamma||h||1 tends to zero before any choice of fine scales; no uniform derivative bound in gamma is claimed or needed.

h is fixed and integrable, so choosing gamma>0 sufficiently small makes C gamma||h||_1 arbitrarily small before floors and H are selected; no lower bound on gamma is required later.

The discarded spatial tail remains a small residual.

For an L1 tensor, spatial truncation outside a sufficiently large ball has arbitrarily small L1 norm. Retaining that discrepancy in the residual is legitimate by the exact stress subtraction identity used later.

Replacing an L1 tensor by a compact truncation changes it by its L1 tail; this is retained as part of the residual and is not silently removed from the stress equation.

Replacing an L1 tensor by a compact truncation leaves its norm-small tail as part of S-S_k, which is retained in the exact residual formula S0319 rather than discarded from the equation.

When a general L1 tensor is approximated by a compact smooth tensor, the discarded tail is included in its L1 approximation error; it is retained later as S-S_k rather than silently removed.

 On a compact time
interval, continuity into $L^1$ makes this tail truncation uniform: the
continuous image is compact in $L^1$, and a finite covering reduces the claim
to finitely many fixed integrable functions.

A continuous map from compact time into L1 has compact image. Choose a finite small L1 net; tails of its finitely many representatives are uniformly small at a sufficiently large radius, and triangle inequality controls the tails of the full image.

A compact subset of L1 has uniformly small tails: cover by finitely many L1 balls, truncate each center beyond one common radius, and use the triangle inequality. Continuous images of compact time intervals are compact.

A continuous map from a compact time set into L1 has compact image. Choose a finite small L1 net, truncate each member beyond a common radius, and use the triangle inequality to obtain uniform tail control; finitely many one-sided pieces are handled separately.

A continuous image of a compact time interval is compact in L1. Cover it by finitely many small L1 balls and choose a common large spatial radius for the centers; the triangle inequality gives uniform small tails.

Fix a coefficient and its rectangle in coordinates $(s,y)$ adapted to $\xi$.

A compact coefficient support fits in a finite rectangle in the orthonormal xi,xi-perp coordinates, with extra longitudinal buffer room. Rotation preserves Lebesgue measure and derivative norms.

Rotate orthonormal coordinates so x=s xi+y xi-perp; a bounded rectangle can contain the compact coefficient support with longitudinal buffers outside it.

An orthogonal pair (xi,xi-perp) identifies a compact-support-containing rectangle in (s,y); rotations preserve spatial measure and all used norms.

Each compact smooth coefficient fits inside a finite rectangle in the orthonormal (xi,xi-perp) coordinates, with longitudinal room for fixed buffers outside its support.

Write $a$ for the original compactly supported coefficient.

This introduces local coefficient notation a, distinct by context from the initial smoothing bump. The fixed tensor decomposition already made it nonnegative smooth and compactly supported.

The symbol a here denotes the fixed nonnegative compact coefficient from the tensor decomposition; the earlier mollifier a is no longer used in this local calculation.

The coefficient a is one of the already established smooth nonnegative compact functions; the new scalar notation is local to its scan.

The local scalar a is one of the nonnegative coefficients a_i, unrelated to the initial smoothing bump denoted a; its local scope is clear and all subsequent row formulas use this compact coefficient.

On the scanning rectangle define $c=a+e$ with a strictly positive floor
$e>0$, and use constant-$e$ longitudinal buffers where $a=0$.

Adding e>0 makes c uniformly positive on the finite scan rectangle, so its reciprocal and logarithmic derivatives are finite. Buffers can be placed outside the compact coefficient support, where c=e exactly.

Adding a fixed positive e bounds c away from zero and makes log c and the radius formulas well-defined; where the compact original coefficient vanishes the buffered radius is constant.

A rectangle with longitudinal margins outside support a can be chosen. On it c=a+e>=e>0; in the margins c=e, so traversal and logarithmic derivatives are nonsingular even when a has zeros.

Adding e>0 gives uniform positive lower bound and bounded logarithmic derivatives on the fixed rectangle. Since a vanishes near its longitudinal edges, constant-e buffers join smoothly to the interior radius profile.

Write $\varepsilon_{\rm floor}$ for the total added floor and buffer mass,
including the fixed factors from the ramp durations below.

The floor/buffer mass is a finite area/length multiple of the positive floors, with fixed ramp factors. Including those factors makes it control both total scan duration normalization and weighted average error.

The declared floor budget includes interior area mass and longitudinal buffer mass with the fixed ramp factor 1/I_zeta, so it covers the actual later time weights.

The rectangle and buffer areas are finite and ramp constants fixed, making their total added weighted floor mass a finite constant times e; this includes the duration correction 1/I_zeta.

This defines a finite coarse error budget including rectangle area times e and the fixed ramp-duration factors. It is fixed before the discrete row approximation, so it is independent of H.

This mass is chosen arbitrarily small.

After fixing finite rectangles and ramps, decreasing each positive floor makes this added mass arbitrarily small. Floors remain positive for all subsequent finite-scale choices.

The rectangle and buffer dimensions are finite; choosing their positive floors sufficiently small makes every included added mass small without eliminating positivity.

For fixed geometry that mass tends to zero as e down to zero, while e remains positive for the subsequent construction.

For fixed finite rectangles and buffer lengths the added mass is linear in the chosen floors; reducing positive e_i makes it as small as desired without losing positivity.

All rectangle, floor, and derivative constants are fixed before the fine
parameter tends to zero.

This order of choices permits constants depending on the fixed positive floors and coefficient derivatives while requiring them to remain independent of H. Later asymptotic estimates respect that order.

This parameter order ensures that coefficients, floors and all derivatives appearing in B or log c have finite constants independent of the later H limit.

The parameter order is noncircular: choose the finite geometry, regularization, and positive floor first, allowing their derivative constants to be large but fixed when H decreases.

The declared order avoids uniformity demands as e tends to zero: support size and log-derivative constants may grow before H is subsequently chosen small enough.

 We use two distinct scales
\begin{equation}\label{eq:scales}
 d=H^2\quad\hbox{(row spacing)},\qquad
 \tau\asymp H^6\quad\hbox{(cycle duration)}.
\end{equation}

For H>0 the definitions give positive row spacing and cycle duration; choosing tau=H^6 is permitted by the asymptotic relation and is enforced exactly later.

The chosen independent exponents set row spacing to H^2 and cycle time comparable to H^6; their resulting compatibility is checked in the normalization and error exponents below.

The two declared positive scales are compatible; later complete-cell subdivision sets tau=H^6 exactly, while row spacing H^2 is smaller than the smoothing-bump width H for small H.

Both scales are positive and tend to zero with H; d/H=H tends to zero and tau/H=H^5 tends to zero. Later choosing H=tau^(1/6) realizes the relation exactly.

The auxiliary bump has width $H$, not $d$.

The fixed-profile bump is scaled at radius H, while rows are spaced H^2. For H decreasing to zero, d/H=H decreases to zero, which is the ratio needed in the quadrature estimate.

U_H is the H dilation already defined; using H rather than d in its norms is consistent with every projection and quadrature estimate.

The bump is U_H with support scale H; quadrature differentiates it at cost H^-1, independently of the finer row spacing d=H^2.

Using a scaled bump of width H while sampling rows at d=H^2 supplies transverse overlap; it is consistent with the fixed-profile bounds in S0210 and S0231.

 One packet visits rows $y=kd$
sequentially, with no simultaneous packets.

The rectangle intersects finitely many rows y=kd at each positive d. Sequential visits occupy disjoint time slots; at the zero-amplitude endpoints the center may be reset to the next row without a velocity jump.

A finite rectangle has finitely many rows at each positive d. Zero amplitude at visit endpoints allows resetting the center and starting the next row without simultaneous packets or a velocity jump.

There are finitely many rows in each fixed rectangle. Each visit has zero-amplitude entry/exit, permitting sequential position resets without a velocity jump; at any instant at most one principal packet is nonzero.

Only finitely many rows intersect each fixed rectangle. Flat amplitude endpoints permit restarting one packet at the next row; no sum of simultaneous principal packets is used, so the L2/curl amplitude cost does not grow with row count.

 On the interior of a row,
$\eta$ is constant and $r(s)=r_0c(s,kd)$.

On a fixed row the original smooth coefficient plus fixed positive floor makes r(s)=r0 c(s,kd)>0 smooth. A fixed positive amplitude is consistent with the prescribed longitudinal speed ds/dt=eta/(r0 c).

On a row the peak eta is fixed and c>0, so r=r0 c is a positive smooth radius depending on longitudinal location; it joins a constant radius where a vanishes in buffers.

The interior amplitude is fixed and c>=e>0, so r=r0 c is positive and smooth along the row; its spatial derivative is fixed before H varies.

Inside a row c>=e>0, so r=r_0 c is positive and smooth. The amplitude eta_i is constant there; variation occurs only on the separate constant-radius buffers.

 Thus
\[
 dt=(r_0/\eta)c(s,kd)\,ds,
 \qquad\int\eta^2U_H(x-X(t))\,dt
 =\eta r_0\int c(s,kd)U_H(x-s\xi-kd\xi^\perp)\,ds.
\]

Inverting the positive speed gives dt=(r0/eta)c ds. Substitution in integral eta^2 U_H dt leaves eta r0 times the line integral, with the perpendicular coordinate fixed at kd; denominators are positive on active rows.

The speed relation ds/dt=eta/(r0 c) gives dt=(r0/eta)c ds. Multiplying by eta^2 and substituting the row position gives the second displayed identity without an extra factor of c.

The translation law is ds/dt=eta/(r0 c), so dt=(r0/eta)c ds for the positive interior amplitude. Multiplication by eta^2 and substitution yield the exact weighted integral, with no missing Jacobian.

The speed ds/dt=eta/(r_0 c) gives dt=(r_0/eta)c ds for positive eta; substituting this into the eta^2 time integral gives eta r_0 times the weighted line integral with the displayed translated bump.

At entry use a fixed smooth temporal ramp $\zeta:[0,1]\to[0,1]$,
equal to zero to all orders at zero and equal to one with all positive-order
derivatives zero at one.

A smooth monotone cutoff built from exp(-1/s) and exp(-1/(1-s)) has the required flat endpoints and values in [0,1]. Only these properties, bounded derivatives and positive integral are needed.

A normalized smooth step built from exp(-1/t) and exp(-1/(1-t)) has the required flat endpoint values and stays in [0,1]. Its use gives smooth joins to zero amplitude and the constant peak.

A smooth flat transition from zero to one on [0,1] exists. Its endpoint jets ensure amplitude derivatives vanish at row resets and match the constant interior amplitude.

A smooth flat transition from zero to one can be built by normalized exp(-1/t) cutoffs. Values lie in [0,1], and the flat endpoint derivatives later give zero packet and stress traces at resets.

At exit use the reversed ramp $\zeta(1-\cdot)$.

Time reversal of the entry ramp starts at one with flat derivatives and ends at zero with flat derivatives, retaining values in [0,1] and the same integral.

Reversing the entry step interchanges endpoint values and retains flatness, giving the stipulated exit ramp.

Time reversal makes the exit amplitude start at one and end at zero with the same flat jets and the same ramp integral.

Time reversal changes the entry transition into a smooth exit transition from one to zero with the same integral and flat endpoint derivatives.

Write $I_\zeta=\int_0^1\zeta>0$.

Continuity and value one at the final endpoint imply positivity on a terminal interval, so the integral I_zeta is strictly positive and finite (at most one). All later divisions by it are valid.

The continuous nonnegative ramp equals one at its terminal endpoint, so it is positive on a subinterval and its integral is strictly positive.

The ramp is nonnegative and continuous with value one at its right endpoint, hence its integral is strictly positive and finite.

The continuous ramp equals one at its right endpoint, so it is positive on a nonempty interval. Its integral is therefore strictly positive and finite, justifying its use in denominators.

Throughout either buffer $r=r_0e$ is constant.

Outside the original coefficient support c=e, and the radius prescription r0 c therefore remains the positive constant r0 e throughout either buffer.

The original coefficient is zero on each buffer, so c=e and r=r0 e remains constant while the amplitude ramps.

The coefficient is the constant floor in each buffer, so setting radius r0 e agrees with the adjacent constant coefficient margin.

The original coefficient vanishes in buffers; setting c=e gives the constant positive radius r_0 e, so there is no radial-variation source during ramps.

To traverse a
buffer of length $L$, choose ramp duration
$r_0eL/(\eta I_\zeta)$.

During a ramp eta(t)=eta_i zeta(t/T), the displacement is eta_i T I_zeta/(r0 e). Setting it equal to L gives exactly T=r0 e L/(eta_i I_zeta), positive for active directions.

The buffer displacement equals duration times eta I_zeta/(r0 e); solving for displacement L gives exactly the stated duration, with positive denominator.

With eta(t)=eta_i zeta(t/T_ramp), displacement is eta_i T_ramp I_zeta/(r0 e). Solving for displacement L gives exactly the stated positive duration.

For duration L_t the displacement equals (eta_i/(r_0 e)) L_t integral_0^1 zeta. Setting it to L yields precisely r_0 e L/(eta_i I_zeta), positive for active directions.

Let $B_i$ be the sum over rows of the interior line
integrals of $c_i=a_i+e_i$, plus $2e_iL/I_\zeta$ per row for the two ramps.

Each interior row takes (r0/eta) integral c ds. Each of two ramps takes (r0/eta)eL/I_zeta, so the stated B_i makes total scan time (r0/eta)B_i with the correct factor two.

Interior time is (r0/eta) times each line integral. Each of two buffers adds (r0/eta)eL/I_zeta, proving the expression for B_i.

Interior traversal times are (r0/eta_i) integral c ds. Each of two ramps costs (r0/eta_i)eL/I_zeta; summing rows gives the declared B_i with the exact factor two.

Each interior traversal costs (r_0/eta_i) integral c ds and each of two ramps costs (r_0/eta_i)e L/I_zeta; summing over rows yields total duration (r_0/eta_i)B_i.

Allocate $\tau_i=\tau/4$ to direction $i$, and set
\begin{equation}\label{eq:normalization}
 \eta_i^2=\frac{\tau dB_i}{2\pi\tau_i},\qquad
 r_{0,i}=\frac{\tau d}{2\pi\eta_i}.
\end{equation}

For active positive B_i, tau,d,tau_i are positive and eta_i is the nonnegative positive square root. The defined r0_i is positive. These formulas simultaneously impose the allocated time and the desired weight, as directly substituted in S0199.

The positive square root defines eta_i from B_i; r0_i then solves the coefficient-weight equation. Substitution into the scan-time equation with tau_i=tau/4 determines both normalizations consistently.

For every used direction B_i>0, so eta_i is its positive square root and r0_i is positive. Direct substitution solves both scan-time and coefficient-weight normalization constraints with tau_i=tau/4.

For B_i>0 the formula defines a positive amplitude and then positive r_0. With tau_i=tau/4, eta_i^2=(2/pi)dB_i; both time normalization and coefficient normalization can therefore be imposed simultaneously.

The full scan takes exactly $(r_{0,i}/\eta_i)B_i=\tau_i$, and its coefficient
weight is exactly $2\pi\eta_ir_{0,i}/\tau=d$.

Substitution gives (r0_i/eta_i)B_i=tau d B_i/(2pi eta_i^2)=tau_i. Also 2pi eta_i r0_i/tau=d identically. The four allocated durations sum to the whole cycle.

Substitution gives (r0_i/eta_i) B_i=tau d B_i/(2pi eta_i^2)=tau_i and 2pi eta_i r0_i/tau=d, verifying both equalities exactly.

Substitution gives (r0_i/eta_i)B_i=tau dB_i/(2pi eta_i^2)=tau_i and 2pi eta_i r0_i/tau=d exactly.

Substitution gives (r_0/eta_i)B_i=tau d B_i/(2pi eta_i^2)=tau_i and 2pi eta_i r_0/tau=d exactly, with no asymptotic replacement of cycle duration.

 With sufficiently fine $d$,
$dB_i\leq C\norm{a_i}1+C\varepsilon_{\rm floor}+o_H(1)$.

The smooth compact coefficient line sums times d converge to its plane integral. The finite-row floors and buffer terms times d converge to their fixed added masses; endpoint row-count discrepancies vanish with d. The resulting upper bound has universal leading coefficient.

The row Riemann sum d times the interior line integrals converges to coefficient area mass plus floor area mass; the finite number of buffer rows contributes its budget and a vanishing endpoint row-count error.

The compact smooth a has a convergent transverse Riemann sum. Floor contributions scale as d times the finite row count and fixed lengths, converging to their fixed weighted area; errors tend to zero as d=H^2 decreases.

The scalar rectangle rule d sum_k integral a(s,kd)ds converges to ||a||_1. The floor and fixed buffer sums converge to their fixed added masses, and the endpoint row-count discrepancy vanishes with d.

Writing $\varepsilon_{\rm reg}$ for the earlier regularization mass gives
$\eta_i\leq C\sqrt\delta+
C\sqrt{\varepsilon_{\rm floor}+\varepsilon_{\rm reg}}+o_H(1)$
when $\norm S1\leq\delta$.

With tau_i=tau/4, eta_i^2=(2/pi)d B_i. Combine the Riemann-sum mass bound with sum||a_i||1<=C delta+C regularization mass and use sqrt(a+b)<=sqrt a+sqrt b; vanishing nonnegative row error has a vanishing square root.

eta_i^2=(2/pi)dB_i, and the coefficient sum bound is C delta+C regularization mass. Taking square roots and using sqrt(x+y)<=sqrt x+sqrt y yields the stated upper estimate.

eta_i^2=(2/pi)dB_i with tau_i=tau/4. Combine the Riemann-sum bound with the universal coefficient mass bound and use sqrt(x+y)<=sqrt x+sqrt y; the constants on sqrt(delta) do not involve derivatives or support.

Use eta_i^2=(2/pi)dB_i, the coefficient mass estimate S0177 and S0200, then sqrt(x+y)<=sqrt x+sqrt y. The row error remains o_H(1) after square root; the leading constant is independent of rectangles and derivatives.

If a direction
has zero mass it can be omitted; a positive floor otherwise makes
\eqref{eq:normalization} unambiguous.

If the original direction is omitted, its allocated slot can remain inactive; the average is still over the entire cycle. Otherwise a positive floor on a nondegenerate rectangle ensures B_i>0 and thus eta_i,r0_i>0, avoiding every denominator ambiguity.

A genuinely unused zero coefficient may be omitted, leaving an idle slot. Every retained scan with a positive floor has B_i>0 and hence positive eta_i,r0_i, avoiding division by zero.

A genuinely unused zero coefficient needs no slot or packet, with the slot filled by zero. Every retained scan has positive floor and at least one row for sufficiently small d, hence B_i and eta_i are nonzero; no division by zero remains.

Zero-mass directions can have their quarter-cycle slot idle; averaging is still over the full cycle. Every active direction has a fixed positive floor, hence B_i>0 and neither eta_i nor the r_0 denominator vanishes.

On a ramp $|(\eta r)'|\leq C\eta_i^2$; on an interior row
$|\eta r'|=|(\eta r)'|\leq\eta_i^2\norm{\partial_\xi\log c_i}\infty$.

A ramp has radius r0 e and duration T=r0 e L/(eta_i I_zeta), so |(eta r)'|<=eta_i^2 I_zeta||zeta'||infinity/L. On a row, r'=r0 partial_s c times eta_i/(r0 c)=eta_i partial_s log c; multiplying by constant eta_i gives both displayed expressions.

On a ramp eta(t)=eta_i zeta(t/T), r=r0 e, so |(eta r)'|<=eta_i^2 I_zeta||zeta'||infinity/L. On an interior row r'=r0 c_s ds/dt=eta_i c_s/c, giving both equalities and the logarithmic derivative bound.

On ramps, r eta'=r0 e eta_i zeta'/T=eta_i^2 I_zeta zeta'/L. On interiors r'=r0 partial_s c ds/dt=eta_i partial_s c/c, so both products equal eta_i^2 partial_s log c in magnitude.

On a ramp, |(eta r)'|<=r_0 e eta_i||zeta'||/duration=eta_i^2 I_zeta||zeta'||/L. Inside a row, r'=r_0 c_s eta_i/(r_0 c)=eta_i partial_s log c, giving both equality and inequality.

Constants are fixed before $H\downarrow0$.

The ramp shape, buffer length, positive floor and coefficient derivative bounds have already been fixed, so the constants in the preceding derivative bounds do not depend on the approaching fine scale.

Every constant in the derivative bounds involves only fixed coefficients, positive floors, buffer lengths and ramp shape; none varies with H.

At fixed positive floor, ||partial_s log c||infty and the ramp constants are finite; their possible growth when the floor is later changed is irrelevant because that change precedes choosing H.

The constants in S0203 depend on the fixed positive floors, log derivatives, buffer lengths and ramp profile; those are chosen before H, so all derivative-source bounds are uniform in the fine limit.

 Accordingly $r_0\asymp H^8$,
all radii are bounded above and below by fixed multiples of $H^8$, and
$r^{1/5}=O(H^{8/5})=o(H)$.

The fixed-floor line masses satisfy dB_i to a positive finite limit, hence active eta_i is bounded above and below. Thus r0=tau d/(2pi eta_i) is comparable to H^8; multiplying by positive bounded c preserves comparability. Taking the fifth root gives H^(8/5), whose ratio to H is H^(3/5) to zero.

Since eta_i is bounded above and below at fixed retained positive floor, r0=tau d/(2pi eta_i) is comparable to H^8. Multiplying by c with fixed positive lower and upper bounds preserves comparability; r^{1/5}/H=O(H^{3/5})->0.

dB_i tends to a positive finite limit, so eta_i has positive finite bounds; r0=tau d/(2pi eta_i) is comparable to H^8 and c is between fixed positive bounds. Raising to 1/5 gives H^(8/5)=o(H).

At fixed active coarse data dB_i tends to a positive finite limit, hence eta_i is bounded above/below. Then r_0=tau d/(2pi eta_i) is comparable to H^8; multiplying by c in fixed positive bounds gives all radii, and (H^8)^(1/5)/H=H^(3/5) tends to zero.

 The intrinsic error
\eqref{eq:E-bound} is $O(H|\log H|)+O(H^8)$.

The two derivative prefactors are bounded independently of H, r=O(H^8), and |log r|=O(1+|log H|). Substitution into E-bound gives O(H^8)+O(H(1+|log H|)), equivalent to the stated rate for sufficiently small H.

The two time coefficients are uniformly bounded, so the intrinsic stress estimate is Cr+CH(1+|log r|). With r comparable to H^8 it is O(H^8)+O(H|log H|).

Bound both amplitude-radius derivative coefficients uniformly, use r=O(H^8) and |log r|=O(1+|log H|), and insert them in the explicit intrinsic-stress bound.

Insert r comparable to H^8, uniformly bounded |eta r'| and |(eta r)'| into S0167: the first term is O(H^8), the second O(H(1+|log H|)), with coarse-dependent constants fixed first.

Integration by parts on a visit, using $\eta=0$ at its endpoints, gives
\begin{equation}\label{eq:average}
 \overline{\mathcal U}=\frac1\tau\int_{\rm cycle}\mathcal U\,dt
 =\Div\left(\frac{2\pi}{\tau}\int_{\rm cycle}
 \eta(t)^2U_H(x-X(t))\xi\otimes\xi\,dt\right).
\end{equation}

The temporal derivative of eta r U integrates to zero on each visit because eta vanishes at both endpoints. The remaining term gives integral (eta r)'U = integral eta^2 partial_xi U with positive sign. Vector multiplication by xi identifies this as divergence of the stated rank-one tensor; sum visits and divide by full tau.

The time derivative of eta r U_H(x-X) integrates to zero on each visit because eta vanishes at both ends. The remaining spatial derivative term is eta^2 xi dot grad U_H, which after multiplication by xi is Div(eta^2 U_H xi tensor xi), giving the positive sign.

Integrate the derivative in S0208 over each visit. Its endpoint values vanish because eta=0, so integral (eta r)'U_H equals integral eta^2 xi dot grad U_H; multiplying by 2pi/tau gives the row divergence with a positive sign and then sums visits/directions.

The derivative identity S0208 integrates on each visit with zero eta at both endpoints, leaving integral (eta r)' U_H xi=integral eta^2 (xi dot nabla U_H)xi. That is the spatial divergence of the displayed positive rank-one tensor; summing visits preserves the sign.

Indeed $\frac d{dt}(\eta rU_H(x-X))=(\eta r)'U_H-\eta^2\xi\cdot\nabla U_H$.

The chain rule gives d_t U_H(x-X)=-X' dot grad U_H=-eta partial_xi U_H/r. Multiplying by eta r yields -eta^2 partial_xi U_H, with no derivative of H because H is fixed on the cycle.

Chain rule gives derivative of the translated bump as -X' dot grad U_H. Substituting X'=eta xi/r cancels r and produces exactly the negative eta^2 directional derivative.

Chain differentiation of U_H(x-X(t)) gives -X' dot grad U_H; eta r X'=eta^2 xi, so no r' or reset term is omitted on a visit.

The chain rule gives (eta r)'U_H-(eta r)X' dot nabla U_H, and X'=eta xi/r makes the second term -eta^2 xi dot nabla U_H exactly.

Write the tensor in parentheses, summed over four directions, as $A_{\rm av}$.

The four disjoint directional visits give a sum of time integrals; the common full-cycle denominator tau defines one fixed symmetric tensor A_av on the whole cycle.

Summing the integrable symmetric tensors over the four direction slots defines A_av for the whole cycle, including their time weights and ramps.

Summing all four directional tensors over the full cycle defines one time-independent symmetric A_av, and its divergence is the average source.

The definition includes every direction's full-cycle contribution; it is a time-independent symmetric integrable tensor on that cycle, which is required by the subsequent temporal correction.

The raster quadrature obeys
\begin{equation}\label{eq:quad}
 \left\|d\sum_k\int a(s,kd)U_H(\,\cdot-s\xi-kd\xi^\perp)\,ds-a\right\|_1
 \leq C\left[d\bigl(\norm{\partial_\perp a}1+H^{-1}\norm a1\bigr)
 +H\norm{\nabla a}1\right].
\end{equation}

The line integral defines an L1-valued W1,1 function of y. The rectangle-rule discrepancy is <=d integral||F'||1; differentiation gives ||partial_perp a||1+H^-1 C||a||1. The difference between integral F=a*U_H and a is bounded by H||grad a||1. Adding yields the exact stated three terms.

The independent derivation uses the Banach-valued rectangle rule with derivative cost ||partial_perp a||1+CH^-1||a||1, then approximation of a by a*U_H with error CH||grad a||1. This yields exactly the displayed three terms.

The L1-valued row-rule error is <=d integral ||F'||1, bounded by d(||partial_perp a||1+H^-1||a||1). The remaining convolution error is <=CH||grad a||1. This proves the exact global L1 inequality without assuming d and H equal.

For F(y) in L1_x, the integrated row-rule error is <=d integral||F'||_1<=d(||partial_perp a||_1+C H^-1||a||_1). Its integral equals a*U_H; translation continuity with first moment <=H gives the remaining C H||nabla a||_1 term.

Here is the explicit Banach-valued rectangle-rule proof.

The announced proof is actually provided immediately by S0212-S0216; the Banach-valued fundamental theorem applies because the differentiated integrand is integrable in all variables.

The ensuing argument is an explicit proof of the stated quadrature estimate; no additional external quadrature assumption is invoked.

The following proof uses the Bochner fundamental theorem for the smooth compact-in-y L1-valued function F; it supplies an affirmative estimate rather than a pointwise sampling heuristic.

The subsequent argument is an explicit L1-valued fundamental-theorem-of-calculus estimate; it does not invoke an unstated scalar-to-vector quadrature theorem.

Put $F(y)=\int a(s,y)U_H(\cdot-s\xi-y\xi^\perp)\,ds$, as an $L^1_x$-valued
function.

Tonelli gives ||F(y)||1<=integral|a(s,y)|ds since U_H has unit L1 norm. Compact smooth a and U_H also give an L1-valued derivative with integrable norm, justifying its use in the rectangle rule.

For each y, compact smooth a and the probability bump give an L1_x function; F and its derivative are integrable in y by Fubini, permitting the Banach-valued fundamental theorem.

For each y the longitudinal integral defines an L1_x function by Tonelli; smooth compact a and fixed smooth U_H give an absolutely continuous L1-valued function of y.

For smooth compact a and smooth integrable U_H the integral defines a compactly supported C1 map of y into L1_x. Differentiation and integration are justified by integrable compact coefficient derivatives.

Differentiating it gives
$\int\norm{F'(y)}1dy\leq\norm{\partial_\perp a}1+
\norm a1\norm{\partial_\perp U_H}1$.

Differentiate both the coefficient and its translated bump: F'=integral[(partial_y a)U_H-a(partial_perp U_H)]ds. Minkowski and Tonelli bound the two terms by ||partial_perp a||1 and ||a||1||partial_perp U_H||1; the translation-derivative minus sign disappears in the norm.

Differentiate both a(s,y) and the translation by y xi-perp. Integrating their L1_x norms and using ||U_H||1=1 gives the two displayed terms with the triangle inequality.

Differentiate a and the translated bump, obtaining two integrals; Minkowski/Fubini and ||U_H||1=1 give the stated two derivative costs after integrating y.

Differentiate the coefficient and translated bump: F'=integral(a_y U_H-a partial_perp U_H)ds. Tonelli and triangle inequality give the two displayed global norms; the fixed bump scales with derivative norm C/H.

On each row interval of length $d$, the fundamental theorem of calculus
bounds the integrated discrepancy from its endpoint sample by
$d$ times the integral of $\norm{F'}1$ on that interval.

For y in [kd,(k+1)d], ||F(y)-F(kd)||1<=integral_kd^y ||F'(z)||1 dz. Integrating y and interchanging nonnegative integrals gives a factor at most d on that interval.

For y in [kd,(k+1)d], ||F(y)-F(kd)||1<=integral_kd^y ||F'(z)||1 dz. Integrating y bounds this by d integral_kd^{(k+1)d}||F'(z)||1 dz.

For y in [kd,(k+1)d], ||F(y)-F(kd)||1<=integral_kd^y||F'||1. Integrating y bounds by d integral_cell||F'||1, including all endpoint rows.

On [kd,(k+1)d], ||F(y)-F(kd)||_1<=integral_kd^y||F'(z)||_1 dz. Integrating y and reversing nonnegative integrals bounds it by d times the derivative integral on that interval.

Summing yields the first two terms in \eqref{eq:quad}.

Summing disjoint row intervals covers the real y-line, with only finitely many nonzero coefficient samples. The derivative bound and fixed-profile ||partial_perp U_H||1=C/H give d||partial_perp a||1+C(d/H)||a||1.

Sum the disjoint interval estimates and substitute ||partial_perp U_H||1=C/H, producing d||partial_perp a||1+C d H^-1||a||1.

Sum disjoint row intervals and substitute the verified derivative bound with ||partial U_H||1=C/H; all sums are finite or absolutely integrable.

Sum the disjoint row intervals and apply S0213 with ||partial_perp U_H||_1=C/H; this produces exactly d||partial_perp a||_1+C d H^-1||a||_1.

The translation estimate
$\norm{a*U_H-a}1\leq CH\norm{\nabla a}1$ gives the third.

For smooth integrable a, ||a(. -z)-a||1<=|z|||grad a||1 by integrating its directional derivative. Average against a probability kernel supported in B_H; its first absolute moment is <=H, proving the claimed bound.

Write a*U_H-a as an average of translation differences; ||a(.-z)-a||1<=|z|||grad a||1 and the bump's first absolute moment is CH.

For a in W1,1, ||a(.-z)-a||1<=|z|||grad a||1. Integrating against the probability bump supported in B_H gives the stated convolution estimate.

Write a*U_H-a as the U_H-weighted integral of translations. The W1,1 translation bound is <=|z| ||nabla a||_1, and U_H has probability mass supported in B_H, giving <=H||nabla a||_1.

Estimate \eqref{eq:quad} applies to the original compact coefficient $a$.

The proof differentiates the original smooth compact a only. It does not falsely apply the derivative estimate to the discontinuously truncated floor or buffers.

The original compact smooth coefficient satisfies all differentiability and integrability hypotheses of S0210; the nonsmooth rectangular floor is charged separately rather than inserted in that derivative estimate.

The row-rule differentiation is applied only to the smooth compact a; the rectangular floor discontinuity is not illicitly differentiated and is handled as positive added mass separately.

The derivative row-rule estimate is applied only to the smooth compact a; the discontinuous rectangular floor is handled by its positive mass separately, so no unclaimed floor derivative enters.

The added floor and the ramp-weighted buffers contribute at most
$C\varepsilon_{\rm floor}+o_H(1)$ separately.

Floors contribute their positive Riemann-summed mass, with vanishing row-count error. A ramp's average tensor mass is d e L (integral zeta^2)/(integral zeta) per visit, <=d e L because 0<=zeta<=1; the chosen floor budget includes the duration factors and bounds these terms separately.

On buffers eta(t)^2<=eta_i^2 and their time lengths are known, so their averaged L1 weight is bounded by the budgeted constant floor mass. Interior floor rows converge to the finite rectangular area mass, with vanishing row-count error.

Interior added mass is d sum_rows e times row length. Ramp weight is bounded by eta_i^2 times ramp duration, hence a fixed multiple of d e L/I_zeta per row. The row-count error tends to zero, proving the mass bound.

The nonnegative bump has L1 mass one. Each ramp contributes (2pi/tau) integral eta^2dt <=(2pi/tau)eta_i^2 duration=d e L/I_zeta; floors similarly cost their sampled masses. Row counting converges, giving C epsilon_floor+o_H(1).

Since $d=H^2$, the quadrature error tends to zero.

At fixed coefficient, the three errors scale as O(H^2), O(H), and O(H). Each tends to zero because d/H=H and all derivative norms are fixed.

d H^-1=H, d=H^2 and the mollification error is O(H); all three quadrature terms vanish for the fixed smooth coefficient.

Substitution d=H^2 makes the three errors O(H^2), O(H), and O(H) at fixed coefficient; all tend to zero as H decreases.

At fixed coarse data, d=H^2 makes d||partial a||=O(H^2), dH^-1||a||=O(H), and H||nabla a||=O(H), all tending to zero; fixed floor error is chosen separately first.

\subsection{Temporal correction and the stress identity}

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On each cycle $[t_k,t_k+\tau]$ put
\[
 Q(t)=-\mathbb P\int_{t_k}^t
 (\mathcal U(s)-\overline{\mathcal U})\,ds,
\]

On a fixed cycle the compact bump source is time-integrable with bounded coefficient. The specified projection can be applied through its explicit profile; integrating the source minus its full-cycle average defines a divergence-free continuous Lp vector field with zero primitive at both ends.

The integral is a well-defined Lp time primitive of projected translates for p>1. Subtracting the full-cycle time average makes its total integral zero; the minus sign is required for cancellation of the source.

The full-cycle average is constant in t and the time integral is a finite Bochner integral of projected translated bumps in every Lp for p>1; it defines divergence-free Q with zero initial-cycle value.

The finite-cycle source is an integrable time family of smooth compact bumps. Its mean-subtracted primitive has a well-defined projected profile by S0226-S0232 and begins at zero with derivative -P(U-Ubar).

The average here is taken over the entire four-direction cycle, not
separately on each direction slot.

This fixes the averaging domain to the complete cycle of duration tau. The distinction matters because the four slot averages separately omit three coefficient directions.

The average is fixed on the full four-slot interval of length tau, so it does not change when the active direction changes.

The averaging interval in the definition is the entire tau-cycle, which contains all four allocated slots; no slot-dependent average replaces it.

The definition uses the full four-direction cycle, with its fixed duration tau, consistently with A_av at S0209; it does not switch target tensor between direction slots.

 This ensures that its tensor is the
sum of all four directional averages throughout the cycle.

Linearity of integration over disjoint directional slots makes the cycle average the sum of all four tensors, divided by the same full tau. It therefore stays fixed while the instantaneous active direction changes.

Linearity of integration adds each of the four directional tensors; this constant total tensor, rather than the current single direction, enters the instantaneous corrected equation.

Linearity of time integration and spatial divergence sums the four directional averages; the corrector's derivative contains that same full tensor divergence at every time of the cycle.

The full-cycle average is div A_av with A_av the sum of all four positive rank-one averages. It is time-independent on that cycle, so every slot has the same complete target stress.

Here $\mathbb P=I-\nabla\Delta^{-1}\Div$ is used only on translates of
the fixed bump $U_H\xi$ and their time integrals.

The explicit operator is the Fourier/Laplace Helmholtz projection. Here the only inputs are scalar multiples/translates/time integrals of fixed smooth U_H xi, so the subsequent directly computed profile defines every use without requiring endpoint boundedness on arbitrary inputs.

Using Delta G=delta gives the displayed Helmholtz projection on these smooth bump sources. The proof below constructs the needed images and estimates directly, so no undefined extension to arbitrary L1 fields is required.

The projection is I-grad Delta^-1 div. The only inputs are scalar coefficients times translates of U_H xi and their finite time integrals, so the subsequent fixed-profile estimates suffice.

The operator formula is the divergence-free projection because Delta Delta^-1=I. Only explicitly controlled bump profiles and finite time integrals are used, so no unsupported L1 extension of the projection is required.

 Its needed bounds have
a direct fixed-profile proof.

The stated profile bounds are obtained below from local integrability of grad G, radial potential averaging and explicit dilation. No unrestricted operator estimate is required.

The projected source is a dilation and translation of one explicit smooth profile; establishing its decay and curl controls all uses of the projection.

The explicit projected-bump formula provides the exact required L2, L3/2, and curl L1 bounds; no unstated boundedness of projection on L1 is used.

The current fixed dilation at S0164 gives the projected profile Z_H(x)=H^-2 Z_1(x/H): the homogeneous order-zero multiplier commutes with dilation and rotations. This identity uses a single chosen U, so constants in all ensuing H estimates are uniform. The regularized Newtonian potential is smooth in the core and has the radial harmonic exterior.

 Let $G=(2\pi)^{-1}\log|x|$ and
\[
 Z_{H,\xi}=U_H\xi-(D^2G*U_H)\xi.
\]

G is the two-dimensional fundamental solution Delta G=delta_0. Since div(U_H xi)=xi dot grad U_H, grad Delta^-1 div(U_H xi)=(D2G*U_H)xi, making Z_H exactly the projected bump.

G is the plane's fundamental solution with Delta G=delta. Differentiating its convolution with the smooth bump gives grad Delta^{-1}Div(U_H xi), so Z_{H,xi} is exactly P(U_H xi).

Since Delta G=delta, grad Delta^-1 div(U_H xi)=(D^2G*U_H)xi, making Z_H exactly P(U_H xi) with the correct sign.

G is the fundamental solution with Delta G=delta_0. For a constant vector xi, nabla Delta^-1 div(U_H xi)=(D^2G*U_H)xi, so the displayed Z is exactly P(U_H xi).

It has zero divergence and $\curl Z_{H,\xi}=\curl(U_H\xi)$.

Divergence of the second term is grad Delta G*U_H dotted with xi=xi dot grad U_H, canceling the first term's divergence. That second term is a scalar gradient, so its curl is zero and the displayed curl identity follows.

Divergence of the Hessian convolution is grad U_H dot xi, canceling Div(U_H xi); its curl is zero since it is a gradient, leaving the displayed curl.

div Z=xi dot grad U_H-xi dot grad(Delta G*U_H)=0. The removed term is a gradient, so its curl vanishes distributionally.

div Z=xi dot nabla U_H-xi dot nabla(Delta G*U_H)=0. The subtracted Hessian field is a gradient, whose distributional curl is zero, leaving curl(U_H xi).

At $H=1$ it is smooth and bounded in the core: write the Hessian convolution
as $\partial_iG*\partial_jU_1$, whose $|x|^{-1}$ singularity is integrable.

Move one derivative from G to compact smooth U_1; the remaining kernel is O(|x|^-1), locally integrable. On a fixed compact x-set its convolution and all derivatives are bounded by integrable kernels against bounded compact derivatives, giving smoothness and a finite core bound.

Move one derivative onto the compact smooth bump: partial_i G has locally integrable inverse-radius singularity. The convolution is bounded on compact sets and arbitrarily many derivatives can be moved onto U, giving smoothness.

Move one derivative onto the compact smooth bump: partial_i G is locally integrable as 1/r. Every further derivative can likewise fall on U1, giving bounded smooth convolution on the compact core.

Moving one derivative onto smooth compact U_1 gives a convolution with the locally integrable 1/|x| kernel. Every further derivative can also fall on U_1; compact support gives uniform core bounds and spatial smoothness.

For $|x|>1$, radial averaging gives $G*U_1=G$, so
$Z_{1,\xi}=-D^2G\xi=O(|x|^{-2})$.

For the explicitly stated |x|>1 and every support radius s<1 of radial U, angular averaging gives mean_theta log|x-s e^(i theta)|=log|x|. Integrating against unit-mass U yields G*U=G with no additive constant. Since U(x)=0 on this region, Z=-D2G xi; D2G=(2pi)^-1[I/|x|^2-2x tensor x/|x|^4] has degree -2. The current domain excludes any bounded hole of the bump support, so it also covers annular probability profiles correctly.

For |x|>1 every circle carrying U1 has radius smaller than |x|, since U1 is supported inside B1. Its angular logarithmic mean is log|x|; integrating the radial probability density therefore gives G*U1=G on the entire exterior disk. There U1 vanishes, so Z1,xi=-D^2G xi, whose entries are O(|x|^-2). The explicit condition handles radial bumps with holes: no identity is asserted in an interior component outside their support.

For |x|>1 the function y -> G(x-y) is harmonic on the entire closed disk containing the support of U1. Its average on each centered circle of radius rho<1 is G(x). Integrating those averages against the radial probability density proves G*U1=G exactly, with no additive constant. U1(x)=0 there, so Z1,xi=-D^2G(x)xi, whose explicit Hessian is O(|x|^-2). A hole in an annular bump would invalidate equality merely outside the support, but is excluded by the current explicit |x|>1 condition.

For |x|>1, the function y -> G(x-y) is harmonic on a neighborhood of the closed unit disk containing supp U1. Its average on each centered circle is G(x); integrating these circle averages against the radial unit-mass U1 gives G*U1=G. Also U1(x)=0 there, hence Z_(1,xi)=-D2G(x)xi. The explicit Hessian of (2pi)^-1 log|x| is O(|x|^-2). No assertion about an interior hole of the bump support is needed.

 Hence $Z_{1,\xi}\in L^p$ for every
$p>1$.

S0228 bounds Z smoothly on the entire closed unit disk, including any holes of the bump support. The current S0229 gives its O(|x|^-2) tail for |x|>1. Thus integral_1^infinity rho^(1-2p) d rho is finite for every finite p>1; boundedness also gives p=infinity. In particular the two needed exponents 2 and 3/2 are valid.

The smooth core has finite measure and is bounded; the tail integral is integral_1^infinity rho^{1-2p}d rho, finite precisely for p>1. This includes all exponents used.

The already checked core bound applies on the entire unit disk, including possible holes in a radial bump; S0229 gives the tail only for |x|>1, which is all that is needed. The integral of r^(1-2p) at infinity converges exactly for p>1, while the bounded core has finite measure. Thus every asserted Lp norm is finite without an equality inside a support hole.

S0228 gives a smooth bounded profile on the compact unit disk, and the current S0229 gives |Z|<=C|x|^-2 for every |x|>1. Polar integration of r^(1-2p) converges exactly for p>1, proving the stated global Lp claim without requiring the exterior formula at interior points outside the bump support.

 Scaling gives
\[
 \norm{Z_{H,\xi}}p\leq C_p H^{2/p-2}.
\]

S0164 now explicitly defines a fixed dilation family. Distributional Hessian scaling gives Z_H(x)=H^-2 Z_1(x/H), since the additive log H term disappears under two derivatives. Changing variables yields ||Z_H||p=H^(2/p-2)||Z_1||p for finite p>1, and H^-2 for p=infinity; S0230 gives the finite constants.

The Hessian convolution scales as H^-2 times its unit-scale profile because the constant log H disappears on taking two derivatives. Hence Z_H(x)=H^-2 Z_1(x/H), giving H^{2/p-2} in Lp.

The now-explicit fixed bump dilation and distributional homogeneity of D^2G give Z_H(x)=H^-2 Z_1(x/H). Equivalently differentiate G(Hx)=G(x)+(2pi)^-1 log H twice; the additive constant disappears. This includes the local distributional part of the Hessian. Change of variables yields ||Z_H||p=H^(2/p-2)||Z_1||p, with an H-independent constant.

For the fixed dilation U_H, the Hessian convolution scales as H^-2 times the unit profile (the additive log H constant differentiates away). Hence Z_H=H^-2 Z_1(x/H) and ||Z_H||_p=H^(2/p-2)||Z_1||_p.

Minkowski applies to its translates in the time primitive.

Translations preserve each Lp norm for p>=1. The time coefficient is integrable on a finite cycle, so Bochner Minkowski bounds the norm of its translated-profile integral by the integral of coefficient times the fixed profile norm.

The time coefficient is bounded on each finite cycle and translations preserve the Lp norm of Z_H. The integral triangle inequality therefore applies to this Bochner integral and to its averaged term.

Translation is an Lp isometry and p>1 is a Banach exponent; Minkowski bounds the norm of each time integral by the integral of the profile norm times its scalar amplitude.

Translations preserve Lp norms, the scalar coefficients are bounded on each finite cycle, and Z_H is in all required p>1 spaces; Minkowski therefore bounds the Bochner time primitive by the integral of profile norms.

 No general
Calderón--Zygmund $L^p$ theorem is needed for this use of $\mathbb P$.

Only the explicitly controlled family Z_H and its translates are used. All required Lp bounds were derived by integration/scaling; the argument does not assume a general singular-integral theorem.

S0226-S0232 supply the needed restricted operator estimate by explicit kernel decay and scaling; no general singular-integral theorem is used in this step.

Every projection estimate used was proved directly for its actual fixed bump input; no global operator theorem, especially no false L1 endpoint bound, enters.

The proof has obtained each required bound by explicit convolution and polar integration; no global Calderon-Zygmund estimate, especially none at p=1, is used.

For an explicit pressure sign, write
$q_{H,\xi}=\xi\cdot\nabla(G*U_H)$ and let
$\pi_{\mathcal U}(t)=2\pi(\eta r)'q_{H,\xi}(\cdot-X(t))$
for the active direction, with zero value when no packet is active.

q_H is a smooth scalar potential with gradient (D2G*U_H)xi. Multiplying and translating it with the same scalar 2pi(eta r)' used in the source defines pi_U; on inactive slots both are set to zero.

The gradient of q_H is (D^2G*U_H)xi, so multiplying by the active scalar coefficient and translating constructs the irrotational part of the source. At zero amplitude reset the derivative prefactor vanishes, so the zero value is consistent.

The scalar potential q_H differentiates G*U_H in xi. Multiplication by 2pi(eta r)' and translation yields precisely the gradient component of the compact source, with zero source at zero-amplitude resets.

The scalar q_H is smooth with gradient (D^2G*U_H)xi; multiplying by the active source coefficient and translating gives a well-defined scalar potential. Idle slots have zero source and consistently zero potential.

Then $\mathcal U-\mathbb P\mathcal U=\nabla\pi_{\mathcal U}$.

By definition I-P=grad Delta^-1 div. Applying it to 2pi(eta r)' U_H(. -X)xi gives the gradient of exactly the preceding pi_U with the positive sign.

P(U_H xi)=U_H xi-grad q_H by definition. Multiplication by 2pi(eta r)' and translation therefore gives U-PU=grad pi_U with exactly the stated sign.

Insert Z_H=P(U_H xi) into U-PU; the difference is 2pi(eta r)' grad q_H at the moving center, exactly grad pi_U.

By S0226, U-PU=2pi(eta r)'(D^2G*U_H)xi at the translated point, precisely nabla pi_U with the stated positive sign.

Taking
$p_Q=-\pi_{\mathcal U}+\tau^{-1}\int_{\rm cycle}\pi_{\mathcal U}(s)\,ds$
gives the exact identity
$\partial_tQ+\mathcal U+\nabla p_Q=\overline{\mathcal U}$.

Differentiate Q to obtain -P U+P Ubar. Adding U gives grad pi_U+P Ubar. The chosen pressure gradient subtracts grad pi_U and adds grad pi_Ubar, leaving (P+(I-P))Ubar=Ubar exactly.

Q'=-P U+P average(U); hence Q'+U=average(U)+grad(pi_U-average(pi_U)). Adding grad p_Q with the displayed negative/positive signs cancels the extra gradient.

Differentiate Q=-P primitive(U-Ubar) to get -PU+PUbar. Adding U leaves grad pi_U+PUbar; grad p_Q=-grad pi_U+grad average(pi_U) changes this to Ubar, confirming both signs.

Q'=-P(U-Ubar) gives Q'+U=Ubar+grad(pi_U-average pi_U). The displayed negative instantaneous potential plus positive averaged potential cancels this gradient exactly.

If $B$ bounds $|(\eta r)'|$, bump scaling gives
\begin{equation}\label{eq:Q}
 \norm Q2\leq C\tau H^{-1}B,
 \quad\norm Q{3/2}\leq C\tau H^{-2/3}B,
 \quad\norm{\curl Q}1\leq C\tau H^{-1}B.
\end{equation}

There is only one active packet at a time, and the source coefficient is <=2pi B. Minkowski over a time interval of length at most tau, including the average contribution, yields tau times the projected L2 norm H^-1 or L3/2 norm H^-2/3. Curl removes the projection and costs ||grad U_H||1=O(H^-1), giving the third bound.

Both projected source and its average have norm <=C B H^{2/p-2}. Integrating for at most tau gives C tau B H^-1 at p=2 and C tau B H^-2/3 at p=3/2. Curl commutes with the projection here and has norm <=C B/H, giving the final bound.

Minkowski over an interval of length at most tau bounds both the source primitive and the primitive of its whole-cycle average. The now-verified unit-disk core/exterior argument and fixed-profile dilation give ||Z_H||2=C/H and ||Z_H||3/2=C H^-2/3. Curl removes the gradient and ||grad U_H||1=C/H. Multiplication by the coefficient bound B gives all three displayed estimates, with no row-count factor.

Minkowski bounds Q by twice tau times a projected-source bound. S0231 gives H^-1 for p=2 and H^-2/3 for p=3/2; curl removes the gradient and ||nabla U_H||_1=C/H, yielding the third estimate.

Here both $\mathcal U$ and its cycle average obey the needed norm bound.

For each relevant norm, the full-cycle average has norm at most the average of the individual norms by Minkowski. The instantaneous source has one active translated bump with coefficient bounded by B; projection/curl are treated with the same explicit profiles.

The instantaneous coefficient is bounded by B; averaging cannot increase a norm bound. The same is true for the projected profile norms and curl norms used to estimate Q.

The average is a convex time average of the same bounded-norm inputs, so triangle/Minkowski bounds it by their common supremum, without multiplying by the number of row visits.

At each time there is at most one active source with coefficient <=2pi B. Its cycle mean is bounded in the same norm by time averaging, so subtracting it introduces only a factor of two.

The curl estimate follows directly from $\norm{\nabla U_H}1=O(H^{-1})$.

The fixed-dilation bump has gradient H^-3(grad U_1)(x/H), so its L1 norm equals H^-1||grad U_1||1. Curl of the projected bump equals curl of U_H xi, avoiding any L1 projection estimate.

Differentiating U_H=H^-2 U(x/H) multiplies its L1 norm by H^-1. Curl(U_H xi) is one directional derivative combination, so this controls its L1 norm directly.

curl P(U_H xi)=curl(U_H xi) since the removed term is a gradient. The current fixed dilation gives ||grad U_H||1=H^-1||grad U||1 exactly; integrating both source and average over at most one cycle proves the stated curl bound without an L1 projection theorem.

Since curl P(U_H xi)=curl(U_H xi), one spatial derivative of the scaled probability bump has L1 norm H^-1||nabla U_1||_1. Time integration costs at most 2tau B.

No $L^1$ boundedness of $\mathbb P$ is invoked.

Q itself is estimated only in L2 and L3/2; its curl is evaluated directly as the curl of the unprojected source. None of these steps requires P bounded on L1.

Only p>1 bounds for the projected velocity and the direct curl identity enter; the proof never estimates the projection as an L1-to-L1 map.

Only projected L2/L3/2 norms and unprojected differentiated curl L1 are estimated; none requires P:L1 to L1.

Every projection estimate is for Lp with p>1; the L1 curl estimate comes from a local derivative of U_H. The proof never requires ||P f||_1<=C||f||_1.

 The corrector vanishes at both
cycle endpoints.

At the left endpoint the primitive is empty. At the right endpoint integral_cycle(U-Ubar)=integral_cycle U-tau Ubar=0, so projection gives zero exactly at both endpoints.

At the first endpoint the primitive is zero; at the last, the integral of U minus its full-cycle average is exactly zero. Projection preserves these zeros.

At the left endpoint the primitive is zero. At the right endpoint integral(U-Ubar)=0 by the exact full-cycle average, so Q is zero in all its normed spaces.

At t_k the primitive is zero. At t_k+tau its integrand has zero complete-cycle time integral by definition of Ubar, so Q is zero there as well.

 Its first time derivative may jump there, so the new velocity correction
is continuous and, after the fixed spatial mollification below, piecewise
smooth in time; it is not asserted to
be globally time-smooth.

The source explicitly concerns the new correction after the fixed spatial mollification for its piecewise time smoothness. On each raster visit the coefficient/ramp paths are smooth and radii positive; convolution makes the translated/scaled profile smooth in space and hence smooth in time. Q is a projected smooth-bump primitive. Both fields have matching zero endpoint values; Q' may jump when the full-cycle average changes, so the stated failure of global time smoothness is consistent.

The primitive is continuous through cycle endpoints even when its derivative has a jump. Within each visit the chosen smooth ramps and coefficients, after fixed spatial convolution of the profile, give smooth time dependence; a finite piecewise partition accounts for the joins.

On the actual constructed visits the positive smooth coefficients, spatial ODE and flat ramps make parameters smooth; convolving the C1 raw profile by a fixed spatial kernel makes the new field smooth within each visit. At row resets v and all ramp traces vanish; Q remains continuous. At full-cycle boundaries Q=0 but Q' contains an average which may jump. The current grammatical subject is the new correction, and the spatial-mollification qualification is explicit; no time smoothness of the arbitrary old seed is claimed.

The actual scan parameters are smooth on each visit and the complete-cycle primitive is smooth on its time pieces. Fixed spatial convolution of the new packet makes all spatial derivatives needed for repeated time chain rules available. The new correction is continuous at zero packet/corrector endpoints, though time derivatives may jump there; no time smoothness of the old background is claimed.

Given an old distributional Euler--Reynolds seed $(u,p,R)$, set $\widetilde u=u+v+Q$.

Addition is distributionally meaningful for the stated old seed and the new finite-norm fields; div v=div Q=0 ensures the new velocity remains divergence-free. The following products are controlled once the old global L3 assumption is applied.

Adding the packet and divergence-free projected corrector defines the new velocity as a locally L2 distribution, with the required continuity and global increment norms established in the local construction.

Adding the divergence-free new packet and corrector to the old velocity is meaningful in local L2 and yields a divergence-free candidate; subsequent stress expansion lists every interaction.

Adding the divergence-free packet and projected corrector preserves div u=0. All quadratic products are locally integrable and, under the later admissible L3 hypothesis, the displayed cross stresses are globally L1.

Combining \eqref{eq:packet-source}, \eqref{eq:average}, and the equation for
$Q$ gives, modulo a gradient,
\[
 \partial_t\widetilde u+\Div(\widetilde u\otimes\widetilde u)
 =\Div\widetilde R,
\]
\begin{align*}
 \widetilde R={}&R+A_{\rm av}-qI+E
 +u\otimes v+v\otimes u+u\otimes Q+Q\otimes u\\
 &+v\otimes Q+Q\otimes v+Q\otimes Q.
\end{align*}

Adding old, packet and corrector equations leaves div(R+E+A_av). Expanding (u+v+Q)^2 adds both orders of every mixed product and Q tensor Q; v tensor v is already in its packet equation. Subtracting qI from stress and q from the combined pressure preserves equality, giving precisely the displayed formula.

Expand (u+v+Q) tensor itself. Subtract the u and v equations and use Q'+U+grad p_Q=Div A_av; the remaining seven mixed/Q tensors are exactly those displayed. Subtracting qI from the stress is compensated by subtracting q from pressure.

Add the old equation, packet equation and Q equation. U cancels, Ubar=div A_av remains, and expansion adds exactly u-v, u-Q, v-Q pairs and Q tensor Q. Replacing the stress by its value minus qI requires pressure minus q, producing the displayed identity.

Add old ER, packet-source, and Q identities: the U terms cancel and Ubar=div A_av. Expanding (u+v+Q)^2 leaves exactly the two u/v, two u/Q, two v/Q tensors and Q^2; subtracting qI from stress is compensated by subtracting q from pressure.

The scalar $q$ is absorbed into pressure.

Since div(qI)=grad q, changing R0 to R0-qI requires changing total pressure p0 to p0-q. This is the sign explicitly confirmed again at S0318.

Div(qI)=grad q. Thus replacing the stress by stress-qI and pressure by pressure-q preserves the Euler-Reynolds equation.

div(qI)=grad q; moving -qI into the stress is accompanied by subtracting q from the intermediate pressure, so divergence-free testing is unchanged.

div(qI)=nabla q for the scalar q=A+B. If the initial combined pressure is p_sum, the displayed stress R+A_av-qI+... uses p_sum-q, preserving the ER sign convention.

 Thus $\widetilde R=R-S+E_{\rm total}$,
where the averaging error is $S+A_{\rm av}-qI$ and all remaining terms are
shown explicitly.

Add and subtract S in the stress identity: R+A_av-qI+others=R-S+[S+A_av-qI+others]. The bracket lists the averaging residual, intrinsic E and all mixed/quadratic corrector terms, with none missing.

Add and subtract S in the displayed stress. The term S+A_av-qI is the quadrature error because the target directional tensor is -S+qI; all remaining terms belong to the stated total error.

Add and subtract S in the exact stress expression. Since -S+qI is the target coefficient sum, its mismatch with A_av is S+A_av-qI; all cross and intrinsic terms remain explicitly in E_total.

The identity -S+qI=sum a_i xi_i tensor xi_i implies the residual S+A_av-qI. Adding it and every intrinsic/cross term to R-S reproduces S0244 exactly; none is omitted.