==== TABLE OF CONTENTS ====
Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//00-introduction.tex
Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//01-geometric.tex
Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//02-entropy.tex
Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//03-projection.tex
Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//04-aspect.tex
Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//05-packets.tex
Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//06-extremal.tex
Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//07-global.tex
Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//08-consequences.tex
Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//adjacency-radii.tex
Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//capacity-tests.tex
Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//packet-proof-map.tex
Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//paper.tex
Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//preamble.tex
Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//references.bib
===========================

==== BEGIN Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//00-introduction.tex ====
\begin{abstract}
We prove the bounded-data Fourier restriction conjecture for the sphere in
three dimensions: the Fourier extension operator maps $L^\infty(S^2)$
boundedly into $L^p(\R^3)$ for every $p>3$. This is the full conjectured
open range, and the threshold $p=3$ is sharp.
\end{abstract}

\section{Introduction}
\label{sec:introduction}

Let \(S^2=\{\omega\in\R^3:|\omega|=1\}\), and let \(\sigma\) be surface
area measure, with \(\sigma(S^2)=4\pi\). For a bounded measurable
function \(g:S^2\to\C\), define
\[
  Eg(x)=\int_{S^2}g(\omega)e^{2\pi i x\cdot\omega}\,d\sigma(\omega).
\]
The bounded-data restriction problem asks for the exponents $p$ for which
this extension operator maps $L^\infty(S^2)$ boundedly into
$L^p(\R^3)$. We prove the full conjectured open range.

\begin{theorem}\label{thm:main}
For every real \(p>3\), there is a finite constant \(C_p\), depending
only on \(p\), such that
\[
  \|Eg\|_{L^p(\R^3)}
  \le C_p\|g\|_{L^\infty(S^2,\sigma)}
\]
for every bounded measurable complex-valued function \(g\) on \(S^2\).
\end{theorem}

The exponent range is sharp. For the constant function \(g=1\),
direct integration gives
\[
  E1(x)=\frac{2\sin(2\pi |x|)}{|x|}\qquad (x\ne0).
\]
The radial integral of \(|E1|^p\) diverges for \(0<p\le3\).
Thus Theorem~\ref{thm:main} proves exactly the conjectured open range;
it makes no endpoint assertion.

\subsection{The restriction problem and earlier methods}

The restriction problem originates in Stein's work, and the
Tomas--Stein theory gives the classical $L^2(S^2)\to L^4(\mathbb R^3)$
extension estimate~\cite{Stein1979,Tomas1975,Tomas1979}.
The oscillatory-integral work of Carleson--Sj\"olin and H\"ormander
\cite{CarlesonSjolin1972,Hormander1973}, Fefferman's ball-multiplier
obstruction~\cite{Fefferman1971}, and C\'ordoba's geometric multiplier
estimates~\cite{Cordoba1975} established important parts of the
analytic and geometric background. Bourgain's connection between
restriction and tube geometry~\cite{Bourgain1991Besicovitch} made
concentration of wave packets a central issue: curvature separates
packet directions, but many tubes can still pass through a small
region of space.

The bilinear framework of Tao--Vargas--Vega
\cite{TaoVargasVega1998} isolates interactions between separated
frequency pieces and reassembles them across separation scales by
rescaling and almost orthogonality. Wolff's cone theorem
\cite{Wolff2001} and Tao's paraboloid theorem~\cite{Tao2003}
developed this approach through wave packets and induction on scale.
In Tao's argument, a constraint from the Fourier analysis restricts
the directions of the packets that must be counted geometrically;
this supplies the additional structure needed to adapt Wolff's
counting argument to the paraboloid.

The multilinear theorem of Bennett--Carbery--Tao
\cite{BennettCarberyTao2006} controls interactions among surface
pieces whose normals uniformly span the ambient space. In three
dimensions it bounds the geometric mean of three extensions in
\(L^3\) on a ball, using the \(L^2\) norms of the three inputs and
an arbitrarily small positive power of the radius.
Bourgain--Guth~\cite{BourgainGuth2011} made this useful for the
linear problem by separating transverse interactions from
concentration in one small frequency region and from coplanar
interactions. Their argument treats the latter configurations by
rescaling and square-function estimates. This progression makes
the concentration of packet directions, as well as their
transverse intersections, a central part of the restriction problem.

Guth's polynomial-partitioning argument~\cite{Guth2016} divides the
extension integral between cells and a neighborhood of an algebraic
surface. Away from that neighborhood, each packet tube meets few of the
trimmed cells, allowing induction; the remaining concentration is
organized by tubes tangent to the surface. This gives the bounded-data
estimate for $p>13/4$ on a compact smooth positively curved surface $S$.
Shayya's weighted estimates~\cite{Shayya2017} give estimates with finite
$L^q(S)$ input norms in this output range. Kim~\cite{Kim2017} treats
these finite-input estimates through rescaling and interpolation. Wang's broom
method~\cite{Wang2022Brooms} then exploits a further constraint on the
truncated paraboloid: tubes that collect near one region spread toward
others, which limits repeated concentration near separated surface
neighborhoods. Combined with polynomial partitioning and a two-ends
argument, it yields the bounded-data estimate for $p>42/13$.

Wang and Wu~\cite{WangWu2022,WangWu2024} combine incidence estimates
with refined decoupling. The latter controls oscillatory interaction
using the number of packets meeting each small ball, while the incidence
argument retains how the occupied part of a tube is distributed along
its length. Their two-ends Furstenberg inequalities use this spacing
information to improve the multiplicity bound. In~\cite[Theorem~0.2]{WangWu2024}
they obtain the diagonal estimate
$L^p(S)\to L^p(\mathbb R^3)$ for $p>22/7$ on compact positively curved
$C^2$ surfaces, including boundary. The input norm matters in this
comparison: a diagonal estimate controls all $L^p(S)$ data, whereas
Theorem~\ref{thm:main} is stated for bounded sphere data. The latter
reaches the full open output range $p>3$; its sphere-specific
factorization consequence is discussed separately below.

\subsection{Elliptic capacity and the geometric obstacle}

The geometric and oscillatory estimates use the same eccentricity-biased
weight. For an ellipse \(E\subset\R^2\) with semiaxes \(L\ge w>0\), set
\[
 W(E)=L^{1+\kappa}w^{1-\kappa},\qquad 0<\kappa<1/10.
\]
A joint test for two vector coordinates uses translates of the same
ellipse, with the two centers chosen independently. It therefore
measures concentration of their joint law without assuming independence.
Equivalently, $W(E)=Lw(L/w)^\kappa$: the usual area weight is
multiplied by a power of the eccentricity. Differently oriented tests
intersect in smaller rectangles. In the hairbrush estimate of
Lemma~\ref{geo:hairbrush}, the extra exponent $\kappa$ turns the
resulting overlap count into a power saving. The propagation argument
uses that saving to control how concentration can change as a line moves.

Let \(M\) be a large dyadic scale, and consider a joint law of uniformly
bounded line parameters \((U,V)\) and a time \(T\in[0,1)\), with
\(X_t=V+tU\). Suppose its initial joint tests obey
\(\Pr(U\in u+E,V\in v+E)\le K W(E)\). The time hypothesis is
conditional on a base label that determines \(U,V\): at interval length
\(M^{-y}\), for \(0\le y\le1\), both the number of occupied time bins and the largest bin
probability have upper bounds with exponents \(sy\) and \(-sy\),
respectively, up to a small error. Here \(s\in(0,1]\).

At the terminal resolution let \(T_1\) be the left endpoint of the
time bin of length \(M^{-1}\) containing \(T\). A propagation exponent
\(q\) means that a dominated sublaw \(\nu\) satisfies
\[
 \nu(T_1=t,\ U\in u+E,\ X_t\in x+M^{-1}E)
       \le M^{-q+o(1)}K W(E)
\]
simultaneously for all terminal bins and independent centers \(u,x\).
The retained mass is \(M^{-o(1)}\), meaning that it exceeds every fixed
negative power for sufficiently large \(M\). Theorem~\ref{geo:main}
supplies every strict exponent below \(1+2s-10\kappa\) when the time
error is sufficiently small. That theorem specifies the input and
terminal ellipse scales and all conditional quantifiers.

\paragraph{Inputs and methodological precedents.}
The planar input belongs to the projection and Furstenberg-set
tradition of Marstrand and Kaufman~\cite{Marstrand1954,Kaufman1968},
Bourgain's discretized projection theorem~\cite{Bourgain2010}, and
Oberlin's work on exceptional projections~\cite{Oberlin2012}.
Packing-dimension advances of Orponen and Shmerkin
\cite{Orponen2020Packing,Shmerkin2022Packing}, their Hausdorff-dimension
improvement~\cite{OrponenShmerkin2023Hausdorff}, and their work relating
projections, Furstenberg sets, and the $ABC$ sum-product problem
\cite{OrponenShmerkin2026ABC} provide further context for Ren--Wang's
sharp finite incidence theorem~\cite{RenWang2023}. In the finite form
used here, each point of a planar family is incident to a comparably
sized family of thin tubes. Counts in smaller balls and angular
intervals control concentration of the points and tube directions.
Ren--Wang's theorem gives a lower bound for the total number of
distinct tubes. Drawing lines of constant projection through the
points converts this into a bound for projected supports. We prove
the weighted, conditional, and joint-coordinate consequences needed
for our laws.

The proof uses finite entropy in the sense of Shannon
\cite{Shannon1948} and the entropy rules organized by Tao
\cite{Tao2010Entropy}. The local-entropy approach of
Hochman--Shmerkin~\cite{HochmanShmerkin2012} and Hochman's inverse
principles~\cite{Hochman2014} are related methodological precedents;
our correlated conditional laws are handled by the finite arguments
below. The radial step is closely related to the reciprocal thin-tube
bootstrap of Orponen--Shmerkin--Wang
\cite{OrponenShmerkinWang2024Radial}. Hairbrush intersection counting
\cite{Wolff1995Kakeya,TaoVargasVega1998} and the density and factoring
arguments of Wang--Zahl and Guth--Wang--Zahl
\cite{WangZahl2025Convex,GuthWangZahl2026} supply further geometric
context. The independently translated elliptic capacity and its
propagation through arbitrary finite complex arrays and coordinate
masks are the specific estimates established here. The external
analytic input is Bourgain--Demeter's $\ell^2$ decoupling theorem
\cite{BourgainDemeter2015}. Section~\ref{sec:packets} proves the
canonical-packet refinement needed here, following the grouping and
rescaling method of Guth--Iosevich--Ou--Wang~\cite{GIOW2020} and
the localized extension formulation of Wang--Wu~\cite{WangWu2024}.
Besides these decoupling and finite-incidence inputs, the proof uses
classical almost-everywhere differentiation
\cite{Tao2011Measure,Heinonen2005Lipschitz}.

\subsection{The finite packet theorem}

The packet estimate underlying the sphere theorem applies to arbitrary
finite complex coefficient arrays. We summarize its definitions before
stating the uniform bound; Section~\ref{sec:packets} develops the frame
identities and norm properties in detail.

On a graph patch of the sphere, absorbing the surface-area density
into the data gives the phase \(x\cdot\xi+t\phi(\xi)\), with
\(\xi\in\R^2\). Rescaling a physical region of size \(N^2\) gives
the factor \(e^{2\pi iN^2(x\cdot\xi+t\phi(\xi))}\). A packet near
frequency \(\nu\), centered at \(z\) at time \(t_I\), is concentrated
near the line
\[
 x=z+(t-t_I)U_\nu,\qquad U_\nu=-\nabla\phi(\nu).
\]
Thus its frequency and position labels specify the velocity and
center tested by the elliptic capacity. The phase is continued
smoothly beyond the patch as specified in the theorem below.

For a fixed dyadic \(N\), a dyadic factor \(q\in[1,N]\) determines the
frequency width \(\theta=q/N\), time-bin length \(r=q^{-2}\), and
spatial mesh parameter \(w=(qN)^{-1}\). Choose a fixed smooth,
nonnegative, compactly supported window \(\chi\) whose integer
translates satisfy \(\sum_{n\in\mathbb Z^2}\chi(\xi-n)^2=1\).
Let \(L_{\mathrm f}\) be a fixed side length containing its support.
A bin \(I\) of length \(r\), with left endpoint \(t_I\), has packet
labels \((\nu,z)\in\theta\mathbb Z^2\times
L_{\mathrm f}^{-1}w\mathbb Z^2\). With
\(\chi_\theta^\nu(\xi)=\chi((\xi-\nu)/\theta)\), the exact maps for
the fixed phase \(\phi\) in the theorem below are
\begin{align*}
 (\mathsf P_{q,I}f)_{\nu,z}
 &=\int f(\xi)\chi_\theta^\nu(\xi)
       e^{2\pi iN^2(z\cdot\xi+t_I\phi(\xi))}\,d\xi,\\
 \mathsf S_{q,I}c(\xi)
 &=(L_{\mathrm f}\theta)^{-2}\sum_{\nu,z}
 c_{\nu,z}\chi_\theta^\nu(\xi)
       e^{-2\pi iN^2(z\cdot\xi+t_I\phi(\xi))}.
\end{align*}
Analysis thus takes unnormalized Fourier coefficients in each window;
synthesis has the corresponding inverse Fourier-series factor.

For an array \(c=(c_v)\) in one bin, indexed by
\(v=(\nu_v,z_v)\), put
\begin{align*}
 K(c)&=\theta^2\sup_{E,u,z}\frac{1}{W(E)}
       \sum_{\substack{U_{\nu_v}\in u+E\\z_v\in z+rE}}|c_v|^2,\\
 G(c)^3&=w^2\Bigl(\sum_v|c_v|^2\Bigr)K(c)^{1/2},\qquad
 A(c)=\inf_{|c|\le\sum_{j=1}^k|a_j|}\sum_{j=1}^kG(a_j).
\end{align*}
The supremum uses ellipses with both semiaxes at least \(\theta\),
and independent centers \(u,z\); the infimum uses finite arrays and
coordinatewise domination. The squared coefficients place mass
\(|c_v|^2\) at the velocity--position pairs \((U_{\nu_v},z_v)\).
Thus \(K\) measures joint concentration,
\(G\) combines that concentration with squared coefficient mass, and
the atomic norm \(A\) allows finite decompositions into arrays of
controlled \(G\)-cost. Subscripts below specify the scale and bin.

\begin{theorem}[Uniform propagation of finite packet arrays]\label{thm:intro-packets}
Fix \(0<\kappa<1/10\), a smooth elliptic continuation \(\phi\) of a
spherical graph patch allowed in Section~\ref{sec:packets}, a bounded
frequency-label region \(\Omega\subset\R^2\), and the fixed window system
used in \eqref{pkt:analysis}--\eqref{pkt:synthesis}.  The continuation is
defined on \(\R^2\), has uniformly definite bounded Hessian, and has
bounded derivatives of every order at least two.

For every finite \(B\ge1\) and every \(\varepsilon>0\), there is a finite
constant \(C\) with the following property.  Let \(m,N\) be dyadic with
\(1\le m\le N\le m^B\), and let \(d\) be any finite complex array on the
initial packet grid, at factor \(1\).  For each interval \(I\) in the
partition of \([0,1)\) into intervals of length \(m^{-2}\), let \(M_I\)
be any coordinate projection retaining finitely many entries of the
factor-\(m\) packet grid.  Assume that every initial index in
\(\operatorname{supp}d\) and every retained terminal index has frequency
label \(\nu\in\Omega\) and position label \(|z|\le m^B\).  Set
\[
 (T_md)_I=M_I\,\mathsf P_{m,I}\mathsf S_{1,[0,1)}d.
\]
Write \(A_1,G_1\) for the atomic norm and capacity gauge at the initial
scale
\[
 (\theta,r,w)=(N^{-1},1,N^{-1}),
\]
and \(A_{m,I}\) for the same atomic norm in the terminal bin \(I\), at
scale
\[
 (\theta,r,w)=(m/N,m^{-2},(mN)^{-1}).
\]
Then
\[
 m^{-2}\sum_{\substack{I\subset[0,1)\\ |I|=m^{-2}}}
       A_{m,I}\bigl((T_md)_I\bigr)^3
 \le C m^{10\kappa+\varepsilon}A_1(d)^3
 \le C m^{10\kappa+\varepsilon}G_1(d)^3,
\]
where the sum is over the specified partition.  The constant is uniform
in \(m,N,d\) and the coordinate projections.  It may depend on
\(B,\kappa,\varepsilon,\phi,\Omega\), and the fixed windows.
\end{theorem}

The maps retain the complete transition matrices between the selected
coordinate sets.  Each projection deletes coordinates; it does not
delete selected summands within a retained coefficient.  The initial
array need not be the analysis array of a function.  Frequency labels
may lie anywhere in the fixed region \(\Omega\) of the chosen
continuation, including its transition annulus.  The phase and its
derivative bounds are fixed before \(m,N\) vary.

Theorem~\ref{pkt:main} proves this estimate, with the atomic-input
formulation supplied by Lemma~\ref{pkt:atomic}.  The scale subscripts
above only specify which of the norms from Section~\ref{sec:packets}
is being used.

\subsection{Consequences and related estimates}

For the sphere, Bourgain's factorization theorem
\cite{Bourgain1991Besicovitch}, in the weak-type and interpolation
form explained in~\cite[Section~1.1, (1.1)--(1.2)]{Buschenhenke2024},
upgrades the full family in Theorem~\ref{thm:main} to
\[
 \|Eg\|_{L^q(\R^3)}\lesssim_q\|g\|_{L^q(S^2)}
 \qquad(3<q<\infty).
\]
This controls finite-$L^q$ sphere data as well as bounded data.
Interpolation also gives the open mixed-norm range in
Section~\ref{sec:consequences}.
The factorization is sphere-specific; the general-surface application
below uses the packet theorem itself.

\paragraph{A Kakeya maximal consequence.}
The sphere conclusion gives another route to the strong maximal estimate
proved independently in~\cite[Theorem~1.1]{OpenAIKakeya2026}.
For \(a\in\R^3\), \(\omega\in S^2\), and \(0<\delta<1\), put
\begin{align*}
 T_\delta(a,\omega)
 &=\{a+t\omega+u:|t|\le\tfrac12,\ u\perp\omega,\ |u|\le\delta\},\\
 K_\delta f(\omega)
 &=\sup_{a\in\R^3}\frac1{\pi\delta^2}
                  \int_{T_\delta(a,\omega)}|f(x)|\,dx.
\end{align*}
Thus the tubes have unit length and volume \(\pi\delta^2\).

\begin{corollary}\label{cor:sphere-kakeya-maximal}
For every \(\varepsilon>0\) there is a finite \(C_\varepsilon\) such that,
for every \(0<\delta<1\) and \(f\in L^3(\R^3)\),
\[
 \|K_\delta f\|_{L^3(S^2,\sigma)}
 \le C_\varepsilon\delta^{-\varepsilon}\|f\|_{L^3(\R^3)}.
\]
\end{corollary}

Sphere factorization, weighted tube duality, and interpolation give
this corollary; the complete transfer is proved in
Section~\ref{sec:consequences}. The supremum over tube locations is
part of the assertion, so it controls translated tube averages with
no restriction on where the tubes meet. The scale loss is an arbitrary
positive power. In particular, the standard implication from maximal
estimates gives Hausdorff dimension $3$ for every set containing a
unit line segment in every direction~\cite[Theorem~2.12]{MattilaFourierNotes}.
That dimension conclusion was already proved by Wang and
Zahl~\cite[Theorem~1.1]{WangZahl2025Convex}; their result also gives
Minkowski dimension $3$.

\paragraph{The full packet theorem on other surfaces.}
The full array formulation also has a separate application.
The companion article~\cite[Theorem~1.1]{OpenAIDiagonal2026} combines it with a
translated-tube Kakeya maximal estimate to prove
$L^p(\Sigma)\to L^p(\mathbb R^3)$ extension for every $3<p<\infty$
on each compact smooth positively curved surface, possibly with smooth boundary.
For spheres and quadratic graphs, bounded-data-to-diagonal
factorization already has an established history
\cite{Bourgain1991Besicovitch,Buschenhenke2024}; the companion treats
general elliptic charts using the full packet theorem and an explicit
phase continuation. Recent smooth Alpert testing formulations
\cite{Sawyer2026TestingComparison,RiosSawyer2026ConvolutionTesting}
study extension through specified testing conditions. The conclusions
here concern deterministic complex arrays and all bounded measurable
sphere data, with the fixed phase, window, and complexity parameters
specified in the theorem.

\paragraph{An independent Bochner--Riesz route.}
The bounded-data spherical estimate in Theorem~\ref{thm:main} also
follows independently from the Bochner--Riesz multiplier theorem
in~\cite[Corollary~12.4]{OpenAIBochnerRiesz2026}, through Tao's
implication~\cite{Tao1999Epsilon}. This implication does not recover
the capacity and packet estimates proved here.

\subsection{The argument}

The eccentricity bias in \(W\) limits the mass that can concentrate
simultaneously in differently oriented tests. The proof develops this
fact for conditional laws, applies the resulting propagation theorem
to packet arrays, and finally removes the spatial power loss.

\paragraph{Geometric propagation.}
Suppose the retained-law bound above fails. Extremizing over relative
spatial scales selects tests whose mass is large compared with their
capacity weight at successive time resolutions. Writing their radii as
powers of \(M\) gives two logarithmic width-depth functions of time
depth. The hairbrush bound
controls differences between their axes, assigning each sample of
the joint law a reference axis on a short interval of scales.
Near a differentiability point of the two width-depth functions, their
rates describe whether the aspect ratio is stationary, increasing,
or decreasing. This is the calibrated width path constructed in
Section~\ref{sec:geo}.

We record conditional Shannon entropies of velocity, position, time,
and the reference axis, rounding these variables at compatible
resolutions. Time and axis may be correlated. The laws are first
regularized on finite grids, so the limiting entropy increments
retain the support counts and probability bounds required by the
projection theorems. Those theorems force lower bounds on entropy
growth. In each of the three aspect cases, the bounds contradict the
concentration required by failure of geometric propagation.

\paragraph{Propagation of wave-packet bounds.}
The exact frame maps compose between packet scales, and the atomic
norm lets us combine estimates despite intermediate coordinate
restrictions. Insert finitely many intermediate scales and form a
layered graph whose vertices are the retained packet indices. An edge
records that a child frequency and center lie in a parent's
localization neighborhood. A path is a sequence of such indices from
an initial packet to a terminal one; the coefficients still evolve
by the full masked matrices.

Suppose amplification has a growth exponent greater than
\(10\kappa\). Among families approaching the largest exponent, first
take the infimum \(s\) of the exponents \(\sigma\) for which each
initial index reaches at most \(m^{2\sigma+o(1)}\) terminal time bins.
This infimum is positive; it need not be attained.
Then minimize the number of paths per original initial index, again
at the level of scale exponents. The second extremum forces later
coordinate selections to retain a subpower fraction of those paths
whenever they preserve nearly maximal amplification. Uniform counting
on selected paths then gives both conditional prefix counts and
maximal prefix probabilities with exponent \(s\) in the geometric base
\(M=m^2\). The initial packet index is the base label and determines
the line parameters \((U_\nu,z)\). This is the law to which geometric
propagation applies.

The geometric bound controls output capacity. Refined decoupling
controls a fourth moment, while the frame identity gives a second
moment bound. Their combination cancels the possible time-support
deficit and leaves a growth exponent at most \(10\kappa\).
Figure~\ref{fig:packet-proof-map} shows how the two extrema lead to
the conditional time law and the capacity--mass cancellation.

\paragraph{Removing the spatial loss.}
The atomic norm controls the normalized sum of cubed packet
coefficients. For actual graph-patch data, terminal coefficients are
samples of the localized extension, up to rapidly decaying truncation
errors. Averaging the sampling lattice therefore gives a cubic
integral estimate. Taking \(\kappa\) small makes its spatial power
loss arbitrarily small. On widely separated balls, a joint frame
estimate bounds the total initial coefficient mass without a factor
for the number of balls.
A covering of the large-value set, followed by Fourier reproduction
and layer integration, yields the global estimate for every \(p>3\).
This last step works directly for bounded measurable complex data.

\paragraph{Structural statements within the proof.}
The geometric and entropy arguments also yield structural statements.
The matrix form of the ellipse weight
obeys the composition inequality \eqref{geo:submultiplicativity}.
Corollary~\ref{geo:axis-prediction} determines a sample's axis, up to
the corresponding aspect precision, from its exact line parameters
and time prefix. Lemmas~\ref{ent:peeling} and~\ref{ent:signed-envelope} describe
additive parts of the entropy's local limits and lower bounds when
some resolutions increase along a path and others decrease. They
also apply to the specified paths on boundary faces of the admissible
resolution domain.

\subsection{Organization and conventions}

Sections~\ref{sec:geo}--\ref{sec:aspect} prove the geometric
propagation theorem. Section~\ref{sec:geo} constructs the calibrated
path; Section~\ref{sec:entropy} develops its entropy profiles and
tangent rules; Section~\ref{sec:projection} proves the projection
consequences of the planar Furstenberg theorem; and
Section~\ref{sec:aspect} treats the three possible directions of
aspect change.
Sections~\ref{sec:packets} and~\ref{sec:extremal} prove the
wave-packet estimate, and Section~\ref{sec:global} concludes the proof
of Theorem~\ref{thm:main}. Section~\ref{sec:consequences} gives the
sphere-to-Kakeya transfer and the mixed-norm sphere estimates.

Functions may be complex-valued, and probability laws may have
arbitrary correlations, unless a more restrictive hypothesis is
explicitly stated. Ellipse containments suppress independent translations
when two spatial coordinates are tested. Fixed multiplicative
constants are harmless in scale exponents. The geometric and
packet sections specify their limiting conventions separately;
in each case finite grids and fixed complexity are chosen before
the large-scale limit. A \emph{subpower} loss denotes a factor
\(M^{o(1)}\), or \(m^{o(1)}\), in the base currently under discussion.
==== END Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//00-introduction.tex ====

==== BEGIN Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//01-geometric.tex ====
\section{Geometric propagation and a calibrated path}\label{sec:geo}

\subsection{Ellipse capacity and the propagation exponent}

We parametrize lines by \(P=(U,V)\in\R^2\times\R^2\), and write
\[
X_v=V+vU.
\]
Throughout this section, two tests using the same ellipse may have
independently chosen centers.  Thus \(U\in E,\ V\in E\) abbreviates
\(U\in u+E,\ V\in v+E\), with arbitrary \(u,v\in\R^2\); it does not
require the two centers to coincide.

Fix \(0<\kappa<1/10\).  For a centered ellipse \(E\) with radii
\(L\ge w>0\), define
\[
W(E)=L^{1+\kappa}w^{1-\kappa}.
\]
We call \(W(E)\) its capacity weight.  Rectangles with the same
semiaxes may replace ellipses in upper bounds, at the cost of an
absolute constant.  The same is true of dilations by bounded factors.
In particular, a constant enlargement of a tested ellipse is covered
by boundedly many tests at the original widths.

Let \(M=2^n\), with \(n\to\infty\).  A time test at depth \(y\) uses
the dyadic intervals of length \(2^{-\lfloor yn\rfloor}\).  For
\(T\in[0,1)\), let \(T_y=[T]_y\) denote the left endpoint of its
interval at this depth.  Dyadic roundings change the scales below
by bounded factors only.

\Needspace{4\baselineskip}
\begin{definition}\label{geo:large-sublaw}
A \emph{sublaw} of a probability measure \(\mu\) is a finite measure
\(\nu\le\mu\).  Along a sequence with base \(M\to\infty\), a sublaw
has \emph{large mass} if
\[
\nu(\text{whole space})=M^{-o(1)}.
\]
Normalizing such a sublaw changes every upper mass bound by at most
a factor \(M^{o(1)}\).  We use either its unnormalized masses or its
normalized probabilities, recording this factor when necessary.
\end{definition}

All probability spaces may be taken to be standard Borel.  Auxiliary
labels are retained under restriction and pushforward; only the
displayed spatial and time tests enter the mass bounds.  At each
finite stage, compact coordinate windows are discretized by finite
dyadic grids.  Orientations at widths \(L,w\) are discretized at
spacing comparable to \(w/L\).  A translated ellipse of arbitrary
orientation is covered by boundedly many constant enlargements of
grid cells at these matched scales.  Conversely, each grid cell is
covered by boundedly many continuous tests.  All assignments of
heavy cells can therefore be made from finite lists.

Fix \(s\in(0,1]\), and initially allow a positive error \(\epsilon\)
in the time distribution.  There is a base label determining
\(P=(U,V)\), such that, conditionally on this label, almost surely,
\begin{equation}\label{eq:source-1}
\begin{split}
\max_{I\in\mathcal D_j}
 \Pr(T\in I\mid\text{label})
 &\le M^\epsilon 2^{-sj},\\
\#\{I\in\mathcal D_j:
 \Pr(T\in I\mid\text{label})>0\}
 &\le M^\epsilon 2^{sj},
\qquad 0\le j\le n,
\end{split}
\end{equation}
where \(\mathcal D_j\) is the dyadic partition at depth \(j\).
Testing a sufficiently fine fixed grid of ratios \(j/n\) is
equivalent to testing every depth, with a corresponding arbitrarily
small increase of \(\epsilon\): interpolate the upper weight bound
from the preceding grid depth and the support bound from the
following one.

Suppose that \(P\) remains in a bounded set, uniformly along the
sequence, and that for a fixed finite \(D\ge1\),
\begin{equation}\label{eq:source-2}
\Pr(U\in E,\ V\in E)\le K W(E),
\qquad M^{-D}\le w\le L\le1.
\end{equation}
The constant \(K\) may depend on the instance.  For an exponent \(q\),
the propagation assertion is the existence of a large sublaw on
which, simultaneously for all indicated times, centers, and ellipses,
\begin{equation}\label{eq:source-3}
\Pr(T_1=v,\ U\in E,\ X_v\in M^{-1}E)
\le M^{-q+o(1)}K W(E),
\qquad M^{-(D-1)}\le w\le L\le1.
\end{equation}
Figure~\ref{fig:capacity-tests} depicts the input and terminal
tests.  The shared ellipse orientation imposes no independence
condition on the law of the two spatial variables.
\input{figures/capacity-tests}

Let \(q(\epsilon,D)\) be the supremum of the exponents for which this
assertion holds universally along sequences satisfying
\eqref{eq:source-1} and \eqref{eq:source-2}.  Define
\[
q_*=\lim_{\epsilon\downarrow0}\inf_{1\le D<\infty}q(\epsilon,D).
\]
The limit exists by monotonicity in the admissible time error.
Constants in the coordinate bounds, or bounded changes in the
coordinate ranges, are immaterial by covering and isotropic
normalization.

\Needspace{3\baselineskip}
\begin{theorem}[Geometric propagation]\label{geo:main}
For \(0<s\le1\) and \(0<\kappa<1/10\),
\begin{equation}\label{eq:source-4}
q_*\ge1+2s-10\kappa.
\end{equation}
\end{theorem}

The estimate concerns arbitrary joint laws of \(U,V,T\), with the
base label determining \(U,V\).  Both bounds in
\eqref{eq:source-1} are conditional on that label and hold at every
dyadic depth.  The capacity in \eqref{eq:source-2} is one joint test,
with independent centers and a shared ellipse shape.  The conclusion
allows a dominated sublaw; it need not hold for the original law.
For example, if \(U\) is uniform on a fixed disk, \(V=0\), and \(T\)
is independently uniform, a unit test in the first time bin has mass
of order \(M^{-1}\).  Restricting \(T\) to \([1/2,1)\) removes this
focusing event and gives the corresponding \(M^{-3}W(E)\) bound.

\paragraph{Route through the geometric proof.}
We argue by failure of Theorem~\ref{geo:main}.  The following four stages
organize the argument.
\begin{enumerate}[label=\textup{(\roman*)}]
\item In this section, heavy-box counts and angular exclusion reduce
failure to a calibrated path of ellipse widths and reference axes.
The initial complexity \(D\) may increase as the time error decreases;
the bounded-complexity reduction precedes the local limit.
\item Section~\ref{sec:entropy} encodes that obstruction by finite
sheared states.  Regularization supplies hereditary support and weight
bounds before entropy profiles and their tangents are formed.
\item Section~\ref{sec:projection} derives scalar and joint-coordinate
rate inequalities from the planar incidence theorem, including the
case of correlated time and axis parameters.
\item Section~\ref{sec:aspect} excludes every possible aspect velocity
by a backward comparison whose gain exceeds the calibrated rate.
\end{enumerate}
The heavy ellipses and their calibrated path belong to this
counterfactual failure construction.  The universal conclusion is the
retained-law estimate \eqref{eq:source-3} at every strict exponent
\(q<q_*\).

\subsection{Sequence conventions and conditional time profiles}

We first make the order of limits precise.  A statement valid for
every sequence, with subpower retention and subpower losses, is
uniform with arbitrarily small fixed power losses once the base
is sufficiently large and the other parameters are fixed.
Otherwise, selecting a violating instance at each of arbitrarily
large bases would give a violating sequence.  Conversely, the
fixed-power statements yield subpower losses by a diagonal choice.
This applies simultaneously to any fixed finite collection of
restrictions or estimates.  Passing to subsequences is permitted
throughout.

For clarity, the recurring limit choices are summarized here.  Each
row fixes a finite problem before the underlying base is increased;
the local proofs below give the quantitative choices.
\begin{center}
\small
\begin{tabular}{@{}p{.23\linewidth}p{.72\linewidth}@{}}
\toprule
Stage & Order of choices \\
\midrule
Failure sequence & Fix a positive failure margin, finite complexity,
mesh, and list of restrictions; then increase the base. \\
Calibration & Fix the first-order scale and stretched window, then
choose finite-stage errors negligible in \(\delta\log M\), with
\(\delta\log M\to\infty\). \\
Tangent realization & Fix the tangent radius and finite physical
cutoffs; make previous errors negligible in the new logarithmic unit
before shrinking that radius. \\
Projection tests & Fix a positive output gap, then smaller input
nonconcentration and density losses; finally let the block base tend
to infinity. \\
Backward comparison & Fix the tangent problem and any large velocity;
choose clipping and endpoint errors afterward. \\
\bottomrule
\end{tabular}
\end{center}
In particular, for each \(q<q_*\), the definition of \(q_*\) gives
one sufficiently small time-error threshold valid for every finite
\(D\).  The sufficiently large base threshold may depend on that
fixed \(D\).  We will justify the sequence-to-uniform use of this
statement below and retain this distinction in the packet application.

The input bound immediately gives \(q_*\ge0\).  Indeed, a terminal
test in \eqref{eq:source-3} implies
\(V=X_v-vU\in c+O(E)\), since \(0\le v<1\), and hence its mass is
at most a constant multiple of \(KW(E)\).
Suppose for contradiction that
\[
q_*<1+2s-10\kappa.
\]
All exponents under consideration are then bounded; in particular,
they may be taken less than \(4\).

Choose \(\epsilon_i\downarrow0\), finite complexities \(D_i\), and
exponents tending to \(q_*\) from above, for which the universal
assertion fails.  For each fixed \(i\), the complexity \(D_i\) is
fixed and \(n\to\infty\).  We distinguish errors \(o_i(1)\), which
tend to zero as \(i\to\infty\), from \(o_n(1)\), which tend to zero
along the sequence with \(i\) fixed.

We may choose these sequences with a fixed-power margin at each
fixed \(i\).  More precisely, after leaving room above the relevant
supremum, there are positive tolerances tending to zero with \(i\)
such that even a slightly weaker tested exponent cannot be obtained
while retaining at least the corresponding fixed negative power
of \(M\).  To see this, negate the uniform fixed-power formulation
of the sequence assertion.  If every fixed positive tolerance
worked at every sufficiently large base, a diagonal would give
the sequence assertion itself.  Shrinking a tolerance preserves
failure, so the tolerances can be chosen as small as needed.
Consequently, restrictions retaining \(M^{-o_n(1)}\) mass and changes
of \(M^{o_n(1)}\) in the bounds preserve the chosen failure.  This
also explains why a subsequence on which vanishing-loss conclusions
hold cannot remain inside the selected failing sequence.

\begin{lemma}[Reuse of the time profile]\label{geo:time-conditioning}
Suppose a reference law satisfies \eqref{eq:source-1} with error
\(\epsilon>0\).  Fix a finite collection of prefix depths.  After
any fixed finite succession of restrictions retaining
\(M^{-o_n(1)}\) mass, one may remove a negligible part of the
retained measure so that the following statements hold.

Conditionally on each surviving base label, the retained time law
satisfies \eqref{eq:source-1} with error \(O(\epsilon)\).
Conditionally on a surviving base label and a prefix at depth
\(j_0\), the descendant law has maximum weights and support counts
of exponent \(s\), with error \(O(\epsilon)\) in the original
\(\log M\) units.  Further conditioning on parent cells determined
by the base label and prefix does not change this conclusion.
\end{lemma}

\begin{proof}
All comparisons are made directly with the original reference law.
For a fixed base label, let \(p_I\) be the probability of a prefix
\(I\in\mathcal D_{j_0}\), and let \(N_I(j)\) be its number of
supported descendants at depth \(j\ge j_0\).  The hypotheses give
\[
\max_Ip_I\le M^\epsilon2^{-sj_0},
\qquad
\sum_I N_I(j)\le M^\epsilon2^{sj}.
\]
Thus, when the prefix is sampled according to its original law,
\[
\sum_I p_I N_I(j)
\le M^{2\epsilon}2^{s(j-j_0)}.
\]
For a sufficiently large fixed constant \(C\), Markov's inequality
allows the removal of prefixes for which
\[
N_I(j)>M^{C\epsilon}2^{s(j-j_0)},
\]
at original mass at most \(M^{-(C-2)\epsilon}\).
Summing over \(j\le n\) remains power-small, since \(n=M^{o_n(1)}\).

The prefixes for which
\[
p_I<2^{-sj_0}M^{-C\epsilon}
\]
have total original probability at most
\(M^{-(C-1)\epsilon}\), by the prefix support bound.  First remove
these prefixes and the support-count exceptions at every prescribed
depth.  Then, from the deepest prescribed depth to the shallowest,
discard each label-prefix fiber whose current retained fraction,
relative to its original mass, is less than \(M^{-C\epsilon}\).
Finally apply the same retained-fraction test to the base-label
fibers.  Each retained-fraction stage removes at most
\(M^{-C\epsilon}\) times the original total mass.  A later stage
deletes whole fibers at every
previously checked depth, so the retained fraction in each surviving
fiber keeps its lower bound.  The finite union of these losses and
the original-law exceptions is negligible compared with
\(M^{-o_n(1)}\).

Measured in the original conditional law given a surviving base
label, the final retained mass of a surviving prefix is at least
\[
p_I M^{-C\epsilon}\ge 2^{-sj_0}M^{-2C\epsilon}.
\]
Dividing the original descendant weight bound by this retained
denominator therefore gives
\[
\Pr(T\in J\mid\text{retained label and prefix})
\le M^{O(\epsilon)}2^{-s(j-j_0)}
\]
for a descendant \(J\) at depth \(j\).  Its support count is bounded
by the preceding Markov truncation.  Parent cells determined by
the label and prefix select entire such fibers.  Finally, making
every truncation against the reference law keeps the absolute
constants from accumulating under successive restrictions; only
a fixed finite union of power-small exceptional sets is removed.
\end{proof}

On a fixed interval of time-depth length \(\tau>0\), the new base is
\(M^{\tau+o_n(1)}\), and the time error in its log units is
\(O(\epsilon_i/\tau)\).  Since
\(\inf_D q(\epsilon,D)\to q_*\), the universal propagation estimate
is therefore available on that interval with exponent
\(q_*-o_i(1)\).  A sufficiently small time-error threshold works
for every finite complexity at an exponent strictly below \(q_*\).
The sufficiently large base threshold may depend on the fixed
complexity, which is all that is needed here.

When the estimate is applied in many parents, fixed-parameter
uniformity supplies the same subpower relative retention and loss
in all of them.  Width complexities and dyadic base ratios that
vary along a sequence, but remain bounded for fixed \(i\), cause
no difficulty: pass to convergent logarithmic ratios, or use a
finite fine power net.  Apply the estimate at a fixed complexity
slightly larger than required.  If this asks for input tests just
below the available floor, round their widths up to that floor;
the arbitrarily small slack changes the input constant by an
arbitrarily small power only.  No application uses a complexity
escaping to infinity along an \(n\)-sequence with \(i\) fixed.

\begin{lemma}[Isolation of terminal widths]\label{geo:fixed-widths}
The failing families may be chosen so that the terminal widths
\(L,w\) are fixed in logarithmic units along each sequence, up to
arbitrarily small prescribed rounding errors.  The orientation
and the two centers remain unrestricted.  The input constant is
still the original \(K\), up to subpower factors, and the time
profile retains error \(O(\epsilon_i)\).
\end{lemma}

\begin{proof}
For each fixed \(i\), choose a finite power net in the permitted
width range, including its endpoints.  Its step tends to zero
as \(i\to\infty\), and is small compared with the margin in the
selected failure.  Round both widths upward.  This enlarges the
test and increases \(W\) by only the allowed small power.

Seek the desired sublaw successively for each pair in this finite
net, always testing against the original \(K\).  If all steps
succeeded, their successive subpower restrictions would give
\eqref{eq:source-3} for every width, contradicting the selected
failure.  Some step therefore fails on a retained law.  Successful
steps need only proceed along a subsequence, and there are finitely
many steps for fixed \(i\).  Lemma~\ref{geo:time-conditioning}
preserves the time profile, and the fixed-power failure margin
absorbs the finite net and normalization losses.
\end{proof}

\subsection{A hairbrush bound and angular uniqueness}

The time profile is now reusable after the stated trims, and the
terminal widths can be isolated.  We next count heavy boxes and
control all their axes through each surviving point.  The proof uses
the intersection-counting mechanism of classical hairbrush arguments;
see Wolff~\cite{Wolff1995Kakeya} and the discussion in
\cite[arXiv version~1, Section~3.7]{TaoVargasVega1998}.
The joint ellipse estimate required here is proved in full below.

Here a \emph{particle} is an outcome of the full joint law, including
any auxiliary labels.  A \emph{lag} is a time depth \(y\in[0,1]\).
For a possible value \(v\) of \(T_y\), independent centers \(u,x\in\R^2\),
and a centered ellipse \(E\), the corresponding box is the event
\[
C_y(v;u,x;E)
=\{T_y=v,\ U\in u+E,\ X_v\in x+M^{-y}E\}.
\]
Its radii are those of \(E\), measured in the variables
\((U,M^yX_v)\), and its mass includes the time-bin probability.
Auxiliary labels do not enter the membership test.  A source box is
the time-free event \(\{U\in u+E,\ V\in x+E\}\).

\Needspace{3\baselineskip}
\begin{lemma}[Hairbrush estimate]\label{geo:hairbrush}
Let \(\mu\) be a subprobability law on the particles.  Fix a lag
and suppose that every box in a chosen time bin, with independently
translated spatial tests of the same ellipse \(F\), satisfies
\(\mu(C)\le H W(F)\).  Fix radii \(L\ge w>0\), and put
\(A=HW(E)\), where \(E\) has radii \(L,w\).  Assume the cap is
available at all radii between \(w\) and \(L\), including bounded
dilations.

Let \(C_0\) be a box of radii \(L,w\).  Suppose each particle
\(z\) in a measurable set \(S\subset C_0\) belongs to an assigned
box \(C_z\) in the same bin, of the same radii, such that
\[
\mu(C_z)\ge A M^{-e},
\qquad
\theta\le\angle(C_z,C_0)\le2\theta,
\qquad
\theta\gtrsim w/L.
\]
Angles are distances between unoriented long axes.  At fixed
polynomial scale complexity, \(m=\mu(S)\) satisfies
\begin{equation}\label{eq:source-5}
m\le M^{2e+o(1)}H W(E)
       \left(\frac{w}{L\theta}\right)^{2\kappa}.
\end{equation}
The same assertion holds for source boxes, with time omitted.
\end{lemma}

\begin{proof}
Replace ellipses by comparable rectangles.  Within the chosen bin,
write \(Z=(Z^{(1)},Z^{(2)})\) for the two spatial variables; at lag
\(y\), these are \((U,M^yX_v)\).  Translations in the two components
remain independent.  Set
\[
\eta=\frac{w}{L\theta}.
\]
The bounded-factor cases with \(\eta\gtrsim1\) follow directly from
\(\mu(S)\le\mu(C_0)\lesssim A\).  We may therefore suppose
\(\eta\lesssim1\).

For \(z,z'\in S\), let \(dL\) be the larger longitudinal separation
of \(Z^{(j)}(z),Z^{(j)}(z')\), measured in the axis of \(C_0\).
For \(d\gtrsim\eta\), the particles \(z'\) at separation at most
\(dL\) lie in a product rectangle of radii \(O(dL),O(w)\).
Consequently their mass is at most
\[
A d^{1+\kappa}.
\]
Suppose \(d\gg\eta\) and \(C_z\cap C_{z'}\ne\varnothing\).
Choose a component realizing this longitudinal separation.  The
transverse separation of its anchors in the axis of \(C_0\) is
\(O(w)\).  Projection onto a normal to the axis of \(C_z\), using
a common point of the two rectangles and their bounded lengths,
gives
\[
dL\theta\lesssim w+L\gamma,
\qquad
\gamma=\angle(C_z,C_{z'}).
\]
Hence \(\gamma\gtrsim d\theta\).

In each component the intersection is contained in a rectangle
with the axis of \(C_z\) and radii
\[
O\!\left(\frac{w}{d\theta}\right)=O(L\eta/d),
\qquad O(w).
\]
The two component intersections use the same orientation, though
their centers may differ.  The assumed cap thus gives
\[
\mu(C_z\cap C_{z'})
\lesssim A(\eta/d)^{1+\kappa}.
\]
Every radius used lies between \(w\) and \(L\), up to bounded
factors.  This argument only uses that a box contains its anchor;
the anchor need not be its center.

For fixed \(z\), the close pairs \(d=O(\eta)\) contribute at most
\(A^2\eta^{1+\kappa}\) to the overlap integral.  Each remaining
dyadic shell contributes at most
\[
\bigl(A d^{1+\kappa}\bigr)
\bigl(A(\eta/d)^{1+\kappa}\bigr)
=A^2\eta^{1+\kappa}.
\]
There are \(O(1+\log(L/w))\) shells, so
\[
\int_S\int_S\mu(C_z\cap C_{z'})\,d\mu(z')\,d\mu(z)
\lesssim mA^2\eta^{1+\kappa}\bigl(1+\log(L/w)\bigr).
\]
The union of the \(C_z\) lies in a product rectangle of radii
\(O(L),O(\theta L)\), of mass at most
\(O(A\eta^{-(1-\kappa)})\).  For
\[
f(p)=\int_S\mathbf1_{C_z}(p)\,d\mu(z),
\]
heaviness and Fubini give
\[
\int f\,d\mu\ge mAM^{-e},
\qquad
\int f^2\,d\mu
=\int_S\int_S\mu(C_z\cap C_{z'})\,d\mu(z')\,d\mu(z).
\]
Cauchy--Schwarz on the enclosing box yields
\[
m^2A^2M^{-2e}
\lesssim A\eta^{-(1-\kappa)}
\,mA^2\eta^{1+\kappa}\bigl(1+\log(L/w)\bigr).
\]
After cancellation this is
\[
m\lesssim A M^{2e}\eta^{2\kappa}\bigl(1+\log(L/w)\bigr).
\]
The logarithm is subpower at fixed polynomial complexity, proving
\eqref{eq:source-5}.  Nothing else in the proof uses time.
\end{proof}

\begin{corollary}[Counting heavy boxes]\label{geo:heavy-count}
Suppose the hypotheses of Lemma~\ref{geo:hairbrush} hold in every
time bin with the same \(H\).  Fix radii \(L,w\), and use an
orientation grid of spacing comparable to \(\rho=w/L\), with
translated grids having bounded overlap at each orientation.
Let \(\mathcal H\) be the family of boxes of mass at least
\(HW(E)M^{-e}\).  Then, over all time bins,
\begin{equation}\label{eq:source-6}
\#\mathcal H
\le \bigl(HW(E)\bigr)^{-1}M^{O_\kappa(e)+o(1)}.
\end{equation}
\end{corollary}

\begin{proof}
Write \(A=HW(E)\) and \(c_\kappa=4/\kappa\).  Choose
\(\tau_M\downarrow0\) sufficiently slowly that all bounded and
logarithmic factors are \(M^{o(\tau_M)}\) and
\(\tau_M\log M\to\infty\).  Set
\[
\Theta_0=\rho M^{c_\kappa(e+\tau_M)}.
\]
From each \(C\in\mathcal H\), remove the particles belonging to
another heavy box whose axis differs from that of \(C\) by more
than \(\Theta_0\).  Summing \eqref{eq:source-5} over angular shells
bounds the removed mass by
\[
A M^{2e-2\kappa c_\kappa(e+\tau_M)+o(\tau_M)}
=A M^{-6e-8\tau_M+o(\tau_M)}
=o(AM^{-e}).
\]
If the cutoff exceeds the angular range there is nothing to
remove.  The remaining set \(G_C\subset C\) has mass at least
\(\tfrac12 AM^{-e}\).

A particle belonging to some \(G_C\) can belong to heavy boxes
only at orientations within \(\Theta_0\) of \(C\).  It therefore
belongs to at most
\[
O(1+\Theta_0/\rho)
\le M^{c_\kappa(e+\tau_M)+o(\tau_M)}
\]
grid boxes: there are this many possible orientations, bounded
translated-grid overlap at each orientation, and only one time
bin.  Summing the masses of the \(G_C\) and using total mass at
most one gives
\[
\tfrac12\,\#\mathcal H\,AM^{-e}
\le M^{c_\kappa(e+\tau_M)+o(\tau_M)}.
\]
In particular,
\[
\#\mathcal H
\le A^{-1}M^{(1+c_\kappa)e+c_\kappa\tau_M+o(\tau_M)},
\]
which proves \eqref{eq:source-6}.
\end{proof}

\Needspace{3\baselineskip}
\begin{corollary}[Pointwise angular exclusion]\label{geo:angular-exclusion}
Let \(\mathcal H\) be the family in
Corollary~\ref{geo:heavy-count}.  Suppose a large sublaw
\(\nu\le\mu\) carries assigned boxes \(C(z)\in\mathcal H\), with
\(z\in C(z)\).  Assume the cap is also available at all enlarged
widths used below.  After removing \(o(\nu(\text{whole space}))\)
mass, the following implication holds pointwise:
\[
z\in M^d D,\quad D\in\mathcal H
\quad\Longrightarrow\quad
\angle(C(z),D)
\le\frac{w}{L}M^{C_\kappa(e+d)+o(1)}.
\]
Here \(D\) is in the particle's time bin, and dilation acts on
each spatial rectangle about its own center.  One retained set
works simultaneously for the dyadic dilations in any fixed
power range \(d\ge0\).
\end{corollary}

\begin{proof}
Keep \(A=HW(E)\), \(\rho=w/L\), \(c_\kappa=4/\kappa\), and take
\[
C_\kappa=\frac{c_\kappa+10}{2\kappa}.
\]
Write \(\nu(\text{whole space})=M^{-r_M}\), with \(r_M\to0\).
Choose \(\tau_M\downarrow0\) slowly enough that
\(r_M=o(\tau_M)\), all counting and logarithmic error exponents
are \(o(\tau_M)\), and \(\tau_M\log M\to\infty\).

Fix a dilation \(Q=M^d\).  Its baseline is
\[
A_Q=HW(QE)=Q^2A=M^{2d}A.
\]
Every enlarged heavy box has mass at least
\[
AM^{-e}=A_QM^{-(e+2d)}.
\]
For each \(C\in\mathcal H\), consider its particles that belong
to some \(QD\), \(D\in\mathcal H\), at angle greater than
\[
\Theta_d=\rho M^{C_\kappa(e+d+\tau_M)}.
\]
Since \(C\subset QC\), apply Lemma~\ref{geo:hairbrush} to the
central box \(QC\), the tilted boxes \(QD\), baseline \(A_Q\),
and tolerance \(e+2d\).  Summing angular shells bounds this
exceptional mass by
\[
A M^{2e+6d-2\kappa C_\kappa(e+d+\tau_M)+o(\tau_M)}.
\]
The application uses the caps between \(Qw\) and \(QL\), with
bounded dilations.

Summing over the original heavy-box list, using the explicit
count in the preceding proof, gives total exceptional mass at
most
\[
\begin{split}
&M^{(3+c_\kappa)e+6d+c_\kappa\tau_M
       -2\kappa C_\kappa(e+d+\tau_M)+o(\tau_M)}\\
&\hspace{2cm}
=M^{-7e-(c_\kappa+4)d-10\tau_M+o(\tau_M)}.
\end{split}
\]
The dyadic dilations in a fixed power range have
\(M^{o(\tau_M)}\) members after the choice of \(\tau_M\).  Their
union has exceptional mass
\[
M^{-10\tau_M+o(\tau_M)}
=o(M^{-r_M}).
\]
Removing this union proves the pointwise implication; its
remaining tolerance \(C_\kappa\tau_M\) is \(o(1)\).
\end{proof}

The heavy list and capacity bounds refer to the fixed prelaw.
The retained set has the stronger pointwise exclusion property for
every relevant high box.  Later conditioning will use this property,
rather than a bound only for a randomly selected pair of boxes.

\subsection{Reduction to bounded spatial complexity}

\begin{lemma}[Bounded complexity]\label{geo:bounded-complexity}
In the contradiction families for Theorem~\ref{geo:main}, the
complexities \(D_i\) may be bounded by a constant depending only
on \(\kappa\).
\end{lemma}

\begin{proof}
Work at the fixed failing widths supplied by
Lemma~\ref{geo:fixed-widths}.  Choose a fixed large integer \(J\),
depending only on \(\kappa\), with \(2\kappa J>30\), and put
\[
f=\min(L,wM^J),\qquad F=(L,f).
\]
Here \(F\) denotes the ellipse of those radii in the orientation
that will be specified for each group.

First suppose \(L>wM^J\).  Call a source box of widths \(L,w\)
\emph{high} if its mass is at least
\[
KW(E)M^{-10}.
\]
Use boundedly enlarged grids so that every terminal test lies
in a source test with these widths up to bounded factors:
from \(U\in E,\ X_v\in M^{-1}E\) we obtain
\(V=X_v-vU\in c+O(E)\).

Consider the particles whose \(P\) lies in two high source boxes
at angular distance greater than \(M^Jw/L\).  Inside any tested
source box, at least one of those two boxes makes angle
\(\gtrsim M^Jw/L\) with the test.  Applying
Lemma~\ref{geo:hairbrush} at the source with \(e=10\), and
summing angular shells, bounds this sublaw's mass in the test by
\[
KW(E)M^{20-2\kappa J+o_n(1)}.
\]
Our choice of \(J\) makes this better than every failing exponent
under consideration.  The sublaw of particles in no high source
box likewise has mass at most \(KW(E)M^{-10}\) in each source
test.  Each of these sublaws therefore satisfies the required
terminal bounds.  Neither can have large mass in the selected
counterexample family.  After their removal, a large sublaw
remains, and all high source boxes through any one surviving
\(P\) have axes within \(O(M^Jw/L)\).

Choose one such axis as a function of \(P\), round it at spacing
comparable to \(f/L=wM^J/L\), and partition by the rounded
direction and by translated \(U,V\) grid boxes of widths \(F\)
in that direction.  This partition depends on \(P\) alone.
When \(L\le wM^J\), instead use isotropic \(F=(L,L)\), with
no angular partition.

Any terminal test that violates the pre-partition threshold
supplies a high source box through all of its retained
particles.  In the eccentric case, their rounded directions
must therefore lie within \(O(f/L)\) of its direction.  There
are boundedly many such rounded directions.  In either case,
the terminal test meets boundedly many translated \(U,V\)
group cells at each allowed direction, because it is contained
in a source \(O(E),O(E)\) test.  Thus each violating terminal
test meets boundedly many groups.  Tests already below the
threshold remain below it under further restrictions.

We now normalize a group.  Let \(A_F\) be the linear map taking
the unit disk to \(F\), and subtract the two group centers
before applying \(A_F^{-1}\).  For a nonsingular linear map
\(A\), extend the notation by \(W(A)=W(AB_2)\).  The singular
value formula is
\[
W(A)=|\det A|^{1-\kappa}\|A\|^{2\kappa},
\]
and consequently
\begin{equation}\label{geo:submultiplicativity}
W(AB)\le W(A)W(B).
\end{equation}
If the group has mass \(m_G\), its normalized input cap is thus
at most
\[
\frac{KW(F)}{m_G}.
\]
This bound holds for every orientation of the normalized test.

For a previously violating terminal test at the selected widths
that meets the group, its target ellipse \(E\) makes angle
\(O(f/L)\) with the group axes, by the preceding partition argument.
If \(A_E\) maps the unit disk to that ellipse, then
\[
\|A_F^{-1}A_E\|\asymp1,
\qquad
|\det(A_F^{-1}A_E)|=w/f.
\]
Its pullback therefore has radii comparable to \(1,w/f\).
Moreover,
\[
W(F)W(A_F^{-1}E)
\lesssim
L^{1+\kappa}f^{1-\kappa}(w/f)^{1-\kappa}
=W(E).
\]
The input floor needed for this target is a constant multiple
of \(M^{-1}w/f\).  Under \(A_F\), the smallest radius of any
such normalized input ellipse is at least
\[
f(M^{-1}w/f)=M^{-1}w\ge M^{-D_i}.
\]
The original input cap is therefore available.  The normalized
complexity is
\[
1+\log_M(f/w)\le J+1.
\]
Bounded coordinate normalizations and endpoint roundings can
be handled with arbitrarily small positive slack in complexity
and the corresponding arbitrarily small loss in the input
constant, as described above.  For all sufficiently large
indices, complexity at most \(J+3\) suffices.

If the limiting worst exponent for complexities bounded by
\(J+3\) were strictly greater than \(q_*\), its universal bound
would improve all these normalized groups beyond the selected
failing threshold.  Fixed-parameter uniformity supplies a
large sublaw in every group with the same subpower losses.
On multiplying back by group masses, the factors \(m_G^{-1}\)
cancel, and the preceding inequality for the capacity weights
returns the original \(KW(E)\).  Recombining uses only boundedly
many groups per previously violating test, while all other
tests already obey the threshold.  The recombined sublaw has
large mass, a contradiction.

The limiting worst exponent at bounded complexities is
therefore \(q_*\), and counterexample families may be chosen
there.  Increasing the fixed bound if necessary proves the
lemma.
\end{proof}

\begin{lemma}[Extension of the width range]\label{geo:range-extension}
After Lemma~\ref{geo:bounded-complexity}, one may extend the
input cap to any prescribed larger bounded power range of
widths, preserve failure at the selected terminal widths, and
place those widths a fixed power below the upper boundary of
the permitted range.
\end{lemma}

\begin{proof}
Put \(\rho=M^{-D_i}\).  Add independent uniform disk noises of
radius \(\rho\) to \(U\) and \(V\), and append those noises to
the base label.  Their independence of the original law
preserves the conditional time profile.

For an ellipse \(E\) of radii \(L,w\), let \(E^+\) have the
same axes and radii \(\max(L,\rho),\max(w,\rho)\).
For one noise variable, the probability of entering a
translated \(E\), given the original point, is at most a
constant times \(|E|/|E^+|\), and vanishes unless that point
lies in a bounded enlargement of the corresponding
translate of \(E^+\).  Independence of the two noises yields
\[
\Pr(U_{\rm sm}\in E,\ V_{\rm sm}\in E)
\lesssim
KW(E^+)\left(\frac{|E|}{|E^+|}\right)^2
\lesssim KW(E).
\]
The final inequality follows because contracting a semiaxis
gets exponent \(2\) from dilution, whereas its exponent in
\(W\) is either \(1+\kappa\) or \(1-\kappa\), both less than
\(2\).  This extends the cap to smaller widths.  Bounded
support extends it to larger widths by clipping the long
radius at a fixed support bound and covering by boundedly
many tests.  If both radii exceed that bound, the total-mass
bound suffices: a fixed covering of the original bounded
support by unit source boxes already gives \(K\gtrsim1\),
with a constant depending only on that support bound.

Failure persists under smoothing.  A good sublaw of the
smoothed joint law pushes back to a sublaw of the original
law with the same mass.  The perturbations of \(U\) and
\(X_v\) are \(O(\rho)\), and the smallest selected terminal
position radius is \(M^{-1}w\ge\rho\).  Thus every original
terminal test maps into boundedly enlarged smoothed tests,
and their good estimates would give the original ones.

Finally, an isotropic power shrinking of both spatial
variables puts the relevant widths a fixed power below the
upper endpoint.  If the shrinking factor is \(B^{-1}\),
the transformed constant is \(KB^2\), while the ellipse
weight becomes \(W(E/B)=B^{-2}W(E)\); their product, and
therefore the failing comparison, is unchanged.  After the
bounded-complexity reduction, all required ranges and
shrinking powers remain bounded.
\end{proof}

From now on we use these extended ranges for universal upper
bounds and angular exclusions.  Factoring through width bands
will use only the original available range, shifted by the
isotropic normalization.  This distinction avoids requiring
an unavailable input cap at a band endpoint.

We have reduced the spatial complexity without changing the
retained-law obstruction.  The backward construction uses two
complementary facts at each selected epoch: a simultaneous cap bounds
all allowed tests, while the finite heavy-box list prevents a large
later sublaw from improving the selected-width bound.

\subsection{Backward construction of heavy ellipses}

Fix a finite dyadic mesh of positive lags \(y\le1\), including
\(1\).  At each lag, apply the universal estimate with exponent
\(q_*-o_i(1)\), propagating from depth \(0\) to depth \(y\).
Lemma~\ref{geo:time-conditioning} and fixed-parameter uniformity
permit these finitely many applications successively.  We
obtain a large normalized \emph{prelaw}, still carrying the
selected failure, with simultaneous bounds
\begin{equation}\label{geo:prelaw-caps}
\Pr(T_y=v,\ U\in E,\ X_v\in M^{-y}E)
\le KW(E)M^{-q_*y+o_i(1)+o_n(1)}
\end{equation}
throughout the required extended width ranges.  Dyadic
roundings are understood up to bounded factors.  For each
fixed mesh, all displayed \(o_i(1)\) errors tend to zero
before the mesh is subsequently refined.

At \(y=1\), call a prelaw box at the selected failing widths
high if its mass is at least
\[
KW(E)M^{-q_*y-e_y},
\]
where \(e_y=o_i(1)>0\) is chosen to dominate the upper-cap
errors and the small margin in the selected failure.
Particles lying in no such box already satisfy a stronger
terminal cap, so they cannot carry large mass.  Assign a
high box to each remaining particle.

Corollary~\ref{geo:heavy-count} bounds the resulting list by
\[
\bigl(KW(E)M^{-q_*y}\bigr)^{-1}
M^{O_\kappa(e_y)+o_n(1)}.
\]
This creates a hereditary obstruction: any later sublaw
carried by the assigned list and satisfying a cap improved
by a sufficiently larger multiple of \(e_y\) has power-small
total mass, simply by summing over the list.  Thus a later
large sublaw cannot make that improvement.  Importantly,
the boxes are high and counted in the prelaw; no later
restriction changes the validity of their covering list.

\Needspace{4\baselineskip}
\begin{lemma}[Backward width band]\label{geo:width-band}
Suppose that at a mesh epoch \(y\) the current large sublaw
has assigned boxes of widths \(L,w\) and the preceding
hereditary obstruction.  Let \(y'<y\) be the previous mesh
epoch.  After a further large restriction, the particles
have assigned high prelaw boxes at \(y'\), with a common
pair of dyadic widths both belonging to
\begin{equation}\label{eq:source-7}
\bigl[wM^{-(y-y')},L\bigr],
\end{equation}
up to bounded endpoint roundings.  The tolerance \(e_{y'}\)
may be chosen to tend to zero sufficiently slowly relative
to \(e_y\) and the propagation errors.
\end{lemma}

\begin{proof}
Call a prelaw box at epoch \(y'\), with both widths in
\eqref{eq:source-7}, high when its mass is at least
\[
KW(F)M^{-q_*y'-e_{y'}}.
\]
Suppose a large sublaw consists of particles belonging to
none of these high boxes.  Every test in this band then
has the improved cap with factor \(M^{-e_{y'}}\): a test
above that threshold would itself be a high prelaw box
containing those particles.  The finite grids and bounded
coverings make this statement simultaneous over all
centers, orientations, and band widths.

Partition this contrary sublaw by the prefix \(v'=T_{y'}\)
and by isotropic parent cells of widths \(L\) for \(U\)
and \(h'L\) for \(X_{v'}\), where
\[
h'=2^{-\lfloor y'n\rfloor}.
\]
Let a parent have centers \(u_Q,x_Q\) and mass \(m_Q\).
Normalize it by
\[
U'=\frac{U-u_Q}{L},\qquad
V'=\frac{X_{v'}-x_Q}{h'L},\qquad
S=\frac{T-v'}{h'}.
\]
The residual time \(S\) lies in \([0,1)\), and its terminal
partition has base
\[
M_{\rm res}
=2^{\lfloor yn\rfloor-\lfloor y'n\rfloor}
=M^{y-y'+o_n(1)}.
\]
If \(s_0\) is the left endpoint of a tested residual interval of
length \(M_{\rm res}^{-1}\), then on that event
\(v=v'+h's_0=T_y\), and
\[
V'+s_0U'
=\frac{X_v-x_Q-(v-v')u_Q}{h'L}.
\]
Both translations are fixed by the parent and this residual prefix.
Moreover, \(h'/M_{\rm res}=2^{-\lfloor yn\rfloor}\asymp M^{-y}\),
so the residual terminal position scale agrees with the original
one, up to the declared dyadic rounding.

A normalized terminal test has ellipse widths \(1,w/L\),
so the smallest input width required for its propagation
is a constant multiple of
\[
M_{\rm res}^{-1}w/L.
\]
In original units this is exactly the lower endpoint
\(wM^{-(y-y')}\) of \eqref{eq:source-7}, up to a bounded
factor.  The band cap hence supplies the full normalized
input bound, with constant
\[
\frac{K L^2M^{-q_*y'-e_{y'}}}{m_Q}
\]
up to subpower factors, because \(W(LF)=L^2W(F)\).
The parent time profile is supplied by
Lemma~\ref{geo:time-conditioning}.  Its complexity is
bounded for the fixed mesh and fixed \(i\).

Apply universal propagation on the residual interval.
Multiplication by the original parent mass cancels
\(m_Q^{-1}\); multiplication by \(W(E/L)\) cancels the
factor \(L^2\).  Thus the resulting sublaw in this parent
satisfies the original terminal test with bound
\[
KW(E)M^{-q_*y-e_{y'}+o_i(1)+o_n(1)}.
\]
The relative retained masses and errors may be taken
uniformly in the parents.

Recombination costs only a bounded factor.  A terminal
bin determines its prefix, and
\[
X_{v'}=X_v+(v'-v)U,\qquad |v'-v|\lesssim M^{-y'}.
\]
As \(U\) varies in a terminal ellipse of long radius \(L\),
the rebased position varies by \(O(M^{-y'}L)\).
The terminal cell consequently meets boundedly many of
the isotropic parent cells in both variables.

Choosing \(e_{y'}\) larger than the relevant multiples of
\(e_y\) and the propagation errors contradicts the
hereditary obstruction at \(y\).  The contrary particles
cannot have large mass.  We may assign high boxes to the
remaining large sublaw and select a common dyadic pair
of widths.  The number of such pairs is subpower, so
this selection retains large mass.  For a fixed finite
mesh the tolerances can be chosen in this backward
order while all still tend to zero.  Bounded rounding
of the band endpoints contributes a vanishing error
in logarithmic units.
\end{proof}

Iterating Lemma~\ref{geo:width-band} backward gives assigned
heavy ellipses at every epoch of the mesh.  The same
list-count obstruction is inherited at each newly assigned
epoch, so the induction can continue.  All restrictions
still compare directly to the prelaw for spatial caps
and to the reference law for the conditional time profile.

\subsection{Width compactness and first-order calibration}

Write the selected radii at epoch \(y\) as
\[
L(y)=M^{-x(y)},\qquad
w(y)=M^{-x(y)-d(y)},\qquad d(y)\ge0,
\]
and write \(\operatorname{axis}_y(z)\) for a particle's
assigned long axis.

The band relation implies, for successive \(y'<y\),
\[
x(y')\ge x(y),\qquad
x(y')+d(y')+y'\le x(y)+d(y)+y,
\]
up to the vanishing rounding errors.  The width depths
remain in a bounded range by
Lemma~\ref{geo:bounded-complexity} and
Lemma~\ref{geo:range-extension}.

Choose successively finer dyadic meshes, with refinement
slow enough that the accumulated errors tend to zero.
On each finite mesh first take sufficiently large \(i\),
then sufficiently large \(n\).  Compactness for the two
monotone functions, or a diagonal argument on dyadic
rationals, gives limiting functions of bounded variation,
still denoted \(x,d\), such that
\[
x\ \text{is nonincreasing},\qquad
x+d+y\ \text{is nondecreasing}.
\]
The finite-stage depths converge at continuity points.  The two
bounded-variation functions have ordinary derivatives almost
everywhere; we use the monotone and bounded-variation differentiation
theorems in \cite[Theorem~1.6.25 and Corollary~1.6.35]{Tao2011Measure}.
This differentiability concerns the limiting path.  The finite paths
will approximate its first-order window by the separate choices in
Lemma~\ref{geo:calibration}.

\Needspace{4\baselineskip}
\begin{lemma}[Stability of neighboring axes]\label{geo:axis-stability}
The large sublaws may be chosen so that neighboring mesh
epochs satisfy
\begin{equation}\label{eq:source-8}
\begin{split}
\angle\bigl(\operatorname{axis}_{y}(z),
            \operatorname{axis}_{y'}(z)\bigr)
&\le M^{-d(y)+C_\kappa\Delta+o(1)},\\
\Delta&=|y'-y|+|x(y')-x(y)|\\
&\hspace{1.5cm}
+|(x+d)(y')-(x+d)(y)|.
\end{split}
\end{equation}
They also satisfy the pointwise single-epoch exclusions of
Corollary~\ref{geo:angular-exclusion}, including the allowed
dilations.  These assertions are needed only when
\(\Delta\) is small.
\end{lemma}

\begin{proof}
Let \(\mu\) be the prelaw and \(y'<y\).  Put
\[
\begin{split}
h&=y-y',\qquad a_0=|x(y)-x(y')|,\\
b_0&=|(x+d)(y)-(x+d)(y')|,\\
B_0&=(1+\kappa)a_0+(1-\kappa)b_0.
\end{split}
\]
A later box has a single earlier prefix \(v'\), and
\[
M^{y'}X_{v'}=M^{-h}M^yX_v+M^{y'}(v'-v)U.
\]
The last coefficient is bounded, so its particles fit an earlier-bin
box with the same axis and the later widths, up to bounded factors.

Let \(F\) have radii \(L_F=\max(L(y),L(y'))\) and
\(w_F=\max(w(y),w(y'))\).  Enlarge the transported box and the
earlier assigned box along their own axes to these common widths.
For \(j\in\{y,y'\}\), let \(E_j\) have radii \(L(j),w(j)\), and set
\[
A_j=KW(E_j)M^{-q_*j},\qquad
A_F=KW(F)M^{-q_*y'}.
\]
The capacity ratios and the later box's original heaviness give,
up to the existing vanishing errors,
\[
\frac{A_F}{A_{y'}}\le M^{B_0},\qquad
\mu(\text{transported later box})
 \ge A_FM^{-e_y-q_*h-B_0},\qquad
\frac{w_F}{L_F}\le M^{-d(y)+b_0}.
\]
The extended width range supplies the caps at these common widths
and at the intermediate widths required by Lemma~\ref{geo:hairbrush}.

Apply that lemma inside each enlarged earlier box, assigning to
each exceptional particle its transported later box.  Use angular
cutoff \(M^{-d(y)+C_\kappa\Delta+\sigma}\), where \(\sigma>0\)
will tend to zero, and sum the angular shells.  Sum next over the
\emph{original} earlier assigned list, whose count is at most
\(A_{y'}^{-1}M^{(1+4/\kappa)e_{y'}+o(1)}\) by the proof of
Corollary~\ref{geo:heavy-count}.  The factor \(A_F\) in the
hairbrush bound cancels this list baseline up to
\(A_F/A_{y'}\le M^{B_0}\).  Thus the total exceptional mass is at
most
\[
M^{3B_0+2q_*h+2\kappa b_0+(1+4/\kappa)e_{y'}+2e_y
      -2\kappa C_\kappa\Delta-2\kappa\sigma+o(1)}.
\]
Since \(B_0,h,b_0=O(\Delta)\), a sufficiently large
\(C_\kappa\) absorbs their contributions.  Write the current
retained mass as \(M^{-r_M}\), with \(r_M=o_n(1)\).  At each
fixed mesh, choose \(\sigma=\sigma_i\downarrow0\) slowly enough
that \(e_y,e_{y'}\) and all \(o_i(1)\) errors are \(o(\sigma_i)\).
With \(i\) fixed, take \(n\) sufficiently large that \(r_M\), the
\(o_n(1)\) errors, and logarithmic counting losses are negligible
compared with \(\sigma_i\), while \(\sigma_i\log M\to\infty\).
The exceptional mass is then \(o(M^{-r_M})\), as required.

Perform these removals for each neighboring pair and apply
Corollary~\ref{geo:angular-exclusion} at each epoch, including its
dyadic dilation grid.  At each fixed mesh the number of requirements
is subpower.  Choose the vanishing margins also to dominate the
logarithm of this number divided by \(\log M\); the union of the
exceptional sets still has negligible relative mass.
\end{proof}

\Needspace{4\baselineskip}
\begin{lemma}[Calibrated widths and a reference axis]
\label{geo:calibration}
Let \(y_0\in(0,1)\) be a point of ordinary differentiability
of both limiting width depths, and put
\[
x_0=x(y_0),\qquad d_0=d(y_0),\qquad
a=-x'(y_0),\qquad b=-(x+d)'(y_0),\qquad
\lambda=a-b.
\]
Then \(a\ge0\), \(b\le1\), and
\(\lambda\ne0\) implies \(d_0>0\).

There are thicknesses \(\delta\downarrow0\), stretches
\(H_\delta\uparrow\infty\), and finite-stage families
whose local mesh epochs \(y=y_0+\delta v\),
\(|v|\le H_\delta\), satisfy uniformly
\begin{equation}\label{geo:affine-widths}
\begin{split}
x(y)&=x_0-a\delta v+o(\delta),\\
x(y)+d(y)&=x_0+d_0-b\delta v+o(\delta).
\end{split}
\end{equation}
Their neighboring increments in \eqref{eq:source-8}
have \(\Delta=o(\delta)\).  All cap and list errors are
\(o(\delta)\), the mesh spacing is \(o(\delta)\), and
\(\log n/n=o(\delta)\), while \(\delta\log M\to\infty\).

For each particle, choose as reference the assigned
axis at the end of the stretched interval having
the deeper affine aspect.  Then
\begin{equation}\label{geo:reference-axis}
\angle\bigl(\operatorname{axis}_{\rm ref}(z),
            \operatorname{axis}_{y_0+\delta v}(z)\bigr)
\le M^{-d_0-\delta\lambda v+o(\delta)}
\end{equation}
uniformly on the local mesh.  If \(\lambda=0\), any
one local epoch may be used as reference.  When
\(d_0>0\), a further constant-mass restriction permits
the reference directions to be represented by
\((1,R^\circ)\), with \(R^\circ\) uniformly bounded.
\end{lemma}

\begin{proof}
The two monotonicities give \(a\ge0\) and \(b\le1\).
Also \(d'=\lambda\).  Since \(d\ge0\), a differentiable
interior point with \(d_0=0\) is a local minimum and
has derivative zero.  This proves the assertion
when \(\lambda\ne0\).

Differentiability gives a common modulus
\(\omega(h)\to0\) for the two depth functions, with
affine approximation error at most
\(|h|\omega(|h|)\).  Choose \(H_\delta\to\infty\)
so slowly that \(\delta H_\delta\to0\) and
\[
H_\delta\omega(\delta H_\delta)\to0.
\]
The errors are then \(o(\delta)\) uniformly for
\(|h|\le\delta H_\delta\).

To transfer this conclusion to the finite-stage
families, first fix \(\delta\) and a finite accuracy
grid of continuity points.  Convergence at these
points, together with the monotonicity of \(x\)
and \(x+d+y\), traps intermediate width values
between nearby ones.  Next take sufficiently
advanced meshes, sufficiently large \(i\), and
sufficiently large \(n\).  A diagonal choice gives
\eqref{geo:affine-widths}, makes every cap, list,
and rounding error \(o(\delta)\), and makes the
neighboring depth increments \(o(\delta)\).  For the angular
bounds, first fix \(\delta\) and prescribe a margin \(o(\delta)\).
Choose the mesh and \(i\) so that \(\sigma_i\) lies below that
margin and every \(e_y\) and outer error is \(o(\sigma_i)\); then
choose \(n\) so that \(r_M\), the finite-base errors, and
\(\log n/n\) are \(o(\sigma_i)\).  Choose the single-epoch
angular-exclusion margins with the same hierarchy.
We may also impose
\(\log n/n=o(\delta)\),
\(\delta\log M\to\infty\), and a number of epochs
subpower in \(M^\delta\).

If \(\lambda>0\), use the latest local epoch as
reference; if \(\lambda<0\), use the earliest.
Along the chain from a given epoch to that
reference, every intermediate aspect depth is
at least \(d_0+\delta\lambda v-o(\delta)\),
by \eqref{geo:affine-widths}.  Summing
\eqref{eq:source-8} along this chain gives
\eqref{geo:reference-axis}: the worst angular
error determines the logarithmic scale, and the
number of links costs \(M^{o(\delta)}\).
For \(\lambda=0\), all aspect depths are
\(d_0+o(\delta)\), so the same argument works
with any reference epoch.

Finally cover the space of unoriented directions
by finitely many sectors.  On one sector a large
constant fraction of the law remains; after a
fixed rotation, its slopes are uniformly bounded.
This gives the stated representation of the
reference axis.
\end{proof}

\Needspace{4\baselineskip}
\begin{corollary}[Pointwise axis prediction]\label{geo:axis-prediction}
In the families of Lemma~\ref{geo:calibration}, two
retained particles with the same \(P=(U,V)\) and the
same time prefix at a local epoch have reference
axes agreeing to that epoch's aspect precision,
up to a factor \(M^{o(\delta)}\).
In a bounded-slope sector, the number of possible
reference-slope bins at depth
\(d_0+\delta\lambda v\), given the base label and
the time prefix at \(y_0+\delta v\), is at most
\(M^{o(\delta)}\).
\end{corollary}

\begin{proof}
The two particles have exactly the same displayed
point \((U,M^yX_{T_y})\) at the epoch in question.
Each therefore lies in the other's assigned high
prelaw box.  The pointwise exclusion of
Corollary~\ref{geo:angular-exclusion} bounds the
distance between their assigned axes by the
aspect ratio times \(M^{o(\delta)}\).
Equation~\eqref{geo:reference-axis} transfers the
same bound to their reference axes.  Angle and
slope are comparable in the fixed sector.

This is a pointwise membership argument against
all high prelaw boxes.  It does not require the
two particles to carry the same auxiliary label,
or their coincidence to have positive pair
probability.  Since the base label determines
\(P\), the asserted conditional bin count follows.
\end{proof}

The geometric reduction supplies a first-order width path and a
reference axis with subpower error in the local logarithmic unit
\(\delta\log M\).  The exact line and time prefix predict that axis
at the aspect precision of each local epoch.  The capacity bounds,
heavy-box lists, and pointwise angular exclusions remain available
after any further restriction of relative mass
\(\exp(-o(\delta\log M))\); normalization then costs
\(\exp(o(\delta\log M))\).  For each fixed finite list of later
restrictions whose relative masses are subpower with \(\delta\)
fixed, choose the sufficiently large finite stage before decreasing
\(\delta\), so their normalization losses are negligible in this
local unit.
The next section records these properties in finite measured states
before passing to entropy limits.
==== END Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//01-geometric.tex ====

==== BEGIN Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//02-entropy.tex ====
\section{Entropy profiles and their tangents}\label{sec:entropy}

We retain the calibrated path constructed in Section~\ref{sec:geo}.
At its differentiability point $y_0$, write
\[
x_0=x(y_0),\qquad d_0=d(y_0),\qquad
a=-x'(y_0),\qquad b=-(x+d)'(y_0),\qquad \lambda=a-b.
\]
Thus $a\geq0$, $b\leq1$, and the path widths, on the scale
$y=y_0+\delta v$, have depths
$x_0-\delta av+o(\delta)$ and
$x_0+d_0-\delta bv+o(\delta)$.
The first-order logarithmic unit is
\[
L_{\log}=\log\mathcal M=\delta\log M\longrightarrow\infty.
\]
All the errors from the path construction are $o(L_{\log})$ in
logarithmic units.  The stretched interval in
Lemma~\ref{geo:calibration} contains every fixed bounded interval of
offsets in the limit.  In particular, choosing an earlier or later
\emph{fixed} offset will not leave the available path.

The task in this section is to preserve the cap, heavy-list, and
axis-prediction information in finite conditional laws.  The profile
rules used later come from those laws and their hereditary
regularization.

We use discrete Shannon entropy, denoted by $\Ent$, with natural
logarithms; see Shannon~\cite{Shannon1948}.  The standard chain rule,
conditioning, and submodularity rules are collected, for example,
in \cite[arXiv:0906.4387v5, Appendix~A]{Tao2010Entropy}.
An expression such as $\Ent(Z\mid\mathcal B)$ is a conditional entropy; the base label $\mathcal B$ need not be discrete.
It determines $P=(U,V)$.  Every measured variable below is discretized,
and its range at a finite stage has at most $\exp(O(\log M))$ elements.

\subsection{Admissible sheared states}

Suppose first that $\lambda\ne0$, so $d_0>0$.  Let $R^\circ$ be the
reference slope chosen for each particle and used throughout the
calibrated local epochs in Lemma~\ref{geo:calibration}, in a fixed
bounded slope chart.  For offsets $z=(C,B,D,A,Y,R)$, put
$t=T_{y_0+\delta Y}$ and $r=[R^\circ]_{d_0+\delta R}$.
The state $Z(z)$ is the ordered tuple of bins of the following
measured coordinates, at the corresponding absolute depths:
\begin{equation}\label{eq:source-9}
\begin{array}{c|c|c}
\text{offset}&\text{measured coordinate}&\text{absolute depth}\\ \hline
C&U_1&x_0+\delta C\\
B&U_2-rU_1&x_0+d_0+\delta B\\
D&(X_t)_1&x_0+y_0+\delta D\\
A&(X_t)_2-r(X_t)_1&x_0+y_0+d_0+\delta A\\
Y&t&y_0+\delta Y\\
R&r&d_0+\delta R
\end{array}
\end{equation}
The notation for a depth includes the
dyadic rounding convention of Section~\ref{sec:geo}.
The domain of admissible offsets is the closed convex set
\begin{equation}\label{eq:source-10}
\Omega=\{B\leq C+R,\quad D\leq C+Y,\quad
                  A\leq B+Y,\quad A\leq D+R\}.
\end{equation}
Full alignment means equality in all four constraints.  It can be
parametrized by $(C,Y,R)$, with
$B=C+R$, $D=C+Y$, and $A=C+Y+R$.

We write $z\leq z'$ when every offset of $z$ is at most the
corresponding offset of $z'$.  The meet $z\wedge z'$ and join
$z\vee z'$ are their coordinatewise minimum and maximum.
A comparison rectangle has corners $z\wedge z'$, $z$, $z'$,
and $z\vee z'$; all four must belong to $\Omega$.

\begin{lemma}[Finite-state comparison]\label{ent:finite-states}
On any fixed bounded subset of $\Omega$, comparable states are nested
up to bounded ambiguity.  A refinement costs at most the logarithm of
the product of its scalar resolution ratios, up to an additive
constant.  For a coordinate rectangle whose four corners belong to
$\Omega$, the two side states together determine the upper state up
to bounded ambiguity, and each determines the bottom state up to
bounded ambiguity.  Consequently their entropies satisfy monotonicity,
the scalar-coordinate Lipschitz bounds, and submodularity, with
additive bounded errors.

If only the parameter cutoffs increase, the entropy increment is at
least the corresponding parameter entropy increment conditional on
$\mathcal B$, again up to a bounded error.
\end{lemma}

\begin{proof}
Write the spatial entries in the order of
\eqref{eq:source-9} as $(x,h,z,w)$.  Changing the truncated parameters
by $(\Delta t,\Delta r)$ gives the exact update
\begin{equation}\label{ent:shear-update}
(x,h,z,w)\longmapsto
\bigl(x,h-\Delta r\,x,z+\Delta t\,x,
w+\Delta t\,h-\Delta r\,z-\Delta t\Delta r\,x\bigr).
\end{equation}
For component widths $\beta_x,\beta_h,\beta_z,\beta_w$ and parameter
widths $\beta_t,\beta_r$, the errors in this update at the starting
component resolutions are bounded precisely when
\[
\beta_x\beta_r\lesssim\beta_h,\qquad
\beta_x\beta_t\lesssim\beta_z,\qquad
\beta_t\beta_h\lesssim\beta_w,\qquad
\beta_r\beta_z\lesssim\beta_w.
\]
These are \eqref{eq:source-10}, including the absolute background
depths.  They also bound the doubly sheared error, since
$\beta_t\beta_r\beta_x\lesssim\beta_t\beta_h\lesssim\beta_w$.
The shifts of bin centers are known from the parameter labels and
can be subtracted exactly.  Only the within-bin errors require these
inequalities; large absolute center coordinates cause no additional
loss.

A finer state supplies finer parameter bins and finer spatial
measurements.  Applying \eqref{ent:shear-update} recovers a coarser
state with a bounded number of possible bin labels.  Conversely,
inside a coarse state, each newly tested scalar coordinate and each
parameter has at most a constant times its resolution ratio many
refinements.  This proves the first assertions.

For a coordinate rectangle, the side states supply the finer pair of
parameter labels.  Each finest spatial component is already measured
by at least one side.  Update that side to the finer parameter pair at
its own component accuracy, using the inequalities valid at that
side.  This recovers the upper state.  The same update proves that
either side recovers the bottom state.  Applying the usual entropy
submodularity inequality to the two side labels proves the asserted
inequality with an additive bounded error.

Finally let $Q_0,Q_1$ be the coarse and fine parameter pairs in a
parameter-only refinement, and let $Z_0,Z_1$ be its states.  The
nesting just proved gives
\begin{align*}
\Ent(Z_1)-\Ent(Z_0)
&\geq \Ent(Q_1\mid Z_0)-O(1)\\
&\geq \Ent(Q_1\mid Z_0,\mathcal B)-O(1)\\
&=\Ent(Q_1\mid Q_0,\mathcal B)-O(1).
\end{align*}
For the last equality, $Z_0$ is determined by $\mathcal B$ and $Q_0$,
whereas $Z_0$ includes $Q_0$.  No independence between the two
parameters, or between either parameter and $\mathcal B$, is used.
\end{proof}

\subsection{Regularization before taking a profile}

Entropy values alone do not give the probability bounds needed in a
projection test.  We first control finite support sizes and maximal
cell weights uniformly under the restrictions that follow; only then
do we pass to an entropy profile.  A subpower restriction here means
a restriction of relative mass $\exp(-o(L_{\log}))$.

\begin{lemma}[Hereditary regularization]\label{ent:regularization}
For any fixed finite family of states, their joint labels, and
conditional parameter labels given $\mathcal B$, one may restrict to
a subpower-density law with the following properties.  At every
unconditional state of absolute entropy $h$,
\begin{equation}\label{ent:absolute-flatness}
\#\supp Z\leq\exp\bigl(h+o(L_{\log})\bigr),\qquad
\max_z\Pr(Z=z)\leq\exp\bigl(-h+o(L_{\log})\bigr).
\end{equation}
Analogous conditional bounds hold on the retained base-label fibers.
For comparable states of absolute entropies $h_0,h_1$, typical coarse
bins have fine support at most
$\exp(h_1-h_0+o(L_{\log}))$ and maximal conditional fine weight at
most $\exp(-(h_1-h_0)+o(L_{\log}))$.

These assertions can hold simultaneously on any prescribed
countable family of successively finer finite grids, interpreted
diagonally.  Subsequent subpower restrictions preserve the referenced
entropy profiles and these estimates, after discarding exceptional
parent or base-label fibers.  If lower weights are required in a
finite collection of tests, bins of smaller weight than the
corresponding reciprocal-support exponent can be discarded with
subpower slack; their retained lower weights refer, in particular, to
the law before that last discard.
\end{lemma}

\begin{proof}
Fix a finite family first.  The number of bins in each of its labels
is at most $\exp(O(\log M))$.  Tails of probabilities smaller than an
appropriately large negative power of $M$ have negligible total mass,
by this count.  Partition the remaining probabilities into dyadic
classes.  There are $O(\log M)$ relevant classes per label.  Do the
same for all the prescribed joint labels and, pointwise on the base
label, for the prescribed conditional parameter probabilities.
Pigeonholing selects common classes on an event of relative mass
\[
\exp\bigl(-O_J(\log\log M)\bigr)
       =\exp\bigl(-o(L_{\log})\bigr),
\]
where $J$ is the fixed number of tests.  We choose all diagonal
parameters slowly enough that this remains true as the grids grow.

Suppose a selected unconditional class has original weight comparable
to $w$.  It contains at most $O(w^{-1})$ bins.  After restriction and
normalization by a subpower density, its maximal weight is at most
$w\exp(o(L_{\log}))$.  Its new absolute entropy is therefore
$-\log w+o(L_{\log})$, by the upper support bound and the lower bound
on entropy supplied by the maximal atom.  This proves
\eqref{ent:absolute-flatness}.  For conditional classes, discard
base-label fibers on which the relative retained mass is smaller
than a sufficiently slowly vanishing power.  Such fibers have
negligible total retained mass.  The identical support and maximal
weight argument then applies on every remaining fiber.  This
procedure is also valid when $\mathcal B$ takes values in a standard
Borel space: the dyadic class of a conditional probability is a
measurable finite label, and no count of base labels is involved.

Here is the conditional-parent deduction explicitly.  Replace the
coarse and fine labels by their joint label when necessary.  By
Lemma~\ref{ent:finite-states}, this changes the fine support and
entropy bounds by bounded factors and additive constants.  Write the
errors in \eqref{ent:absolute-flatness} as $\varepsilon L_{\log}$.
If a parent has more than
$\exp(h_1-h_0+\eta L_{\log})$ descendants, then counting all joint
cells and using the parent maximal weight bounds the total mass of
such parents by $O(\exp(-(\eta-2\varepsilon)L_{\log}))$.
Parents of weight below $\exp(-h_0-\eta L_{\log})$ have total mass
at most $\exp(-(\eta-\varepsilon)L_{\log})$.  Outside these two
exceptions, the conditional support and maximal weight bounds are
the asserted ones, with errors $O((\eta+\varepsilon)L_{\log})$.
Choose $\eta\downarrow0$ much more slowly than $\varepsilon$.
The discarded fractions can thereby be made negligible compared
with any finite list of subpower densities needed later.

The support bound also shows that bins of absolute weight below
$\exp(-h-\eta L_{\log})$ have negligible total mass.  For a finite
list of joint labels, one may remove these bins successively, or
iteratively remove bins whose remaining weight is below the chosen
threshold.  Charge each removed bin when it is first deleted.  The
sum of all charges is bounded by the sum of the corresponding
support counts times their thresholds, and is still negligible if
the slack dominates the existing errors.  Dividing joint weights by
parent weights gives the corresponding conditional lower-weight
statements.

After any further restriction of density $\rho=\exp(-o(L_{\log}))$,
the support bounds remain valid and maximal weights grow by at most
$\rho^{-1}$.  Thus the absolute entropies change by $o(L_{\log})$.
The same parent and base-fiber trims recover the conditional
statements.  This proves heredity.  Finally apply the construction on
larger finite grids, with their size increasing sufficiently slowly.
Off-grid tests are sandwiched by admissible finer and coarser grid
states.  The domain has interior shift vectors with every coordinate
positive, and others with every coordinate negative, so such
sandwiches exist with arbitrarily small offset errors.  The
Lipschitz estimates of Lemma~\ref{ent:finite-states} complete the
diagonal extension.
\end{proof}

\subsection{The calibrated profile and conditional axis prediction}

\begin{proposition}[Calibrated entropy profile]\label{ent:states}
After the regularization of Lemma~\ref{ent:regularization} and passage
to a subsequence, the functions
\[
\frac{\Ent(Z(z))-\Ent(Z(0))}{L_{\log}}
\]
converge locally uniformly on $\Omega$ to a continuous Lipschitz
function $F$.  Each scalar partial lies in $[0,1]$ where defined.
The function is coordinatewise nondecreasing, and its off-diagonal
distributional second partials are nonpositive in the interior.
The same assertions hold for disjoint groups of coordinates advanced
together with nonnegative speeds, on every valid comparison
rectangle.  At full alignment,
\begin{equation}\label{eq:source-11}
 F(C,B,D,A,Y,R)\geq k_C C+k_B B+q_*Y,
 \qquad k_C=1+\kappa,\quad k_B=1-\kappa,
\end{equation}
with equality on
\[
\gamma(Y)=(-aY,-bY,(1-a)Y,(1-b)Y,Y,\lambda Y).
\]
\end{proposition}

\begin{proof}
The finite-state comparisons give equicontinuity, after reference
subtraction, with errors tending to zero in the present log unit.
One first takes a subsequence on a countable dense grid, and then
uses the admissible coordinatewise joins and bounded comparable
shifts to obtain local uniform convergence on the full domain.
Monotonicity and Lipschitz bounds pass to the limit.  The entropy
rectangle inequalities give submodularity.  Equivalently,
$-\partial_{ij}F$ is a nonnegative distribution, and hence a
nonnegative Radon measure, for $i\ne j$ in the interior.
Pinning coordinates or summing disjoint nonnegative coordinate
increments preserves those inequalities.  Comparisons on faces
follow by translating the valid rectangle into the interior and
passing to the limit; convexity of the domain makes these
translations available.

To verify the normalization in \eqref{eq:source-11}, put
\[
B_0=-\log K+
 \bigl(2x_0+(1-\kappa)d_0+q_*y_0\bigr)\log M.
\]
At full alignment, a state atom is contained in a tested ellipse
cell, with its displayed time bin and axis.  The simultaneous cap
from Section~\ref{sec:geo} therefore gives
\[
\Ent(Z(z))\geq
B_0+L_{\log}(k_C C+k_B B+q_*Y)-o(L_{\log}).
\]
For a nonmesh time use a nearby earlier mesh time, with the rebasing
and width errors controlled by \eqref{eq:source-8}.
On the calibrated path, Corollary~\ref{geo:heavy-count} supplies the
opposite support estimate.  Given an assigned high box and its time
bin, the reference-axis approximation at path aspect accuracy leaves
only $\exp(o(L_{\log}))$ possibilities for the labels
\eqref{eq:source-9}.  The same is true of the slight width
mismatches and time rebasing.  Thus the support logarithm is at most
the displayed lower bound plus $o(L_{\log})$.  In particular,
$\Ent(Z(0))=B_0+o(L_{\log})$.  Subtract this equality and divide by
$L_{\log}$.  Equality first holds on the fine mesh and then
everywhere on the line by continuity.  All the cap and list
estimates survive the subpower restriction used to regularize.
\end{proof}

We also track the referenced conditional entropy of $(t,r)$ given
$\mathcal B$ on the same realizing sequence.  Its limiting profile
will be denoted by $\Psi(Y,R)$.  The time-only profile is $sY$, since
the upper weights and upper support counts in \eqref{eq:source-1}
give both entropy bounds, with errors negligible in the present
unit.

\begin{lemma}[Conditional prediction of the axis]\label{ent:predictability}
For $\lambda>0$,
$\Psi(Y,R)=sY$ on $R\leq\lambda Y$.
For $\lambda<0$, this equality holds for every finite $(Y,R)$.
Parameter-coordinate increments of $F$ dominate those of $\Psi$.
In particular, if $\lambda<0$, then $\partial_YF\geq s$.

In the latter case set $c=b-a=-\lambda>0$.  In the region
\begin{equation}\label{eq:source-12}
\begin{gathered}
C>aR/c,\qquad B>bR/c,\qquad Y>-R/c,\\
D>(a-1)R/c,\qquad A>(b-1)R/c
\end{gathered}
\end{equation}
inside $\Omega$, $F$ is locally independent of $R$.
\end{lemma}

\begin{proof}
Given $P$ and a time prefix at a local epoch, all particles in that
fiber have the same measured line state at that epoch.  Their
assigned high boxes therefore pass through that state.  The
pointwise exclusion in Corollary~\ref{geo:angular-exclusion} applies to
\emph{all} high prelaw boxes, irrespective of subsequent restrictions
or additional labels.  It implies that the assigned axes in the
fiber have only subpower many possibilities at path aspect
precision.  Corollary~\ref{geo:axis-prediction} transfers this conclusion
to the common reference axis.  The conditional entropy excess of
the axis over the time prefix is consequently $o(L_{\log})$ at that
precision.

If $\lambda>0$, the epoch of offset $Y$ predicts the reference axis
at offset depth $\lambda Y$, hence also at every $R\leq\lambda Y$.
If $\lambda<0$, for any fixed $(Y,R)$ choose an earlier fixed epoch
$v\leq Y$ with $\lambda v>R$.  Its time prefix is determined by the
one at $Y$, and its aspect precision is finer than the requested
axis cutoff.  The common reference epoch is still earlier, on the
deeper-aspect end of the stretched interval.  Chaining
\eqref{eq:source-8} from that reference to $v$ loses only the worst
aspect precision, namely the precision at $v$.  Thus this epoch
also predicts the common reference axis to the requested accuracy.
This proves the asserted zero excess entropy, including its
absolute $o(L_{\log})$ bound before reference subtraction.
Lemma~\ref{ent:finite-states} proves the increment comparison with
$F$, and hence $\partial_YF\geq s$ in the decreasing-aspect case.

We give the spatial prediction argument for
\eqref{eq:source-12} with all its buffers.  Fix a state in that open
region and choose a sufficiently small $\eta>0$.  Test the path epoch
\[
v_*=-\frac{R+\eta}{c}.
\]
Two particles with the same displayed state have a common truncated
slope $r$ and time $t_Y=T_{y_0+\delta Y}$.  Their differences satisfy
\begin{align*}
|\Delta U_1|&\lesssim M^{-x_0-\delta C},&
|\Delta U_2-r\Delta U_1|
 &\lesssim M^{-x_0-d_0-\delta B},\\
|\Delta(X_{t_Y})_1|&\lesssim M^{-x_0-y_0-\delta D},&
|\Delta(X_{t_Y})_2-r\Delta(X_{t_Y})_1|
 &\lesssim M^{-x_0-y_0-d_0-\delta A}.
\end{align*}
Since $Y>v_*$, they also have the same epoch-$v_*$ prefix.  Choose
the assigned axis $r_*$ of either particle at that epoch.  Its
distance from $r$ is at most
$M^{-d_0-\delta R+o(\delta)}$: compare first with that particle's
reference axis at depth $R+\eta$, and then with its common prefix
$r$ at depth $R$.

In the $r_*$ frame, the velocity errors have the two offset depths
\[
C,\qquad \min(B,C+R),
\]
with absolute background depths $x_0$ and $x_0+d_0$.  If
$t_*=T_{y_0+\delta v_*}$, then
$|t_*-t_Y|\lesssim M^{-y_0-\delta v_*}$.
Using $X_{t_*}=X_{t_Y}+(t_*-t_Y)U$, the position errors in that
frame have offset depths
\[
\min(D,C+v_*),\qquad
\min(A,D+R,B+v_*,C+R+v_*),
\]
with background depths $x_0+y_0$ and $x_0+y_0+d_0$.
The path targets at this epoch are
\[
-av_*,\quad -bv_*,\quad (1-a)v_*,\quad (1-b)v_*.
\]
Every displayed error depth is strictly finer than its target for
small $\eta$.  Indeed this follows from the five inequalities in
\eqref{eq:source-12}; the additional comparisons
$C+R>bR/c$ and $D+R>(b-1)R/c$ follow from $c=b-a$.

Thus peers in this state lie in bounded enlargements of one
another's assigned epoch boxes.  The all-high-box angular exclusion
forces their assigned axes to agree up to a subpower factor at
offset depth $R+\eta$.  Comparison with each particle's reference
axis gives the same conclusion for the finer reference-axis bins.
The state therefore determines this extra axis precision with
entropy cost $o(L_{\log})$.  After refining the parameter, its
spatial labels at the original cutoffs can be updated with bounded
ambiguity by \eqref{eq:source-10}.  Hence increasing $R$ a little
at fixed other offsets costs zero in the limiting profile.  The
strict margins hold throughout a neighborhood, proving local
independence.
\end{proof}

\paragraph{The stationary reference axis as a conditioning label.}
When $\lambda=0$, let $R_0$ be the reference slope truncated at
absolute depth $d_0$; take a constant label if $d_0=0$.  Retain
$R_0$ and measure the spatial coordinates in its frame.  Normalize
the velocity widths by $M^{-x_0},M^{-(x_0+d_0)}$ and the position
widths by the same factors times $M^{-y_0}$.  Use common offset
depths $u$ for both velocity components, $z$ for both position
components, and $Y$ for time.  The domain is $z\leq u+Y$.

Write $W_\zeta$ for these spatial and time bins at
$\zeta=(u,z,Y)$, and $\widehat Z_\zeta=(R_0,W_\zeta)$ for the
recorded state.  We use the referenced conditional entropy
\begin{equation}\label{ent:stationary-axis-cancellation}
\frac{\Ent(W_\zeta\mid R_0)-\Ent(W_0\mid R_0)}{L_{\log}}
=\frac{\Ent(\widehat Z_\zeta)-\Ent(\widehat Z_0)}{L_{\log}}.
\end{equation}
This is an exact finite-stage identity.  The size of $\Ent(R_0)$
is unrestricted: it cancels, rather than being charged as a
subpower loss.  Include $R_0$ and the needed joint labels in the
regularization list.  Thus we work with conditional blocks of a
recorded label, without selecting a single axis bin at a supposed
subpower cost.  The resulting profile has the following properties.

\begin{proposition}[Stationary-aspect profile]\label{ent:stationary-profile}
There is a hereditary Lipschitz submodular profile $F(u,z,Y)$ on
$z\leq u+Y$, with scalar-group Lipschitz bounds $2,2,1$, satisfying
at alignment
\begin{equation}\label{eq:source-13}
\begin{gathered}
F(u,u+Y,Y)\geq2u+q_*Y,\\
F(-aY,(1-a)Y,Y)=-2aY+q_*Y,\qquad \partial_YF\geq s.
\end{gathered}
\end{equation}
At alignment, arbitrary ellipse tests in the normalized frame are
also available.  For their two width depths $l_1,l_2$, the width
term $2u$ is replaced by
\[
(1+\kappa)\min(l_1,l_2)+(1-\kappa)\max(l_1,l_2).
\]
Any additional axis label required for such a lower-bound test may
be included on the measured side.
\end{proposition}

\begin{proof}
\textbf{Conditional capacity and the heavy list.}
The finite-state and regularization arguments apply with the two
spatial components grouped and with $R_0$ recorded.  Conditional
entropy inherits the same comparison inequalities.  To check the
baseline, at alignment set
\[
b_\zeta=B_0+L_{\log}(2u+q_*Y),
\]
with $B_0$ as in the proof of Proposition~\ref{ent:states}.
The joint cap gives maximal atom weight
$\exp(-b_\zeta+o(L_{\log}))$ for $\widehat Z_\zeta$.
On a positive fiber $R_0=r$ of probability $p_r$, normalization
therefore gives
\[
\Ent(W_\zeta\mid R_0=r)
\ge b_\zeta+\log p_r-o(L_{\log}).
\]
Averaging over $r$ gives the lower bound
$b_\zeta-\Ent(R_0)-o(L_{\log})$.  On the calibrated path,
the joint high-box list bounds
$\Ent(\widehat Z_\zeta)\le b_\zeta+o(L_{\log})$.
Indeed, an assigned high box and time bin leave only
$\exp(o(L_{\log}))$ possibilities for the depth-$d_0$ reference-axis
bin, by Lemma~\ref{geo:calibration} with $\lambda=0$.
Consequently
\[
\Ent(W_\zeta\mid R_0)
=b_\zeta-\Ent(R_0)+o(L_{\log}).
\]
The upper estimate is an averaged conditional entropy statement;
no uniform support bound in every untrimmed axis fiber is needed.
The common $\Ent(R_0)$ term cancels in
\eqref{ent:stationary-axis-cancellation}, giving the alignment
lower bound and calibrated equality in \eqref{eq:source-13}.
Typical-parent regularization gives the conditional support and
weight bounds within the recorded $(R_0,\text{parent})$ fibers,
so these comparisons have the required hereditary realizations.

\paragraph{Conditional time increments.}
For brevity put $t_Y=T_{y_0+\delta Y}$.  The same single-epoch
prediction as in Lemma~\ref{ent:predictability} gives
$\Ent(R_0\mid\mathcal B,t_Y)=o(L_{\log})$ at every fixed
local cutoff.  The exact chain rule reads
\[
\Ent(t_Y\mid\mathcal B,R_0)
=\Ent(t_Y\mid\mathcal B)
 +\Ent(R_0\mid\mathcal B,t_Y)-\Ent(R_0\mid\mathcal B).
\]
Subtract this identity at two cutoffs.  The last term cancels,
and the middle terms are negligible.  Thus the time increment
retains slope $s$, and the finite-state conditional comparison
gives $\partial_YF\ge s$.  No bound on the unconditional entropy
of $R_0$, or on $\Ent(R_0\mid\mathcal B)$, is asserted.

\paragraph{Elliptic tests in the normalized frame.}
For an ellipse in the normalized frame, apply
the submultiplicativity of $W$ under the fixed background linear
normalization, as in Section~\ref{sec:geo}.  Its weight in offset
units is the negative exponential of the stated width term.  Each
cell, with its axis label included if necessary, is contained in
one of these ellipse tests up to bounded covering, proving the
same entropy lower bound.
\end{proof}

\subsection{Realizing tangents and conditional blocks}

A tangent at a point $z_0$ of a profile is a locally uniform limit,
on the corresponding rescaled domain, of
\[
F_{z_0,h}(z)=\frac{F(z_0+hz)-F(z_0)}{h},\qquad h\downarrow0.
\]
We may take several profiles to their tangents simultaneously.
Centers may lie on an alignment face or on an affine sheet in that
face.  Constraints strict at the center disappear locally after
rescaling; binding linear constraints retain their exact tangent
form.

\begin{lemma}[Realization of tangents]\label{ent:tangent-realization}
Every tangent used below, and every finite succession of such
tangents at finitely many sites, has a realizing sequence of the
same sheared states, with hereditary support and weight estimates
in its final log unit.  Required common physical cutoffs at
different sites can be kept identical.  Exact alignment identities
are preserved, before bounded dyadic rounding, whenever the center
and the tested cutoffs lie on the corresponding alignment.
\end{lemma}

\begin{proof}
First fix a radius $h$ and a finite test grid for the desired
tangent.  Approximate the original profile on the corresponding
grid of absolute cutoffs to an error much smaller than $h$.
Choose the original stage sufficiently far out that
$hL_{\log}\to\infty$, and that every previous logarithmic error
and every probability-class loss is $o(hL_{\log})$.
Lemma~\ref{ent:regularization} can be imposed on this finite list
and on its joint and conditional labels.  Now pass to a diagonal
sequence as $h\downarrow0$ and the grid grows.  This realizes the
tangent in its new log unit.  Repeating the construction handles
iterated tangents.  A finite collection of sites is treated by
putting all its cutoffs in the same list.  Coordinates prescribed
to be physically common are represented by the same absolute
cutoff, rather than by separately rounded approximations.

Inside a common parent bin, subtract the known bin centers and use
its spatial and parameter widths as units.  Formula
\eqref{ent:shear-update} still applies.  For instance a time
increment changes position by its normalized scalar increment
times normalized velocity, with known center terms subtracted
exactly.  At an aligned triple the position unit equals the
velocity unit times the time unit.  The defining affine equalities
of these depths are preserved under a centered rescaling through
alignment.  Dyadic rounding changes the units only by bounded
factors.

For comparable parent and refinement states, the conditional
support and maximal-weight estimates now use their entropy
difference, by Lemma~\ref{ent:regularization}.  This applies as
well after a fine parameter bin is included in the parent tuple:
the old spatial labels can be rebased to that parameter bin with
bounded ambiguity.  Lower typical weights, when needed, are
obtained by the same trims.

To see the resulting product density, fix a typical parent and
let $\mathsf A,\mathsf B$ be the supported bin sets of two
prescribed refinement labels.  Let $\mathsf E\subset
\mathsf A\times\mathsf B$ consist of pairs with positive
conditional joint probability.  Use joint labels with the parent
where nesting has bounded ambiguity; this does not change the
exponents.  Write $L$ for the final logarithmic unit.  If the
two marginal entropy exponents are $\alpha,\beta$ and the joint
exponent is $\alpha+\beta$, regularization gives
\begin{gather*}
|\mathsf A|\leq e^{(\alpha+o(1))L},\qquad
|\mathsf B|\leq e^{(\beta+o(1))L},\\
\max_{(a,b)\in\mathsf E}\Pr(a,b\mid\text{parent})
\leq e^{-(\alpha+\beta-o(1))L}.
\end{gather*}
The conditional law has mass one, so
$|\mathsf E|\geq e^{(\alpha+\beta-o(1))L}$ and hence
\[
\frac{|\mathsf E|}{|\mathsf A|\,|\mathsf B|}
\geq e^{-o(L)}.
\]
This is subpower density of supported bin pairs, with no
independence assertion.  All these statements hold at every
fixed finite power grid, and hence on successively finer grids
by the same diagonal choice.
\end{proof}

In projection arguments we will use the actual realized profiles.
A full-alignment test is based in a full profile, or in a tangent
centered through full alignment at each intervening step.  A slice
with other coordinates moved to create slack need preserve only
the aligned triples it tests.

\subsection{Peeling at an affine sheet}

\begin{lemma}[Peeling]\label{ent:peeling}
Let a Lipschitz, coordinatewise nondecreasing, submodular profile be
defined on a convex domain given locally by linear inequalities,
and let $\Sigma$ be an affine sheet in one of its faces.  Suppose
a group of coordinates is held fixed by a tangent direction to
$\Sigma$ that strictly increases all the coordinates still
attached outside that group.  Coordinates separated at earlier
steps may vary arbitrarily in this direction.  At almost every
point of $\Sigma$, a first tangent splits that group as an
independent additive summand.  A finite succession of such
separations can be made in the same first tangent.

Suppose that, after these separations, each group varies on
$\Sigma$ by one proportional family of linear forms, and that the
nonzero forms of different groups are pairwise nonproportional.
Then each group's restriction to its sheet diagonal is affine.
In particular, a separated singleton with nonconstant form has a
constant full partial derivative throughout the tangent.
\end{lemma}

\begin{proof}
We prove the assertion with an interior mollification first.  Write
$\mu_{ij}=-\partial_{ij}F\geq0$ for $i\ne j$.
Choose a compact sheet patch, a smooth cutoff on a slightly larger
patch, and an offset compact set lying strictly inside the tangent
domain.  Translate the sheet patch by $hu$, with $u$ in that
offset set and $h>0$ small.  If $i$ belongs to the group being
separated and $w$ is the chosen sheet-tangent direction, then
$w_i=0$ and
\[
-\partial_{iw}F=\sum_{j\ne i}w_j\mu_{ij}.
\]
The integral of the left side on the translated patch against
the cutoff is bounded uniformly in $h$ and $u$: integrate by
parts in the sheet direction $w$ and use the bounded first
derivative $\partial_iF$.  Terms from previously separated
groups have bounded absolute integrals by the estimates at their
earlier separation steps, using symmetry of mixed derivatives.
All the remaining terms outside the current group are
nonnegative, with strictly positive coefficients $w_j$.
Consequently each such $\mu_{ij}$ has a uniformly bounded
integral on the translated sheet patch.  Use nested, slightly
larger patches at successive steps.  This establishes all the
required cross-pair bounds before any tangent is taken.

Average these bounds also over compact interior offsets $u$.
For the rescaled function $F_{x,h}$ the corresponding mixed
derivative has density $h\mu_{ij}(x+hu)$ in the offset variable.
Thus its averaged mass on an offset compact set, integrated over
sheet centers $x$, is $O(h)$.  These bounds are summable for
dyadic $h$.  By positivity, summability, and exhaustion by
countably many compact subsets, at almost every sheet center all
the cross-pair measures tend to zero on compact subsets in every
limiting dyadic tangent.

For an unmollified profile, the off-diagonal negative mixed
derivatives are Radon measures.  Mollify at a radius much smaller
than $h$ times the strict interior margin of the offset compact
set.  Enlarge that compact set slightly when testing the measure.
The same averaged bounds pass to the measures and give the
conclusion just obtained.  This argument never takes a Hessian
trace on a binding face.

Vanishing cross derivatives give additive separation on the
convex open tangent domain.  One way to see this is to mollify
locally again: the gradient in a given group is then constant
along each fiber of all the other coordinates.  Connected
convex fibers show that it depends only on that group's
coordinates.  The resulting conservative field has a primitive
on their convex projection.  Patching these primitives gives a
sum of group functions, up to a constant; Lipschitz continuity
extends the identity to the closed domain.

Apply Rademacher's theorem in intrinsic Euclidean coordinates on
the relative interior of $\Sigma$
\cite[Theorem~3.1]{Heinonen2005Lipschitz}.  At almost every sheet center
the restriction of the original profile to $\Sigma$ is differentiable.
Its locally uniform linear blowup shows that the restriction of any
tangent there is affine on the whole tangent sheet.  This uses
intrinsic sheet measure, not ambient almost-everywhere differentiability.
After the preceding splitting it is a sum of Lipschitz functions
of the distinct group forms.  Such a sum can be affine only if
each summand is affine along its form.  For completeness, fix
one form and apply successive finite differences in directions
annihilating each of the other forms but not the fixed one.
Additional differences annihilate the affine right side.  The
result says that a sufficiently high distributional derivative
of the chosen one-variable function vanishes.  It is a
polynomial, and its global Lipschitz bound makes its degree at
most one.  A zero group form is constant on the sheet and
requires no assertion beyond that.  For a singleton with
nonzero form, its projected coordinate runs through the real
line on the sheet, so its full summand is affine and its full
partial is constant.
\end{proof}

\begin{remark}\label{ent:peeling-forms}
A useful sufficient condition for successive peeling is that the
distinct nonzero group forms are positive on a common vector.
They generate a pointed polyhedral cone.  An extreme ray has an
exposing functional zero on that ray and strictly positive on
the other generating rays.  This is the required sheet
direction for that group.  Remove it and repeat.  Zero forms
can first be separated as the group held fixed by the common
positive direction.  Proportional forms must be combined
before this procedure.

The tangent choices can be made measurably when their rates are
integrated over a sheet.  The dyadic blowups, jointly for any
countable list of profiles, form a precompact family in a
separable metrizable space of locally uniform convergence.
On a patch of fixed binding constraints, choose nested balls
from countable nets that are visited infinitely often.  This
selects a limiting tangent and a subsequence measurably.
Tangential derivatives of the original restriction agree for
all choices at its differentiability points.
\end{remark}

\subsection{Signed envelopes along a path}

The next lemma separates the cost of the coordinates that decrease
along a path from the remaining coordinates. It constructs a tangent
of the full profile and an auxiliary function of the decreasing
coordinates, obtained from tangents of slices with the remaining
cutoffs fixed at finer levels. The auxiliary function gives lower
bounds for ordered increments and an upper comparison that is exact
at the starting point. In the applications, projection tests continue
to use actual entropy profiles and their realized tangents with the
required alignments preserved. The auxiliary function is used only
through the increment comparisons and overlap identities proved below.

\begin{lemma}[Signed envelope]\label{ent:signed-envelope}
Let $F_0$ be a Lipschitz, coordinatewise nondecreasing, submodular
profile on a convex polyhedral cutoff domain $\mathcal D$.  All
increment comparisons below are required to have valid corners.
Let
\[
\gamma(y)=(N(y),Q(y)),\qquad y\in J,
\]
be a Lipschitz path in a boundary sheet of $\mathcal D$.  Suppose
each coordinate of $N$ is strictly decreasing, with its speed
bounded above and below in magnitude, and each coordinate of $Q$
is nondecreasing.  Constant coordinates may belong to $Q$.
Assume
\[
(N(\alpha),Q(\beta))\in\mathcal D\qquad(\alpha\leq\beta).
\]
Assume also the following exterior-slack property.  On compact
subintervals of $J$, one can choose a vector $e$ with strictly
positive $Q$-coordinates such that replacing $Q(\beta)$ by
$Q(\beta)+\varepsilon e$ preserves admissibility and opens every
constraint involving $Q$ by a positive constant times
$\varepsilon$.  At a path point, fix a strictly exterior value
of $Q$ supplied by this property.  Define the pure tangent
domain $\mathcal P$ to be the tangent domain of that fixed
slice at the path's $N$-coordinates.  The inequalities involving
$Q$ are strict, so $\mathcal P$ is given by the active
inequalities involving only $N$ and is independent of the chosen
exterior value.  Assume that $\mathcal P$ is either full space or
the three-dimensional half-space
$\{(C,B,R)\in\R^3:B\leq C+R\}$; in the latter case the pure
group consists exactly of $N=(C,B,R)$.

At almost every $y_0\in J$, put
\[
v=-N'(y_0),\qquad w=Q'(y_0),\qquad
n(t)=-tv,\qquad q(t)=tw.
\]
Thus $v_i>0$ and $w_j\geq0$.  There are a first tangent $F$ of
$F_0$ at $\gamma(y_0)$ and a Lipschitz, nondecreasing,
submodular function $G$ on the pure tangent domain $\mathcal P$,
both referenced to vanish at the origin, with the following
properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item
The function $F-G$ is nondecreasing under valid ordered increases
of $N$ at fixed $Q$.  Moreover,
\[
F(n(\alpha),q(\beta))-G(n(\alpha))
\]
is independent of $\alpha$ whenever $\alpha\leq\beta$.

\item
On any compact test region, if $H$ is sufficiently large that
$n(-H)\geq N$ and the comparison tuples are valid, then
\begin{equation}\label{eq:source-14}
F(N,Q)\leq
G(N)+F(n(-H),Q)-G(n(-H))
=G(N)+F^+(Q).
\end{equation}
Equality holds at the origin.  There is also equality for pure
increments in the overlap region: if $Q\geq q(\beta)$ and
$N\leq N'$ satisfy $N_i>n_i(\beta)$ for every $i$, and all four
tuples obtained from $N,N'$ and $Q,q(\beta)$ are valid, then
\[
F(N',Q)-F(N,Q)=G(N')-G(N).
\]

\item
If $\mathcal P=\R^d$, then $G$ is linear on every strict ordering
region of the normalized depths $x_i=N_i/v_i$.  There is a number
$m$ such that $G(n(t))=-mt$, and in every ordering region
\[
\sum_i v_i\partial_iG=m.
\]
When $x_i$ is strictly largest, $\partial_iG$ is a constant
marginal rate; this rate is a global upper bound for that partial.

If instead $\mathcal P$ has the constraint $B\leq C+R$, the same
affine rate holds on the tied line.  Write the corresponding
speeds as $a,b,c>0$, so $b=a+c$.  The $C$-partial is constant on
\[
\frac Ca>\max\left\{\frac Bb,\frac Rc\right\},
\]
and the $R$-partial is constant on
\[
\frac Rc>\max\left\{\frac Ca,\frac Bb\right\}.
\]
Each constant is a global upper bound for its partial,
interpreted distributionally in the interior.
\end{enumerate}
The tangent and all genericity requirements can be imposed
simultaneously for countably many ancillary profiles and exterior
slices.
\end{lemma}

\begin{proof}
\textbf{Fixed exterior slices and their rates.}
For each fixed later parameter $\beta$, the function
$y\mapsto F_0(N(y),Q(\beta))$ is Lipschitz on $y<\beta$.
Choose a countable dense family of parameters containing all
dyadic partition endpoints used below.  At common
differentiability points of the corresponding restrictions, define
\[
r_\beta(y)=-\frac{d}{dy}F_0(N(y),Q(\beta)),\qquad
m(y)=\lim_{\beta\downarrow y}r_\beta(y),
\]
where the limit is taken through that countable family.
Submodularity and monotonicity of $Q$ show that $r_\beta(y)$
increases as $\beta$ decreases to $y$.  All these rates are
bounded, so $m$ is bounded and measurable.

\paragraph{Finite increments with a moving tied parameter.}
The Lipschitz integral formula
\cite[Theorem~3.3]{Heinonen2005Lipschitz} applies to each fixed
restriction.  Its bounded rates are locally integrable.  Choose
$y_0$ to be a Lebesgue point of $m$ and of every applicable
fixed-$\beta$ rate, using \cite[Theorem~1.6.19]{Tao2011Measure}
after a cutoff on a larger compact interval.  There are only
countably many fixed restrictions.  A shrinking interval with fixed
rescaled endpoints has average oscillation bounded by a fixed
multiple of a centered interval's average, so the same Lebesgue-point
conclusion applies below.  Fix
$\alpha_1<\alpha_2<\beta_0$, and put
\begin{align*}
I_h={}&F_0\bigl(N(y_0+h\alpha_1),Q(y_0+h\beta_0)\bigr)\\
&-F_0\bigl(N(y_0+h\alpha_2),Q(y_0+h\beta_0)\bigr).
\end{align*}
For each fixed $\beta_*>y_0$ in the countable family, and all
sufficiently small $h>0$, ordered increment comparisons give
\[
\int_{y_0+h\alpha_1}^{y_0+h\alpha_2}r_{\beta_*}(s)\,ds
\leq I_h\leq
\int_{y_0+h\alpha_1}^{y_0+h\alpha_2}m(s)\,ds.
\]
These are finite-increment comparisons on valid rectangles,
or their limits along the ordered path.  If the moving parameter
$y_0+h\beta_0$ is outside the countable family, approximate it
by parameters in that family still later than the entire
increment interval.  Lipschitz continuity of $F_0$ and $Q$
passes the bounds to the limit; no continuity of derivatives
in the conditioning value is required.  Divide by $h$, apply
Lebesgue differentiation, and then let $\beta_*\downarrow y_0$ to obtain
\[
\lim_{h\downarrow0}\frac{I_h}{h}
=m(y_0)(\alpha_2-\alpha_1).
\]
Differentiability of the path at $y_0$ and the Lipschitz bound
permit its replacement by the linearizations $n,q$.  Hence
every tangent at this point has pure tied rate $m(y_0)$ when
the pure cutoffs are strictly finer than the fixed $Q$-path
time.  Continuity includes the endpoint of the ordered region.

\paragraph{Approximating the trace by exterior slices.}
We next obtain the same rate from strictly exterior slices.
Work on a compact path interval with a fixed exterior vector
$e$.  For a dyadic partition of mesh $\delta_k$, set, on each
interval $I=[a_I,b_I]$,
\[
Q^I=Q(b_I)+\delta_k e,
\qquad
m_k(y)=-\frac{d}{dy}F_0(N(y),Q^I),\quad y\in I.
\]
The entire countable family of these constants is fixed before
choosing generic centers.  Ordered comparison gives
$0\leq m_k\leq m$.  We claim that $m_k\to m$ in $L^1$ on
compact subintervals.

Fix a coarser dyadic interval $[a,b]$.  For every fine
$I\subset[a,b]$, one has $Q^I\leq Q(b)+\delta_k e$, with all
comparison tuples valid.  Sum the corresponding pure increments
to obtain
\[
\int_a^b m_k(y)\,dy
\geq F_0(N(a),Q(b)+\delta_k e)
      -F_0(N(b),Q(b)+\delta_k e).
\]
First let the fine mesh tend to zero while keeping $[a,b]$
fixed.  Lipschitz continuity removes the exterior shift.
Then refine the coarse partition.  Its fixed-endpoint rates
converge almost everywhere to the defining monotone trace,
so dominated convergence makes the sum of its tied increments
tend to $\int m$.  The upper bound $m_k\leq m$ proves the
claim.  After a subsequence of partitions, $m_k(y)\to m(y)$
almost everywhere.  The fine limit preceding the coarse limit
is essential here; it also handles constant coordinates of $Q$
without continuity assumptions on derivatives in those
coordinates.

\paragraph{The pure limit and overlap equality.}
Choose $y_0$ outside all exceptional sets for the profile, the
countable exterior slices, and the restrictions used below.
Blow up $F_0$ at $(N(y_0),Q(y_0))$ and, simultaneously, each
applicable slice $N\mapsto F_0(N,Q^I)$ at $N(y_0)$.
Equicontinuity and a diagonal subsequence give a first tangent
$F$ and slice tangents $G_k$.  For each fixed exterior slice,
strict mixed slack makes its rescaled domain contain every
compact subset of $\mathcal P$ eventually.  Differentiability
along the tied path gives
\[
G_k(n(t))=-m_k(y_0)t.
\]
For fixed $k$, its exterior $Q^I$ is strictly finer than the
$Q$-coordinates of any fixed compact blowup test once the
blowup radius is small.  Submodularity therefore gives
\[
\partial_iF\geq\partial_iG_k,\qquad i\in N,
\]
in distributions on the interior.  Valid face comparisons
follow by interior translation and continuity.  Take a locally
uniform subsequential limit of the referenced $G_k$, denoted by
$G$.  Then
\[
\partial_iF\geq\partial_iG,
\qquad G(n(t))=-m(y_0)t.
\]
Thus $F-G$ is nondecreasing under valid ordered pure increments.
The finite-increment trace calculation makes its tied pure
increments vanish for $\alpha<\beta$, and continuity includes
$\alpha=\beta$.  This proves (i) and
\eqref{eq:source-14}, including equality at the origin.

For overlap equality, first fix $q(\beta)$.  A pure tuple
strictly finer than $n(\beta)$ is bounded above and below by
two tied tuples with path times strictly below $\beta$.
They are valid.  Monotonicity of $F-G$ and its equal values at
the tied tuples imply that $F(N,q(\beta))-G(N)$ is constant
throughout the comparison region.  The pure increment identity
therefore holds at $q(\beta)$.  Increasing $q(\beta)$ to $Q$
can only decrease a pure increment of $F$, whereas (i) bounds
it below by the corresponding increment of $G$.  These bounds
agree, proving (ii).

\paragraph{Order sectors of the pure profile.}
It remains to identify the pure slice tangents.  Suppose first
that $\mathcal P$ is full space.  Fix an exterior slice $Q^I$
and a nonempty proper coordinate subset $S$.  For each later
parameter $\beta$ in the countable dense family, define on the
local valid restrictions
\begin{align*}
r^S_\beta(y)&=-\frac{d}{dy}
 F_0(N_S(y),N_{S^c}(\beta),Q^I),\qquad y<\beta,\\
r_S(y)&=\lim_{\beta\downarrow y}r^S_\beta(y).
\end{align*}
As $\beta$ decreases, the complementary cutoffs become finer,
so submodularity makes these rates decrease.  Require
differentiability of all these fixed restrictions and take
$y_0$ to be a common Lebesgue point of their rates and traces,
for every exterior slice and subset.

For $\alpha_1<\alpha_2<\beta_0$, put
\begin{align*}
I_h^S={}&F_0(N_S(y_0+h\alpha_1),N_{S^c}(y_0+h\beta_0),Q^I)\\
&-F_0(N_S(y_0+h\alpha_2),N_{S^c}(y_0+h\beta_0),Q^I).
\end{align*}
For fixed $\beta_*>y_0$ in the dense family, sufficiently close
to $y_0$ that the fixed-complement restriction is valid near
$y_0$, and for sufficiently small $h$, the reversed
ordered-increment bounds are
\[
\int_{y_0+h\alpha_1}^{y_0+h\alpha_2}r_S(s)\,ds
\leq I_h^S\leq
\int_{y_0+h\alpha_1}^{y_0+h\alpha_2}r^S_{\beta_*}(s)\,ds.
\]
As before, approximate a moving complementary parameter by
dense parameters later than the increment interval, and use
Lipschitz continuity to pass these finite-increment bounds to
the limit.  Divide by $h$, let $h\downarrow0$, and then let
$\beta_*\downarrow y_0$.  Lebesgue differentiation gives
$I_h^S/h\to r_S(y_0)(\alpha_2-\alpha_1)$.  Thus the slice
tangent has constant tied-$S$ rate $r_S(y_0)$ whenever $S$
is strictly finer than its complement.  The exterior value
$Q^I$ stays fixed throughout this blowup.

The complement need not be tied: bound its normalized depths
above and below by tied, strictly coarser depths.  Full pure
space and the fixed exterior slack make all these corners
valid locally.  Ordered increments sandwich the tied-$S$
rate between the same two constants.  For the slice tangent
$G_k$, abbreviate its rate $r_S(y_0)$ by $r_S$.

Write $x_i=N_i/v_i$ and order them as
$x_{(1)}\leq\cdots\leq x_{(d)}$.  Start with all normalized
depths equal to $x_{(1)}$, increase the remaining coordinates
together, and freeze each as its specified depth is reached.
The tied-subset calculation yields
\[
G_k(N)=r_{[d]}x_{(1)}+
\sum_{j=2}^{d}r_{S_j}\bigl(x_{(j)}-x_{(j-1)}\bigr),
\qquad S_j=\{i:x_i\geq x_{(j)}\},
\]
where $r_{[d]}=m_k(y_0)$.  Repeated depths cause no ambiguity,
since their differences vanish.  This proves linearity on
strict ordering regions.  Translating all normalized depths
by the same amount proves the identity for the weighted sum
of partials.  If $x_i$ is strictly largest, its partial is the
singleton rate $r_{\{i\}}/v_i$.  At any other point, make all
complementary coordinates sufficiently coarser to put $i$ in
this finest sector.  Submodularity shows that the original
partial is no larger than this singleton rate.  The finitely
many coefficients are bounded, so these assertions pass to
the locally uniform limit $G$.

For the pure constraint $B\leq C+R$, the function $B-C-R$ along
the path is nonpositive and vanishes at the center. Its derivative
there is zero, giving $b=a+c$. The full tied trace and
the singleton traces for $C,R$ still have valid comparisons.
For example, if
$C/a>\max(B/b,R/c)$, tied complements
$(B,R)=(bT,cT)$ with $T<C/a$ are valid, because
$bT<C+cT$.  Tied complements above and below the given
normalized complement sandwich its singleton increments.
Nonnegative connecting segments remain in the domain by
convexity and validity of their endpoints.  Thus the
$C$-partial is constant in the finest sector.  Taking a
sufficiently coarse tied complement at any other interior
point gives the global upper bound.  The argument for $R$
is identical.  These statements pass to $G$ in distributions.

All exceptional sets arise from differentiability and Lebesgue
differentiation for countably many fixed restrictions.
Intersect their full-measure sets and diagonalize the
equicontinuous blowups.  This proves the simultaneous assertion.
\end{proof}

The pure auxiliary profile need not inherit projection statements.
At a center on the boundary of a strict order sector, a
pure-gradient lower bound for the actual profile passes
distributionally to the part of its tangent domain consisting of
directions into that sector.  If this sector cone meets the tangent
interior and peeling makes the relevant singleton partial constant
throughout the tangent, its value on that open portion is the same
constant used everywhere else.  The sector bound therefore bounds
that constant.  The transfer uses
interior ordered-increment inequalities and their continuous
extension to valid boundary rectangles.

\Needspace{4\baselineskip}
\begin{remark}[Boundary increments and inherited flatness]
\label{ent:signed-envelope-boundary}
At a binding constraint, overlap equality applies in particular
to a strictly positive pure direction with valid comparison
tuples.  If individual pure-gradient lower bounds hold and a
subsequently tested tangent has constant partials in these
coordinates, equality of that weighted sum forces equality in
each lower bound: every weight is strictly positive and every
excess over its lower bound is nonnegative.  First pass each
bound to the interior part of its sector cone and identify the
constant partial there.  A two-sided coordinate perturbation
at the boundary origin is unnecessary.

Flatness of the exterior slices on a common strict comparison
region passes to their tangents and then to $G$.  In particular,
the flatness in \eqref{eq:source-12} is inherited wherever its
offset conditions hold for those slices.  Pure lower profiles
are used only through the properties just proved; projection
tests continue to use actual entropy profiles.

The exterior-slack vectors have explicit choices in the
applications.  On the calibrated path with $\lambda>0$ and
$a>0$, the negative coordinates are $C$ and whichever of
$B,D$ strictly decrease.  Large positive shifts in $R,Y$,
with smaller positive shifts in the other nonnegative
coordinates, open the mixed constraints.  For $\lambda<0$,
the negatives are $R,B$, and also $C$ if $a>0$.  Positive
shifts in $Y,D,A$, successively smaller in that order, open
the mixed constraints; add a positive shift in $C$ when it
is static.  In the scalar domain $Z\leq X+y$ with decreasing
$X$, shift $y$ more than $Z$.

The tied-subset argument also applies with signs reversed on
a full domain along a line whose speeds are all strictly
positive.  It gives the piecewise-affine order slopes of
unconditional or conditional two-variable profiles used below.
\end{remark}

The final lemma turns local comparisons into one finite path. Its
almost-everywhere hypothesis is along every admissible path, as its
statement specifies.

\subsection{From local gains to a finite path}

\begin{lemma}[Local gains and path control]\label{ent:control}
Fix a starting state and a duration $T>0$, and consider paths
from that state with velocities in a compact convex box.
The states may be restricted to a closed linear half-space;
assume there is an available velocity
following its boundary.  Let the potential be continuous and
Lipschitz on the compact region reachable by these paths.
If the potential depends on time, include time as a state
coordinate with its prescribed unit velocity.
Suppose that along \emph{every} admissible Lipschitz path, at
almost every interior time, arbitrarily short admissible
straight steps increase the potential at a rate exceeding
$g_0+\varepsilon$, where $\varepsilon>0$ is fixed.  Then a
reachable terminal state has potential at least $g_0T$ greater
than its value at the prescribed start.

The local hypothesis is satisfied at a point if a tangent
there has a strict admissible finite-step comparison from the
tangent origin with the stated margin.  A sufficiently small
admissible perturbation of its velocity or endpoint is harmless
within a strict margin.
\end{lemma}

\begin{proof}
Include time among the state coordinates, so the terminal
face is part of the compact reachable set.  Call a state good
if it admits a positive-length straight step before the
terminal time with gain rate greater than
$g_0+\varepsilon/2$, and call all other states bad.  Goodness
is relatively open.  Indeed a strict comparison persists
under small perturbations.  If a step ends on the terminal
face or on the boundary of the half-space, shorten it
slightly; the comparison remains strict.  A step starting
on the half-space boundary either enters the interior or
follows the boundary, and remains admissible when its start
is moved slightly into the half-space.  For an interior
start, shortening or a small convex perturbation toward a
boundary-following velocity keeps the perturbed step
admissible.  Thus the bad set is closed; it includes the
terminal face.

By hypothesis, the bad set occupies zero time on every
admissible path.  Its shrinking neighborhoods consequently
have uniformly vanishing occupation time over all such
paths.  To prove this, otherwise choose shrinking radii and
paths with a fixed positive occupation time.  The paths have
a common Lipschitz bound.  A uniformly convergent subsequence
has an admissible limit, because the velocity box is compact
and convex and the half-space is closed.  Uniform convergence
shows that the limiting path spends at least that positive
time in every fixed neighborhood of the closed bad set.
Continuity from above of Lebesgue measure then gives positive
time in the bad set itself, a contradiction.

Fix a small neighborhood of the bad set.  Outside it, compactness
and openness of goodness give finitely many comparison steps,
each valid on an open set of starting states and with its length
bounded below by a positive number.  At a state outside the
neighborhood, take one of these good steps.  Inside the
neighborhood, take an arbitrary viable straight step of a
fixed sufficiently small length, clipped at terminal time.
Choose that length so that such a step stays in a slightly
larger neighborhood, using the common speed bound.  This
constructs a path reaching the terminal time in finitely many
steps.  The total time of its nongood steps is at most the
occupation time of the larger neighborhood, hence is
uniformly small.  Its good steps gain at least
$g_0+\varepsilon/2$ per unit time, whereas the Lipschitz
bound gives a uniform lower rate on the remaining steps.
Taking the neighborhood small enough makes the total gain
at least $g_0$ times the full duration.

Finally, a strict endpoint comparison from the tangent origin
passes back along its defining blowup sequence: the endpoint error
is little-oh of the step length, while the gain margin is
a fixed positive multiple of that length.  Admissibility under
the linear domain constraint is retained in this passage.
Any additional perturbation must remain admissible and have
velocity in the given box; sufficiently small such perturbations
preserve the strict margin.  This gives the last assertion.
\end{proof}
==== END Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//02-entropy.tex ====

==== BEGIN Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//03-projection.tex ====
\section{Projection tests and correlated parameters}
\label{sec:projection}

The only external projection statement used here is the finite
incidence estimate below.  We derive its weighted consequence, a
radial improvement, and the correlated-product estimate needed when
two spatial coordinates have the same rate.  Each test uses the
hereditary finite realizations of Section~\ref{sec:entropy}.

Throughout
this section, $m\to\infty$ denotes the base of the block being tested; a
fixed positive block length is absorbed into this base. A probability law
in a bounded Euclidean set is \emph{Frostman of exponent $d$ at this
scale} if
\[
 \mu(B(v,h))\le m^{o(1)}h^d,\qquad m^{-1}\le h\le 1.
\]
The analogous terminology for directions uses angular intervals. It
suffices to impose these bounds on arbitrarily fine fixed grids of powers
of $m$, allowing an arbitrarily small power loss. Between grid points one
uses monotonicity. A restriction is \emph{dense} if its probability is
$m^{-o(1)}$. At a fixed power scale, \emph{power-small} means bounded by
$m^{-c}$ for some $c>0$; the constant may depend on that scale and on the
strict exponent gaps in the assertion. All limits below keep these gaps
fixed before letting the input losses tend to zero. Bounded enlargements
of bins and boundedly overlapping grids do not affect an exponent.

\subsection{The discretized input and its weighted projection consequence}

We state the external incidence theorem in the form needed here. A
$(\delta,d,C)$-set is a nonempty finite family $P$ of dyadic $\delta$-squares such
that
\[
 |\{p\in P:p\cap B(v,h)\ne\varnothing\}|
 \le C h^d |P|,\qquad \delta\le h\le 1,
\]
with harmless constant enlargements at the mesh scale. Tubes are counted
by their dyadic cells in a bounded slope--intercept chart of line space.
Finitely many charts cover the directions that occur.

\Needspace{5\baselineskip}
\begin{theorem}[Discretized planar Furstenberg estimate]
\label{proj:furstenberg}
Let $0<k\le2$ and $0<\sigma\le1$. For every $\varepsilon>0$ there is
$\eta=\eta(\varepsilon,k,\sigma)>0$ such that the following holds for
sufficiently small dyadic $\delta$. Suppose $P$ is a
$(\delta,k,\delta^{-\eta})$-set of pins in a fixed bounded planar
region, normalized to $[0,1]^2$. For every $p\in P$, let
$\mathcal T(p)$ be a family of $\delta$-tubes meeting $p$, with
$|\mathcal T(p)|\asymp M$, whose directions form a
$(\delta,\sigma,\delta^{-\eta})$-set. Then
\[
 \left|\bigcup_{p\in P}\mathcal T(p)\right|
 \gtrsim_{\varepsilon}
 \delta^{-\min\{k,(k+\sigma)/2,1\}+\varepsilon}M.
\]
Consequently the union has lower exponent at least
$\min\{k+\sigma,(k+3\sigma)/2,1+\sigma\}$ as the nonconcentration
losses tend to zero.
\end{theorem}

This is Theorem~4.1 of Ren--Wang~\cite{RenWang2023}, in the
explicitly cited arXiv version~2, with the notation and nonemptiness
convention of their Definitions~1.3, 2.3--2.4, and 3.1. Their theorem is already a
finite-scale estimate for incident tube pencils. In particular, no
discretization of a Hausdorff-dimension conclusion is being used. The
last assertion follows from $M\gtrsim\delta^{-\sigma+\eta}$, which is
the direction nonconcentration bound at radius $\delta$. Comparable
pencil cardinalities are obtained, when necessary, by a dyadic
pigeonhole. There are only logarithmically many such classes in our
bounded, polynomial-size grids. The dependence of $\eta$ on
$\varepsilon$ is part of the theorem and is retained in every use below.

\begin{lemma}[Weighted projection test]
\label{proj:projection}
Let $\mu$ be a planar Frostman $k$ probability law and let $\zeta$ be a
Frostman $\sigma$ law of directions, where $0<k\le2$ and
$0<\sigma\le1$. Let $E$ be a set of point--direction pairs of dense
$\mu\times\zeta$ measure. For a direction $\omega$, write $E_\omega$
for the corresponding point fiber. If the projections of these fibers
have at most $m^{u+o(1)}$ mesh bins for every retained direction, then
\begin{equation}
 u\ge \min\{k,(k+\sigma)/2,1\}.
 \label{eq:source-15}
\end{equation}
The same conclusion applies to dense point restrictions chosen
separately for each direction. It is enough to have the stated ball
bounds with losses that can be made arbitrarily small.
\end{lemma}

\begin{proof}
First discard direction fibers and pin fibers whose relative masses are
too small, using a threshold with exponent tending to zero sufficiently
slowly. Normalize the restrictions of the reference laws $\mu$ and
$\zeta$ to the retained pin and direction sets, and keep $E$ as a
subset of their product. We continue to denote these reference laws by
$\mu,\zeta$; their Frostman bounds lose only subpower factors, and $E$
still has dense product measure. We now reduce these weighted laws to
the finite sets in Theorem~\ref{proj:furstenberg}.

Sample the reference direction law $\zeta$ \emph{globally}, before
drawing the pencils, so every pencil uses the same sampled population.
For small fixed $0<\rho<\sigma$, take
$n\asymp m^{\sigma-\rho}$ independent samples. Choose a permitted
empirical loss $m^a$ with $a>\rho$, with both exponents below the
input allowance in Theorem~\ref{proj:furstenberg}. The required count
in an interval of radius $h$ is at most $m^a n h^\sigma$.
At $h=m^{-1}$ this threshold is of order $m^{a-\rho}$, whereas
the expected count is at most $n h^\sigma m^{o(1)}$.
Binomial tails therefore give the count bound simultaneously over
the polynomial family of intervals with mesh endpoints. Enlargement
then gives it for all angular intervals. The expected proportion of
samples sharing a mesh bin with another sample is at most
\[
 O\bigl(m^{\sigma-\rho}\sup_\omega
             \zeta(B(\omega,O(m^{-1})))\bigr)
 \le m^{-\rho+o(1)}.
\]
Thus repetitions can be removed at a cost negligible compared with
the dense incidence. The tail failures can also be made negligible
relative to that density. Averaging gives a sample on which dense
incidence survives, with dense direction fibers at a dense set of
pins. Apply the same construction to the reference pin law, using
$m^{k-\rho}$ samples after also requiring $\rho<k$; mesh balls have
a polynomial-size covering family. Restriction to a dense fiber
divides a normalized ball bound by at most a subpower factor.
Finally pigeonhole the pencil cardinalities. Their exponent is
$\sigma-\rho-o(1)$, and the pin exponent is $k-\rho-o(1)$;
the normalized nonconcentration losses remain arbitrarily small.

For each retained point--direction pair, draw the line of constant
projection through that point. The sampled directions number at most
$m^{\sigma+o(1)}$ in total. Hence the projection support hypothesis gives
the global line-count upper bound $m^{\sigma+u+o(1)}$. On the other
hand, Theorem~\ref{proj:furstenberg}, with output error
$\varepsilon$, gives lower exponent
\[
 \sigma-\rho+\min\{k,(k+\sigma)/2,1\}-\varepsilon-o(1).
\]
To make this comparison quantitative, first prescribe an output error,
then choose all sampling, restriction, and Frostman losses below the
input allowance in that theorem, and finally let $m$ tend to infinity.
Sending the output error and $\rho$ to zero proves
\eqref{eq:source-15}.

In a bounded pin chart, the map from a direction to the line through a
given pin is uniformly bi-Lipschitz in its slope coordinate. The passage
between angular bins, lines, and dyadic tubes therefore changes only
bounded covering constants. If angular bounds are initially known only
for balls centered at surviving directions, an arbitrary ball meeting
the support is contained in the ball of twice its radius centered at
one of those directions. This also proves the stated variants.
\end{proof}

The weighted reduction above uses one common sampled direction
population to obtain its line-count upper bound. Its point fibers
may depend on the direction; the dense incidence hypothesis supplies
the needed comparison of laws. In the radial bootstrap below, only
the Furstenberg lower bound is needed, and its direction pencils
will instead be sampled separately at each pin.

\subsection{A radial consequence}

The dispersed/concentrated-tube division below is closely related to
the reciprocal thin-tube bootstrap of Orponen--Shmerkin--Wang
\cite[arXiv:2209.00348v1, Section~1.3, Lemma~2.8, and
Corollary~2.18]{OrponenShmerkinWang2024Radial}.  We prove the required
finite-scale bounds and power-small exceptions here, retaining the
conditional-law hypotheses needed in later applications.

For distinct planar points $x,y$, let $[y-x]$ denote the unoriented
direction of their joining line. For a planar law $\nu$, put
\[
 A_\beta(x,y)
 =\nu\{z\ne x:\operatorname{angle}([z-x],[y-x])\le\beta\}.
\]
Using oriented directions would make only a bounded difference.

\Needspace{3\baselineskip}
\begin{lemma}[Radial nonconcentration away from strips]
\label{proj:radial}
Let $0<s_0\le1$ and let $\nu$ be a planar Frostman $s_0$ law at
resolution $m^{-1}$. Suppose that, for every fixed $\chi>0$, every
strip of width $m^{-\chi}$ has power-small $\nu$ mass, uniformly over
the strip. For every $\sigma<s_0$ and every fixed $h\in(0,1]$, with
$\beta=m^{-h}$, the set of pairs for which
\[
 A_\beta(x,y)>\beta^\sigma
\]
has power-small $\nu\times\nu$ measure. Near-diagonal pairs may be
included in the exceptional set. The assertion holds simultaneously
on any fixed finite grid of scale powers.
\end{lemma}

\begin{proof}
At a fixed power scale $\beta$, all original subpower losses are also
$\beta^{o(1)}$. It is enough to rule out a sequence on which the bad
pair mass is $\beta^{o(1)}$: failure of any power bound would supply
such a subsequence. We use angular grids with bounded overlap, choosing
strict exponent margins so that fixed constants in the definition of
$A_\beta$ are harmless.

We first prove the assertion for $2\sigma<s_0$. At any pin there are
at most $O(\beta^{-\sigma})$ angular bins of mass at least
$\beta^\sigma$. Suppose that pairs in these bins have dense incidence.
Cauchy--Schwarz, applied to target fibers, gives two pins with a dense
common target set. They can be required to be at distance at least
$\beta^\alpha$, for any fixed sufficiently small $\alpha>0$, because
the Frostman estimate makes closer pin pairs power-small. Likewise,
discard the strip of width $\beta^\alpha$ about their joining line.
The strip hypothesis makes the total discarded triple mass
power-small, whereas the common-target triple mass is the square of
the dense incidence and is still subpower.

For a target outside this strip, the two joining directions make an
angle whose sine is at least a constant times $\beta^{2\alpha}$.
Indeed the cross product is the pin separation times the distance to
their joining line, and all distances in the denominator are bounded.
An intersection of two pencil tubes is therefore contained in a ball
of radius $O(\beta^{1-2\alpha})$. The common targets lie in at most
$O(\beta^{-2\sigma})$ such balls. Their total mass is at most
\[
 \beta^{-2\sigma+(1-2\alpha)s_0-o(1)},
\]
which is power-small when $\alpha$ is sufficiently small. This
contradicts dense common incidence and proves the initial range.

For the improvement step, suppose the assertion is already known at
an exponent $\sigma<s_0$, at every fixed power scale. Choose
$\alpha,\eta>0$ so that
\begin{equation}
 3\eta+\alpha<\frac{s_0-\sigma}{2},
 \qquad
 \eta<\frac{\alpha(s_0-\sigma)}2.
 \label{proj:radial-margins}
\end{equation}
We prove it at $\sigma+\eta$. A sufficiently fine finite power grid,
the previous assertion in both orders of a pair, and removal of
power-small exceptions give the following inherited estimates: at
every retained pair the angular masses, at all radii from $\beta$ to
$1$, are bounded by $\beta^{-\tau}$ times the radius to the power
$\sigma$. Here $\tau>0$ is arbitrarily small and is chosen after the
strict margins in \eqref{proj:radial-margins}. Bounds centered at
surviving directions extend to all angular balls by enlargement.

Suppose that angular bins of unrestricted target mass at least
$\beta^{\sigma+\eta}$ carry dense incidence even after this removal.
Call such a bin \emph{dispersed} if its containing tube has
$\nu\times\nu$ mass at least $\beta^{2\sigma+3\eta}$ in pairs
separated by at least $\beta^\alpha$. If the dispersed bins carry
dense incidence, discard pins whose surviving dispersed target fiber
has less than a subpower fraction of the target mass. The normalized
remaining pin law satisfies the planar $s_0$ bounds. At each such pin
$x$, normalize its surviving target fiber and push it forward by
$y\mapsto[y-x]$. The inherited angular bounds make this pin's
direction law Frostman $\sigma$, with arbitrarily small restriction
losses. Sample the pins and then sample each of these direction laws,
using the sampling and collision estimates above; choose comparable
pencil cardinalities. Every selected tube is dispersed, up to a
bounded enlargement, and the direction sets may vary with the pin.
These are the pin set and incident pencils in
Theorem~\ref{proj:furstenberg}. As its input losses tend to zero, it
produces at least
\[
 \beta^{-(s_0+3\sigma)/2+o(1)}
\]
distinct dispersed tubes. Here
$\min\{s_0+\sigma,(s_0+3\sigma)/2,1+\sigma\}
=(s_0+3\sigma)/2$, since $\sigma<s_0\le1$.

A separated subcollection of tubes gives the opposite bound. A pair
of points at separation at least $\beta^\alpha$ belongs to at most
$O(\beta^{-\alpha})$ tubes of this subcollection: the admissible
slopes occupy an interval of length $O(\beta/|z-z'|)$, and for each
slope only boundedly many intercept bins are possible. Integrating
the separated-pair masses of the tubes therefore gives at most
\[
 O(\beta^{-2\sigma-3\eta-\alpha})
\]
tubes. The first strict inequality in
\eqref{proj:radial-margins}, followed by sufficiently small input
losses in Theorem~\ref{proj:furstenberg}, contradicts the lower bound.

It remains to consider nondispersed bins. If such a bin has mass
$b\ge\beta^{\sigma+\eta}$, averaging the separated-pair mass inside
the bin gives a ball of radius $O(\beta^\alpha)$ containing all but
\[
 O(\beta^{2\sigma+3\eta}/b)
 \le O(\beta^{\sigma+2\eta})
\]
of its unrestricted target mass. There are at most
$O(\beta^{-\sigma-\eta})$ heavy bins per pin, so removing these
outside portions costs $O(\beta^\eta)$ in pair mass. Dense incidence
survives.

At a target $y$ with a dense reverse fiber, each surviving pin $x$
now determines a tube carrying unrestricted target mass
$\gtrsim\beta^{\sigma+\eta}$ inside $B(y,C\beta^\alpha)$:
the surviving $y$ lies in the ball chosen for its bin, and doubling
that ball about $y$ retains the same original target mass. If $z$
belongs to the same angular bin from $x$, boundedness of the ambient
region gives
\[
 \left|\det\left(z-y,\frac{y-x}{|y-x|}\right)\right|
 \lesssim\beta.
\]
For fixed $z$, the admissible reverse directions $[x-y]$ therefore
lie within $O(\beta/|z-y|)$ of $[z-y]$, with the trivial bound
when $|z-y|\lesssim\beta$. Apply the inherited reverse angular
estimate to this interval and integrate in $z$. Normalizing the
dense reverse fiber costs only a subpower factor, so the average is
at most
\begin{align*}
 &\beta^{-2\tau}
 \int_{|z-y|\lesssim\beta^\alpha}
       \min\{1,(\beta/|z-y|)^\sigma\}\,d\nu(z)\\
 &\hspace{25mm}\lesssim
       \beta^{\sigma-3\tau+\alpha(s_0-\sigma)}.
\end{align*}
To obtain the last line, split the integral into the ball of radius
$\beta$ and dyadic annuli between $\beta$ and $C\beta^\alpha$,
then use the planar Frostman bound; its exponent exceeds $\sigma$.
Taking $\tau$ sufficiently small, the second inequality in
\eqref{proj:radial-margins} makes this upper bound power-smaller than
$\beta^{\sigma+\eta}$. This is again a contradiction.

The averaging in the nondispersed case uses the original targets
inside each angular bin; thus its discarded mass estimate does not
presuppose that a restricted bin retains its original weight. On a
dense reverse fiber the prior centered angular estimates remain
valid after normalization, with only a subpower loss. These facts
justify both removals used above.

The improvement $\eta$ can be chosen bounded below while $\sigma$
stays a fixed distance below $s_0$. Starting from any exponent below
$s_0/2$ and iterating finitely many times proves the assertion at
every fixed exponent below $s_0$. At each step the desired output
gap is fixed first, then the old angular losses and the
Furstenberg-theorem input losses are chosen. A finite union of the
exceptional sets proves the simultaneous-grid assertion.
\end{proof}

\subsection{Scalar consequences for entropy tangents}

In the order of the coordinates in \eqref{eq:source-9}, write the
full partial derivatives as
\[
 (p,n_1,e,g,t,r)
 =(\partial_C F,\partial_B F,\partial_D F,
   \partial_A F,\partial_Y F,\partial_R F).
\]
Each of these six partials lies in $[0,1]$.
Here $t=\partial_Y F$ and $r=\partial_R F$ are entropy rates; in
\eqref{eq:source-9} the same letters denote truncated parameters.
To avoid imposing a false conclusion when the
input rate vanishes, define, for $z>0$,
\[
 f_z(0)=0,\qquad f_z(l)=\min\{1,z+l\}\quad(l>0).
\]

\Needspace{3\baselineskip}
\begin{proposition}[Scalar projection inequalities]
\label{proj:scalar}
At almost every point of the full alignment there is a tangent with
constant full partials, representing the derivatives along the
alignment, for which
\begin{equation}
 e\ge f_t(p),\qquad g\ge f_t(n_1),\qquad
 n_1\ge f_r(p),\qquad g\ge f_r(e),
 \label{eq:source-16}
\end{equation}
whenever the corresponding subscript is positive.
\end{proposition}

\begin{proof}
On the full alignment the six coordinate forms are
\[
 C,\quad C+R,\quad C+Y,\quad C+Y+R,\quad Y,\quad R.
\]
They are pairwise nonproportional and positive on a common vector.
The peeling rule of Lemma~\ref{ent:peeling}, applied successively to
extreme rays of the cone generated by these forms, makes each
coordinate a singleton summand in a tangent. Hence the tangent is
affine with the stated full partials.

We give the normal-velocity/time test in detail. Choose a block length
$\ell>0$ and a parent state $z_0$ on $A=B+Y$, with each of the
other three slacks in \eqref{eq:source-10} larger than $\ell$.
Keep $C,D,R$ fixed. Writing $\mathbf e_i$ for the coordinate vectors,
compare these four refinements of $z_0$:
\[
\begin{array}{c|c}
 \text{state}&\text{entropy increment from }z_0\\ \hline
 z_{\rm sp}=z_0+\ell(\mathbf e_B+\mathbf e_A)&\ell(n_1+g)\\
 z_{\rm par}=z_0+\ell\mathbf e_Y&\ell t\\
 z_{\rm joint}=z_0+\ell(\mathbf e_B+\mathbf e_A+\mathbf e_Y)
      &\ell(n_1+g+t)\\
 z_{\rm out}=z_0+\ell(\mathbf e_A+\mathbf e_Y)&\ell(g+t).
\end{array}
\]
The spatial and output refinements preserve $A=B+Y$; the parameter
and joint states lie on its admissible side. The chosen slacks make
all four states admissible. Their increments are those of the affine
tangent.

Inside a common parent cell, subtract the known bin centers and
normalize by the parent units. Let $v,w$ be the two skew-normal
spatial entries in the starting parameter frame, and let $a$ be the
normalized time increment. The matched equality makes the position
unit the product of the velocity and time units, so the output
measurement is the scalar shear $w+av$. The spatial state measures
$(v,w)$ and the parameter state measures $a$.
Lemma~\ref{ent:finite-states} identifies their joint labels with
$z_{\rm joint}$ up to bounded ambiguity: the fine time label permits
rebasing with the finely measured velocity. Center-dependent terms
are known translations, and \eqref{eq:source-10} bounds the
remaining errors.

Lemmas~\ref{ent:regularization} and~\ref{ent:tangent-realization}
turn these increments into hereditary support and weight bounds.
At every fixed block subresolution, the spatial law is planar
Frostman $n_1+g$, the coefficient law is Frostman $t$, and their
joint exponent is $n_1+g+t$.

We make the comparison with the actual marginal laws explicit.
Let $\mu_1,\mu_2$ be the marginals of the same joint law, with
support exponents $\alpha,\beta$ and maximal joint-bin weight
$m^{-\alpha-\beta+o(1)}$. In any incidence of joint mass
$\rho=m^{-o(1)}$, discard marginal bins with reference weights
below $m^{-\alpha-o(1)}$ and $m^{-\beta-o(1)}$, choosing the
threshold slack to tend to zero sufficiently slowly. The support
bounds make their marginal masses, and hence the joint mass incident
to them, $o(\rho)$. Since the paired labels encode the joint cells
up to bounded multiplicity, at least
$\rho m^{\alpha+\beta-o(1)}$ pairs remain. Each has
$\mu_1\times\mu_2$ weight at least
$m^{-\alpha-\beta-o(1)}$. Their total product mass is therefore
at least $\rho m^{-o(1)}=m^{-o(1)}$. For the present labels take
$\alpha=n_1+g$ and $\beta=t$. This proves dense incidence for
the actual Frostman marginals, also after the required subpower
joint restrictions.

Conditioning $z_{\rm out}$ on $z_{\rm par}$ leaves exponent $g$,
so the projections occupy at most $m^{g+o(1)}$ bins for typical
fine time labels, where $m$ absorbs the block length. The parent
and time-bin trims can be made negligible relative to this dense
incidence. If $n_1,t>0$,
Lemma~\ref{proj:projection} gives
\[
 g\ge\min\{n_1+g,(n_1+g+t)/2,1\}.
\]
If $g<1$, the first and third entries of this minimum exceed $g$,
so $g\ge n_1+t$. If $g=1$, the desired inequality is automatic.
This proves $g\ge f_t(n_1)$; the case $n_1=0$ is also automatic
by the definition of $f_t(0)$.

The other three tests use the following triples and shear maps:
\[
\begin{array}{c|c|c}
 \text{cutoffs}&\text{matched face}&\text{projection}\\ \hline
 (C,D,Y)&D=C+Y&z+ax\\
 (C,B,R)&B=C+R&y-ax\\
 (D,A,R)&A=D+R&w-az.
\end{array}
\]
Here the unused cutoffs are fixed with strict slack, and $a$ denotes
the parameter increment in the corresponding normalized units. The
same joint-count and conditional-support calculation proves the
remaining assertions. The shear update
\eqref{ent:shear-update} controls the labels at the unused cutoffs; it
does not require the normalized spatial law to be independent of
the coefficient law. Only dense product incidence is used.

The tangent is affine throughout its admissible domain. Placing a
test on its matched face with slack at the other constraints leaves
all six rates unchanged. The zero-input and saturated-output cases
are automatic; in every other case the projection minimum gives the
linear inequality that the output rate is at least the sum of the
input and parameter rates. Thus each displaced test proves the
assertion for the original constant rates.
\end{proof}

\begin{proposition}[Conditional parameter profile at an increasing tie]
\label{proj:parameter-profile}
Suppose $\lambda>0$. At a generic point of the calibrated line one
may take a further joint tangent in which the conditional parameter
profile, referenced to zero at the center, is
\begin{equation}
 K(Y,R)=sY+\theta(R-\lambda Y)_+,
 \qquad 0\le\theta\le s/\lambda.
 \label{eq:source-17}
\end{equation}
Consequently $t\ge s$ below the tie, while $r\ge\theta$ and
$t\ge s-\lambda\theta$ above it. The simultaneous parameter
advance $(Y,R)=(u,\lambda u)$ has rate at least $s$.

Let $\Gamma$ be the plane $R=\lambda Y$ inside full alignment. If
$\lambda\ne1$, the spatial partials can simultaneously be made
singleton constants at generic points of $\Gamma$. There
\eqref{eq:source-16} holds with time subscript $s$, and also with
tilt subscript $\theta$ when $\theta>0$.
\end{proposition}

\begin{proof}
The predictability statement of Section~\ref{sec:entropy} gives
$K(Y,R)=sY$ for $R\le\lambda Y$. Apply the increasing-coordinate
trace rule proved with Lemma~\ref{ent:signed-envelope} to the two
conditional parameter coordinates. A generic tangent along their
increasing tie is affine on each of the two ordering half-planes.
Continuity along $R=\lambda Y$ forces its upper-side expression to
be $sY+\theta(R-\lambda Y)$. Monotonicity in $R$ and $Y$ gives
$\theta\ge0$ and $s-\lambda\theta\ge0$, respectively. This proves
\eqref{eq:source-17}. Conditional parameter increments lower-bound
the corresponding full-state increments, giving all the asserted
rate bounds without an independence assumption.

On $\Gamma$ the spatial forms are
\[
 C,\qquad C+\lambda Y,\qquad C+Y,
 \qquad C+(1+\lambda)Y.
\]
For $\lambda>0$, $\lambda\ne1$, they are distinct,
nonproportional forms, and the only remaining proportional group
consists of the two parameters. Peeling separates the four spatial
singletons, whose rates are constant throughout this tangent.
At a generic full-alignment point on the lower side $R<\lambda Y$,
take a further realized tangent. It has the same four spatial rates,
and the conditional-parameter increment bound gives its time rate
at least $s$. Proposition~\ref{proj:scalar} therefore gives the time
inequalities with subscript $s$. At a generic full-alignment point
on the upper side $R>\lambda Y$, the same construction gives tilt
rate at least $\theta$ and the tilt inequalities with that subscript.
The two further tangents may have different parameter partials, but
both conclusions concern the original four constant spatial rates.
They therefore hold simultaneously at the tie.
\end{proof}

\subsection{Products of correlated parameter differences}

At $\lambda=1$ the forms $B$ and $D$ coincide on $\Gamma$.
Accordingly the preceding scalar tests alone do not separate their
rates. We first prove the parameter fact needed for this case.

\begin{lemma}[Product coefficients]
\label{proj:product}
Let $\nu$ be a law of pairs $d=(r_d,t_d)$ in a bounded planar set.
Assume its time marginal is Frostman $s$, its tilt marginal is
Frostman $\theta$ for some $\theta>0$, and $0<s\le1$. Assume also
that its simultaneous two-coordinate refinements have an affine
hereditary profile of dimension at least $s$: in typical
intermediate cells the normalized unrestricted increments satisfy
the Frostman $s$ bounds at all subsequent fixed block
subresolutions. For a pin $f=(r_f,t_f)$ define
\[
 Q_f(d)=(r_d-r_f)(t_d-t_f).
\]
The following alternatives suffice at every fixed finite grid of
resolutions.
\begin{enumerate}[label=(\roman*)]
\item If some strip of width $m^{-\chi}$, with fixed $\chi>0$,
has dense $\nu$ mass, restrict $\nu$ to that strip. At any fixed
resolution $m^{-h}$ with $0<h<\chi$ sufficiently small, the
coefficients $Q_f(d)$ have Frostman exponent $s$, up to subpower
loss, on dense pairs away from the time diagonal.
\item If every strip of each fixed power width has power-small
mass, then for every $\sigma<s$ and every fixed $h\in(0,1]$,
pairs lying in coefficient bins of conditional mass greater than
$m^{-h\sigma}$ have power-small total mass. After removal of
these exceptions on a sufficiently fine finite grid, the pinned
coefficient laws satisfy Frostman bounds with exponent
arbitrarily close to $s$ and arbitrarily small power loss.
\end{enumerate}
\end{lemma}

\begin{proof}
\textbf{The dense-strip alternative.}
Suppose first that a strip of width $m^{-\chi}$ has dense mass.
Its slope $a=dr/dt$ satisfies
\[
 m^{-o(1)}\le |a|\le m^{o(1)}.
\]
Indeed, if $|a|\le m^{-\varepsilon}$ along a subsequence, its
intersection with the bounded parameter region lies in a tilt
interval of length $O(m^{-\varepsilon}+m^{-\chi})$. The positive
tilt Frostman exponent makes its mass power-small. If
$|a|\ge m^\varepsilon$, including the vertical case, use the time
marginal instead. Since this applies to every fixed
$\varepsilon>0$, it proves the claim. Conditioning on the strip
costs only a subpower factor in either marginal bound.

On the strip write
\[
 r=at+b+e,
 \qquad |e|\le m^{-\chi}\sqrt{1+a^2}=m^{-\chi+o(1)}.
\]
Uniformly in its pins and targets,
\[
 Q_f(d)=a(t_d-t_f)^2+O(m^{-\chi+o(1)}).
\]
The time Frostman bound permits removal of pairs with
$|t_d-t_f|<m^{-o(1)}$, with the cutoff decreasing sufficiently
slowly that the lost mass is negligible compared with every dense
incidence being retained. On each sign branch of $t_d-t_f$ the
quadratic has derivative of magnitude between $m^{-o(1)}$ and
$m^{o(1)}$. At radii $\rho\ge m^{-h}$, with fixed $h<\chi$,
the inverse image of a coefficient interval is therefore contained
in at most two time intervals of length $m^{o(1)}\rho$. The time
marginal bound proves (i), also at all the intermediate radii
needed for a projection test.

\paragraph{Power-light strips and the conditional query law.}
For (ii), the planar law is Frostman $s$, by the simultaneous
refinement hypothesis. Lemma~\ref{proj:radial} therefore applies.
Fix $\sigma<s$ and $h>0$. Suppose that pairs landing in bins of
$Q_f$ of conditional mass at least $m^{-h\sigma}$ carry dense
incidence $E$. Condition the product law on this incidence. For
each pin the conditioned output occupies at most
$O(m^{h\sigma})$ bins, so, with $P_h$ the output bin,
\begin{equation}
 \Ent(P_h\mid f)\le h\sigma\log m+O(1).
 \label{proj:product-upper}
\end{equation}
All entropies in the rest of this proof are under the conditioned
law unless explicitly described as unrestricted.

Choose $\sigma<s'<s$ and divide $[0,h]$ into $J$ equal blocks
of length $\eta=h/J$. The integer $J$ will be large but fixed.
Let $D_j$ be the target parameter cell at depth $j\eta$, and
let $P_j$ be the nested bin of $Q_f$ at that depth. For
$j\ge3$ we claim that, apart from a negligible conditioned mass,
\begin{equation}
 \Ent(P_{j+1}\mid f,D_j)
 \ge (s'-\varepsilon)\eta\log m-o(\log m),
 \label{proj:product-block}
\end{equation}
where $\varepsilon>0$ is arbitrarily small. We verify carefully
the conditioning and direction estimates behind this assertion.

Use the hereditary parent estimates on this fixed block grid under
the original target law $\nu$, before conditioning on $E$. Choose
their errors and parent-trim thresholds as in
Lemma~\ref{ent:regularization}, so that the omitted original cell
mass is negligible compared with the dense incidence being retained.
For a remaining value $C$ of $D_j$, let $c$ be its center and let
$\mu_C$ be the normalized original restriction $\nu|_C$, expressed
in the cell's increment units. This \emph{unrestricted} law is
Frostman $s$ at all the required block subresolutions. The following
density trims compare the conditioned query and target fibers to
this law. The map
$\psi_C(v)=c+m^{-j\eta}v$ returns an increment to its physical
target cell. Define the pulled-back incidence and its target fibers by
\[
 E_C=\{(f,v):(f,\psi_C(v))\in E\},\qquad
 E_{C,f}=\{v:(f,v)\in E_C\}.
\]
The pin law $\nu$ remains in the original parameter coordinates.
Set
\[
 \alpha_C=(\nu\times\mu_C)(E_C).
\]
After discarding cells with too small an incidence density,
$\alpha_C=m^{-o(1)}$ in cells carrying almost all of the
conditioned mass. To see this, the total incidence in cells with
$\alpha_C<\rho$ is at most $\rho$; choose the subpower threshold
$\rho$ much smaller than the original dense incidence. The query
law in such a cell is
\begin{equation}
 q_C(df)=\frac{\mu_C(E_{C,f})}{\alpha_C}\,\nu(df)
 \le m^{o(1)}\nu(df).
 \label{proj:query-domination}
\end{equation}
Another density trim ensures $\mu_C(E_{C,f})=m^{-o(1)}$ for almost
all contributing queries. The same reasoning applies after any
of the finitely many required power-small exceptional pair sets
has been removed.

\paragraph{Radial control of the query directions.}
The gradient of $Q_f$ at $c$ is
\[
 \nabla Q_f(c)=(t_c-t_f,r_c-r_f).
\]
Thus its direction is the radial direction from $f$ to $c$ with
the coordinates interchanged. Remove the exceptional pairs from
Lemma~\ref{proj:radial}, in both orders and at a fine grid of
angular powers, using exponent $s'$, for the physical pairs
$(f,\psi_C(v))$. This removes a negligible
fraction of the dense incidence. Direction estimates may be
transferred from an actual surviving target $d\in C$ to the
center $c$. For comparison pins at distance at least
$m^{-\eta}$ from $c$, this changes the direction by
\[
 O(m^{-j\eta}/m^{-\eta})
   =O(m^{-(j-1)\eta}),
\]
which is below the block angular mesh for $j\ge3$. Comparison
pins closer than $O(m^{-\eta})$ have mass at most
$m^{-\eta s+o(1)}$ by planar Frostman. The surviving queried
pin itself can be required to be at distance at least
$m^{-o(1)}$ from the target, so its direction also changes by
less than the block mesh. For a ball centered at a surviving
query direction, choose its corresponding good target in $C$
and use its radial estimate. Enlarging the angular ball by the
above errors bounds all the far comparison pins; the near ones
have just been bounded separately. Re-centering balls extends
the conclusion to every angular interval. In view of
\eqref{proj:query-domination}, the law of gradient directions
under $q_C$ is consequently Frostman $s'$, with any prescribed
small power loss down to the block mesh. The grids and prior
losses are chosen sufficiently fine and small for that purpose.

\paragraph{Collision bounds inside a target cell.}
Here is the elementary projection-energy calculation in the
normalized cell. Let $v,v'$ be independent with law $\mu_C$,
and let $\omega$ have the gradient-direction law. A projection
collision at mesh $m^{-\eta}$ requires $\omega$ to lie in
boundedly many angular intervals of total length
$O(m^{-\eta}/|v-v'|)$. Therefore its averaged probability is
bounded by
\begin{align}
 &m^{\varepsilon_0}
 \iint\min\{1,(m^{-\eta}/|v-v'|)^{s'}\}
                       \,d\mu_C(v)d\mu_C(v')
 \notag\\
 &\hspace{35mm}\le m^{-\eta s'+\varepsilon_1}.
 \label{proj:collision}
\end{align}
The last estimate follows by summing annuli and using the
Frostman $s$ bound, with $s>s'$. Both
$\varepsilon_0,\varepsilon_1$ can be chosen arbitrarily small
compared with $\eta$. Markov's inequality gives this collision
bound, with another arbitrarily small loss, for almost all
queries. Restriction to $E_{C,f}$ multiplies the collision
probability by at most $\mu_C(E_{C,f})^{-2}=m^{o(1)}$. Thus the
estimate applies to the target fibers actually present in the
conditioned incidence, rather than just to unrestricted
increments.

The gradient magnitude is at least $m^{-o(1)}$ after the
near-diagonal removal. Normalizing it changes the mesh to
$m^{-\eta+o(1)}$, an admissible subpower change in
\eqref{proj:collision}. Taylor expansion gives, for bounded $v$,
\[
 Q_f(c+m^{-j\eta}v)
 =Q_f(c)+m^{-j\eta}\nabla Q_f(c)\cdot v
      +m^{-2j\eta}v_rv_t.
\]
The remainder is below the physical next resolution
$m^{-(j+1)\eta}$. Hence a bin of the nonlinear output is
covered by boundedly many bins of the linearized output at the
relevant normalized mesh. Finally, for bin probabilities $a_i$,
\[
 -\sum_i a_i\log a_i\ge-\log\sum_i a_i^2.
\]
The collision estimate yields \eqref{proj:product-block}.
The negligible removed conditioned mass is restored using
entropy concavity. All bounds are averaged with respect to the
actual conditional query laws, so no independence after
conditioning has been assumed.

\paragraph{Summing the output entropy increments.}
Because $Q_f$ is uniformly Lipschitz on the bounded parameter
region, $D_j$ together with $f$ determines $P_j$ up to bounded
ambiguity. Nested output bins also give
\begin{align}
 \Ent(P_{j+1}\mid f)-\Ent(P_j\mid f)
 &\ge
 \Ent(P_{j+1}\mid f,D_j)-\Ent(P_j\mid f,D_j).
 \label{proj:product-chain}
\end{align}
Indeed the left side is
$\Ent(P_{j+1}\mid f,P_j)$, and conditioning further on $D_j$
can only decrease it. The subtracted right-hand term is $O(1)$.
Sum \eqref{proj:product-chain} for $j=3,\ldots,J-1$ and use
\eqref{proj:product-block}. This gives
\[
 \Ent(P_h\mid f)
 \ge (J-3)(s'-\varepsilon)\eta\log m-o(\log m).
\]
Choose $J$ so large, and then $\varepsilon$ and all other losses
so small, that $(1-3/J)(s'-\varepsilon)>\sigma$. This contradicts
\eqref{proj:product-upper}.

We have ruled out a subpower sequence of bad pair masses, and
therefore proved a power-small bound at each fixed coefficient
scale. Remove these exceptions on a finite fine grid. The
conditional mass bound at the bin containing any surviving
coefficient gives the centered Frostman bound there; an interval
meeting the retained support is covered by a bounded enlargement
centered at one of its points. Normalizing dense pin fibers and
interpolating between grid powers costs an arbitrarily small
power. This proves (ii).
\end{proof}

\subsection{The tied spatial pair}

We now return to the coincident spatial forms.  The middle pair
$(B,D)$ remains one joint measurement.  Overlaps of full spatial
fibers will produce scalar projections with coefficients supplied
by Lemma~\ref{proj:product}; no separate rates for the two tied
coordinates are assumed.

\begin{proposition}[The equal-slope tie]
\label{proj:tied}
Suppose $\lambda=1$ and $\theta>0$ in
\eqref{eq:source-17}. At a generic point of $\Gamma$ there is a
tangent with singleton spatial rates $p,g$, an affine diagonal
rate $H$ for the pair $(B,D)$, and an affine diagonal parameter
rate. These rates satisfy
\begin{equation}
 g\ge f_s(p),\qquad H\ge f_s(p).
 \label{eq:source-18}
\end{equation}
\end{proposition}

\begin{proof}
On $\Gamma$ the coordinate forms are $C$, $C+Y$ twice,
$C+2Y$, and $Y$ twice. Peeling separates the singleton groups
$C,A$ and the paired groups $(B,D)$ and $(Y,R)$; each group
has an affine restriction to its diagonal. This gives the
claimed form. Write $J(B,D)$ for the separated middle-pair summand
of this actual tangent. On lower-side full alignment, $R<Y$ is
equivalent to $B<D$. Further full-alignment tangents there preserve
$p$ and have time rate at least $s$, so the scalar time test gives
$\partial_DJ\ge f_s(p)$ almost everywhere in $B<D$. For $h>0$,
monotonicity in $B$ and integration of this $D$-bound give
\[
 Hh=J(b+h,b+h)-J(b,b)
 \ge J(b,b+h)-J(b,b)\ge h f_s(p).
\]
The integration first holds on almost every vertical segment;
continuity includes every such segment and its boundary endpoint.
This proves the inequality for $H$ without assigning separate
$B,D$ partials at the tie. If $p=0$,
both assertions in \eqref{eq:source-18} are immediate. Assume
from now on that $p>0$.

Take a block in a typical fully aligned parent on the tie. In normalized units
write the spatial entries in its starting frame as $x,y,z,w$,
and fine parameter pairs as $d=(r_d,t_d)$ with law $\nu$.
The parameter diagonal rate is at least $s$. Heredity makes
its law Frostman of that diagonal dimension, also inside
intermediate parameter cells at every later fixed
subresolution. The time marginal is Frostman $s$, and the
tilt marginal has exponent at least $\theta>0$, by
\eqref{eq:source-17} and the conditional-parameter lower
bound. Thus the parameter laws satisfy the hypotheses of
Lemma~\ref{proj:product}.

Denote the parent by $z_0$ and the block length by $\ell>0$.
With $\mathbf e_i$ denoting the coordinate vectors, set
\[
\begin{aligned}
 z_{\rm sp}&=z_0+\ell(\mathbf e_C+\mathbf e_B+\mathbf e_D+\mathbf e_A),\\
 z_{\rm par}&=z_0+\ell(\mathbf e_Y+\mathbf e_R),\\
 z_{\rm joint}&=z_{\rm par}
                  +\ell(\mathbf e_C+\mathbf e_B+\mathbf e_D+\mathbf e_A).
\end{aligned}
\]
The spatial increment of $z_{\rm sp}$ has exponent $p+H+g$.
Group additivity gives the same exponent for $z_{\rm joint}$
conditional on $z_{\rm par}$. The marginal-weight comparison in
the proof of Proposition~\ref{proj:scalar}, with spatial exponent
$p+H+g$ and the parameter diagonal exponent, therefore gives dense
incidence under the product of the actual spatial and parameter
marginals. Given the fine
parameter label $d$, write the spatial entries in its frame as
$x,y_d,z_d,w_d$. The two further states
\[
 z_{\rm par}+\ell(\mathbf e_C+\mathbf e_A),\qquad
 z_{\rm par}+\ell(\mathbf e_B+\mathbf e_D+\mathbf e_A)
\]
have increments $\ell(p+g)$ and $\ell(H+g)$ from $z_{\rm par}$.
The remaining refinement from the first of these states to
$z_{\rm joint}$ costs $\ell H$. All these states satisfy
\eqref{eq:source-10}; in particular the fine parameters provide
the slack for these partial spatial refinements.

Lemma~\ref{ent:finite-states} rebases the full spatial labels using
the known fine parameters, with bounded ambiguity.
Lemmas~\ref{ent:regularization} and~\ref{ent:tangent-realization},
applied to these state pairs at every required block subresolution,
make the conditional $(x,w_d)$ law planar Frostman $p+g$.
Its additional $(y_d,z_d)$ fibers have support upper exponent $H$,
and the output triples $(y_d,z_d,w_d)$ have support upper exponent
$H+g$. We use these conditional laws before any overlap restriction.

We spell out how the dense spatial overlaps carry the weights
needed for Lemma~\ref{proj:projection}. This also ensures that
subsequent choices of parameters do not replace a weighted
projection problem by a mere cardinality assertion. Let $m'$
be the base of the positive-length block presently used; it
may be a fixed fractional block. In a typical parent, let
$S_f$ be the coarse-frame spatial bins incident to a fine
parameter bin $f$. Coarse-frame and fine-frame full spatial
bins determine one another up to bounded coverings when $f$
is given. The exact truncated parameters are used in
subtracting center-dependent translations, so large absolute
centers create no new error.

Put $L=p+H+g$. The common spatial universe is the support of the
spatial label at $z_{\rm sp}$, of size at most $(m')^{L+o(1)}$.
For almost every
tested $f$, $S_f$ has size $(m')^{L+o(1)}$, and every retained
joint cell has pre-trim conditional weight at least
$(m')^{-L-o(1)}$ given $f$. These conclusions follow from the support
and maximal-weight bounds for $z_{\rm joint}$ conditional on
$z_{\rm par}$, followed by discarding cells below the
reciprocal-support threshold. The lower weights refer to this
conditional law before that final discard. The exceptional fraction
can be made negligible compared with any prescribed dense
parameter restriction, by taking the threshold slack to zero
sufficiently slowly. In particular the entire sets $S_f$
remain available after such a restriction.

For any dense restriction of the parameter law, Cauchy--Schwarz
in the common spatial universe gives
\begin{align*}
 \iint |S_f\cap S_d|\,d\nu(f)d\nu(d)
 &=\sum_x\left(\int\mathbf1_{S_f}(x)\,d\nu(f)\right)^2\\
 &\ge
 \frac{\left(\int |S_f|\,d\nu(f)\right)^2}
      {|\text{spatial universe}|}
 \ge (m')^{L-o(1)}.
\end{align*}
Here $x$ in the sum denotes a full spatial cell, not only its
first coordinate. Since the overlap upper bound is
$(m')^{L+o(1)}$, a dense set of parameter pairs has overlaps
of full exponent $L$. Power-small exceptional parameter pairs
can be removed before selecting these overlaps.

For one such pair $f,d$, put
$\Delta r=r_d-r_f$ and $\Delta t=t_d-t_f$. The shear update
gives, to mesh accuracy,
\begin{align}
 y_d&=y_f-\Delta r\,x,
 &z_d&=z_f+\Delta t\,x,
 \notag\\
 w_d-\Delta t\,y_d+\Delta r\,z_d
     &=w_f+\Delta t\Delta r\,x.
 \label{proj:overlap-identity}
\end{align}
The last identity follows by substituting the first two into
the full shear update, including its mixed term. Bounded bin
errors produce only bounded mesh errors in normalized units.

There are at most $(m')^{p+g+o(1)}$ $(x,w_f)$ bins in the
overlap. Discard those carrying fewer than
$(m')^{H-o(1)}$ additional full cells, with the slack chosen
sufficiently slowly. They account for a negligible fraction
of an overlap of exponent $L$. Call the remaining input bins
rich. Inside one rich input bin, the different full cells map
to different triples $(y_d,z_d,w_d)$ up to bounded
multiplicity: once $x$ is fixed to mesh accuracy, the frame
change can be inverted from that triple. Thus a scalar value
on the right of \eqref{proj:overlap-identity}, attained in
one rich input bin, uses at least $(m')^{H-o(1)}$ output
triples. Scalar values separated by a sufficiently large
mesh constant use disjoint triples, because the left side
of that identity is determined to mesh accuracy by a triple.
The total output support has size at most
$(m')^{H+g+o(1)}$. Consequently the scalar projection
\[
 (x,w_f)\longmapsto w_f+\Delta r\Delta t\,x
\]
of the rich subset has at most $(m')^{g+o(1)}$ bins.

These rich subsets are dense for the original conditional
$(x,w_f)$ law. Indeed they retain $(m')^{L-o(1)}$ joint cells,
each with pre-trim conditional weight at least
$(m')^{-L-o(1)}$. Their total weight is therefore
$(m')^{-o(1)}$, and their image consists of the corresponding
coarse-accuracy input bins. The planar Frostman bound is the
one before this final restriction, obtained from the matched
$C,A$ advances. The rich restriction costs only a subpower
factor. This is the required passage from overlap counts to
weighted point fibers.

We now obtain enough coefficient directions. If $\nu$ has
dense mass in a strip of width $m^{-\chi}$ for some fixed
$\chi>0$, first restrict the parameters to this strip and
repeat the overlap argument with the full spatial sets of
the typical parameter bins. Use a fixed positive fractional
block ending at depth $h<\chi$ sufficiently small.
Lemma~\ref{proj:product}(i) gives a Frostman $s$ law of
$\Delta r\Delta t$ on dense pairs. The removed time-diagonal
pairs have negligible mass compared with the dense overlaps.
All spatial rate, fiber, and weight statements above apply
at this fractional depth with $m'=m^h$.

If there is no such strip restriction along the realizing
sequence, pass to a subsequence on which every strip of each
fixed power width has power-small mass. This dichotomy is
obtained by inspecting the maximum strip mass on countably
many fixed power grids; bounded mesh enlargements suffice.
Lemma~\ref{proj:product}(ii) gives coefficient exponents
arbitrarily close to $s$ off power-small exceptional pairs.
Remove these pairs on the finite fine grid of coefficient
resolutions before choosing a pin with dense overlap.

In either case, averaging selects $f$ with densely many
targets $d$ that have the coefficient Frostman bounds and
the rich projected-input subsets just constructed. Parameter
binning changes the coefficient only by mesh order. If
several target bins produce the same coefficient bin, choose
one corresponding rich input subset for that bin; each has
the required dense mass for the same pre-trim conditional
$(x,w_f)$ law. Thus the coefficient marginal together with
these rich subsets supplies dense product incidence for
Lemma~\ref{proj:projection}. The map from a bounded scalar
coefficient to the associated projection direction is
bi-Lipschitz in a bounded chart.

Applying that lemma and letting all coefficient and
restriction losses tend to zero yields
\[
 g\ge\min\{p+g,(p+g+s)/2,1\}.
\]
Since $p>0$, either $g=1$, or the first and third terms
exceed $g$ and the middle term gives $g\ge p+s$. In both
cases $g\ge f_s(p)$, completing the proof.
\end{proof}
==== END Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//03-projection.tex ====

==== BEGIN Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//04-aspect.tex ====
\section{Excluding calibrated aspect paths}\label{sec:aspect}

We complete the proof of Theorem~\ref{geo:main}. Suppose, to the contrary, that
\begin{equation}\label{asp:contradiction}
q_*<1+2s-10\kappa.
\end{equation}
The preceding construction supplies a calibrated entropy profile and finite
parameters
\[
a\ge0,\qquad b\le1,\qquad \lambda=a-b.
\]
We may take further simultaneous tangents at generic points of the calibrated
line. The support inequality, its equality on that line, the conditional
parameter profiles, and the predictability statement survive these operations
by Lemmas~\ref{ent:predictability} and~\ref{ent:tangent-realization}.
All actual entropy profiles below have been regularized as in
Lemma~\ref{ent:regularization}.
Thus a ``dense'' restriction means a restriction of subpower relative mass,
and support exponents used in a conditional block have the corresponding
typical-bin weight bounds. These conventions also apply to iterated tangents.

The cases below exhaust the finite parameters supplied by the
calibration.  The table records their roles; every comparison is
proved in the corresponding subsection.
\begin{center}
\small
\begin{tabular}{@{}p{.23\linewidth}p{.30\linewidth}p{.40\linewidth}@{}}
\toprule
Aspect & Further distinction & Comparison mechanism \\
\midrule
$\lambda=0$ & $a>0$ or $a=0$ & Isotropic profile, radial test,
and strip-axis reduction. \\
$\lambda>0$, $a>0$ & $\theta=0$; or $\theta>0$ with
$\lambda\ne1$ or $\lambda=1$ & Longitudinal comparisons,
pure-order costs, and the joint-coordinate tie. \\
$\lambda>0$, $a=0$ & Zero or positive full $C$-slope; for
$\theta>0$, causal interior or boundary & Static-coordinate separation and
a local constrained comparison. \\
$\lambda<0$ & $a>0$ or $a=0$ & Spatial axis prediction,
constrained pure orders, and longitudinal comparisons. \\
\bottomrule
\end{tabular}
\end{center}
Here $\theta$ is the additional conditional axis rate in
\eqref{eq:source-17}, not a spatial slope.  The backward gain
identity and reusable scalar comparisons are collected after the
stationary case.  All projection bounds below are applied to actual
realizing profiles; pure envelopes provide only the ordered costs
and overlap identities of Lemma~\ref{ent:signed-envelope}.

\subsection{Stationary aspect}

\begin{proposition}\label{asp:stationary}
There is no calibrated profile satisfying \eqref{asp:contradiction} with
\(\lambda=0\).
\end{proposition}

\begin{proof}
Here \(0\le a=b\le1\), and we use the isotropic profile
\(F(u,z,Y)\) of \eqref{eq:source-13}, on \(z\le u+Y\).
The reference-axis label is retained with the exact entropy
cancellation \eqref{ent:stationary-axis-cancellation}; the later
choice of a strip axis will also retain its earlier parent.
Suppose first that \(a>0\). Take a generic tangent on the calibrated line and
apply Lemma~\ref{ent:signed-envelope}, with \(u\) the only negative coordinate.
Its pure profile is \(Pu\), where \(P\) is constant and
\(\partial_uF\ge P\) throughout the domain. For \(T>0\), run backwards
from \((0,0,0)\) to \((T,0,-T)\), keeping \(z=0\), with the frozen
velocity cutoff in the upper envelope sufficiently fine for this
segment. The remaining time rate is at least \(s\), so the support
inequality and upper envelope give
\[
(2-q_*)T\le F(T,0,-T)\le(P-s)T.
\]
Consequently
\[
q_*\ge s+2-P,
\qquad\text{and hence}\qquad P>1-s.
\]

On the early side of the calibrated line, namely
\(u+aY<0\), \(z=u+Y\), the support inequality and equality on the line imply
an average bound \(\partial_uF+\partial_zF\le2\) when integrating toward
the line at fixed \(Y\). Since \(\partial_uF\ge P>1-s\), there is a generic
aligned point on this side with position rate
\(E:=\partial_zF<1+s\). Lemma~\ref{ent:peeling} gives an affine tangent in
the three isotropic coordinates there; denote its time rate by
\(\tau\ge s\). Simultaneously retain a tangent at the calibrated point with
the same velocity depth \(u\), whose time depth is \(-u/a\). This is later
than the early point and has finer position depth. The two sites have the
same physical velocity units, and the later profile still satisfies
\(\partial_uF\ge P\).

If \(a=0\), first peel off the constant coordinate on the line
\(u=0,z=Y\). In a tangent,
\[
F(u,z,Y)=J(u)+H_0(z,Y),
\]
where the second factor is defined on the full projected \((z,Y)\)-plane.
At a further generic point on the line, the two-coordinate order rule makes
\(H_0\) affine on each order half-plane. On \(z\ge Y\), write
\[
H_0(z,Y)=Ez+\tau Y.
\]
Then \(\tau\ge s\) and \(E+\tau=q_*\), so \(E<1+s\).
At \(Y=0,u=z>0\), the support inequality gives
\begin{equation}\label{asp:J-prefix}
J(u)\ge(2-E)u.
\end{equation}
The following projection block is based at the origin in this case.
Choose any fixed point on \(u=0,z=Y\) with \(Y>0\) as the later site;
the tests near it will use velocity depths near zero.

We give the common block argument. Write \(m\) for its base and let
\(b_0\in\mathbb R^2\) be normalized velocity. Let
\((d,c_0)\in\mathbb R\times\mathbb R^2\) be normalized time and position
in the fine-time frame, after subtracting known translations. Position in the
parent frame is therefore
\[
A_{b_0}(d,c_0):=c_0-d b_0
\]
to mesh accuracy. The profile and its regularization give the following
properties inside typical parents.
\begin{enumerate}[label=(\roman*)]
\item Velocity has a Frostman ball bound of some exponent strictly greater
than \(1-s\). For \(a>0\) this follows from \(\partial_uF\ge P\); for
\(a=0\) it follows from \eqref{asp:J-prefix} at every tested prefix depth.
\item Velocity cells and time-position cells have dense product incidence.
Indeed the refinements \((u,z,Y)=(1,0,0)\), \((0,1,1)\), and
\((1,1,1)\) have additive joint exponents. The time-position exponent is
\(E+\tau\).
\item Given a fine velocity bin \(b_0\), the image under \(A_{b_0}\) has
a Frostman \(E\) bound and at most \(m^{E+o(1)}\) bins. The fiber in one
image bin has at most \(m^{\tau+o(1)}\) time-position cells. These follow
by testing \(u=1\), \(0\le z\le1\), \(Y=0\), and the joint refinement,
rebasing with the known velocity.
\end{enumerate}

Suppose that every strip of power-small width has power-small velocity mass.
By Lemma~\ref{proj:radial}, applied with a slightly smaller ball exponent
at most one, directions \(b-b_0\) have an angular Frostman exponent
\(\sigma>1-s\) outside removable exceptional pairs. Product density and
Cauchy--Schwarz give a pin \(b_0\) and a dense set of such \(b\)'s that
share a dense set of time-position cells with \(b_0\). We may also arrange
\(|b-b_0|\ge m^{-o(1)}\).

Fix one of these overlaps. Keep its rich \(A_b\)-bins, each containing at
least \(m^{\tau-o(1)}\) cells. The support upper bound for \(A_b\) shows
that these retain full overlap exponent \(E+\tau\). Inside a fixed
\(A_b\)-bin the different cells give, up to bounded mesh multiplicity,
different \(d\)-bins: once \(d\) and \(A_b\) are given, \(c_0=A_b+db\)
is known to the tested accuracy. Moreover
\[
A_{b_0}=A_b+(b-b_0)d.
\]
Since \(|b-b_0|\ge m^{-o(1)}\), the rich bin therefore yields at least
\(m^{\tau-o(1)}\) distinct \(A_{b_0}\)-bins in one thick row parallel
to \(b-b_0\). Select perpendicular row labels separated by a sufficiently
large mesh constant. Their collections of point bins are disjoint up to
bounded multiplicity. There are at most \(m^{E+o(1)}\) point bins in all,
so the perpendicular projection of the retained \(A_{b_0}\)'s has at most
\(m^{E-\tau+o(1)}\) bins.

This restriction is dense for the Frostman image law, not merely large in
cardinality. Indeed full overlap exponent \(E+\tau\), together with the
fiber upper bound \(\tau\), gives \(E-o(1)\) image exponent. The typical
image bins have pre-trim weights at least \(m^{-E-o(1)}\), by
Lemma~\ref{ent:regularization}. Thus the image restriction has subpower
mass. We can consequently apply Lemma~\ref{proj:projection} to these
direction-dependent restrictions. It is contradicted by
\[
E-\tau<E,\qquad E-\tau<1,\qquad
E-\tau<(E+\sigma)/2.
\]
The first inequality uses \(\tau\ge s>0\), the second uses
\(E<1+s\), and the third follows from
\(E-2\tau<1-s<\sigma\). If \(E-\tau<0\), the support upper bound itself
is already impossible.

It follows that, in a dense family of parents, a dense amount of velocity
mass lies in strips of width \(m^{-\chi}\) for some fixed \(\chi>0\),
after subselection. Here the choice need not use one exceptionally rare
parent. To see this, average the maximal conditional strip mass over the
parents, using finite grids and bounded strip enlargements. If no fixed
width power had subpower average along a subsequence, each such average
would be power-small. Markov's inequality and diagonal selection over
countably many width powers would produce typical parents with all strips
power-light, contrary to the block argument. We retain the selected strip
mass across the parents, choosing its axis separately in each parent.

Choose a fixed neighborhood of the later site in which the time and
position cutoffs remain finer than those of the earlier parent. Use the
resulting parallel and normal coordinates there, and include the earlier
parent label. At later isotropic states
with positive velocity refinement, this label is recoverable up to bounded
ambiguity. Both time and position are finer than at the earlier parent;
rebasing position backwards multiplies velocity uncertainty by at most the
earlier time resolution. The label includes the earlier time bin, spatial
bins, and any fixed original slope label. Once it is known, the strip axis
is fixed, and switching isotropic grids costs only bounded multiplicity.
Thus the later isotropic profile and calibration are preserved under the
dense restriction.

The conditional time rate is also preserved. Given the base label, the
fixed original slope label, and a later time prefix, the earlier parent
time and its spatial bins, hence its selected axis, are already determined.
Conditioning on these extras therefore consumes no further time increment.
The linear frame change in velocity is used for position in the matched
units as well: position units are velocity units times the corresponding
time unit. In the two-site construction only the time and position units
change between the sites. This retains the scalar shear relation.
Lemma~\ref{ent:tangent-realization} allows these tests to be realized
simultaneously, with errors negligible in the final block units.

Now take mixed velocity depths \(l_1,l_2\in(0,\chi)\), with position
depths \(l_i+Y\). Given the parent, the normal velocity rate is zero
throughout this depth window. The parallel velocity rate is at least \(P\)
if \(a>0\), and at least \(J'(l_1)\) almost everywhere if \(a=0\).
For completeness, this lower bound is valid also at the aligned faces used
in projection tests. Compare a short parallel increment with the same
increment after making time and both position cutoffs finer, with the two
position cutoffs equal and with strict slack. At the finer state change
the normal velocity cutoff to the parallel cutoff at both endpoints; this
costs zero while both depths remain in \((0,\chi)\). The resulting
isotropic increment has the asserted lower rate. Dropping the extra
conditioning can only increase that increment by submodularity. Valid
interior comparisons pass to the aligned tangents by continuity.

Whenever this parallel rate exceeds \(1-s\), its aligned position rate
equals one by the scalar projection test \eqref{eq:source-16}, now in the
fixed axes. The distinct velocity, position, and time forms can be peeled
at generic mixed aligned states, so these are full partial rates there.

For \(a>0\), start on the later calibrated path with equal velocity
depths \(l_1=l_2=l\in(0,\chi)\). Choose the backward duration small
enough to stay in the chosen neighborhood and to keep both depths in
\((0,\chi)\). Hold the parallel depth \(l_1\) fixed and increase the
normal depth \(l_2\) at speed one. Then the normal position depth
\(l_2+Y\) is fixed, while the parallel position depth decreases at
speed one. The normal velocity has zero rate, so time contributes at
least \(s\) and parallel position contributes one. The advancing normal
depth is the larger width depth and therefore has coefficient
\(1-\kappa\) in the ellipse support term. Thus the profile minus
the ellipse width term decreases backwards at rate at least
\begin{equation}\label{asp:stationary-gain}
2+s-\kappa.
\end{equation}
To integrate generic derivatives, perturb the isotropic starting depth and
starting time slightly. The resulting segments sample all three mixed
alignment parameters. Integrate on generic segments and pass back by
Lipschitz continuity. Parent recoverability holds throughout because the
starting velocity depths remain inside the strip window. The initial
equality and support now contradict \eqref{asp:contradiction}.

For \(a=0\), \eqref{asp:J-prefix} supplies arbitrarily small positive
\(l\) with \(J'(l)>1-s\), on a set of positive measure. Fix a positive
comparison duration within the chosen later neighborhood and smaller
than \(\chi/2\). For each sufficiently small such generic \(l\), set
both initial depths to \(l\) and use generic later starting times.
The error of the initial isotropic state against the calibrated value
is \(O(l)\), and
including the parent adds no error in exponent. Sending \(l\) to zero
with the duration fixed again contradicts \eqref{asp:contradiction}, since
\(2+s-\kappa>1+2s-10\kappa\) for \(s\le1\).
\end{proof}

\subsection{Gain and scalar backward comparisons}

Henceforth \(\lambda\ne0\). Set
\[
u_*=1-s,\qquad h=(1+s)/2,\qquad
k_C=1+\kappa,\qquad k_B=1-\kappa.
\]
Testing states are fully aligned unless indicated otherwise:
\[
R=B-C,\qquad D=C+Y,\qquad A=B+Y.
\]
Put \(\mathcal H=F-k_CC-k_BB\), and write \(z(y)\) for a fully aligned
state with \(Y=y\). A backward segment of duration \(T\) starts at
\(z(0)\) and ends at \(z(-T)\); its \emph{gain} is
\(\mathcal H(z(0))-\mathcal H(z(-T))\).
Prescribe velocities by \(C'=-v_1\), \(B'=-v_2\), where primes denote
forward \(y\)-derivatives. Recall that
\[
(p,n_1,e,g,t,r)
=(\partial_CF,\partial_BF,\partial_DF,\partial_AF,
  \partial_YF,\partial_RF).
\]
The forward \(y\)-derivative of \(\mathcal H(z(y))\) is
\begin{equation}\label{eq:source-19}
t+(v_1-v_2)r-v_1p-v_2n_1
 +(1-v_1)e+(1-v_2)g+k_Cv_1+k_Bv_2.
\end{equation}
Indeed \(R'=v_1-v_2\), \(D'=1-v_1\), and \(A'=1-v_2\).
Integrating this derivative over \([-T,0]\) gives the gain. With the
increasing backward clock \(\tau=-y\), the potential in
Lemma~\ref{ent:control} is \(-\mathcal H(z(-\tau))\).
At a calibrated start, referenced to zero, \eqref{eq:source-11} gives
\(\mathcal H(z(-T))\ge-q_*T\), so the gain is at most \(q_*T\).
If an upper envelope \(U\ge F\) from \eqref{eq:source-14} agrees with
\(F\) at the start, its endpoint gain is a lower bound for the actual
gain. An initial error tending to zero is harmless.

We make precise how rates on generic sheets will be used. A singleton
rate obtained by peeling is a constant full partial in that tangent; a
group rate is the derivative along its specified diagonal. To integrate
an inequality valid on a sheet-open region, first integrate on almost all
slightly translated parallel segments and then pass to the chosen segment
by Lipschitz continuity. This also transfers an inequality to a boundary
approached from the indicated side. Subsequent off-sheet tests of a
singleton-affine tangent retain that singleton rate.

In particular, let \(\Gamma\) be the fully aligned plane
\(R=\lambda Y\) when \(\lambda>0\). For \(\lambda\ne1\), its spatial
partials are singleton constants in a generic first tangent, and the
parameter diagonal \((Y,R)\mapsto(Y+z,R+\lambda z)\) has rate at least
\(s\). The latter lower bound can be tested with spatial cutoffs fixed
in the interior of the separated tangent. Projection tests on its two
sides supply the subscripts \(s\) and \(\theta\) of
\eqref{eq:source-17} simultaneously. When \(\lambda=1\), the corresponding
grouped conclusion is \eqref{eq:source-18}. These facts hold in tangents
based anywhere on the tie, not only on the equality line.

\begin{lemma}[Scalar comparison]\label{asp:scalar}
Suppose a local restriction of an inherited profile has grouped coordinates
\(X,y,Z\), with domain \(Z\le X+y\), and no other local constraints.
The groups consist of disjoint nonnegative coordinate increments of the
underlying profile; additional fixed coordinates are allowed. At generic
aligned points write their rates as \(d_X,d_y,d_Z\). Suppose, also in the
tangent tests at the path points under consideration, that
\begin{equation}\label{eq:source-20}
d_y\ge t_0,
\qquad d_X>u_*/2\ \Longrightarrow\ d_Z\ge h.
\end{equation}
For \(k\in\{k_C,k_B\}\), the gain of this scalar profile minus \(kX\)
can reach rate \(t_0+h-\kappa\), with arbitrarily small further loss.
More precisely, along every path on \(Z=X+y\), parametrized by \(y\)
and with \(-X'(y)\in[\rho,1-\rho]\), almost every point admits arbitrarily short
straight backward comparisons with this rate, up to a loss tending to zero
with \(\rho\), and with velocity in the same interval.
\end{lemma}

\begin{proof}
Apply Lemma~\ref{ent:signed-envelope} separately along the path under
consideration. The negative coordinate is \(X\); \(y,Z\) strictly
increase. Slack is obtained by shifting \(y\) more than \(Z\).
At almost every path point the resulting tangent has pure slope \(p_0\),
and \(d_X\ge p_0\). If \(p_0>u_*/2\), velocity zero gives rate at least
\(t_0+h\) by \eqref{eq:source-20}.

If \(p_0\le u_*/2\), use velocity one. In centered offsets its ray is
\(X=-y,Z=0,y<0\). If \(m\in[\rho,1-\rho]\) is the original path speed,
choose \(\beta\in(y/m,y)\). Then
\[
-y>-m\beta,\qquad y>\beta,\qquad0>(1-m)\beta.
\]
These inequalities give the overlap condition of
Lemma~\ref{ent:signed-envelope}, locally up to alignment. Thus
\(d_X=p_0\) on the comparison ray, and integration gives rate at least
\[
t_0+k-p_0\ge t_0+1-\kappa-u_*/2=t_0+h-\kappa.
\]
Perturb velocities zero and one into the permitted interval. The loss is
arbitrarily small by the Lipschitz bounds. The strict comparison in the
tangent gives arbitrarily short comparisons in the original profile.
Any additional linear contribution to the gain is included in these same
endpoint calculations.
\end{proof}

\begin{lemma}[Longitudinal comparisons]\label{asp:longitudinal}
The following two comparisons are available locally, and in inherited
path tangents, under the stated derivative and conditional-time bounds.
\begin{enumerate}[label=(\roman*)]
\item \emph{Low longitudinal rate.} Fix \(v_1=1\), so \(D=d_0'\) is
constant. Suppose the backward cost of the varying \(C\)-term has been
bounded by \(P\le u_*/2\) and removed. This may be done in the pure upper
envelope when \(B\) is not negative, or where \(\partial_CF=P\) exactly.
If \(t\ge s\) in the local region, the total gain can reach
\(1+2s-\kappa\), with arbitrarily small further loss.
\item \emph{High longitudinal rate.} Fix \(v_1=0\), so \(C=c_0'\) is
constant. Suppose \(p\ge P>u_*/2\) in neighborhoods of the aligned states
being visited, up to the domain constraints, and \(t\ge s\). The same
gain \(1+2s-\kappa\) is available.
\end{enumerate}
\end{lemma}

\begin{proof}
For the low comparison use the scalar groups
\[
X=B,\qquad y=Y,\qquad Z=A=R+d_0'.
\]
For the upper envelope \eqref{eq:source-14}, the remaining function is
\[
F^+(Q)=F(n(-H_0),Q)-G(n(-H_0)).
\]
Its rates are those of the actual frozen-cutoff slice
\(Q\mapsto F(n(-H_0),Q)\), since the subtracted term is constant.
Choose \(H_0\) sufficiently large for all the following compact tests;
in particular \(C\) is fixed sufficiently fine in this slice.
Any additional negative \(D\) is held fixed in the pure part;
in the remaining part it is either \(d_0'\) or fixed finer, with
\(D<C+Y\). In the exact-slope version \(F-PC\) is locally independent
of \(C\). Freeze \(C\) at a slightly finer value to obtain strict slack
for \(B<C+R\) and \(D<C+Y\); on alignment this auxiliary slice agrees
with the remainder up to a constant.

To verify \eqref{eq:source-20}, retain \(R,A\) separately with
\(A\le D+R\), \(A\le B+Y\), while the other faces have slack. At a
generic scalar aligned point in any path tangent, peel off the singleton
rates for \(B,Y\); the \((R,A)\)-summand has affine diagonal rate.
Increase \(R\) separately to open \(A<D+R\), retaining \(A=B+Y\).
The time-projection block for \((B,Y,A)\) gives
\(g\ge f_s(n_1)\). Its fixed \(D\)-measurement rebases under time
refinement with bounded ambiguity because \(C\) is fine. The two marginal
laws and their dense product relation are the comparable-cutoff laws of
Lemma~\ref{ent:regularization}; thus this is the actual sheared block
test. If \(d_X=n_1>u_*/2\), it follows that \(g\ge h\).
The direction increasing \(B,R,A\) together preserves \(A=B+Y\)
and has rate \(n_1+r+g\ge n_1+h\). Integrate this inequality on
the displaced matching slices and pass back by continuity to the
\((R,A)\)-grouped slice. There the same direction has rate
\(n_1+d_Z\). The peeled singleton \(n_1\) is constant on both slices,
so subtraction gives \(d_Z\ge h\). The conditional parameter
increment with \(C,D\) fixed gives \(d_y\ge s\).

These constructions also apply after the signed-path tangent: take
simultaneous tangents of the indicated auxiliary slices, which have actual
realizing states. When freezing \(C\) is only local, cover the open path
portion by countably many neighborhoods with a single valid frozen value
on each. Lemma~\ref{asp:scalar} with \(t_0=s\) now gives total rate at least
\[
s+h-\kappa+k_C-P\ge1+2s-\kappa.
\]

For the high comparison the scalar groups are
\[
X=B=R+c_0',\qquad y=Y=D-c_0',\qquad Z=A.
\]
At full-aligned generic states, \(e\ge f_s(p)\ge h\). Hence the
\(y\)-group has rate at least \(s+h\). If \(g<h\), the projection
inequalities imply \(n_1\le u_*/2\). They also force \(r=0\): otherwise
\(g\ge f_r(e)\ge h\), a contradiction.

We justify transfer to the grouped slice, including its path tangents.
At a generic scalar aligned point, first peel off the fixed coordinate
\(C\) by a direction increasing every other coordinate; then separate
the \(X,y,Z\) groups. Their distinct forms are positive on a common
vector. The \(A\)-rate is therefore a constant singleton in this first
tangent. Test full-aligned generic points on nearby translated sheets,
allowing \(C\) to vary. The aligned \(y\)-direction also advances \(A\);
subtracting its constant rate transfers \(d_y=t+e\ge s+h\).
If this singleton rate is less than \(h\), the aligned \(X\)-direction,
with the same subtraction, transfers
\(d_X=n_1+r\le u_*/2\). Thus \eqref{eq:source-20} holds with
\(t_0=s+h\), and Lemma~\ref{asp:scalar} gives
\(s+2h-\kappa=1+2s-\kappa\).
\end{proof}

Whenever a prescribed-length comparison is needed in a free region, apply
Lemma~\ref{ent:control} to these pathwise short steps, taking \(\rho\)
small. If the construction occurs inside a tangent, its endpoint comparison
is enough: the effective straight velocity lies in the convex velocity box,
and the comparison passes back to arbitrarily short original steps. A
linear closed half-space constraint passes back as well. Pure backward
cost bounds may be integrated with the remaining gain, or subtracted at
the final endpoint from an upper envelope that is exact at the start.
Choose the additional endpoint, clipping, and control losses so that,
together with the loss already incurred in the selected comparison,
the total is less than \(10\kappa\).

\subsection{Increasing aspect with positive longitudinal speed}

\begin{proposition}\label{asp:increasing-positive}
There is no calibrated profile satisfying \eqref{asp:contradiction} with
\(\lambda>0\) and \(a>0\).
\end{proposition}

\begin{proof}
Take the conditional parameter tangent \eqref{eq:source-17} and a further
simultaneous signed-envelope tangent on the calibrated line. The negative
coordinates are \(C\), together with \(B\) if \(b>0\) and \(D\) if
\(a>1\). Their pure domain is full: the parameters \(R,Y\) in mixed
constraints increase, and negative cutoffs at an earlier path time can be
paired with the other cutoffs at a later time. Large positive shifts in
\(R,Y\) open the required slack. On compact aligned tests all negatives can
be replaced by \(n(-H_0)\) in \eqref{eq:source-14} for sufficiently large
\(H_0\). When relevant, the differences \(C-B\) and \(C-D\) of the
shifted negatives increase strictly, since \(a>b\) and \(a>a-1\).

Let \(P\in[0,1]\) be the pure marginal rate of \(C\). If \(P=0\), test
the upper envelope backwards for unit time with \(v_1=M_0,v_2=0\),
where \(M_0\) is large and fixed. The \(C\)-cost is zero, \(B\) is
constant, and \(Y,A,R\) decrease backwards. The only possible positive
backward cost is that of \(D\), at most \(M_0-1\), whether it belongs
to the pure or remaining profile. The gain is at least
\[
k_CM_0-(M_0-1)=1+\kappa M_0,
\]
contradicting support for large \(M_0\). Comparisons along binding faces
follow by translating the segments into the relative interior first.
Henceforth \(P>0\).

We divide the remaining proof into three cases. When \(\theta=0\),
we use longitudinal comparisons, upper-envelope tests, and full-alignment
projection bounds. When \(\theta>0\) and \(\lambda\ne1\), we use the
projection inequalities on the tied plane and then separate the possible
velocities. When \(\theta>0\) and \(\lambda=1\), the coincident spatial
forms require the grouped projection bounds.

\subsubsection*{No additional conditional axis rate}

First suppose \(\theta=0\), so \(t\ge s\) everywhere. If \(b\le0\)
and \(P\le u_*/2\), apply the low comparison of
Lemma~\ref{asp:longitudinal} to the upper envelope starting at zero.
Its pure cost along \(v_1=1\) is at most \(P\), and any additional
negative \(D\) is constant. If \(b\le0\) and \(P>u_*/2\), use the
high comparison in the actual profile. On its backward paths from
\(C=0\), the pure \(C\)-rate is \(P\): if \(D\) is negative, its
value \(D=y<0\) is strictly coarser than \(C=0\) in normalized negative
units. Thus \(p\ge P\) locally at the visited states. Both comparisons
contradict support.

Suppose \(b>0\), and let \(Q_1\) be the pure marginal of \(B\).
If \(P+Q_1\le u_*\), the upper-envelope test \(v_1=v_2=1\) keeps
\(D,A,R\) fixed and costs at most \(P+Q_1\) in the pure part. The
remaining time rate is at least \(s\), so its gain is at least
\(2+s-(P+Q_1)\ge1+2s\). If \(Q_1=0\) and \(P>u_*\), the pure
profile is independent of \(B\), and the high comparison has \(p\ge P\)
locally at its visited states, with any negative \(D\) strictly coarser.

The remaining possibility is \(P,Q_1>0\) and \(P+Q_1>u_*\).
Consider the aligned sheet
\begin{equation}\label{eq:source-21}
\begin{gathered}
C=-a\alpha,\qquad B=-b\alpha,\qquad R=B-C,\\
Y=y,\qquad D=C+y,\qquad A=B+y .
\end{gathered}
\end{equation}
Initially restrict to \(y<\alpha\). The free \(y\)-direction increases
precisely \(D,A,Y\), so at a generic point peeling splits off
\((C,B,R)\). The remaining three forms are distinct, since
\(a,b>0\) and \(a\ne b\). Therefore \(e,g,t\) are singleton constants
in a first tangent. Keeping \(Y\) fixed, use the two perturbations
\begin{equation}\label{asp:aligned-perturbations}
\begin{aligned}
(C,B,D,A,Y,R)&\longmapsto(C+\delta,B,D+\delta,A,Y,R-\delta),\\
(C,B,D,A,Y,R)&\longmapsto(C,B+\delta,D,A+\delta,Y,R+\delta),
\end{aligned}
\qquad \delta>0.
\end{equation}
They preserve full alignment and make \(C\), respectively \(B\),
strictly finest among these two normalized negative depths. Any negative
\(D\) is strictly coarser because \(y<\alpha\), and remains so for
sufficiently small \(\delta\). The pure lower bounds give
\(p\ge P\) on the first order side and \(n_1\ge Q_1\) on the
second. The full-aligned projection tests therefore give the following
bounds for the same singleton constants \(e,g,t\):
\[
e\ge f_s(P),\qquad g\ge f_s(Q_1),\qquad t\ge s.
\]
The two clipped position bounds sum to at least \(1+s\): if neither
clips their sum is \(2s+P+Q_1>1+s\); if one clips the other is
at least \(s\). At fixed \(\alpha\), the gain is at
least \(1+2s\). Integrate along generic sheet segments and let
\(\alpha\to0\). This gives the contradiction on the backward ray
\(v_1=v_2=0\), \(y<0\).

\subsubsection*{The tied plane for \(\theta>0\), \(\lambda\ne1\)}

Now assume \(\theta>0\), \(\lambda\ne1\). At generic points of
\(\Gamma\), the parameter diagonal rate is at least \(s\), and
\begin{equation}\label{eq:source-22}
e\ge f_s(p),\qquad g\ge f_s(n_1),\qquad
n_1\ge f_\theta(p),\qquad g\ge f_\theta(e).
\end{equation}
Set
\[
K_L=p+e,\qquad K_N=n_1+g,\qquad K_\Sigma=K_L+K_N.
\]
In particular \(K_N\ge K_L\) and \(g\ge e\). Parameterize \(\Gamma\)
by \(L_0=C+ay\) and \(y\), and average over thin strips on each side
of \(L_0=0\) in a fixed time interval. Integrating the support bound
from the equality line gives average \(K_\Sigma\le2\) on the minus
side and average \(K_\Sigma\ge2\) on the plus side. For the gain
on the tie with \(v_1=m,v_2=m-\lambda\), formula
\eqref{eq:source-19} becomes
\begin{equation}\label{eq:source-23}
\mathcal D_\Gamma(m)
=\text{parameter diagonal rate}+e+g+m(2-K_\Sigma)
 +\lambda(K_N-1+\kappa).
\end{equation}
The strip average of \(\mathcal D_\Gamma(a)\) tends to \(q_*\) on
either side as the strip shrinks, by calibration and Lipschitz continuity.
The plus-side average of \(K_N\) is at least one.

Write \(P,N_-,E_-\) for the minus-side pure rates of the negative
coordinates \(C,B,D\), using zero for an absent negative coordinate.
Here the \(C\)-rate is its marginal \(P\), by the ordering of
\(L_0\) divided by the negative speeds. These are lower bounds for the
actual partials. On the plus side let the pure rates be \(p_+,N_+,E_+\).
They are exact actual negative partials there. Indeed all negative
cutoffs are strictly finer than the path at time \(y\), while the
other cutoffs are finer or equal; replacing the latter by their path
values is valid. Increase all negatives strictly, with the \(C\)
increment at least as large as those of any negative \(B,D\).
This preserves the constraints at both tuples. Overlap gives equality
of the increment with its pure value. Each partial already has its
pure affine order slope as a lower bound; in the peeled tangent,
equality of this positive combination forces individual equality.
Interior derivative comparisons pass to these tangents also at alignment.

The weighted pure rate does not depend on the order:
\begin{equation}\label{eq:source-24}
J=aP+b_+N_-+(a-1)_+E_-
 =a p_++b_+N_++(a-1)_+E_+.
\end{equation}
The negative speeds in this use are distinct because \(\lambda\ne1\).
Marginal maximality gives \(p_+\le P\). The minus-side averages imply
\begin{equation}\label{eq:source-25}
P+\max(P+\theta,N_-)\le u_*,
\qquad P+E_-\le1.
\end{equation}
Here is the clipping argument. Since \(K_N\ge K_L\), the average of
\(K_L\) on the minus side is at most one. If \(P+s\ge1\), then
\(K_L\ge P+1>1\), a contradiction. Put
\(l=\max(f_\theta(P),N_-)>0\). Then
\[
K_\Sigma\ge P+(P+s)+l+f_s(l).
\]
If \(l+s\ge1\), this is at least \(2+2P>2\). Thus \(l+s<1\),
which also gives \(P+\theta<1\) and
\(l=\max(P+\theta,N_-)\). The average bound now gives
\(P+l\le1-s\). Finally \(K_L\ge P+E_-\) proves the second
inequality. We will use
\[
2P\le u_*,\qquad P+N_-\le u_*,
\qquad\theta\le u_*-2P.
\]
When \(s=1\), these already contradict \(P>0\). Otherwise the
following cases cover the present parameters.

\paragraph{Case \(a>1\), \(b\le0\).}
On the plus side, direct expansion gives
\[
\mathcal D_\Gamma(a)\ge s+a-J+g+b(1-K_N)+\kappa\lambda.
\]
The term with \(b\) is nonnegative after averaging, because the
plus-side average of \(K_N\) is at least one. Also
\[
J=aP+(a-1)E_-\le a-1+P.
\]
If \(E_+\ge s+P\), then \(g\ge e=E_+\), giving average gain at
least \(1+2s+\kappa\lambda\). If \(E_+<s+P\), then
\(p_++E_+\le2P+s\le1\), and therefore
\[
J=(a-1)(p_++E_+)+p_+\le a-1+p_+.
\]
When \(p_+>0\), \(g\ge e\ge f_s(p_+)=s+p_+\), again giving the
required gain. When \(p_+=0\), choose fixed \(M_0>a\) large and
use \(v_1=M_0,v_2=0\) in the upper envelope. The negative tuple
\((C,D)=(M_0,M_0-1)\) has the plus order, since
\[
\frac{M_0-1}{a-1}>\frac{M_0}{a}.
\]
Its pure cost is at most \(M_0-1\), and the varying remaining
coordinates decrease backwards. The gain is at least
\(1+\kappa M_0\), contradicting support for large \(M_0\).

\paragraph{Case \(a\ge1\), \(b>0\).}
Here \(0<b\le1\). The exact plus-side rates give
\[
\begin{split}
\mathcal D_\Gamma(a)
&\ge s+a+b-J+(1-b)g+\kappa\lambda\\
&\ge s+1-P+b(1-N_-)+(1-b)g+\kappa\lambda,
\end{split}
\]
using \(J\le a-1+P+bN_-\). We have \(1-N_-\ge s+P\).
If \(N_+\ge P\), then \(g\ge f_s(N_+)\ge s+P\) pointwise.
If \(N_+<P\), use \(p_+\le P\), \(e\le g\), and exactness to
get \(K_\Sigma\le2P+2g\). Since its plus-side average is at least
two, the average of \(g\) is at least \(1-P\ge s+P\).
In both alternatives the average gain is at least
\(1+2s+\kappa\lambda\). At \(a=1\), \(D\) is absent from the
negative group and its coefficient is zero, so the same displayed
calculation applies.

\paragraph{Case \(0<a\le1\), \(b\le0\), \(\lambda>1\).}
Only \(C\) is negative, and \(p=P\) exactly on the plus side.
Formula~\eqref{eq:source-23} gives
\[
\mathcal D_\Gamma(a)-\mathcal D_\Gamma(1)
=(a-1)(2-K_\Sigma),
\]
whose plus-side average is nonnegative. Moreover,
\[
\mathcal D_\Gamma(1)
\ge s+1-P+g+(\lambda-1)(K_N-1)+\kappa\lambda.
\]
Here \(g\ge e\ge s+P\), and the last \(K_N-1\) term is
nonnegative in average. This again gives average gain at least
\(1+2s+\kappa\lambda\).

\paragraph{Case \(0<b<a<1\).}
We have \(0<\lambda<1\), and the negatives are \(C,B\).
The plus-side rate \(N_+=Q_1\) is the \(B\)-marginal.
It is positive: if \(Q_1=0\), then \(N_-=0\), and
\eqref{eq:source-24} gives \(p_+=P>0\), contradicting
\(n_1=0\) and \(n_1\ge f_\theta(p)\) on the plus side.

If \(P+Q_1\ge u_*\), use \eqref{eq:source-21} with
\(\alpha<y\). This is strictly causal, since
\(R=\lambda\alpha<\lambda Y\). At a generic point its first
tangent has singleton constants \(e,g,t\). Apply the two perturbations
\eqref{asp:aligned-perturbations}, taking \(\delta<\lambda(y-\alpha)\)
to preserve the strict causal inequality. The two negative orders give
\[
e\ge s+P,\qquad g\ge f_s(Q_1),\qquad t\ge s;
\]
the first bound is unclipped because \(2P\le u_*\).
For increasing \(\alpha,y\) together, the gain rate is at least
\begin{equation}\label{eq:source-26}
a+b+\kappa\lambda+s-J
 +(1-a)(s+P)+(1-b)f_s(Q_1).
\end{equation}
To justify the cost \(J\) at the negative-order tie, perturb within
the tangent to \(B/b>C/a\), equivalently \(C+aR/\lambda>0\)
on alignment. At \(\beta=R/\lambda\) both negatives are
strictly finer than the calibrated path, and all positives are finer
or equal; in particular \(Y>\beta\), with the corresponding
position margins. Replacing positives by their path values is valid,
still with \(B=C+R\). Ordered increases of \(C,B\), with the
\(C\)-increment at least as large, preserve validity. Overlap gives
exact negative rates on this open order side, with weighted sum
\(J\); these comparisons pass to the tangent. The remaining axis
term \(\lambda r\) is nonnegative. Integrate the derivative bound
on perturbed parallel segments that retain strict order, then approach
the tie. This proves \eqref{eq:source-26}. Parallel sheets approaching
\(\alpha=y\) transfer it to the calibrated rate.

If \(Q_1\ge u_*\), subtract \(1+2s\) from
\eqref{eq:source-26}, using \(f_s(Q_1)=1\), to obtain
\[
\begin{split}
&\kappa\lambda+a(u_*-P)-bN_-+(1-a)P\\
&\qquad=\kappa\lambda+a(u_*-P-N_-)+(a-b)N_-+(1-a)P\ge0.
\end{split}
\]
If \(0<Q_1<u_*\), the difference is
\[
\begin{split}
&\kappa\lambda+P+Q_1-u_*+a(u_*-2P)+b(u_*-N_--Q_1)\\
&\quad=\kappa\lambda+(1-b)(P+Q_1-u_*)
 +(a-b)(u_*-2P)+b(u_*-P-N_-)\ge0.
\end{split}
\]
Each term has the stated sign by \eqref{eq:source-25}.

If instead \(P+Q_1\le u_*\), put
\(A_0=u_*-(p_++Q_1)\ge0\). The upper-envelope test
\(v_1=v_2=1\) has pure cost \(p_++Q_1\), because its
negative tuple \(C=B=1\) has the plus order. The remaining
time rate is at least \(s-\lambda\theta\). Its gain minus
\(1+2s\) is therefore at least
\[
A_0-\lambda\theta.
\]
For a second test follow \(\mathcal D_\Gamma(1)\) backwards
from zero. Its ray has \(L_0=(a-1)y>0\) for \(y<0\),
so \(p=p_+\), \(n_1=Q_1\) exactly, and
\(g\ge s+Q_1\), since \(0<Q_1<u_*\).
Formula~\eqref{eq:source-23} gives gain minus \(1+2s\)
at least
\[
A_0-\lambda(u_*-2Q_1)+\kappa\lambda.
\]
One test suffices. If \(Q_1\ge u_*/2\), the second expression
is nonnegative. Otherwise set \(d=u_*-2\max(P,Q_1)\ge0\).
Then
\[
A_0\ge u_*-P-Q_1\ge d,\qquad
0\le\min(\theta,u_*-2Q_1)\le d.
\]
Since \(\lambda<1\), the larger of the two lower bounds is
nonnegative even after discarding \(\kappa\lambda\).

\paragraph{Case \(0<a<1\), \(b\le0\), \(0<\lambda<1\).}
Only \(C\) is negative, with \(0<P\le u_*/2\).
Start at \(C=B=L>0,Y=0\) on alignment, where \(L\) is
arbitrarily small and generic, and run backwards for unit time with
\begin{equation}\label{eq:source-27}
C=L-y,\qquad B=L+x,\qquad
v_2=-x'\in[\rho,1-\rho],\qquad R=x+y\le\lambda y.
\end{equation}
Choose \(\rho\) small also relative to \(1-\lambda\).
These paths satisfy \(0\le x\le-y\). At \(y<0\), the
actual \(C\)-partial is \(P\) locally by overlap.
Indeed \(R\ge y\) and \(\lambda\ge a\) give
\(R/\lambda\ge y/a\). Choose
\(\beta\in((y-L)/a,y/a)\). Then \(C>-a\beta\), and
each of
\[
B=L+x,\quad D=L,\quad A=L+x+y,\quad Y=y,\quad R=x+y
\]
is strictly finer than its path value at \(\beta\).
For \(B\) this uses \(b\le0\); for \(D,A,Y\) it uses
\(a<1,b\le0,\beta<y/a\); for \(R\) it uses the
preceding inequality. The comparison is valid in a neighborhood
up to alignment.

In the strict causal interior \(t\ge s\), so the exact-slope
low comparison gives the pathwise short steps. On the boundary
\(R=\lambda y\), almost every \(y\) is a peeling point of
\(\Gamma\) for generic \(L\): the map
\((L,y)\mapsto(C,Y)=(L-y,y)\) is invertible, so Fubini
applies. This exceptional set in \(y\) is independent of the
path, since its boundary state is determined by \(L,y\).
In the first tangent there, \(p=P\) and \(g\ge s+P\).
Use \(v_1=1,v_2=0\) to enter the strict causal side
backwards. There \(t\ge s,r\ge0\), and the spatial rates
are constant throughout the tangent. The gain reaches
\[
s+k_C-P+g\ge1+2s+\kappa.
\]
Perturb \(v_2\) to \(\rho<1-\lambda\) to retain the
inward direction and the strict gain. Lemma~\ref{ent:control}
gives the endpoint contradiction, since the starting error tends
to zero with \(L\).

\subsubsection*{The coincident tie \(\lambda=1\)}

Finally suppose \(\lambda=1,\theta>0\), with \(P>0\).
Use paths on \(\Gamma\) through zero with
\(v_1=m\in[\rho,1-\rho]\), \(v_2=m-1\).
At \(y<0\) they have \(C>0\), \(D=B<0\).
Thus \(p\ge P\) locally: \(C\) is strictly finest if
\(D,B\) are also negative. Use the grouped domain
\[
B=D=Z_1\le C+Y,\qquad A=Z_2\le Z_1+Y,\qquad R=Y.
\]
Its parameter diagonal rate is at least \(s\).
The \(Z_1\)-diagonal rate \(H\) and the \(Z_2\)-rate
\(g\) satisfy \eqref{eq:source-18} at generic aligned
states and in the relevant inherited tangents. The slice retains
the binding alignment required by the tied projection argument.

Along each path, \(C\) alone decreases, while \(Y,Z_1,Z_2\)
strictly increase. Take its signed tangent, with positive shifts
of these latter coordinates opening both constraints, and let
\(p_0\) be the pure \(C\)-slope. Exactness in overlap
somewhere, together with the inherited \(p\ge P\), gives
\(p_0\ge P>0\). If \(p_0\le u_*\), test
\(v_1=1,v_2=0\) on the tie. Its ray is
\(C=-y,Z_1=0,Z_2=y\); a path time strictly between
\(y/m\) and \(y\) gives exact \(p=p_0\), since
the other grouped cutoffs are strictly finer. Its gain is at least
\[
s+k_C-p_0+g\ge s+k_C-p_0+(s+p_0)=1+2s+\kappa.
\]
If \(p_0>u_*\), use \(v_1=0,v_2=-1\).
The pure lower bound and \eqref{eq:source-18} give
\(H,g\ge1\), so the gain is at least
\[
s+H+2g-k_B\ge2+s+\kappa\ge1+2s+\kappa.
\]
Integrate along nearby generic tie segments and perturb the endpoint
values \(m=0,1\) into \([\rho,1-\rho]\), retaining
\(v_2=m-1\). The tie and velocity box are preserved.
These are the required pathwise control steps, giving the final
contradiction in this case.
\end{proof}

\begin{lemma}[Local constrained comparison]\label{asp:constrained}
Let an inherited full profile be based on the parameter tie, with
\(0<\lambda<1\), \(\theta>0\), the conditional profile
\eqref{eq:source-17}, and the inherited tie projection inequalities.
Suppose its full \(C\)-partial is a constant
\(P\in(0,u_*/2]\). Without requiring calibration of this profile,
there is a backward endpoint comparison from its origin, respecting
\(R\le\lambda Y\), whose gain per unit time is at least
\(1+2s-4\kappa\). Its velocity is in a box with \(v_1=1\)
and \(v_2\in[\rho,1-\rho]\), for sufficiently small \(\rho>0\).
\end{lemma}

\begin{proof}
Use the paths \eqref{eq:source-27} from a small generic
\(C=B=L,Y=0\). In the strict causal interior the constant
full \(C\)-slope and \(t\ge s\) supply the low longitudinal
comparison. At generic boundary points, the tie inequalities give
\(g\ge s+P\), and the spatial singleton rates in the first
tangent are constant. The backward direction \(v_1=1,v_2=0\)
enters the strict causal side and has gain at least
\(s+k_C-P+g\ge1+2s+\kappa\). Perturbing to
\(v_2=\rho<1-\lambda\) retains this strict gain.
The generic-\(L\) argument in Proposition~\ref{asp:increasing-positive}
gives these steps almost everywhere along every admissible path,
including its boundary epochs. Lemma~\ref{ent:control} therefore
gives a fixed-duration endpoint comparison, with arbitrarily small
additional loss.

Translate both endpoints by the same aligned depth shift to move
the start from \(L\) to zero. This changes the gain by \(O(L)\);
it does not alter \(R,Y\), so it preserves the half-space.
Choose \(L\) and the further clipping and control losses so that,
together with the \(\kappa\)-loss in the low longitudinal comparison,
the total loss per unit time is less than \(4\kappa\). The straight endpoint
displacement lies in the convex velocity box and respects the
half-space. It consequently certifies a short step upon passing
back from a tangent.
\end{proof}

\subsection{Increasing aspect with zero longitudinal speed}

\begin{proposition}\label{asp:increasing-zero}
There is no calibrated profile satisfying \eqref{asp:contradiction}
with \(\lambda>0\) and \(a=0\).
\end{proposition}

\begin{proof}
Here \(b=-\lambda\). Take \eqref{eq:source-17} and peel off
the constant coordinate \(C\) in a generic calibrated tangent;
all other coordinates strictly increase. Its additive contribution
is a function of \(C\) alone. Use aligned backward paths through
zero with
\[
\rho\le v_1\le M_0,\qquad -(\lambda+1)\le v_2\le1,
\]
where \(M_0\) is fixed sufficiently large relative to
\(\lambda\) and \(1/\kappa\). If \(\theta>0\), impose
\(R\le\lambda Y\). Along every such path, the \(C\)-summand
is differentiable almost everywhere because \(C\) moves
monotonically with speed at least \(\rho\). In a tangent
there its full slope \(p\) is constant. We verify the control
property in this tangent.

If \(p=0\), take \(v_1=M_0,v_2=0\).
After removing \(C\), the only backward cost is at most
\(M_0-1\), from \(D\), and the gain is at least
\(1+\kappa M_0\). Taking \(M_0\ge\lambda\) also
preserves the causal half-space at its boundary.
Suppose \(p>0\). If \(\theta=0\), or at a strictly
causal state, \(t\ge s\) throughout a neighborhood and its
tangents. Use the low longitudinal comparison for \(p\le u_*/2\)
and the high comparison for \(p>u_*/2\), perturbing
\(v_1=0\) to \(\rho\) in the latter.

It remains to consider boundary states with \(\theta>0\). The path
point need not be generic on \(\Gamma\). Its tangent nevertheless has
constant full \(C\)-slope \(p\). Each of the first three cases below
chooses a fixed tie direction using only \(p,\lambda\). Its gain bound
holds at generic points of the tangent's tie; integrate on nearby
generic parallel tie segments and pass by continuity to the segment
from the tangent origin. This gives the endpoint comparison required
by Lemma~\ref{ent:control}. The fourth case supplies such a comparison
directly.
\begin{enumerate}[label=(\roman*)]
\item For \(\lambda=1\), use \eqref{eq:source-18}.
If \(0<p\le u_*\), the tie test \(v_1=1,v_2=0\)
has gain at least \(s+k_C-p+(s+p)=1+2s+\kappa\).
If \(p>u_*\), the test \(v_1=0,v_2=-1\) has
gain at least \(s+1+2-k_B\). Perturb along the tie;
in the second test use \(v_1=\rho,v_2=\rho-1\).
\item For \(\lambda\ne1\), \(p>u_*/2\), the spatial
inequalities \eqref{eq:source-22} give
\(e,g\ge h\) and \(K_N\ge K_L\ge1\).
Formula~\eqref{eq:source-23} with \(m=0\) gives
at least \(s+2h=1+2s\). A small tie perturbation makes
\(v_1\ge\rho\).
\item For \(\lambda>1\), \(0<p\le u_*/2\), take
\(m=(\lambda+1)/2\) on the tie. Formula~\eqref{eq:source-23}
gives
\[
\mathcal D_\Gamma(m)\ge
s+1-p+g+\frac{\lambda-1}{2}(K_N-K_L)+\kappa\lambda.
\]
Here \(g\ge p+s\), since \(p+s\le h\le1\), and
the difference term is nonnegative. The rate is at least
\(1+2s\), and the velocities fit the chosen box.
\item For \(0<\lambda<1\), \(0<p\le u_*/2\),
use Lemma~\ref{asp:constrained}, with its constant full
\(C\)-slope \(P=p\). This enters or follows the causal
region and gives a strict endpoint comparison in the tangent.
\end{enumerate}
All these comparisons, including any secondary control construction
inside a tangent, retain a fixed positive margin over \(q_*\)
after the chosen losses. Their endpoint displacements lie in the
convex velocity box and respect the constant half-space constraint.
Lemma~\ref{ent:control} therefore contradicts
\eqref{eq:source-11} for the original paths.
\end{proof}

\subsection{Decreasing aspect}

\begin{proposition}\label{asp:decreasing}
There is no calibrated profile satisfying \eqref{asp:contradiction}
with \(\lambda<0\).
\end{proposition}

\begin{proof}
Write \(c=b-a=-\lambda>0\). Then \(0\le a<b\le1\).
We have \(t\ge s\) everywhere. The negative coordinates on the
calibrated line are \(B,R\), and \(C\) when \(a>0\).
Take the signed-envelope tangent with the slack shifts described in
Lemma~\ref{ent:signed-envelope}. In this case it is essential to
retain the additional prediction \eqref{eq:source-12}.

First suppose \(a>0\). The pure domain is \(B\le C+R\).
Let \(P\) be the constant pure \(C\)-partial on
\(C/a>\max\{B/b,R/c\}\), and let \(S\) be the constant pure
\(R\)-partial on \(R/c>\max\{C/a,B/b\}\).
Lemma~\ref{ent:signed-envelope} supplies these constants and makes each
a global upper bound for its corresponding pure partial.
On the aligned sector in which \(C\) is ahead,
\(C/a>R/c\) and \(B=C+R\), the pure profile has the form
\begin{equation}\label{eq:source-28}
G(C,B,R)=PC+Q_2B
\end{equation}
after referencing at zero, with constant \(Q_2\ge0\).
To prove this identity, note first that \(C/a>B/b\) as
well, since \(B/b=(a/b)(C/a)+(c/b)(R/c)\).
The pure \(C\)-rate is consequently \(P\).
The pure \(R\)-rate is zero there. Indeed in the external slices
used to construct the pure profile, \eqref{eq:source-12}
gives flatness in \(R\) when
\(C/a,B/b>R/c\); the other coordinates are fixed strictly
finer. This flatness holds on an open order region of the slice
tangents and passes to \(G\).

Thus varying \(C\) from the order tie at fixed \(B\), with
\(R=B-C\), has slope \(P\). On the tie,
\(C=(a/b)B\), the pure total is affine. Integrating from it
proves \eqref{eq:source-28}. Nonnegativity of the \(B\)-increment
gives \(Q_2\ge0\). The same \(B\)-rate holds locally in the
interior: the \(C,R\) partials there are constant, and one can
approach the aligned boundary keeping \(B\) fixed.
Thus the partials of \(G\) near this aligned sector are
\((P,Q_2,0)\) in the \((C,B,R)\) coordinates. The signed-envelope
lower comparison gives \(p\ge P\) and \(n_1\ge Q_2\) for the actual
profile.

The shift to \(n(-H_0)\) in \eqref{eq:source-14} is valid
on compact aligned tests because \(a,b,c>0\). If \(P=0\),
use \(v_1=M_0,v_2=0\) backwards from zero. The target pure
tuple is \(C=M_0,B=0,R=-M_0\), in the \(C\)-ahead sector,
so \eqref{eq:source-28} gives zero pure cost. The remaining
cost is at most \(M_0-1\), from \(D\); the other varying
remaining coordinates decrease backwards. Thus the gain is at
least \(1+\kappa M_0\), contradicting support for large fixed
\(M_0\).

If \(P+Q_2\le u_*\), use \(v_1=v_2=1\). Its pure tuple
\(C=B=1,R=0\) is again in the \(C\)-ahead sector and
costs \(P+Q_2\). The coordinates \(D,A\) are fixed,
while the remaining time rate is at least \(s\). The gain
is therefore at least \(2+s-(P+Q_2)\ge1+2s\).

If \(P,Q_2>0\) and \(P+Q_2>u_*\), use the sheet
\eqref{eq:source-21} with \(y<\alpha\), and the comparison
\(v_1=v_2=0\). At generic sheet points the same peeling
separates \(e,g,t\): the free \(y\)-direction first splits
off \(C,B,R\), after which the three remaining forms are
distinct. The first perturbation in \eqref{asp:aligned-perturbations}
enters the \(C\)-ahead sector while preserving full alignment.
Formula~\eqref{eq:source-28}, the pure lower bounds,
and the full-aligned projections give simultaneously
\[
e\ge f_s(P),\qquad g\ge f_s(Q_2),\qquad t\ge s.
\]
As above, the two positive input rates with sum greater than
\(1-s\) force \(e+g\ge1+s\). Integrate on generic
parallel segments and pass to \(\alpha=0\), obtaining gain
at least \(1+2s\).

The only remaining possibility with \(a>0\) is
\(P>u_*\), \(Q_2=0\). On the same sheet the
\(C\)-ahead perturbation gives \(e\ge1\).
If \(S>0\), the second perturbation in
\eqref{asp:aligned-perturbations} enters the \(R\)-ahead sector
\(R/c>C/a\), still with \(B=C+R\). There the actual
tilt rate is at least \(S\). Since \(e\) is a constant
singleton in the first tangent, the tilt projection inequality
\(g\ge f_r(e)\) gives \(g\ge1\). Transfer this to the
sheet and use \(v_1=v_2=0\), whose gain is at least
\(s+2\ge1+2s\).

If \(S=0\), the global upper bound for the pure \(R\)-partial makes \(G\)
independent of \(R\). It therefore projects to an ordinary
submodular profile on full \((C,B)\)-space: any \((C,B)\)
is feasible after taking \(R\) sufficiently fine, and the
value does not depend on that choice. If necessary take a further
simultaneous generic tangent on the calibrated line so that this
two-variable pure profile has ordinary order slopes, by the
full-domain order rule of Lemma~\ref{ent:signed-envelope}.
Its lower comparisons with the actual profile and
\eqref{eq:source-28} are preserved. If the resulting \(B\)
marginal is positive, the second perturbation in
\eqref{asp:aligned-perturbations} enters its \(B\)-ahead
order, which is the \(R\)-ahead order on alignment. The
normal time projection gives \(g\ge s\), while the
singleton \(e\ge1\) persists. The same zero-velocity test
then has gain at least \(1+2s\). If the \(B\)-marginal
vanishes, \(G\) depends on \(C\) alone and has global
slope \(P\), by \eqref{eq:source-28}. Thus the actual
profile satisfies \(p\ge P>u_*/2\), and the high
longitudinal comparison contradicts calibration.

It remains to treat \(a=0\), so \(c=b>0\).
Put the static coordinate \(C\) in the nonnegative group.
The external slices now have strict mixed slack also for
\(B\le C+R\), and the pure domain in \(B,R\) is full.
Prediction \eqref{eq:source-12} gives zero pure \(R\)-rate
on \(B>R\). The two-variable pure order rule therefore gives
\begin{equation}\label{asp:decreasing-zero-pure}
G(B,R)=Q_2 B\qquad(B\ge R),
\end{equation}
with \(Q_2\ge0\); the equality at \(B=R\) follows by
continuity. In particular this applies on alignment when
\(C\ge0\). Both negative coordinates can be shifted to
equal finer tied cutoffs in the upper envelope; the other mixed
constraints have large slack after that shift.

Suppose \(Q_2\le u_*/2\). Test the upper envelope
backwards from zero with \(v_2=1\) and
\(v_1\in[\rho,1-\rho]\). At interior negative times
\(C>0\), while \(A\) is fixed. The pure backward cost
per unit time is \(Q_2\), by
\eqref{asp:decreasing-zero-pure}. With \(B,R\) frozen
at their finer tied shift, the remaining scalar coordinates are
\[
X=C,\qquad y=Y,\qquad Z=D,
\]
with \(A\) fixed. Because \(C>0\), the first shear
constraint has slack; the only binding local constraint is
\(D\le C+Y\). The direct longitudinal velocity-time
projection gives \(e\ge f_s(p)\), and the time rate
is at least \(s\). The fixed normal measurements rebase
with bounded ambiguity by \eqref{eq:source-10}, with
\(B,R\) fine. Thus \eqref{eq:source-20} holds with
\(t_0=s\), also in the path tangents. The scalar comparison
gives \(s+h-\kappa\); the additional normal contribution is
\[
k_B-Q_2\ge1-\kappa-u_*/2=h-\kappa.
\]
The total is at least \(1+2s-2\kappa\), before an
arbitrarily small clipping and control loss, contradicting
\eqref{asp:contradiction}.

Finally suppose \(Q_2>u_*/2\). Use actual paths
backwards from zero with \(v_2=0\) and
\(v_1=m\in[\rho,1-\rho]\). At negative times \(C>0\),
so the pure bound gives \(n_1\ge Q_2\) locally and
therefore \(g\ge h\) at generic aligned states and in
their inherited tangents. Along each path, \(C\) alone
decreases; \(B\) is static, while \(R,Y,D,A\) strictly
increase. Apply the signed rule at almost every point of that
path, with pure \(C\)-slope \(p_0\). The required
overlap and slack hold: increase \(R,Y\) substantially
more than the shifts of the other nonnegative coordinates.

If \(p_0>u_*/2\), use \(v_1=v_2=0\) in this
tangent. The pure and projection bounds give \(e,g\ge h\),
and \(t\ge s\), yielding \(1+2s\).
If \(p_0\le u_*/2\), use \(v_1=1,v_2=0\).
We verify exactness of \(p=p_0\) where the gain is
transferred. The centered comparison ray has
\[
C=-y,\qquad B=0,\qquad R=y,\qquad y<0.
\]
Perturb to \(B>0\), \(R=B-C\), with \(C\)
near \(-y\). Away from \(y=0\), choose
\[
\beta\in(-C/m,(B-C)/m),\qquad \beta<y.
\]
This is possible for sufficiently small positive \(B\),
since at \(B=0,C=-y\) both endpoints tend to
\(y/m<y\). Then \(C>-m\beta\), and the
remaining cutoffs are strictly finer than their path values
\[
0,\quad m\beta,\quad\beta,\quad(1-m)\beta,\quad\beta
\]
for \(B,R,Y,D,A\), respectively. In particular
\(R=B-C>m\beta\); \(D=C+y\) is near zero
and exceeds \((1-m)\beta<0\); and
\(A=B+y>\beta\). Overlap therefore gives exact
\(p=p_0\) at generic nearby aligned states.
Integrate on these perturbed segments and pass to the ray.
Discarding the nonnegative \(r\)-term gives gain at least
\[
s+k_C-p_0+h\ge1+2s+\kappa.
\]
Clip endpoint velocities into the prescribed box and use
Lemma~\ref{ent:control}. This proves the contradiction in
the last decreasing-aspect case.
\end{proof}

\subsection{Conclusion of the geometric estimate}

\begin{proof}[Completion of the proof of Theorem~\ref{geo:main}]
If the claimed exponent bound failed, the calibrated construction would
give the profile used at the start of this section.
Proposition~\ref{asp:stationary} excludes \(\lambda=0\).
Propositions~\ref{asp:increasing-positive} and
\ref{asp:increasing-zero} exclude \(\lambda>0\), and
Proposition~\ref{asp:decreasing} excludes \(\lambda<0\).
These exhaust \(a\ge0,b\le1,\lambda=a-b\).

Each comparison is in a fixed tangent problem. Fix the finite parameters
and any large auxiliary velocity bound first. For an upper-envelope
comparison, then choose \(H_0\) sufficiently large for the resulting
compact test region. The admissible depth perturbations and velocity
clipping parameters may subsequently depend on these choices.
The scalar losses are at most \(2\kappa\) in the uses above,
and the local constrained comparison reserves less than \(4\kappa\).
Choose the additional endpoint, clipping, and control losses so that,
together with the scalar or local constrained loss already incurred,
the total in the selected comparison is strictly below \(10\kappa\).
These are alternative comparisons, so their losses are not summed
over the mutually exclusive cases. Static coordinates at \(a=1\)
or \(b=1\) belong to the nonnegative group, as permitted in
Lemma~\ref{ent:signed-envelope}; no division by a zero negative
speed is used. When \(s=1\), the branches requiring
\(0<P\le u_*/2\) are empty, and the zero-rate and high-rate
comparisons cover the remaining possibilities.
The finite realizing grids and large-scale limit are chosen after
these finite tests, as in Lemma~\ref{ent:tangent-realization}.
The strict inequality \eqref{asp:contradiction} is therefore
impossible, proving \eqref{eq:source-4}.
\end{proof}
==== END Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//04-aspect.tex ====

==== BEGIN Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//05-packets.tex ====
\section{Propagation of oscillatory packets}\label{sec:packets}

The principal result of this section is the uniform estimate in
Theorem~\ref{pkt:main} for arbitrary finite complex packet arrays and
coordinate masks.  We define its exact operators and atomic norms,
then prove the composition, localization, and fourth-power estimates
used in Section~\ref{sec:extremal}.  That section derives the conditional
time profile \eqref{eq:source-1} required by geometric propagation from
near-extremal packet arrays themselves.

\subsection{The phase and an exact packet frame}

The Fourier-series construction and changes of packet frame follow the
established wave-packet framework; see
\cite[arXiv:math/0210084v2, Section~4 and Lemma~4.1]{Tao2003} and
\cite[arXiv:2104.11188v1, Section~4.1]{GOWWZ2025Packets}.
We prove the exact frame identities and coefficient estimates used here.

A finite partition of the sphere into sufficiently small graph patches
reduces the local analysis to the phase
\[
 x\cdot\xi+t\phi(\xi),\qquad \xi\in\R^2,
\]
with a smooth elliptic function $\phi$; the surface measure factor is
absorbed into the data.  We fix such a patch and phase throughout this
section.  We may extend $\phi$ to all of $\R^2$ so that its Hessian is
uniformly definite and bounded, and all its derivatives of order at
least two are bounded.  Indeed, on a patch of radius $\rho$, replace
$\phi$ outside the patch by its quadratic Taylor polynomial, using a
cutoff supported on the patch of radius $2\rho$.  The resulting change
in the Hessian is $O(\rho)$, since the Taylor remainder and its first two
derivatives are $O(\rho^3)$, $O(\rho^2)$, and $O(\rho)$, respectively.
Taking $\rho$ small preserves definiteness.  A reflection of the time
coordinate exchanges the positive and negative definite cases.  Write
\[
 U_\xi=-\nabla\phi(\xi).
\]
On each fixed bounded frequency region, $\xi\mapsto U_\xi$ is
quantitatively bi-Lipschitz.  All constants below may depend on the
fixed phase, the frequency region, and the chosen windows.

Once this continuation is fixed, the arrays below may use frequency
labels anywhere in a fixed bounded region, including the transition
region of the cutoff.  Their windows need not lie in the original
spherical patch.  The phase and all its derivative bounds remain fixed
before the packet scales vary.  The global sphere argument will return
to data supported on the original graph patches in
Section~\ref{sec:global}.

Let $N$ be dyadic, put $\delta=N^{-1}$, and use the oscillatory factor
\[
 e^{2\pi iN^2(x\cdot\xi+t\phi(\xi))},\qquad 0\le t<1.
\]
Choose a smooth nonnegative function $\chi$ whose integer translates
form a square partition of unity.  In particular, with
\[
 \chi_\theta^\nu(\xi)=\chi\bigl((\xi-\nu)/\theta\bigr),
 \qquad \nu\in\theta\Z^2,
\]
we have $\sum_\nu(\chi_\theta^\nu)^2=1$.  Fix $L_{\mathrm f}$ so that
each window is supported in a square of side $L_{\mathrm f}\theta$.
For a dyadic factor $1\le m\le N$, set
\begin{equation}\label{pkt:scales}
 \theta=m\delta,\qquad r=m^{-2},\qquad
 w=r\theta=\frac{\delta}{m}.
\end{equation}
Partition $[0,1)$ into intervals $I$ of length $r$, and let $t_I$ be the
left endpoint of $I$.  A packet index at this scale is
$v=(\nu_v,z_v,I_v)$, where
\[
 \nu_v\in\theta\Z^2,
 \qquad z_v\in L_{\mathrm f}^{-1}w\Z^2.
\]
At one time $t_I$, define the analysis and synthesis operators by
\begin{align}
 (\mathsf P_{m,I}f)_{\nu,z}
 &=\int f(\xi)\chi_\theta^\nu(\xi)
       e^{2\pi iN^2(z\cdot\xi+t_I\phi(\xi))}\,d\xi,
       \label{pkt:analysis}\\
 \mathsf S_{m,I}d(\xi)
 &=(L_{\mathrm f}\theta)^{-2}
    \sum_{\nu,z}d_{\nu,z}\chi_\theta^\nu(\xi)
       e^{-2\pi iN^2(z\cdot\xi+t_I\phi(\xi))}.
       \label{pkt:synthesis}
\end{align}
Synthesis is initially defined for finite arrays, and then by continuity
on $\ell^2$.

\Needspace{4\baselineskip}
\begin{lemma}[Frame identity]\label{pkt:frame}
For each $I$,
\[
 \mathsf S_{m,I}\mathsf P_{m,I}f=f,
 \qquad
 \|\mathsf P_{m,I}f\|_{\ell^2}^2
   =(L_{\mathrm f}\theta)^2\|f\|_2^2,
 \qquad
 \|\mathsf S_{m,I}\|_{\ell^2\to L^2}
   =(L_{\mathrm f}\theta)^{-1}.
\]
Consequently, if $I_+\subset I_-$ and
$\ell=m_+/m_-$, the transition
$\mathsf P_{m_+,I_+}\mathsf S_{m_-,I_-}$ has
$\ell^2\to\ell^2$ norm at most $\ell$.
\end{lemma}

\begin{proof}
The identity $N^2w\theta=1$ shows that, as $z$ varies over its grid,
$N^2z$ varies over $(L_{\mathrm f}\theta)^{-1}\Z^2$.  Thus the analysis
entries for one frequency window are the unnormalized Fourier
coefficients on a square of side $L_{\mathrm f}\theta$.  Parseval gives
\[
 \sum_z|(\mathsf P_{m,I}f)_{\nu,z}|^2
  =(L_{\mathrm f}\theta)^2\|f\chi_\theta^\nu\|_2^2.
\]
Sum over $\nu$.  Fourier inversion, with its factor
$(L_{\mathrm f}\theta)^{-2}$, gives
$\mathsf S_{m,I}\mathsf P_{m,I}f
=f\sum_\nu(\chi_\theta^\nu)^2=f$.
Finally, $\mathsf S_{m,I}$ is
$(L_{\mathrm f}\theta)^{-2}$ times the adjoint of
$\mathsf P_{m,I}$.  The operator norm statements follow.
\end{proof}

The initial scale is $m=1$, with the single time interval $[0,1)$.
A \emph{single-step operator} from factor $1$ to factor $m$ consists of
synthesis at factor $1$, followed by analysis in every terminal time
bin, followed by any coordinate mask.  More generally, successive
transitions use synthesis independently in each parent time bin and
analysis in its child bins.  Each finite array has a specified retained
coordinate set, and an index is called \emph{active} when it belongs to
that set, even if its coefficient is zero.  A coordinate mask projects
onto a subset of the retained set.  If \(M_I\) denotes the terminal
coordinate mask, the single-step map is exactly
\[
 (T_m d)_I=M_I\mathsf P_{m,I}\mathsf S_{1,[0,1)}d.
\]
The initial array is arbitrary and need not belong to the range of an
analysis operator.  Every map retains the full transition matrix
between its active index sets.  The localization graphs introduced
later count possible interactions; they do not delete individual
matrix entries or summands.

\subsection{Capacity and the atomic norm}

Recall the biased area
$W(E)=L^{1+\kappa}a^{1-\kappa}$ for an ellipse with radii $L\ge a>0$.
For an array in one time bin at scale $(\theta,r,w)$, define
\begin{equation}\label{eq:source-29}
 \begin{split}
 K(c)&=\theta^2
   \sup_{\substack{E:\text{both radii at least }\theta}}
   \frac{1}{W(E)}
   \sum_{\substack{U_{\nu_v}\in u+E\\ z_v\in z+rE}}|c_v|^2,
       \\
 G(c)^3&=\left(w^2\sum_v|c_v|^2\right)K(c)^{1/2}.
 \end{split}
\end{equation}
The supremum includes both independent translations $u,z\in\R^2$.
There is no upper restriction on the radii.  Define the lattice atomic
norm by
\[
 A(c)=\inf\left\{\sum_{j=1}^qG(a_j):
          |c|\le\sum_{j=1}^q|a_j|,\ q<\infty\right\}.
\]
The inequality in this definition is coordinatewise.  For arrays over
all bins of length $r$, put
\begin{equation}\label{pkt:combined-norm}
 \mathcal A(c)=\left(r\sum_I A(c_I)^3\right)^{1/3}.
\end{equation}

\Needspace{4\baselineskip}
\begin{lemma}[Atomic norm properties]\label{pkt:atomic}
On each finite index set, $A$ is a norm, is monotone under absolute
coordinatewise domination, and is contracted by coordinate masks.
The same assertions hold for $\mathcal A$.  Moreover,
\begin{equation}\label{eq:source-30}
 r w^2\sum_v|c_v|^3\le\mathcal A(c)^3.
\end{equation}
An operator estimate with input $G(d)$ and output a norm remains true
with input $A(d)$ and the same constant, provided the estimate holds
for every input on the specified support.
\end{lemma}

\begin{proof}
A translated disk of radius $\theta$ can include any chosen entry in
both tests in \eqref{eq:source-29}.  Since its biased area is
$\theta^2$, we have
\[
 K(c)\ge\|c\|_\infty^2,
 \qquad
 G(c)^3\ge w^2\|c\|_2^2\|c\|_\infty
          \ge w^2\sum_v|c_v|^3.
\]
The triangle inequality in $\ell^3$ transfers this lower bound to
every atomic decomposition and hence to $A$.  A coordinate atom has
cost $w^{2/3}|c_v|$, so $A$ is finite and positive away from zero.
Homogeneity, the triangle inequality, and absolute monotonicity follow
directly from its definition.  The weighted $\ell^3$ sum in
\eqref{pkt:combined-norm} proves the corresponding statements for
$\mathcal A$ and gives \eqref{eq:source-30}.

Atoms in a decomposition can first be restricted to $\supp d$, since
$G$ is monotone.  If $|d|\le\sum_j|a_j|$, choose, at each coordinate,
complex numbers $d_j$ with the phase of $d$, with
$\sum_jd_j=d$ and $|d_j|\le|a_j|$.  For a linear map $T$ and an output
norm $\mathcal B$, a bound $\mathcal B(Tb)\le C G(b)$ therefore gives
\[
 \mathcal B(Td)\le\sum_j\mathcal B(Td_j)
       \le C\sum_jG(a_j).
\]
Taking the infimum proves the last assertion.  In particular, support
restrictions imposed on the input are inherited by all atoms.
\end{proof}

\subsection{Uniform exponents and composition}

\begin{definition}[Admissible families]\label{pkt:families}
An admissible family has $m\to\infty$, with $m,N$ dyadic and
$m\le N\le m^B$ for a fixed finite $B$.  Its arrays have finite
supports, all their position labels have size at most $m^B$, and all
frequency labels lie in a fixed bounded set.  Enlarging $B$ is allowed;
it must remain fixed along each family.  Constants controlling the
phase and windows are fixed as well.  The same convention applies to
diagrams with a fixed finite number of scales.
\end{definition}

The full position lattices are infinite, but their retained portions
have polynomial size.  At a factor $q\le m$, the frequency and position
bounds permit at most
\[
 C(N/q)^2(m^BqN)^2=Cm^{2B}N^4
\]
indices per bin, and there are $q^2\le m^2$ bins.  The norms are also
comparable to the largest coordinate up to polynomial factors.  For
example, if one bin has $D\ge1$ retained coordinates and
$M=\max_v|c_v|$, the singleton test and the decomposition into coordinate
atoms give
\[
 w^{2/3}M\le A(c)\le D w^{2/3}M.
\]
Likewise, $M^2\le K(c)\le D M^2$, since $W(E)\ge\theta^2$.
These comparisons and the weighted bin norm show that a coordinate
error bounded by an arbitrarily large negative power of $m$ times the
largest input coordinate is negligible in all the norms used here.

Let $\gamma$ be the supremum, over admissible families and subsequences,
of the limiting upper exponents in $m$ of
\begin{equation}\label{eq:source-31}
 \frac{\mathcal A(T_m d)^3}{G(d)^3},\qquad d\ne0,
\end{equation}
where $T_m$ is a single-step operator, including any terminal mask.
By Lemma~\ref{pkt:atomic}, upper bounds have the same meaning with
$A(d)^3$ in the denominator.  An exponent bound is uniform on every
fixed complexity range, with any additional positive power loss:
otherwise a sequence violating that bound would itself be an
admissible family.  We use $m^{o(1)}$ in precisely this sense.  In
particular, it does not assert uniformity when a complexity parameter
escapes to infinity along a single family.

The packet estimate to be proved in Section~\ref{sec:extremal} is the
following.

\begin{theorem}[Uniform propagation of finite packet arrays]\label{pkt:main}
Fix \(0<\kappa<1/10\), the continued phase, a bounded frequency region,
and the window system above.  For every fixed admissible complexity
range and every \(\varepsilon>0\), there is a finite constant \(C\) such
that
\begin{equation}\label{pkt:uniform-main}
 \mathcal A(T_m d)^3
   \le C m^{10\kappa+\varepsilon}G(d)^3
\end{equation}
for every finite complex initial array \(d\) and every terminal
coordinate mask in that range.  The same estimate holds with \(A(d)^3\)
on the right.  The constant is uniform in \(m,N,d\), and the masks; it
may depend on \(\kappa,\varepsilon\), the complexity range, the fixed
phase and frequency region, and the windows.  Equivalently,
\begin{equation}\label{eq:source-32}
 \gamma\le10\kappa.
\end{equation}
\end{theorem}

We prove the theorem in Section~\ref{sec:extremal}.  The next lemmas
give the exact composition and analytic estimates used there.

\begin{lemma}[Composition]\label{pkt:composition}
Fix rational numbers $0=a_0<\cdots<a_J=1$ and dyadic factors
$m_0=1$, $m_J=m$, with $m_j=m^{a_j+o(1)}$.  Put
$\ell_j=m_{j+1}/m_j$.  If normalized transition $j$ has cubed norm
bound $C_j\ell_j^{\beta_j+\varepsilon}$, with $C_j$ independent of
$m$ and $\varepsilon>0$ arbitrary, the composed cubed norm bound is
\[
 \prod_{j=0}^{J-1}C_j\ell_j^{\beta_j+\varepsilon}
   =m^{\sum_{j=0}^{J-1}(a_{j+1}-a_j)\beta_j
          +\varepsilon+o(1)}.
\]
This remains true with arbitrary intermediate coordinate masks.
A better single-step bound restricted to particular input and output
supports can be used at any transition whose supports satisfy that
restriction, provided it is hereditary under restriction of the input
support.
\end{lemma}

\begin{proof}
Consider a parent bin $I'$ at factor $m'$, of length
$r'=(m')^{-2}$ and left endpoint $t_0$.  Make the change
\[
 t=t_0+r'\widetilde t,\qquad
 z=r'\widetilde z,\qquad N'=N/m'.
\]
The factor $e^{2\pi iN^2t_0\phi}$ is removed by rebasing the data.
The phase is again $N'^2(\widetilde x\cdot\xi+
\widetilde t\phi(\xi))$.  At a descendant factor $m''$, the normalized
factor is $\widetilde m=m''/m'$.  Its frequency width is unchanged,
whereas its spatial width and time length become
\[
 \widetilde w=w/r',\qquad \widetilde r=r/r'.
\]
The spatial grids consequently become exactly the grids prescribed
above.  The capacity is unchanged: the condition
$z_v\in z+rE$ becomes
$\widetilde z_v\in\widetilde z+\widetilde rE$.  Thus
\begin{equation}\label{pkt:normalization}
 \widetilde K=K,\qquad
 \widetilde G^3=(r')^{-2}G^3,\qquad
 \widetilde A^3=(r')^{-2}A^3.
\end{equation}
The last identity follows by applying the same common scalar factor
to every atomic cost.

An estimate with normalized cubed norm constant $C_{m''/m'}$ becomes
\[
 \frac{r''}{r'}\sum_{I''\subset I'}A(c_{I''})^3
       \le C_{m''/m'}A(d_{I'})^3.
\]
Multiply by $r'$ and sum over parent bins.  This gives the desired
global cubed bound at that transition.  Iterate it, using mask
contraction at every stage.  Taking logarithms in base $m$ gives the
stated weighted sum of exponents.  In particular, a fixed decrease in
$\beta_j$ decreases the full exponent by that amount times the positive
logarithmic length $a_{j+1}-a_j$.  At each such fixed split, all normalized
position and frequency ranges still have fixed polynomial complexity
in that step's scale ratio.  The uniformity in
Definition~\ref{pkt:families} therefore applies simultaneously to all
bins.  Finally, the last assertion follows from the support-preserving
atomic decomposition in Lemma~\ref{pkt:atomic}.
\end{proof}

\begin{lemma}[Interaction and finite truncation]\label{pkt:locality}
Let $v$ be a parent index and $b$ a child index, with $I_b\subset I_v$.
The transition matrix vanishes unless
$|\nu_b-\nu_v|\le C\theta_+$, and for every integer $L\ge0$ it obeys
\begin{equation}\label{pkt:matrix-decay}
 |T_{bv}|\le C_L
 \left(1+
  \frac{|z_b-z_v-(t_{I_b}-t_{I_v})U_{\nu_v}|}{w_-}
 \right)^{-L}.
\end{equation}
Consequently, for every fixed $\eta>0$, interactions outside the
enlarged parent beam of radius $m^\eta w_-$ give an error smaller than
any prescribed negative power of \(m\), times the largest input
coordinate, on an admissible family.  One may
also insert intermediate frame identities and truncate all intermediate
position grids to sufficiently large polynomial ranges, with the same
type of error.
\end{lemma}

\begin{proof}
The matrix entry is
\[
 (L_{\mathrm f}\theta_-)^{-2}
 \int\chi_{\theta_-}^{\nu_v}(\xi)
      \chi_{\theta_+}^{\nu_b}(\xi)
 e^{2\pi iN^2((z_b-z_v)\cdot\xi+
                    (t_{I_b}-t_{I_v})\phi(\xi))}\,d\xi.
\]
Disjoint frequency supports give the angular assertion.  On overlap,
rescale $\xi=\nu_v+\theta_-\zeta$.  The amplitude and all its
derivatives are uniformly bounded, since $\theta_-\le\theta_+$.
After the constant and linear phase terms are removed, the phase
derivatives are bounded because
\[
 N^2\theta_-^2(t_{I_b}-t_{I_v})
    =\frac{t_{I_b}-t_{I_v}}{r_-}\in[0,1).
\]
The remaining linear frequency is
$(z_b-z_v-(t_{I_b}-t_{I_v})U_{\nu_v})/w_-$.
Repeated integration by parts proves \eqref{pkt:matrix-decay}.

For finite retained arrays, sum the decay bound over the polynomial
number of entries counted after Definition~\ref{pkt:families}.  An
inserted frame identity instead uses the full position lattice; its
infinite tail is controlled by summing the same decay over lattice
annuli.  Fix the number of transitions, the admissible complexity, and
the positive beam-radius power first.  Then enlarge the intermediate
position ranges by fixed polynomial powers, and choose the
integration-by-parts order $L$ for the prescribed final error.
Frequency labels spread by only a bounded amount at each transition,
so a fixed number of insertions and truncations has negligible total
error.  The polynomial norm comparisons transfer this statement to
all the norms in use.  This estimates the error of a finite approximation
to an exact matrix product; it does not replace a transition by an
arbitrarily truncated set of edges.
\end{proof}

\Needspace{5\baselineskip}
\begin{lemma}[Preliminary bounds]\label{pkt:crude}
For a single-step map and each terminal bin,
\begin{equation}\label{pkt:mass-capacity}
 w^2\sum_{v\in I}|c_v|^2\le\delta^2\sum_v|d_v|^2,
 \qquad K(c_I)\le m^{4+o(1)}K(d).
\end{equation}
In particular, $\gamma\le2$.
\end{lemma}

\begin{proof}
The mass estimate is Lemma~\ref{pkt:frame}, since $w=\delta/m$.
For the capacity estimate, fix an output test ellipse $E$, with both
radii at least $\theta=m\delta$, and choose an arbitrarily small fixed
$\eta>0$.  By Lemma~\ref{pkt:locality}, the input entries contributing
to this test, apart from negligible error, have velocities in a
translate of $C m^\eta E$.  Their initial positions lie in another
translate of the same enlarged ellipse.  Indeed, a contributing
position equals its child position minus elapsed time times its
velocity, up to $m^\eta\delta$; the elapsed time is at most one and
both radii of $E$ are at least $\delta$.  Both translations are allowed
in the definition of $K(d)$.

Restrict the input to this one product of two translated tests.  Its
squared sum is at most
\[
 \delta^{-2}K(d)W(Cm^\eta E)
       \le C m^{2\eta}\delta^{-2}K(d)W(E).
\]
Apply the global squared operator norm bound $m^2$ to this restricted
input, and then multiply by the output factor $\theta^2/W(E)$.  Since
$\theta^2=m^2\delta^2$, the result is
$C m^{4+2\eta}K(d)$, with negligible error.  The estimate is uniform
over translations, orientations, and eccentricities.  Tail errors are
uniform too: $\theta^2/W(E)\le1$, and $K(d)\ge\|d\|_\infty^2$.
Let $\eta$ be arbitrarily small to obtain the stated exponent bound.

Finally, $A(c_I)\le G(c_I)$.  Multiply the mass bound by the uniform
upper bound for $K(c_I)^{1/2}$ and sum in time with weight $r$.
There are $r^{-1}$ bins, giving
$\mathcal A(c)^3\le m^{2+o(1)}G(d)^3$.
\end{proof}

\subsection{A fourth-power estimate from decoupling}

We use the ordinary \(\ell^2\) decoupling theorem of Bourgain and
Demeter~\cite[arXiv:1403.5335v3, Theorem~1.1, equation~(2)]{BourgainDemeter2015}.
For a smooth elliptic graph, Fourier thickness \(S^{-1}\), and graph
caps of diameter \(S^{-1/2}\), its critical three-dimensional estimate is
\[
 \Bigl\|\sum_\tau F_\tau\Bigr\|_4
 \le C_\epsilon S^\epsilon
          \Bigl(\sum_\tau\|F_\tau\|_4^2\Bigr)^{1/2}.
\]
The norms here are on all of spacetime. Uniformity over the normalized
phases below follows from the proof for general surfaces in
\cite[Section~7]{BourgainDemeter2015}: the ellipticity bounds are fixed,
and the common third-derivative bound controls the Taylor approximation
on each rescaled cap.

We derive the packet refinement for the precise class used here. The
induction follows Guth--Iosevich--Ou--Wang
\cite[arXiv:1808.09346v1, proof of Theorem~4.2]{GIOW2020}; compare the
weighted extension formulation in
\cite[arXiv:2411.08871v3, Section~4, Theorem~4.4]{WangWu2024}.
Guth--Iosevich--Ou--Wang also record an independent discovery of
the refinement by Du and Zhang. The proof below specifies which tubes count packet activity, including
at the smaller induction scale. The distinction between a packet's
activity region and its counted tube is substantive; a planar
counterexample to a support/counting mismatch is given in
\cite[arXiv:2608.28711v1, Proposition~2.1]{SinghSingh2026Refined}.

\begin{theorem}[Refined decoupling for fixed-profile packets]
\label{pkt:refined-decoupling}
Fix a nonzero, nonnegative \(a\in C_c^\infty(\R^2)\), and a smooth
elliptic phase \(\Phi\) on a fixed compact frequency region. Its
ellipticity and derivative bounds are fixed. A canonical packet at
scale \(R\ge2\) is
\[
 F_T(X,t)=d_T R\int
     a\bigl(\sqrt R(\xi-\xi_T)\bigr)
     e^{2\pi i((X-z_T)\cdot\xi+t\Phi(\xi))}\,d\xi .
\]
The frequency centers range over a fixed bounded set and lie on a grid
of spacing comparable to \(R^{-1/2}\). The compact phase region is
chosen to contain these centers and all their packet supports
\(\xi_T+R^{-1/2}\supp a\).
For each frequency center, the position centers lie on a grid of
spacing comparable to \(\sqrt R\). Assume the
position centers lie in a translate of a box of side \(C R\),
where \(C\) is fixed. Subsets of these grids are allowed.
The coefficients \(d_T\) are arbitrary complex numbers.

Fix \(0<\delta_0<1/8\). The counted tube associated to the packet is
\[
 T=\bigl\{(X,t): |t|\le3R,\quad
 |X-z_T+t\nabla\Phi(\xi_T)|\le R^{1/2+\delta_0}\bigr\}.
\]
The packet has transverse core width \(\sqrt R\), but need not vanish
outside this tube.  On the time slab below, the coefficient-relative
Schwartz bound proved below controls its decay away from the centerline.
The enlarged tube is the region used for counting; at each induction
scale we use the same definition with that scale's core width.
Let \(Y\) be a finite union of cubes from a fixed grid of side
\(\sqrt R\), contained in \(\{|t|\le2R\}\). Suppose each such cube
meets at most \(k\ge1\) counted tubes. For every \(\varepsilon>0\),
\begin{equation}\label{pkt:external-decoupling}
 \Bigl\|\sum_TF_T\Bigr\|_{L^4(Y)}^4
 \le C_{\varepsilon,\delta_0}R^\varepsilon
                         k R^2\sum_T|d_T|^4 .
\end{equation}
All constants may depend on the fixed profile, grid bounds, position
box constant, and phase bounds. The estimate is uniform over finite
subsets and the coefficients. If the weighted packet norms on a box
\(B\) satisfy
\(\|F_T\|_{L^4(w_B)}^4\asymp |d_T|^4R^2\), the right-hand side may
equivalently be written as
\(C_{\varepsilon,\delta_0}R^\varepsilon k
\sum_T\|F_T\|_{L^4(w_B)}^4\).
\end{theorem}

The proof groups the packets into coarser frequency caps and position
squares, then rescales each group to scale $\sqrt R$. Its main task is
to compare the tubes counted at that smaller scale with the original
multiplicity bound $k$. The local decoupling estimates can then be
summed with each packet charged through its fourth-power cost.

\begin{proof}
The grids give at most \(C R^2\) packets. Write
\(\mathcal C_R=R^2\sum_T|d_T|^4\).
For \(|t|\le3R\), integration by parts in the normalized cap gives,
for every fixed \(L\),
\[
 |F_T(X,t)|
 \le C_L|d_T|
 \left(1+\frac{|X-z_T+t\nabla\Phi(\xi_T)|}{\sqrt R}\right)^{-L}.
\]
Outside the counted transverse radius this gains \(R^{-L\delta_0}\).
The polynomial packet count can therefore be included before choosing
$L$ to obtain any prescribed power of decay.  Choose a fixed
band-limited Schwartz function \(\psi\), bounded away from zero on
\([-4,4]\), and put \(P_T=\psi(t/R)F_T\).
The spatial \(L^\infty\) bound is \(C|d_T|\), and Plancherel gives
\(\|F_T(\cdot,t)\|_2^2\le C|d_T|^2R\). Hence
\[
 \|P_T\|_4^4\asymp |d_T|^4R^2.
\]
For the lower bound, use a short interval \(|t|\le cR\) and a
ball of radius \(c\sqrt R\) about the core. The normalized integral
is bounded below there because \(a\) is fixed, nonnegative and
has positive integral. Thus no tail estimate is normalized by the
possibly small norm of an arbitrary oscillatory profile.

Let \(D(R)\) be the best constant in
\[
 \Bigl\|\sum_TF_T\Bigr\|_{L^4(Y)}^4
                       \le D(R)k\mathcal C_R
\]
over the fixed classes just specified. For the induction, strengthen
the allowed time slab slightly to
\(|t|\le(2+R^{-1/16})R\); keep the counted tubes in \(|t|\le3R\).
The same core bounds apply. This harmless buffer accommodates cells
meeting the endpoints after rescaling: the added length is
\(O(R^{3/4}+R^{1/2+\rho})\), and, at \(r=\sqrt R\),
\[
 R^{-1/16}+O(R^{-1/4}+R^{-1/2+\rho})<r^{-1/16}
\]
for sufficiently large \(R\), once \(\rho<1/16\).
For bounded \(R\), the packet count, the preceding cost bound and
H\"older's inequality give a finite bound for \(D(R)\). The zero
array is immediate, so assume \(\mathcal C_R>0\).

Partition the frequency centers into caps \(\tau\) of diameter
\(R^{-1/4}\). For each \(\tau\), partition the time-zero position
centers into squares of side \(R^{3/4}\). Assign each packet once to
its frequency cap and position square; denote the resulting disjoint
packet groups by \(\mathcal W_l\), and set
\(P_l=\sum_{T\in\mathcal W_l}P_T\).
The centerlines in a group remain in a fixed enlargement of the
corresponding coarse cylinder, of radius \(R^{3/4}\) and length
\(O(R)\). At any \(\sqrt R\)-cube, only boundedly many such cylinders
per coarse cap are nearby. Far cylinders have separation comparable
to \(R^{3/4}\) from their \(\sqrt R\) cores and contribute arbitrarily
small relative errors. No packet is multiplied by a coarse spatial
cutoff. The fine cap supports lie in fixed enlargements of their
assigned coarse caps. These enlargements have bounded overlap; a
fixed coloring separates them before ordinary decoupling, at a
constant cost.

Here is the exact normalization of a group. If \(c_\tau\) is its
frequency center and \(z_l\) its position-square center, put
\[
 \xi=c_\tau+R^{-1/4}\zeta,\qquad
 X'=R^{-1/4}\bigl(X+t\nabla\Phi(c_\tau)-z_l\bigr),\qquad
 t'=R^{-1/2}t,\qquad r=\sqrt R.
\]
After removing a common unimodular factor and absorbing each
packet's constant phase into its coefficient, the packets have the
same form at scale \(r\), with the same profile \(a\) and
\(|d'_T|=|d_T|\). Indeed
\(R\,d\xi=r\,d\zeta\), their frequency and position meshes become
\(r^{-1/2}\) and \(\sqrt r\), and the new position centers have size
\(O(r)\). The new phase has derivatives
\(R^{(2-j)/4}D^j\Phi\) for \(j\ge2\), so it stays in the fixed
elliptic class. The cutoff becomes \(\psi(t'/r)\).
The physical Jacobian is \(R\); consequently the inner cost
\(|d_T|^4r^2\), after returning to the original coordinates, is
exactly \(|d_T|^4R^2\).

Apply the inductive estimate at scale \(r\). Its counted radius
\(r^{1/2+\delta_0}\) pulls back to the narrower radius
\(R^{1/2+\delta_0/2}\). A pulled-back inner cube has transverse
size \(R^{1/2}\) and longitudinal size \(R^{3/4}\). Directions in
one coarse cap differ by \(O(R^{-1/4})\), so their relative
displacement across that cube is \(O(\sqrt R)\).
Choose
\[
 0<\rho<\min(\delta_0/8,\,1/16).
\]
If a pulled-back inner cube \(P\) meets a translate of an outer cube
\(Q\) by distance at most \(C R^{1/2+\rho}\), then every inner counted
tube meeting \(P\) belongs to an original tube meeting \(Q\). The
transverse distance is bounded by
\[
 C\bigl(R^{1/2+\delta_0/2}+R^{1/2+\rho}+\sqrt R\bigr)
                                  <R^{1/2+\delta_0}.
\]
This is where the distinction between the core and the counted tube
is used. Each induction step rescales the core and chooses its own
counted radius; it does not rescale the parent's already enlarged tube.

For clarity, we give the local decoupling bookkeeping, including the
weights. For every outer cube \(Q\), choose a translate and dilate
\(\eta_Q\) of one band-limited Schwartz function which is bounded
below on \(Q\). Its spatial and temporal scale is \(\sqrt R\).
Applying ordinary global decoupling to
\(\eta_Q\sum_lP_l\) is legitimate: multiplication broadens Fourier
support by \(O(R^{-1/2})\), exactly the thickness at the coarse
decoupling scale \(S=\sqrt R\). Keep the boundedly many nearby
coarse cylinders per cap. Their fourth-power estimate is
\[
 \Bigl\|\sum_TF_T\Bigr\|_{L^4(Q)}^4
 \le C_\beta R^\beta
       \left(\sum_l\|\eta_QP_l\|_4^2\right)^2
       +\text{relative tail errors},
\]
where \(\beta>0\) is arbitrarily small.

Truncate each local integral to cubes \(Q+j\sqrt R\) with
\(j\in\mathcal J\subset\Z^3\), \(|j|\le C R^\rho\).
The set of shifts is the same for every \(Q\), and
\(\#\mathcal J\le C R^{3\rho}\). On the retained region
\(|\eta_Q|\le C\). For each group \(l\), keep only inner cubes meeting one of these
retained translates, with \(Q\) ranging over the cubes of \(Y\).
The time-buffer inequality above places every such whole inner cube
in the strengthened child slab. Partition these cubes dyadically
by the number of inner counted tubes they meet. Write
\(Y_{l,h}\) for the pulled-back union with between \(h\) and \(2h\)
tubes, where \(h\) ranges over \(O(\log R)\) positive dyadic values.
The zero-count cells give only relative tail errors by the core
bound. Set
\[
 A_{l,j,h}(Q)
   =\int_{Q+j\sqrt R}
             \mathbf1_{Y_{l,h}}\,|P_l|^4.
\]
If this is nonzero, the preceding incidence comparison supplies at
least \(h\) packets from \(\mathcal W_l\) whose outer tubes meet
\(Q\). The groups are disjoint, so there are at most \(Ck/h\)
such groups \(l\). Therefore
\[
 \left(\sum_l A_{l,j,h}(Q)^{1/2}\right)^2
                       \le \frac{Ck}{h}\sum_l A_{l,j,h}(Q).
\]
Two finite Cauchy--Schwarz sums over \(j,h\) cost at most
\(C R^{6\rho}(\log R)^2\).
For each fixed shift \(j\), the cubes \(Q+j\sqrt R\) are disjoint.
Sum their integrals before estimating them by an entire inner-cell
union:
\[
 \sum_Q A_{l,j,h}(Q)\le\int_{Y_{l,h}}|P_l|^4
 \le C D(\sqrt R)\,h R^2
                         \sum_{T\in\mathcal W_l}|d_T|^4.
\]
The last inequality is the child estimate and the exact Jacobian
calculation above; the bounded common cutoff costs only a constant.
The factor \(h\) cancels the preceding \(h^{-1}\), and summation
over the disjoint groups preserves \(\mathcal C_R\). This order
avoids counting one long inner cell once for each outer cube it meets.

All discarded terms are homogeneous in the coefficients. For example,
\(\#\{T\}\le CR^2\) and
\(|\sum_{T\in\mathcal W_l}P_T|^4
 \le(\#\mathcal W_l)^3\sum_{T\in\mathcal W_l}|P_T|^4\).
The Schwartz bound on \(\eta_Q\), summed over the cube grid outside
\(R^\rho Q\), then bounds the off-neighborhood error by
\(C_LR^{8-(4L-3)\rho}\mathcal C_R\), before harmless fixed powers.
Choose its decay order after \(\rho,\delta_0\) and the desired error.
The core bound treats far coarse cylinders and zero-count child cells
in the same way. Summing over space uses integrable decay; no bound
on the number of distant selected cubes is needed.
Thus, for every prescribed \(A>0\), these terms are at most
\(C_A R^{-A}\mathcal C_R\) at the current induction scale.

We have proved the recurrence
\[
 D(R)\le
 C R^{\beta+6\rho}(\log R)^2D(\sqrt R)+C_A R^{-A}.
\]
Choose \(\rho\) smaller if necessary, after \(\delta_0\) and
\(\varepsilon\), and then \(\beta\), so that
\(\beta+6\rho<\varepsilon/4\). At a sufficiently large fixed base,
the logarithm and the fixed constant fit within another
\(R^{\varepsilon/8}\). The resulting exponent, including the
child bound \(R^{\varepsilon/2}\), is less than
\(7\varepsilon/8\). Induction from that base gives
\(D(R)\le C_{\varepsilon,\delta_0}R^\varepsilon\). The additive errors are absorbed
at their own scale; no error at the last small scale is asserted
to be a rapid-decay error in the original scale. The assumption
\(k\ge1\) allows their absorption into \(k\mathcal C_R\).
This proves \eqref{pkt:external-decoupling}.
The weighted version is the stated packet-cost comparison.
\end{proof}

\begin{proposition}[Discrete fourth-power propagation]
\label{pkt:quartic}
Consider one consecutive transition in an admissible family, with
scale ratio $\ell=m_+/m_-=m^{\alpha+o(1)}$, where $\alpha>0$ is fixed.
Fix finite retained parent and child coordinate sets and an adjacency
power $\eta>0$.  Let $d$ be any complex array on the parent sets, and
let $M_{I_+}$ project onto the retained set in child bin $I_+$.  The
output in that bin is
\[
 c_{I_+}=M_{I_+}\mathsf P_{m_+,I_+}
                    \mathsf S_{m_-,I_-}d_{I_-},
 \qquad I_+\subset I_-.
\]
Suppose each active child index has at most $k\ge1$ active parents in
its parent time bin satisfying
\[
 |\nu_b-\nu_v|\le m^\eta\theta_+,
 \qquad
 |z_b-z_v-(t_{I_b}-t_{I_v})U_{\nu_v}|
       \le m^\eta w_-.
\]
Thus $k$ counts nearby pairs of retained coordinates, independently of
whether their coefficients vanish.  The displayed operator still uses
every matrix entry between the retained sets.  Then
\begin{equation}\label{eq:source-33}
 r_+w_+^2\sum_b|c_b|^4
       \le m^{o(1)} k\,r_-w_-^2\sum_v|d_v|^4.
\end{equation}
For each fixed complexity range, $\alpha$, and $\eta$, and every
$\varepsilon>0$, the factor $m^{o(1)}$ may be replaced by
$C_\varepsilon m^\varepsilon$ for all sufficiently large $m$ in the
window
\[
 m^{\alpha/2}\le\ell\le m^{3\alpha/2}.
\]
The constant and the lower threshold within this window are uniform.
A family's entry into the window may depend on the rate at which
$\log_m\ell$ tends to $\alpha$.
\end{proposition}

\begin{proof}
Fix a parent time bin and one child frequency window.  Only parent
windows overlapping this child window contribute.  Select these
parents, but keep their synthesis packets intact.  In particular, we
do not multiply individual parent windows by the child cutoff before
applying refined decoupling.

Let $\nu_+$ be the child frequency center and $t_-=t_{I_-}$.
Set
\[
 \xi=\nu_++\theta_+\zeta,\qquad
 T=\frac{t-t_-}{r_+},\qquad
 X=\frac{x-(t-t_-)U_{\nu_+}}{w_+}.
\]
The identities $N^2w_+\theta_+=N^2r_+\theta_+^2=1$ give the
normalized phase, after removing the constant and linear terms,
\[
 \Phi_+(\zeta)=\theta_+^{-2}
 \bigl(\phi(\nu_++\theta_+\zeta)-\phi(\nu_+)
                 -\theta_+\nabla\phi(\nu_+)\cdot\zeta\bigr).
\]
It has uniformly definite Hessian and bounded higher derivatives on
a fixed compact set.  In these units the parent slab has length
\[
 R=\frac{r_-}{r_+}=\ell^2,
\]
the parent frequency caps have diameter $O(\ell^{-1})$, and the parent
packets have transverse width $\ell$. More precisely, the
normalized synthesis packet has the canonical form in
Theorem~\ref{pkt:refined-decoupling} with the fixed profile
$a=L_{\mathrm f}^{-2}\chi$ and centers
\[
 \zeta_v=\frac{\nu_v-\nu_+}{\theta_+},\qquad
 Z_v=\frac{z_v}{w_+}.
\]
Here the constant phase $e^{-2\pi iN^2z_v\cdot\nu_+}$ is absorbed
into $d_v$, preserving its absolute value.  The position mesh is
$L_{\mathrm f}^{-1}\sqrt R$. The normalized velocities are
bounded, because all selected parent windows overlap the child window.
The synthesis factor in \eqref{pkt:synthesis} cancels the area of a
parent frequency window, so each parent packet has amplitude
$O(|d_v|)$, with no extra scale factor.

For completeness, localize time by a fixed Schwartz function
$\psi(T/R)$ whose Fourier transform is compactly supported and which
is bounded away from zero on an interval containing $[0,1]$ in its
argument.  Denote the resulting parent packets by $P_v$.  Their Fourier
support lies in an $O(R^{-1})$ neighborhood of the normalized graph,
over caps of diameter $O(R^{-1/2})$.  On $|T|=O(R)$, integration by
parts on the parent cap gives transverse Schwartz decay on scale
$R^{1/2}$.  For larger $|T|$, the decay width and constants grow only
polynomially in $1+|T|/R$, which is harmless against $\psi(T/R)$.
Consequently,
\begin{equation}\label{pkt:packet-fourth-cost}
 \|P_v\|_4^4\le C|d_v|^4R^2
                       =C|d_v|^4\ell^4.
\end{equation}
We next pass from the discrete child samples to a spacetime integral.
Write $F=\sum_vP_v$.  In the normalized variables, the child cutoff is
a fixed smooth frequency multiplier.  All spatial and temporal
frequencies of $F$ lie in a fixed compact set.  There is therefore a
Schwartz kernel $H$ on spacetime, with uniform bounds, whose Fourier
transform implements the child multiplier on that set.  An active
child index $b$ has normalized sample location
\[
 p_b=(X_b,T_b)=\left(
   \frac{z_b-(t_{I_b}-t_-)U_{\nu_+}}{w_+},
   \frac{t_{I_b}-t_-}{r_+}\right).
\]
At these retained coordinates, up to a unimodular factor,
\[
 \psi(T_b/R)c_b=(H*F)(p_b).
\]
The time coordinates are integers from $0$ to $R-1$; at each time,
the spatial coordinates lie in a translate of
$L_{\mathrm f}^{-1}\Z^2$.  Thus the retained samples have bounded
multiplicity at unit resolution.  H\"older's inequality and summation
over these samples give
\[
 \sum_b|c_b|^4
       \le C\int |F(q)|^4
                    \sum_b|H(p_b-q)|\,dq.
\]
The summed kernel is uniformly bounded and decays rapidly with the
distance to the active sample set.  Truncate this distance at $m^\beta$,
where $\beta>0$ will be chosen small.  The omitted integral is bounded by
an arbitrarily large negative power times
$R^2\sum_v|d_v|^4$: use the polynomial number of input packets,
\eqref{pkt:packet-fourth-cost}, and the triangle inequality.  Cover the
retained neighborhood by cubes of side $\ell$ and denote their union
by $Y$.  We have reduced the desired estimate to a bound for
$\int_Y|F|^4$.

We check carefully that the degree hypothesis controls the tube
multiplicity on these cubes.  Choose
$\beta<\min(\eta,\alpha)/10$.  Every cube in $Y$ has an associated active
sample within $O(\ell+m^\beta)$ of all its points.  An enlarged parent
tube meeting that cube has, at the time of this sample, transverse
distance at most
\begin{equation}\label{pkt:cube-adjacency}
 C\bigl(\ell R^{\delta_0}+\ell+m^\beta\bigr)
\end{equation}
from the sample: the normalized tube velocities are bounded.
Choose $0<\delta_0<1/8$ with $3\alpha\delta_0<\eta/10$.
In the stated ratio window, $R^{\delta_0}\le m^{3\alpha\delta_0}
\le m^{\eta/10}$.  Then
\eqref{pkt:cube-adjacency} is less than $m^\eta\ell$ for sufficiently
large $m$, which is exactly the allowed parent-beam radius in child
units.  Angular overlap gives a fixed multiple of $\theta_+$, also
less than the permitted $m^\eta\theta_+$.  Each such tube is therefore
one of the at most $k$ parents counted at the associated sample.
This uses the stated adjacency radius, without replacing it by a
larger power.

The sample times satisfy $0\le T_b<R$, so the cube union $Y$ lies in
\[
 -C(\ell+m^\beta)\le T\le R+C(\ell+m^\beta).
\]
In the stated ratio window,
$(\ell+m^\beta)/R\le m^{-\alpha/2}+m^{\beta-\alpha}\to0$.
Thus $Y$ is contained in $\{|T|\le2R\}$ for sufficiently large $m$,
uniformly in that window, as required by
Theorem~\ref{pkt:refined-decoupling}.

Partition space into boxes of side $R$, aligned with the cube grid,
over this time slab.  In each box retain the packets whose centerlines
meet a fixed enlargement of
it.  Their time-zero centers lie in a box of side $O(R)$ because the
normalized velocities are bounded.  The other packets have rapidly
decaying tails there.  Admissibility gives a polynomial number of
boxes meeting $Y$, so choose the integration-by-parts order after this
fixed complexity to make the total discarded contribution negligible.
Each relevant tube is charged in only boundedly many boxes, since its
length and displacement in the slab are $O(R)$.

Write
$P_v=\psi(T/R)\widetilde P_v$, where $\widetilde P_v$ is the
underlying canonical extension packet.  On the selected cubes,
\[
 \left|\sum_vP_v\right|^4
   \le \|\psi\|_\infty^4
                  \left|\sum_v\widetilde P_v\right|^4.
\]
Apply the coefficient-cost estimate of
Theorem~\ref{pkt:refined-decoupling} to these canonical packets in
each box.  Their position centers lie in a box of side $O(R)$, their
coefficients remain arbitrary complex numbers, and their counted-tube
multiplicity is at most $k$.  The common cutoff and bounded box
multiplicity therefore give
\[
 \sum_{\substack{b:\text{fixed child angle}\\
                      I_b\subset I_-}}|c_b|^4
     \le m^{o(1)} k\,\ell^4
          \sum_{\substack{v\in I_-:\text{window overlaps}\ \nu_+}}
                         |d_v|^4.
\]
All tail errors can be absorbed into this bound, since $k\ge1$.
Since $R\le m^{3\alpha}$ in the ratio window, the decoupling loss
can be made smaller than any prescribed positive power of $m$
uniformly there.  For a prescribed final error power, choose first the
error allowed by Theorem~\ref{pkt:refined-decoupling}, then its
localization power $\delta_0$ and the sampling power $\beta$ small enough to
satisfy the preceding constraints.
Their constants are fixed before taking $m\to\infty$.

Each parent window overlaps only boundedly many child windows.  Sum
over child angles and parent bins and multiply by $r_+w_+^2$.  The
scale identity
\[
 r_+w_+^2\ell^4=r_-w_-^2
\]
proves \eqref{eq:source-33}.
\end{proof}

\begin{remark}[Weighted packet costs]
In the normalization of the preceding proof, the canonical packets
also have the weighted norms appearing in comparisons with refined
decoupling formulations.  Write
$P_v=\psi(T/R)\widetilde P_v$ as above.  For a spatial box $B$ used
there, with center $X_B$, put $q_B=(X_B,0)$ and
\[
 w_B(q)\asymp(1+|q-q_B|/R)^{-K_0},\qquad q=(X,T),\quad K_0>4.
\]
For every packet retained in this box,
\[
 \|\widetilde P_v\|_{L^4(w_B)}^4
       \asymp |d_v|^4R^2
       \asymp \|P_v\|_4^4.
\]
Indeed, the frequency-integral bound and Plancherel give
$\|\widetilde P_v(\cdot,T)\|_\infty\lesssim|d_v|$ and
$\|\widetilde P_v(\cdot,T)\|_2^2\lesssim|d_v|^2R$.
Integration with the temporal decay of $w_B$ or $\psi(T/R)$ proves
the upper bounds.  For sufficiently small fixed $c>0$, the
nonnegative profile gives $|\widetilde P_v(X,T)|\gtrsim|d_v|$ on
\[
 0\le T\le cR,\qquad
 |X-Z_v+T\nabla\Phi_+(\zeta_v)|\le c\sqrt R.
\]
This is the short-core argument in the proof of
Theorem~\ref{pkt:refined-decoupling}.  The region has volume
comparable to $R^2$.  Both $|\psi(T/R)|$ and $w_B$ are bounded below
there: the time-zero centers of the retained packets lie within
distance $O(R)$ of $B$, even if a tube meets the box only at a later
time.  This proves the lower bounds.
\end{remark}

\begin{remark}[Vanishing adjacency powers]\label{pkt:vanishing-powers}
Later there will be an outer index $i$, a fixed number $J$ of
transitions, and positive adjacency powers $\eta_i\to0$.  For each
fixed $i$ the scale ratios satisfy
$\ell_j=m^{1/J+o(1)}$, so $R=\ell_j^2=m^{2/J+o(1)}$.
Choose the external error power tending to zero with $i$, and its
localization power $\delta_i$ still smaller, with
$2\delta_i/J<\eta_i/10$.  Choose the sampling power less than
$\min(\eta_i,1/J)/10$.  The cube comparison above then uses precisely
the radius $m^{\eta_i}w_-$, while the coloring and decoupling losses
are $m^{o_i(1)}$.  Here $o_i(1)$ denotes an exponent whose limiting
upper absolute value as $m\to\infty$, with $i$ and $J$ fixed, tends
to zero as $i\to\infty$.  We take the limits in that order.  The
phase, profile, complexity, and positive radius powers remain fixed
in each inner limit.  Constants may depend on $i$; since $J$ is fixed,
the same convention applies to the product of all transition losses.
\end{remark}
==== END Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//05-packets.tex ====

==== BEGIN Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//06-extremal.tex ====
\section{Extremal selection and propagation}\label{sec:extremal}

We prove Theorem~\ref{pkt:main}. We use the packet scales, lattice norms, and
propagation exponent introduced in Section~\ref{sec:packets}. In particular,
the amplification of a diagram with initial array \(d\) and terminal array
\(c\) is
\[
                 \frac{\mathcal A(c)^3}{G(d)^3}.
\]
Every mask below is a coordinate projection on a layer of the diagram.
Between successive masks we retain the entire packet transition matrix.
Graphs will record its significant interactions, but will not replace the
matrix by an edgewise truncation.

We argue by contradiction and assume \(\gamma>10\kappa\).
We use two successive extrema. The first determines the infimum of
the time-support exponents that allow amplification arbitrarily close
to \(\gamma\). The second minimizes, up to a vanishing error, the
asymptotic exponent of the path count per original root among diagrams
with nearly this time support. This makes the path count stable under
the later masks that retain near-extremal amplification, and supplies
the counting law needed for geometric propagation.

\subsection{The least time dimension of a near-extremal family}

For \(0\leq \sigma\leq1\), define \(\gamma_\sigma\) by the same supremum
of limiting amplification exponents as \(\gamma\), with the following
additional restriction. There is a fixed \(\eta>0\) within the family
such that each active initial index touches terminal masked indices in
at most \(m^{2\sigma+o(1)}\) time bins, where a touch means
\[
 |\nu_b-\nu_v|\leq m^\eta\theta,
 \qquad
 |z_b-z_v-t_{I_b}U_{\nu_v}|\leq m^\eta\delta .
\]
Here the initial time is zero. The supremum still ranges over all fixed
positive \(\eta\) and all finite polynomial parameter bounds.

\begin{lemma}[Minimal time dimension]\label{ext:time-dimension}
One has
\begin{equation}\label{eq:source-34}
 \gamma_\sigma\leq 2\sigma,
 \qquad
 s:=\inf\{\sigma\in[0,1]:\gamma_\sigma=\gamma\}>0 .
\end{equation}
For every fixed \(\sigma<s\), the difference
\(\gamma-\gamma_\sigma\) is positive. For every \(\sigma>s\) in
\([0,1]\), one has \(\gamma_\sigma=\gamma\).
Near-extremal families defining any of these exponents may be chosen
with all nonzero initial coordinate amplitudes between \(1\) and \(2\).
\end{lemma}

\begin{proof}
Fix a family satisfying the additional support restriction.
For a terminal bin \(I\), let \(\mathcal R_I\) be the initial indices
touching at least one retained terminal index in \(I\). Every matrix
entry from an initial index outside \(\mathcal R_I\) to a retained
index in \(I\) is a locality tail. Lemma~\ref{pkt:locality} and the
polynomial coordinate count show that replacing the input by
\(d|_{\mathcal R_I}\) changes the output in \(I\) by a negligible
error. Apply the squared-sum contraction of Lemma~\ref{pkt:crude} to
the full transition matrix on this restricted input. Summing over
\(I\) counts each initial index at most \(m^{2\sigma+o(1)}\) times.
Therefore
\[
 r w^2\sum_b |c_b|^2
 \leq m^{-2(1-\sigma)+o(1)}\delta^2\sum_v|d_v|^2.
\]
The same lemma gives \(K(c_I)\leq m^{4+o(1)}K(d)\) in every bin.
Since \(A(c_I)\leq G(c_I)\), multiplication by the square root of this
capacity bound proves amplification at most \(m^{2\sigma+o(1)}\).

The classes of admissible families increase with \(\sigma\).
Consequently \(\gamma_\sigma\) is nondecreasing and at most \(\gamma\);
also \(\gamma_1=\gamma\), since there are only \(m^2\) terminal time
bins. The set on which equality holds is therefore an upper interval,
possibly without its lower endpoint. Its infimum satisfies
\(s\geq\gamma/2>0\), and the other assertions about
\(\gamma_\sigma\) follow. Equality at \(\sigma=s\) is not needed.

To obtain comparable initial amplitudes, first normalize the largest
one. The polynomial coordinate counts and the polynomial comparison
between coordinate norms and the packet norms allow amplitudes below
a sufficiently small fixed negative power of \(m\) to be discarded.
The remaining amplitudes lie in \(O(\log m)\) dyadic classes.
Writing the input as their sum and using linearity followed by the
triangle inequality for the output norm, one class retains at least
an inverse-logarithmic fraction of the output norm. Restriction to
that class can only decrease the initial raw gauge. Its cubed
amplification therefore loses a subpower factor. Rescaling its
amplitudes gives the interval \([1,2]\). The support restriction is
hereditary under this operation.
\end{proof}

An upper bound with exponent \(\gamma_\sigma\) also holds with the
initial atomic norm in place of the raw gauge: decompose the input
into atoms on its given support and apply the triangle inequality.
In particular, Lemma~\ref{pkt:composition} permits this improved bound
at one transition of a masked diagram whenever the two supports of
that transition satisfy the time restriction. We will repeatedly use
the following consequence:
\begin{equation}\label{ext:gap}
 \begin{gathered}
 \text{a fixed improvement below the time exponent \(s\) at one transition}\\
 \text{gives a fixed improvement below \(\gamma\) for the diagram.}
 \end{gathered}
\end{equation}
The improvement in \eqref{ext:gap} is measured in \(m\)-power units
when the transition ratio is a fixed positive power of \(m\).

\subsection{A second extremum: the number of paths}

Fix \(J\geq1\), and let \(m_j\), \(0\leq j\leq J\), be dyadic
roundings of \(m^{j/J}\), with \(m_0=1\) and \(m_J=m\). Put
\[
 \ell_j=\frac{m_{j+1}}{m_j},\qquad
 \theta_j=m_j\delta,\qquad r_j=m_j^{-2},\qquad w_j=r_j\theta_j .
\]
Thus \(\log_m\ell_j\to1/J\). Insert the intermediate frame
identities and truncate to polynomial grids, as permitted by
Lemma~\ref{pkt:locality}. A diagram consists of these exact transitions
and arbitrary vertex masks, including a possible initial mask.
Its input array \(d\) is normalized as in Lemma~\ref{ext:time-dimension},
on a root set \(\mathcal R\); amplification is always measured against
the entire array on this set, even if an additional root mask is used.

For each fixed \(J\), an index \(i\) will label a sequence of families.
Within family \(i\), the polynomial bounds of
Definition~\ref{pkt:families} and all positive radius powers are fixed
before \(m\to\infty\); these bounds may depend on \(i\).
We write \(o_i(1)\) in an exponent for an error whose limiting upper
absolute value as \(m\to\infty\) tends to zero as \(i\to\infty\).
This includes subpower losses within a fixed family. Thus \(J\)
remains fixed while we first take the inner limit in \(m\) and then
the outer limit in \(i\). The construction is valid for every fixed
\(J\); we will later choose one sufficiently large for the geometric
time-profile hypothesis.

For a fixed positive radius power \(\eta\), join an active index \(v\)
at level \(j\) to an active index \(b\) at level \(j+1\) when
\begin{equation}\label{eq:source-35}
 I_b\subset I_v,\qquad
 |\nu_b-\nu_v|\leq m^\eta\theta_{j+1},\qquad
 |z_b-z_v-(t_{I_b}-t_{I_v})U_{\nu_v}|\leq m^\eta w_j .
\end{equation}
A path is a sequence of such edges from an active root to a terminal
leaf.

\paragraph{Locality for error diagrams.}\label{ext:diagram-locality}
We will repeatedly discard a vertex class whose contribution has a
nonadjacent edge in every complete matrix-product summand.
For every prescribed \(A>0\), this contribution is
\(O(m^{-A}\max_{v\in\mathcal R}|d_v|)\), both coordinatewise and in
the packet norms. Indeed, Lemma~\ref{pkt:locality} makes the
nonadjacent coefficient smaller than any prescribed negative power;
the remaining coefficients and the number of summands have polynomial
bounds, and there are only \(J\) layers. Choose the locality decay
order after the graph radius and these fixed family bounds.
This applies, in particular, to roots with no graph path to the tested
output and to intermediate vertices with no suffix to that output.
It bounds the error of the vertex masking; the retained diagram
continues to use the full transition matrices.

\begin{lemma}[Endpoint localization]\label{ext:endpoints}
Along a path from level \(a\) to level \(b>a\), the endpoint angle
difference is \(O_J(m^\eta\theta_b)\), and
\[
 |z_b-z_a-(t_{I_b}-t_{I_a})U_{\nu_a}|
       \leq O_J(m^\eta w_a).
\]
For a fixed root and a fixed terminal leaf, the number of paths is
at most \(m^{O(J\eta)+o(1)}\).
\end{lemma}

\begin{proof}
The angular differences are bounded by a geometric sum of the
successive child widths. For the position bound, write
\(t_j=t_{I_j}\), \(U_j=U_{\nu_j}\), and
\(e_j=z_{j+1}-z_j-(t_{j+1}-t_j)U_j\). Telescoping gives
\[
 z_b-z_a-(t_b-t_a)U_a
 =\sum_{j=a}^{b-1}e_j+
   \sum_{j=a+1}^{b-1}(t_b-t_j)(U_j-U_{j-1}).
\]
The edge errors satisfy \(|e_j|\leq m^\eta w_j\).
For a switch from level \(j-1\) to level \(j\), Lipschitz
continuity of the velocity map gives
\(|U_j-U_{j-1}|\lesssim m^\eta\theta_j\), while nesting gives
\(0\leq t_b-t_j\leq r_j\). Its contribution is therefore
\(O(m^\eta\theta_jr_j)=O(m^\eta w_j)\).
Summing and using \(w_j\leq w_a\) proves the positional endpoint bound.

For fixed endpoints, the intermediate time interval at each level is
the unique ancestor of the terminal interval. At level \(j\), the
angular grid has spacing \(\theta_j\), while the angle lies within
\(O_J(m^\eta\theta_j)\) of the root angle. There are
\(m^{O(\eta)+o(1)}\) choices. Given this angle, the endpoint bound
applied from level \(j\) to the terminal leaf locates \(z_j\) within
\(O_J(m^\eta w_j)\) of a prescribed point. On the position grid of
spacing comparable to \(w_j\), this again gives
\(m^{O(\eta)+o(1)}\) choices. Multiplication over the fixed number
of intermediate levels proves the claim.
\end{proof}

At accuracy \(\epsilon>0\), call a family of diagrams admissible if it
has a fixed radius power \(0<\eta<\epsilon\), amplification at least
\(m^{\gamma-\epsilon}\), and at most \(m^{2(s+\epsilon)}\) terminal
time bins accessible from every root by paths at that radius.
For such a family consider the limiting lower exponent of
\begin{equation}\label{eq:source-36}
                 \frac{\#\{\text{paths}\}}{\#\mathcal R}.
\end{equation}

\Needspace{4\baselineskip}
\begin{lemma}[Minimal path count]\label{ext:path-minimum}
Admissible families exist at arbitrarily small accuracies. The infima
of the limiting lower exponents in \eqref{eq:source-36} have a finite
limit as \(\epsilon\downarrow0\). There are families approaching this
limit, with accuracies tending to zero, for which the following holds.
Call their graphs the high graphs. Choose a smaller fixed positive
radius power in each family, tending to zero with its accuracy, and
call the corresponding graphs low graphs. After any fixed number of
further vertex maskings that retain amplification
\(m^{\gamma-o_i(1)}\), one has
\begin{equation}\label{eq:source-37}
 \#\{\text{remaining low paths}\}
       \geq m^{-o_i(1)}\#\{\text{original high paths}\}.
\end{equation}
\end{lemma}

\begin{proof}
If \(s<1\), choose a support exponent strictly between \(s\) and
\(s+\epsilon\); its propagation exponent is \(\gamma\) by
Lemma~\ref{ext:time-dimension}. If \(s=1\), use the unrestricted
families. In either case choose a near-extremal single-step family
with the desired time sparsity at some fixed enlarged direct radius,
and make its input amplitudes comparable. Insert the intermediate
identities. By Lemma~\ref{ext:endpoints}, a sufficiently smaller fixed
radius power at the intermediate levels ensures that every path
endpoint is a direct touch at the originally chosen radius.
Truncation errors and dyadic losses can be made smaller than the
unused accuracy. This gives the required admissible families.

At each step, a parent has at most
\(O(m^{4\eta}\ell_j^2)\) children in a given time bin: the angular
grid contributes \(O(m^{2\eta})\) choices and the position grid
contributes \(O(m^{2\eta}\ell_j^2)\). There are \(\ell_j^2\) child
time bins. Hence the total path count per root is at most
\(m^{4+4J\eta+o(1)}\). For a lower bound, let
\(\mathcal R'\) be the roots having at least one path. All other
roots contribute a negligible error by the
\hyperref[ext:diagram-locality]{locality observation above}.
Since \(1\leq|d_v|\leq2\),
\[
 G(d|_{\mathcal R'})^3
 \leq C\,\frac{\#\mathcal R'}{\#\mathcal R}\,G(d)^3.
\]
Here the squared mass drops in the displayed proportion, and its
capacity does not increase. Composition with exponent
\(\gamma+o(1)\), together with the lower amplification, gives
\(\#\mathcal R'/\#\mathcal R\geq m^{-\epsilon-o(1)}\).
There is at least one path per root of \(\mathcal R'\), proving
a lower bound for the exponent in \eqref{eq:source-36}.
The admissible classes are nested as \(\epsilon\) decreases, so
their infima have a finite monotone limit; denote it by \(\Lambda_J\).

Choose near-minimizers with accuracies \(\epsilon_i\downarrow0\),
and pass to subsequences on which their path-count exponents
approach \(\Lambda_J\) also from above. Let their fixed high radius
powers be \(\eta_i\), and choose low powers
\(0<\eta_i^{\mathrm{lo}}\ll\eta_i\).
Suppose further masks have amplification \(m^{\gamma-\rho_i+o(1)}\)
with \(\rho_i\to0\). The resulting low graph inherits the original
upper bound on accessible terminal bins. It is therefore an
admissible competitor with accuracy tending to zero, for example
an accuracy slightly larger than
\(\max(\epsilon_i,\rho_i,\eta_i^{\mathrm{lo}})\).
Its limiting lower exponent is at least \(\Lambda_J-o_i(1)\).
The original high graph has exponent at most
\(\Lambda_J+o_i(1)\) on the chosen subsequence. This proves
\eqref{eq:source-37}. All adaptive masks still form families with
fixed polynomial bounds at each \(i\), so this comparison applies
to each of a fixed finite number of such selections.
\end{proof}

Figure~\ref{fig:packet-proof-map} shows how this path-retention property
enters the rest of the proof.
\input{figures/packet-proof-map}

Fix these high and low radii. Between them choose a fixed intermediate
radius power \(\eta_i^{\mathrm{mid}}\) in each family, so that
\(\eta_i^{\mathrm{lo}}<\eta_i^{\mathrm{mid}}<\eta_i\).
The strict gaps absorb bounded constants in neighborhood containments
once \(m\) is sufficiently large. Figure~\ref{fig:adjacency-radii}
illustrates the inclusions used below.

\input{figures/adjacency-radii}

\subsection{Regularizing the graph by vertex masks}

We first record how dyadic selections affect the exact diagram.
Partitioning the active coordinates at any one layer into
\(O((\log m)^C)\) classes writes the exact output as a sum of the
outputs with the corresponding vertex masks inserted. The triangle
inequality for \(\mathcal A\) selects a class losing only a subpower
factor in amplification. This remains true through finitely many
layers. The integer path statistics below have polynomial size, so
their positive values require only logarithmically many dyadic
classes. Vertices with no high suffix have negligible output
contribution by the
\hyperref[ext:diagram-locality]{locality observation} and can be discarded.

Let \(D_j(v)\) be the number of suffix paths from \(v\) in the
\emph{original high graph}. For a vertex \(b\) at level \(j+1\),
let \(\mathcal N_j(b)\) consist of the original high vertices \(b'\)
at level \(j+1\) satisfying
\[
 I_{b'}=I_b,\qquad
 |\nu_{b'}-\nu_b|\leq m^{\eta_i^{\mathrm{mid}}}\theta_{j+1},
 \qquad
 |z_{b'}-z_b|\leq m^{\eta_i^{\mathrm{mid}}}w_j.
\]
Define
\[
 S_j(b)=\sum_{b'\in\mathcal N_j(b)}D_{j+1}(b').
\]
If \(b\) is a low child of \(v\), the triangle inequality shows that
\(\mathcal N_j(b)\) contains every low child of \(v\) in that time
bin and is contained in the high children of \(v\). Indeed, for fixed
\(i\) and sufficiently large \(m\),
\[
 2m^{\eta_i^{\mathrm{lo}}}\leq m^{\eta_i^{\mathrm{mid}}},
 \qquad
 m^{\eta_i^{\mathrm{mid}}}+m^{\eta_i^{\mathrm{lo}}}\leq m^{\eta_i}.
\]
The predicted child position in \eqref{eq:source-35} is the same
for all children in that bin.

\Needspace{18\baselineskip}
\begin{lemma}[Time branching and pruning]\label{ext:branching}
One can retain amplification \(m^{\gamma-o_i(1)}\) by vertex masks such that
\[
                         D_j(v)\asymp d_j,\qquad
                         S_j(b)\asymp s_j'
\]
on the selected coordinates, for \(0\leq j\leq J\) in the
first comparison and \(0\leq j<J\) in the second, with
\begin{equation}\label{eq:source-38}
                         \frac{d_j}{s_j'}
                    \geq \ell_j^{2s}m^{-o_i(1)}.
\end{equation}
One can then prune backwards, retaining this amplification,
so that every surviving vertex at level \(j\)
has at least
\[
                   (m/m_j)^{2s}m^{-o_i(1)}
\]
terminal descendant time bins by low paths. In the pruned graph,
and after any subsequent masks, each root reaches at most
\begin{equation}\label{eq:source-39}
                           m_j^{2s}m^{o_i(1)}
\end{equation}
distinct time bins at level \(j\).
\end{lemma}

\begin{proof}
Select dyadic classes for all the indicated positive statistics.
For each time bin touched by a remaining parent \(v\), select
one remaining low child \(b\). Its intermediate neighborhood
contributes at least a constant times \(s_j'\) to \(D_j(v)\).
These contributions for distinct time bins are disjoint, because
each suffix has a unique child time bin. Thus the number of
touched bins is at most \(C d_j/s_j'\).

If \eqref{eq:source-38} failed by a fixed positive power of \(m\)
along a subsequence, the bound \(C d_j/s_j'\) would, after rebasing
the parent interval, be at most \(\ell_j^{2\sigma+o(1)}\) for some
fixed \(\sigma<s\). Here \(\log_m\ell_j\to1/J\), and the low radius
remains a fixed positive power in the \(\ell_j\) units within each
family. Thus the selected parent and child supports satisfy the
hypothesis of \eqref{ext:gap}. Its strict improvement for the full
transition contradicts the retained amplification, proving
\eqref{eq:source-38}.

For the pruning, begin at level \(J-1\) and proceed backwards.
At level \(j\), delete every parent touching fewer than
\(\ell_j^{2s}m^{-a}\) distinct child time bins in the current child
mask, initially with a fixed \(a>0\). The difference made to the
output is a diagram whose transition at this level starts from
the deleted parent set and ends in that current child mask.
That transition has the strict sparsity improvement just
described. A telescoping decomposition of the successive
mask changes contains only \(J\) such error diagrams. Each has
a fixed exponent gap, and the triangle inequality shows that
their total is negligible compared with a sufficiently accurate
near-extremal diagram.

We may consequently let the deficit \(a=a_i\downarrow0\) slowly
enough that its positive exponent gap still dominates the
amplification errors in the \(i\)-th family. This is a diagonal
choice: first use the gap for each fixed deficit, and then take
the initial accuracies and the family bases sufficiently far
out. No quantitative continuity of \(\gamma_\sigma\) at \(s\)
is required. The resulting pruned graph retains amplification
\(m^{\gamma-o_i(1)}\).

Each surviving parent now has at least
\(\ell_j^{2s}m^{-a_i}\) child time bins with surviving low children.
Choose one child in each such bin and iterate. The descendant
bins associated with distinct child time bins are disjoint.
Induction therefore gives at least
\[
 \prod_{h=j}^{J-1}\ell_h^{2s}m^{-a_i}
             =(m/m_j)^{2s}m^{-(J-j)a_i}
\]
terminal bins from every surviving level-\(j\) vertex.

Fix a root and one reachable vertex in each of its reachable
level-\(j\) time bins. The just constructed terminal descendants
are disjoint for these different time bins, and all are
accessible from the root in the original high graph. In that graph
this root reaches at most \(m^{2(s+\epsilon_i)}\) terminal time bins.
Dividing this bound by the lower descendant count gives
\eqref{eq:source-39}. If later masks remove some descendants,
the original pruned descendants remain valid witnesses in
this counting argument; later masks can only decrease the set
of reachable prefixes.
\end{proof}

\begin{lemma}[Incoming counts and terminal selection]\label{ext:leaves}
Further vertex masks can be chosen retaining amplification
\(m^{\gamma-o_i(1)}\) so that the following hold.
At each transition, every remaining child has between \(k_j\)
and \(2k_j\) low parents in the current parent mask, for a
positive integer \(k_j\). Put \(k=\prod_{j=0}^{J-1}k_j\).
Each terminal leaf has between \(k\) and \(2^Jk\) incoming low
paths, whereas any fixed root--leaf pair has at most
\(m^{o_i(1)}\) such paths. There is \(h_1>0\) such that
\[
                             |c_b|^2\asymp h_1k
\]
on all remaining leaves. Moreover, restricting the output to
any subset occupying a fraction \(m^{-o_i(1)}\) of these leaves
still retains amplification \(m^{\gamma-o_i(1)}\).
\end{lemma}

\begin{proof}
Proceed forwards. At transition \(j\), count the low parents of
each child in the already selected parent mask, discard the
zero-count class by the
\hyperref[ext:diagram-locality]{locality observation}, and select a positive
dyadic class retaining amplification \(m^{\gamma-o_i(1)}\). Subsequent forward
selections do not change any already fixed parent layer.
Thus the asserted incoming degrees hold in the final graph.
Induction from the roots gives the bounds \(k\) and \(2^Jk\)
for the number of incoming paths to each leaf. The bound for
a specified root--leaf pair follows from
Lemma~\ref{ext:endpoints}, because the radius powers tend to zero.

It remains to choose the leaves with the stronger norm property.
On the finite coordinate space, the lattice norm
\(\mathcal A\) has a nonnegative dual functional \(\lambda\) of
dual norm one satisfying
\[
                       \sum_b\lambda_b|c_b|=\mathcal A(c).
\]
One may take the absolute values of a norming functional:
the lattice property preserves its dual norm. Polynomial norm
comparisons and coordinate counts permit tiny coordinate amplitudes
to be discarded with negligible norm, hence negligible pairing with
\(\lambda\). Discarding additionally those contributions
\(\lambda_b|c_b|\) smaller than a sufficiently large negative power
of \(m\) times \(\mathcal A(c)\) loses negligible total pairing, by
the polynomial coordinate count. Partition the remaining leaves jointly into dyadic
classes for these two quantities. There are only a power of
\(\log m\) such classes. Some class \(\mathcal B\) satisfies
\[
 \sum_{b\in\mathcal B}\lambda_b|c_b|
       \geq (\log m)^{-C}\mathcal A(c),
 \qquad
 |c_b|^2\asymp h_1k\quad(b\in\mathcal B)
\]
for a suitable \(h_1>0\). Keep this class as a final leaf mask.
Incoming path counts for its leaves are unaffected.

If \(\mathcal B'\subset\mathcal B\) occupies a fraction \(f\),
comparability of \(\lambda_b|c_b|\) gives
\[
 \mathcal A(c|_{\mathcal B'})
 \geq \sum_{b\in\mathcal B'}\lambda_b|c_b|
 \geq \tfrac12 f\,(\log m)^{-C}\mathcal A(c).
\]
For \(f=m^{-o_i(1)}\), cubing this inequality loses only a
vanishing exponent. The functional need not norm any of the
later restricted outputs.
\end{proof}

\subsection{The conditional time law}

The masks now control prefix counts and incoming leaf degrees while
preserving near-extremal amplification. To apply geometric propagation,
we still need both support counts and maximal probabilities at every
time depth, conditional on each retained root. We obtain these from
uniform counting on a selected set of paths. This counting law is
auxiliary; the coefficients continue to come from the exact masked
matrix maps.

All graph statistics \(D_j\) and \(S_j\) still refer to the
original high graph. All low path counts in this subsection
refer to the final masks of Lemma~\ref{ext:leaves}.

\begin{lemma}[A linear conditional time profile]\label{ext:time-law}
There is a subset \(\Omega\) of the final low paths occupying
a fraction \(m^{-o_i(1)}\) of them such that uniform counting
on \(\Omega\) has these properties:
\begin{enumerate}[label=\textup{(\roman*)}]
\item Its root marginal is comparable to the uniform law on
a set of \(N_0\) roots.
\item Given each root, the terminal time satisfies
\eqref{eq:source-1}, with base \(M=m^2\), target exponent \(s\),
and error \(O(1/J)+o_i(1)\).
\end{enumerate}
The subset is used only to define this law; it does not change
any internal summand of the terminal coefficients.
\end{lemma}

\begin{proof}
Let \(H\) be the original high path count and \(L\) the final
low path count. By Lemma~\ref{ext:path-minimum},
\(L\geq m^{-\alpha_i}H\), where \(\alpha_i=o_i(1)\).
Write \(T^{(j)}\) for the level-\(j\) time interval containing
the terminal time.
For a vertex \(v\), let \(P_H(v)\) and \(P_L(v)\) be its high
and final low prefix counts, and let \(D_L(v)\) be its final
low suffix count. Thus \(P_L(v)\leq P_H(v)\) and
\(D_L(v)\leq D_j(v)\).

Choose \(\beta_i\downarrow0\) sufficiently slowly that
\(\beta_i-\alpha_i\) dominates the other errors and
\(m^{\alpha_i-\beta_i}\to0\) within each family. Call a remaining
level-\(j\) vertex bad if
\[
                         D_L(v)<m^{-\beta_i}D_j(v).
\]
The number of final low paths through bad vertices at this level
is bounded by
\[
 \sum_{v\ {\rm bad}}P_L(v)D_L(v)
 \leq m^{-\beta_i}\sum_v P_H(v)D_j(v)
 =m^{-\beta_i}H.
\]
In the final sum, \(v\) ranges over all vertices of the original
high graph at that level. Its equality to \(H\) holds because
every high path visits exactly one such vertex. Summing over
\(J+1\) levels shows that
the event \(\mathcal G\) that every visited vertex is good has
probability \(1-o(1)\) under uniform final low path counting.

Consider this unrestricted uniform low law, conditional on a
complete vertex prefix ending at a good \(v\) at level \(j\).
For a prescribed child time bin \(I\), its next-bin probability is
\begin{equation}\label{ext:one-bin}
 \frac{\displaystyle
       \sum_{\substack{b\ {\rm low\ child\ of}\ v\\I_b=I}}D_L(b)}
      {D_L(v)}
 \leq m^{\beta_i+o_i(1)}\frac{s_j'}{d_j}
 \leq m^{o_i(1)}\ell_j^{-2s}.
\end{equation}
If the numerator is nonzero, choose one remaining child \(b\)
in that bin. Its intermediate neighborhood contains every
other low child there, and the original high suffix sum on
the neighborhood is comparable to \(s_j'\). This bounds the
numerator. Goodness and \(D_j(v)\asymp d_j\) bound the
denominator, and \eqref{eq:source-38} gives the final inequality.

To iterate \eqref{ext:one-bin}, kill a path as soon as it
visits a bad vertex. Fix a nested sequence of time bins through
level \(j\). Conditional on each surviving \emph{vertex}
prefix, the next prescribed-bin probability is bounded by
\eqref{ext:one-bin}. Averaging over the vertex prefixes having
the same time prefix preserves that bound. Induction, with
the process killed at every step, therefore gives, for every
root \(v\) and level-\(j\) bin \(I\),
\begin{equation}\label{ext:killed}
 \Pr(\mathcal G,\ T^{(j)}=I\mid v)
       \leq m^{o_i(1)}\prod_{h<j}\ell_h^{-2s}
       =m^{o_i(1)}m_j^{-2s}.
\end{equation}
If the root is bad the left side is zero. In writing
\(\mathcal G\) on the left, one may first require survival only
through level \(j\), for which the induction applies directly,
and then impose survival at later levels. This can only
decrease the probability. Multiple vertex prefixes with the
same time prefix introduce no additional count.

Discard roots for which
\(\Pr(\mathcal G\mid v)<1/2\). Their total probability under
the original low root law is at most
\(2\Pr(\mathcal G^c)=o(1)\). Keep the good paths on the other
roots. Dividing \eqref{ext:killed} by their conditional survival
probability costs at most two, so the maximum probability of a
level-\(j\) time bin, conditional on each retained root, is
\(m^{-2s\log_m m_j+o_i(1)}\). The support bound at that level
is \eqref{eq:source-39}.

Between consecutive level depths, monotonicity of probabilities
uses the preceding level for the maximum weight, and monotonicity
of supports uses the following level for the number of occupied
bins. Since the gap in base-\(M=m^2\) depths is
\(1/J+o(1)\), these give \eqref{eq:source-1} at every dyadic
depth with error at most \(s/J+o_i(1)\).
The dyadic roundings change only bounded factors.

The retained measure is still uniform on a subset of paths.
Its positive root weights have polynomial ratios, because the
numbers of paths are positive integers with polynomial upper
bounds. Select a dyadic class of root weights carrying an
inverse-logarithmic fraction of the measure, and keep all
retained paths from those roots. If their number is \(N_0\),
the new root weights are comparable to \(1/N_0\).
This selection of whole root fibers does not change the
root-conditional time laws and retains a fraction
\(m^{-o_i(1)}\) of all final low paths.
\end{proof}

\subsection{Geometry bounds the capacity of the output}

Fix a positive \(\zeta<\gamma-10\kappa\).
Choose \(J\) sufficiently large that the \(O(1/J)\) error in
Lemma~\ref{ext:time-law} lies within the small-error range of
Theorem~\ref{geo:main} for exponent
\(1+2s-10\kappa-\zeta/2\), and then take \(i\) sufficiently large.
All the preceding selections are made for this fixed \(J\).

For a subset \(\mathcal P\) of the final low paths, say that a root
\(v\) is represented in a terminal bin \(I\) if some path of
\(\mathcal P\) starts at \(v\) and ends in \(I\). Count each such
root once in that bin, regardless of the number of its paths.
Call the paths in \(\mathcal P\) represented paths.

\Needspace{4\baselineskip}
\begin{lemma}[Represented roots]\label{ext:represented}
Fix any polynomial upper bound on the radii within each family.
There is a subset \(\mathcal P\), which may depend on this bound,
occupying a fraction \(m^{-o_i(1)}\) of all final low paths with the
following property. In every terminal bin \(I\), the number of
represented roots satisfying
\[
 U_{\nu_v}\in u+F,\qquad z_v+t_IU_{\nu_v}\in x+rF
\]
is at most
\begin{equation}\label{eq:source-41}
 K(d)\delta^{-2}W(F)\,
          m^{-2(1+s-10\kappa-\zeta)+o_i(1)},
 \qquad \text{radii}(F)\geq m^2\delta,
\end{equation}
uniformly over all independent centers \(u,x\in\R^2\) and all radii
up to the chosen upper bound.
One can keep a fraction \(m^{-o_i(1)}\) of the final leaves such
that every kept leaf has at least \(k m^{-o_i(1)}\)
represented incoming paths.
\end{lemma}

\begin{proof}
Put the uniform law of Lemma~\ref{ext:time-law} on its path
subset, take the root as base label, and assign it line parameters
\((U_{\nu_v},z_v)\). The root weights are comparable to \(1/N_0\),
and \(1\leq|d_v|\leq2\). The input capacity definition therefore
gives the density constant
\[
                         K_0\lesssim
                         \frac{K(d)\delta^{-2}}{N_0}
\]
for tests with radii at least \(\delta\).

We fix the width range before applying geometry. Choose a polynomial
cutoff \(H\geq m^2\delta\) bounding the norms of all tested velocities
and all spatial coordinates after division by \(r\), including
the root-line positions and terminal position labels. In the axes of
a rectangular test with half-widths \(e_j\), each coordinate interval
meets the bounded support in a set contained in an interval of
half-width \(\min(e_j,H)\). Choose the two centers independently,
using the same clipped half-widths and axes. The resulting joint test
covers the original test on the support, and its weight is no larger,
up to the fixed ellipse--rectangle comparison. Thus one polynomial
width range controls the full terminal capacity supremum.

Choose \(B=m^A\) larger than this cutoff, the upper width bound in
the statement, and the root coordinate bounds, with a fixed power
margin. This margin also covers the endpoint enlargements. With
\(M=m^2\), we have \(B\geq M\delta\). Rescale both line coordinates
by \(B^{-1}\).
The new input floor is
\[
 \delta'=\delta/B=M^{-D_m},\qquad
 D_m=\tfrac12(\log_mN+A)\geq1.
\]
The density constant becomes \(K_0B^2\), since \(W\) is
homogeneous of degree two. The output weight of a test is
\(W(F/B)=B^{-2}W(F)\), so these factors cancel on returning
to the old coordinates. The new output floor \(M\delta'\)
corresponds exactly to \(M\delta\) in the old coordinates.
This also covers \(m=N\), for which the unnormalized floor
would have given the inadmissible value \(D=1/2\).
To make the complexity fixed, take a subsequence with
\(D_m\to D_0\), and choose \(D_*=D_0+a\), where
\(0<a<\zeta/16\). Eventually
\(0<D_*-D_m<2a\). A radius at least \(M^{-D_*}\) but below
the available input floor \(M^{-D_m}\) can therefore be
rounded up by a factor at most \(M^{2a}\). Doing this to
both radii increases \(W\) by at most \(M^{4a}=m^{8a}\).
Thus the input cap at the fixed complexity \(D_*\) costs
at most this additional factor. The output floor for \(D_*\)
is smaller than the available output floor, so every required
output test is still allowed.

The definition of the geometric exponent as a small-error limit
of an infimum over finite complexities is used here: the
sufficiently small time-error threshold may be chosen for all
finite \(D_*\), although the large-base threshold and subpower
losses can depend on that fixed complexity. Thus the preceding
choice of \(J\), followed by \(i\) and then the family base,
is valid.

Apply Theorem~\ref{geo:main}, with base \(M=m^2\), to obtain a
dominated sublaw of subpower mass. The difference between the
chosen geometric exponent and \(1+2s-10\kappa-\zeta\) absorbs
the arbitrarily small fixed rounding loss just described.
After normalization, its
line-test probabilities satisfy
\begin{equation}\label{eq:source-40}
 \Pr\bigl(T_1=t,\ U\in F,\ X_t\in rF\bigr)
 \leq
 \frac{K(d)\delta^{-2}}{N_0}\,W(F)\,
 m^{-2(1+2s-10\kappa-\zeta)+o_i(1)} .
\end{equation}
As throughout, translations in the two tests are independent.

There are at most \(N_0m^{2s+o_i(1)}\) possible root--terminal-time
pairs. Discard pairs whose probability in this normalized law is
less than
\[
                           N_0^{-1}m^{-2s-\tau_i},
\]
where \(\tau_i=o_i(1)\) tends to zero slowly enough to dominate
the support error. This discards \(o(1)\) mass. Take \(\mathcal P\)
to be the path atoms in the support of the law after this deletion.
Each represented root in a specified terminal bin has pair probability
at least this threshold. Dividing \eqref{eq:source-40} by the threshold proves
\eqref{eq:source-41}, with all normalization factors absorbed
in \(m^{o_i(1)}\).

Let \(L\) again be the final low path count. The starting uniform
law for geometry is on a subset of at least
\(m^{-o_i(1)}L\) paths. Its dominated geometric sublaw retains
subpower mass. After normalization, the weight of any path
atom is consequently at most \(m^{o_i(1)}/L\).
The remaining support therefore represents at least
\(m^{-o_i(1)}L\) paths, even though their new weights need not
be equal.

Every original final leaf has between \(k\) and \(2^Jk\)
incoming low paths. Choose a threshold \(k m^{-\upsilon_i}\),
with \(\upsilon_i=o_i(1)\) sufficiently slow, and keep the
leaves with at least this many represented incoming paths.
The represented paths going to the other leaves are at most
\(k m^{-\upsilon_i}\) times the original leaf count, which is
negligible compared with the total represented path count.
Since no leaf has more than \(2^Jk\) incoming paths, the kept
leaves occupy a fraction \(m^{-o_i(1)}\) of all final leaves.
Lemma~\ref{ext:leaves} ensures that this last output restriction
retains amplification \(m^{\gamma-o_i(1)}\).
\end{proof}

\begin{lemma}[Terminal capacity]\label{ext:capacity}
For the kept coefficients of Lemma~\ref{ext:represented}, in
every terminal time bin one has
\begin{equation}\label{ext:capacity-bound}
 K(c_I|_{\mathrm{kept}})
 \leq h_1K(d)\,
          m^{2(1-s)+20\kappa+2\zeta+o_i(1)}.
\end{equation}
\end{lemma}

\begin{proof}
Consider an output test \(E,rE\), where the ordered radii of
\(E\) are \(e_1\geq e_2\geq\theta=m\delta\).
Let \(F\) have the same axes and radii
\[
                              f_j=\max(e_j,\delta/r).
\]
By Lemma~\ref{ext:endpoints}, the initial root of any represented
path ending in this output test belongs to the corresponding
line test \(F,rF\), enlarged by \(m^{o_i(1)}\).
Indeed the angular endpoint error is at most
\(m^{o_i(1)}\theta\), and the position error from the root line
is at most \(m^{o_i(1)}\delta\). The definition of \(F\) covers
both errors. Its radii satisfy the floor required in
\eqref{eq:source-41}. The harmless power enlargement changes
its weight and the resulting count only by \(m^{o_i(1)}\).

For a fixed represented root at this time, the terminal angular
grid supplies at most \(m^{o_i(1)}\) possible leaf angles.
The leaf positions lie both in the spatial test \(rE\) and in
an enlarged root beam of radius \(m^{o_i(1)}\delta\).
Lattice counting bounds their number by
\[
        m^{o_i(1)}w^{-2}\prod_{j=1}^2\min(re_j,\delta).
\]
No boundary correction is needed beyond a constant factor:
both \(re_j\) and \(\delta\) are at least the position mesh \(w\).
For any root and leaf, the number of paths is
\(m^{o_i(1)}\), by Lemma~\ref{ext:leaves}.

Each kept leaf has \(|c_b|^2\lesssim h_1k\) and at least
\(k m^{-o_i(1)}\) represented incoming paths. Its squared
coefficient is therefore bounded by \(h_1m^{o_i(1)}\) times
that incoming count. Summing represented paths in the test,
then using \eqref{eq:source-41}, gives
\begin{align*}
 K(c_I|_{\mathrm{kept}})
 &\leq m^{o_i(1)}
 \sup_E h_1\theta^2W(E)^{-1}
     w^{-2}\prod_{j=1}^2\min(re_j,\delta)\\
 &\hspace{25mm}\cdot
 K(d)\delta^{-2}W(F)
       m^{-2(1+s-10\kappa-\zeta)} .
\end{align*}
For the products of semiaxes,
\[
 \prod_{j=1}^2\min(re_j,\delta)
       =\delta^2\,\frac{e_1e_2}{f_1f_2},
 \qquad
 \frac{W(F)}{f_1f_2}
       =\left(\frac{f_1}{f_2}\right)^\kappa
       \leq\left(\frac{e_1}{e_2}\right)^\kappa
       =\frac{W(E)}{e_1e_2}.
\]
The inequality uses the reduction of aspect ratio when small
radii are raised to a common floor. Finally
\(\theta^2w^{-2}=r^{-2}=m^4\). Substitution proves
\eqref{ext:capacity-bound}.
\end{proof}

\subsection{Two mass bounds and the contradiction}

The square root of the capacity bound contributes
\(m^{1-s}h_1^{1/2}\), in addition to
\(m^{10\kappa+\zeta+o_i(1)}\). We bound the coefficient mass in
two ways: the frame estimate uses the number of times reached by each
root, and the fourth-power estimate gives a factor \(h_1^{-1}\).
Their minimum cancels \(m^{1-s}h_1^{1/2}\) for every value of \(h_1\).

\begin{lemma}[Terminal mass]\label{ext:mass}
The same kept output satisfies
\begin{equation}\label{ext:mass-bound}
 r w^2\sum_{b\ {\rm kept}}|c_b|^2
 \leq m^{o_i(1)}
       \min\bigl(m^{-2(1-s)},h_1^{-1}\bigr)
       \delta^2\#\mathcal R .
\end{equation}
\end{lemma}

\begin{proof}
For the first bound, fix a terminal time bin \(I\) and let
\(\mathcal R_I\) be the roots having an original high path to
that bin. Roots outside this set give a negligible error by the
\hyperref[ext:diagram-locality]{locality observation}, applied to the
original high graph with output restricted to \(I\).
Along the unique sequence of time ancestors of \(I\), the
successive exact masked transitions satisfy the squared-sum
bound of Lemma~\ref{pkt:frame}, with the parent-bin rebasing
of Lemma~\ref{pkt:composition}. The transition at level \(j\)
has squared operator norm at most \(\ell_j^2\), and masks are
orthogonal coordinate projections. With the scale
normalizations this gives
\[
                    w^2\sum_{b\in I}|c_b|^2
                    \lesssim \delta^2\#\mathcal R_I
\]
apart from negligible error. Summation with weight \(r=m^{-2}\)
counts each root at most \(m^{2(s+\epsilon_i)}\) times by the
original high time-support bound. This proves the first term in
\eqref{ext:mass-bound}.

For the second bound, apply Proposition~\ref{pkt:quartic} at each
actual masked transition. Lemma~\ref{ext:leaves} bounds its
low-parent multiplicity by \(2k_j\). Use
Remark~\ref{pkt:vanishing-powers} with adjacency power
\(\eta_i=\eta_i^{\mathrm{lo}}\): within each family the localization
and sampling powers are chosen below this fixed positive power.
Thus the full masked transition has the required multiplicity bound,
and the product of its localization, coloring, and decoupling losses
over the fixed \(J\) transitions is \(m^{o_i(1)}\).
The factor \(2^J\) is absorbed in the same loss.
Successive applications of \eqref{eq:source-33} yield
\[
 r w^2\sum_{b\ {\rm kept}}|c_b|^4
 \leq k\,m^{o_i(1)}
          \delta^2\sum_{v\in\mathcal R}|d_v|^4
 \lesssim k\,m^{o_i(1)}\delta^2\#\mathcal R .
\]
Since \(|c_b|^2\asymp h_1k\) on the kept leaves, division by
\(h_1k\) proves the second term in \eqref{ext:mass-bound}.
\end{proof}

\begin{proof}[Completion of the proof of Theorem~\ref{pkt:main}]
For each terminal bin, use its entire kept coefficient array
as a single atom in the definition of \(A\). Equations
\eqref{ext:capacity-bound} and \eqref{ext:mass-bound} imply
\begin{align*}
 \mathcal A(c|_{\mathrm{kept}})^3
 &\leq r\sum_I G(c_I|_{\mathrm{kept}})^3\\
 &\leq m^{10\kappa+\zeta+o_i(1)}
     \delta^2\#\mathcal R\,K(d)^{1/2}\\
 &\hspace{10mm}\cdot
   m^{1-s}h_1^{1/2}
      \min\bigl(m^{-2(1-s)},h_1^{-1}\bigr).
\end{align*}
For every positive \(h_1\),
\[
 h_1^{1/2}\min\bigl(m^{-2(1-s)},h_1^{-1}\bigr)
                      \leq m^{-(1-s)}.
\]
Also \(1\leq|d_v|\leq2\) gives
\(G(d)^3\asymp\delta^2\#\mathcal R\,K(d)^{1/2}\).
The cubed amplification of the kept output is consequently
at most \(m^{10\kappa+\zeta+o_i(1)}\).

By Lemmas~\ref{ext:leaves} and \ref{ext:represented}, this same
output has amplification at least \(m^{\gamma-o_i(1)}\).
Our choice \(\zeta<\gamma-10\kappa\), followed by sufficiently
large \(i\) and then sufficiently large \(m\) within that
family, is a contradiction. Thus the assumption
\(\gamma>10\kappa\) is impossible, which proves the propagation
bound and its uniform finite-family formulation.
\end{proof}
==== END Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//06-extremal.tex ====

==== BEGIN Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//07-global.tex ====
\section{Passage to a global estimate}\label{sec:global}

The packet theorem first gives a cubic estimate on widely separated
balls, with an arbitrarily small power loss.  We remove that loss by
covering a finite list of peaks with sparse families of balls.  The
initial finiteness of the peak list follows from an elementary
fourth-power estimate for the original sphere measure.

The choices have a fixed order: first the separation exponent \(C_0\),
then the desired distributional error \(\alpha\), and finally the
power loss \(\epsilon\).  The covering radii may depend on \(\alpha\);
the sparse estimate must therefore permit \(\epsilon\) to be chosen
after those radii have been bounded.

This passage from local estimates to a global bound follows the
sparse-family strategy of Tao~\cite{Tao1999Epsilon}; compare the
bounded-data formulation in
\cite[arXiv:1012.3760v3, Appendix, Lemmas~A1--A2]{BourgainGuth2011}.
We prove the required covering and cubic estimates below.

\subsection{A sparse estimate for graph patches}

Normalize first by $\|g\|_{L^\infty(S^2,\sigma)}=1$. Partition the
sphere into finitely many measurable pieces contained in the graph patches
of Section~\ref{sec:packets}. Subdivide these pieces, if necessary, so
that on each piece one of the terminal windows $\chi_1^{\nu_0}$ is
bounded below by a positive constant. Such a subdivision exists because
the windows form a square partition of unity with bounded overlap.
In the rotated coordinates of one piece, write its extension as
\begin{equation}\label{glob:patch-extension}
 F_0(y)=F_0(x,t)
 =\int_{\R^2}f(\xi)\chi_1^{\nu_0}(\xi)
       e^{2\pi i(x\cdot\xi+t\phi(\xi))}\,d\xi,
 \qquad \|f\|_\infty+\|f\|_2\le C.
\end{equation}
Here $f$ is supported in a fixed compact frequency patch. The smooth
surface-area density and division by $\chi_1^{\nu_0}$ have been absorbed
into $f$. All constants in this section may depend on these finitely many
fixed choices, but are independent of the original bounded measurable
data. In particular, no differentiability of $f$ is assumed.

\Needspace{4\baselineskip}
\begin{lemma}[Sparse cubic estimate]\label{glob:sparse-cubic}
There are constants $C_0>1$ and $\mu_0>0$ such that the following holds.
Let $0<\mu\le\mu_0$, let $R\ge\mu^{-1}$, and let $\mathcal B$ be a
collection of at most $\mu^{-8}$ balls of radius $R$ whose distinct
centers have distance at least $(R/\mu)^{C_0}$. Then, for every
$\epsilon>0$,
\begin{equation}\label{eq:source-42}
             \sum_{B\in\mathcal B}\int_B |F_0(y)|^3\,dy
                    \le C_\epsilon R^\epsilon.
\end{equation}
The constants are uniform over the data in
\eqref{glob:patch-extension} and all translations of the balls.
\end{lemma}

\begin{proof}
Choose a dyadic $N$ with $4R\le N^2<16R$, and put $\delta=N^{-1}$.
For a ball with center $b_B\in\R^3$, set
$a_B=b_B-(0,0,N^2/2)$. The coordinates
$y=a_B+N^2(z,t)$ place that ball inside
$\{|z|\le 1/4,\ 1/4\le t\le3/4\}$.
We shall also translate the physical sampling lattice by
\[
 u\in Q=[0,L_{\mathrm f}^{-1})^2\times[0,1).
\]
For each such $u$, define
\[
 f_{B,u}(\xi)=f(\xi)
       e^{2\pi i(a_B+u)\cdot(\xi,\phi(\xi))}.
\]
These functions have the same $L^\infty$ and $L^2$ bounds as $f$.
All estimates that follow are uniform in \(u\) and in the ball
locations.  Those locations enter only through this modulation of the
input; the retained packet coordinates stay in the same bounded
normalized region.

At factor $1$, retain the analysis coefficients of $f_{B,u}$ with
angular windows meeting the fixed frequency support and positions
\[
 z\in L_{\mathrm f}^{-1}\delta\Z^2,\qquad |z|\le Z_0,
\]
where $Z_0$ is a sufficiently large fixed constant. Denote the resulting
finite array by $d^{B,u}$. For each angle $\nu$, Fourier-series
Parseval gives
\begin{equation}\label{glob:angle-mass}
 \sum_z|d_{\nu,z}^{B,u}|^2
 \le (L_{\mathrm f}\delta)^2
      \int |f(\xi)\chi_\delta^\nu(\xi)|^2\,d\xi
 \le C\delta^4.
\end{equation}
The velocity centers $U_\nu=-\nabla\phi(\nu)$ are separated by at
least $c\delta$, since the Hessian of the extended phase is uniformly
definite. An ellipse $E$ whose radii satisfy $L\ge w\ge\delta$ thus
contains at most $C|E|\delta^{-2}$ such centers: its $O(\delta)$
enlargement has area comparable to $|E|$. Ignoring the positional
restriction in the definition of capacity, we obtain from
\eqref{eq:source-29} and \eqref{glob:angle-mass}
\begin{equation}\label{glob:initial-capacity}
 K(d^{B,u})
 \le C\delta^4\sup_E\frac{|E|}{W(E)}
 \le C\delta^4,
\end{equation}
because $|E|/W(E)$ is a constant multiple of $(w/L)^\kappa\le1$.

We next prove the estimate that saves the number of balls:
\begin{equation}\label{glob:joint-mass}
       \sum_{B\in\mathcal B}\sum_v|d_v^{B,u}|^2
                         \le C\delta^2.
\end{equation}
Let $T_B$ be the retained analysis operator acting on the unmodulated
function $f$, so that $T_Bf=d^{B,u}$, and let $T=(T_B)_{B\in\mathcal B}$.
The frame identity shows that each $T_B$ has operator norm at most
$L_{\mathrm f}\delta$. Hence the diagonal blocks of $TT^*$ have
operator norm at most $C\delta^2$.

An entry in an off-diagonal block is, up to complex conjugation, an
integral of the form
\begin{equation}\label{glob:gram-entry}
 \int A(\xi)e^{2\pi i(q\cdot\xi+s\phi(\xi))}\,d\xi,
 \qquad
 (q,s)=a_B-a_{B'}+N^2(z-z',0),
\end{equation}
where $A$ is a product of two $\delta$-scale windows supported in a fixed
compact set. The common translation $u$ cancels. The position offsets
have size $O(N^2)=O(R)$, so, for $B\ne B'$ and sufficiently small
$\mu_0$,
\[
 H:=|(q,s)|\ge c(R/\mu)^{C_0}.
\]
The amplitude and its derivatives have bounds polynomial in $R$.
There is a fixed exponent $a\ge0$, independent of $\kappa$, $\epsilon$,
and the ball locations, for which
\begin{equation}\label{glob:gram-decay}
 \left|\int A(\xi)e^{2\pi i(q\cdot\xi+s\phi(\xi))}\,d\xi\right|
                       \le C R^aH^{-1/2}.
\end{equation}
For completeness, write $\Psi(\xi)=q\cdot\xi+s\phi(\xi)$.
If $|s|$ is sufficiently small compared with $H$, then
$|\nabla\Psi|\ge cH$ on the fixed support and integration by parts
proves \eqref{glob:gram-decay}. Otherwise $|s|\ge cH$. Uniform
definiteness of the Hessian implies
\[
 |\nabla\Psi(\xi)-\nabla\Psi(\xi')|
                         \ge cH|\xi-\xi'|.
\]
Consequently the set $|\nabla\Psi|\le2H^{3/4}$ has area at most
$CH^{-1/2}$. Insert a smooth cutoff to this set. Its contribution is
bounded by $CH^{-1/2}\|A\|_\infty$. On the complementary region,
integrate by parts with the vector field
$\nabla\Psi/|\nabla\Psi|^2$. This field has size $O(H^{-3/4})$
and divergence $O(H^{-1/2})$; the derivative of the cutoff has size
$O(H^{1/4})$. One integration by parts therefore gives
$CR^aH^{-1/2}$, after increasing the fixed exponent $a$ to cover the
amplitude derivatives. This proves \eqref{glob:gram-decay}.

There are $O(N^2)$ retained angular labels and $O(N^2)$ retained
position labels per ball. Thus each off-diagonal row of $TT^*$ has at
most $C\mu^{-8}R^2$ entries. By \eqref{glob:gram-decay}, its absolute
row sum is at most
\[
 C\mu^{-8}R^{a+2}(R/\mu)^{-C_0/2}.
\]
Choose $C_0$ so large that $C_0/2\ge a+3$ and $C_0/2\ge8$.
This row sum is then $O(R^{-1})=O(\delta^2)$. The same holds for
columns, since the matrix is Hermitian. The Schur bound, together with
the diagonal block bound, proves $\|TT^*\|\le C\delta^2$ and hence
\eqref{glob:joint-mass}. This choice of $C_0$ is made once and does
not depend on the power loss $\epsilon$.

Apply the single-step packet map from factor $1$ to factor $N$ to
$d^{B,u}$. Retain only the terminal angle $\nu_0$ and positions in a
fixed bounded region containing $|z|\le1$, and denote the output by
$c^{B,u}$. At this terminal factor,
\[
       \theta=1,\qquad r=w=N^{-2},\qquad rw^2=N^{-6}.
\]
If the initial positions had not been truncated, the frame identity
would give exactly the samples
\[
       F_0\bigl(a_B+u+N^2(z,t_I)\bigr).
\]
The omitted positions contribute $O_M(N^{-M})$, uniformly on the
retained terminal region, for every fixed $M$.
Indeed, choose $Z_0$ larger than a fixed bound for the retained terminal
positions and the velocities on the frequency patch. If $|z'|>Z_0$,
then the phase in the synthesis--analysis matrix has, on a parent
$\delta$-cap, nonvanishing gradient of size at least
$cN^2(1+|z'|)$. Rescaling that cap and integrating by parts gives a
matrix bound $C_L(1+N|z'|)^{-L}$ for every $L$. There are $O(N^2)$
relevant parent angular labels, and their position mesh is comparable
to $N^{-1}$. Thus, for $L>1$, the squared matrix bounds sum to
\[
 \sum_{\nu\text{ relevant}}\sum_{z':\,|z'|>Z_0}(1+N|z'|)^{-2L}
                         \le C_LN^{4-2L}.
\]
Write $\widetilde d=\mathsf P_{1,[0,1)}f_{B,u}$ for the full initial
analysis array. The frame identity gives
$\sum_v|\widetilde d_v|^2\le C\delta^2$, so Cauchy--Schwarz bounds
each omitted contribution by $C_LN^{1-L}$. Taking $L\ge M+1$ proves
the claimed error, using only smoothness of the windows and the phase.

The supports just described belong to one fixed polynomial complexity
range. Theorem~\ref{pkt:main}, \eqref{eq:source-30}, and
\eqref{glob:initial-capacity} therefore give, for every fixed
$0<\kappa<1/10$ and $\eta>0$,
\begin{align}
 N^{-6}\sum_v|c_v^{B,u}|^3
 &\le \mathcal A(c^{B,u})^3
 \le C_{\kappa,\eta}N^{10\kappa+\eta}G(d^{B,u})^3 \notag\\
 &\le C_{\kappa,\eta}N^{10\kappa+\eta}
                 \delta^4\sum_v|d_v^{B,u}|^2.
 \label{glob:sparse-sampling}
\end{align}
In the last line we used $G(d)^3=\delta^2(\sum|d_v|^2)K(d)^{1/2}$
at the initial factor. Summing \eqref{glob:sparse-sampling} and using
\eqref{glob:joint-mass}, the physical sample sum is at most
\[
 C_{\kappa,\eta}N^6N^{10\kappa+\eta}\delta^4\delta^2
                    =C_{\kappa,\eta}N^{10\kappa+\eta}.
\]
The sampling errors are harmless: there are at most
$C\mu^{-8}N^6\le CN^{22}$ samples, and $M$ may be chosen arbitrarily
large. The inequality $|a+b|^3\le4(|a|^3+|b|^3)$ absorbs their total
cubed contribution.

The terminal physical lattice is
$\Lambda=L_{\mathrm f}^{-1}\Z^2\times\Z$, with fundamental cell $Q$.
The retained lattice points include each $\lambda\in\Lambda$ for which
$\lambda+Q$ meets $B-a_B$, since the ball lies in the interior
normalized region fixed above.
Consequently,
\[
 \sum_{B\in\mathcal B}\int_B|F_0(y)|^3\,dy
 \le\int_Q\sum_{B\in\mathcal B}\sum_{v\text{ retained}}
   |F_0(a_B+u+N^2(z_v,t_{I_v}))|^3\,du.
\]
Apply the uniform sampled bound to the right-hand side. Integration
over $Q$ tiles physical space by $\Lambda$, so it introduces no
$N$-dependent density factor. Finally choose
$\kappa$ and $\eta$ sufficiently small that
$(10\kappa+\eta)/2\le\epsilon$. Since $N^2\asymp R$, this proves
\eqref{eq:source-42}.
\end{proof}

\subsection{A finite list of peaks}

We now use the geometry of the original sphere to obtain a global
estimate at a larger exponent.  The convolution calculation keeps the
normalization of surface measure explicit and applies directly to
bounded measurable complex data.

\begin{lemma}[An elementary fourth-power bound]\label{glob:fourth-power}
For every bounded measurable $h:S^2\to\C$,
\begin{equation}\label{glob:l4-bound}
       \|Eh\|_{L^4(\R^3)}^4\le32\pi^3\|h\|_\infty^4.
\end{equation}
Moreover $Eh$ and its first derivatives are uniformly bounded by
constants times $\|h\|_\infty$.
\end{lemma}

\begin{proof}
The convolution of surface measure with itself has density
\begin{equation}\label{glob:sphere-convolution}
       (\sigma*\sigma)(y)=\frac{2\pi}{|y|}
                               \mathbf1_{\{0<|y|<2\}}.
\end{equation}
To verify the constant, use rotational invariance and fix
$\omega\in S^2$. Under surface area measure in the second variable,
$s=\omega\cdot\nu$ has density $2\pi$ on $[-1,1]$, and
$|\omega+\nu|=(2+2s)^{1/2}$. Thus, for a radial test function $q$,
\[
 \int q(|y|)\,d(\sigma*\sigma)(y)
     =8\pi^2\int_0^2q(r)r\,dr,
\]
which is exactly the integral against the radial density in
\eqref{glob:sphere-convolution}. In particular,
\[
       \|\sigma*\sigma\|_2^2
       =\int_0^2\frac{4\pi^2}{r^2}\,4\pi r^2\,dr
       =32\pi^3.
\]
Let $\lambda=h\sigma$, a finite complex measure. Its convolution
satisfies the total-variation domination
\[
       |\lambda*\lambda|\le\|h\|_\infty^2(\sigma*\sigma),
\]
so it has an $L^2$ density of squared norm at most
$32\pi^3\|h\|_\infty^4$. The function $(Eh)^2$ is the Fourier
transform, with the positive phase convention, of this convolution.
Plancherel proves \eqref{glob:l4-bound}. Finally, differentiating the
defining integral is justified by boundedness of the frequency support
and gives, for example,
\[
       \|Eh\|_\infty\le4\pi\|h\|_\infty,
       \qquad \|\nabla Eh\|_\infty\le8\pi^2\|h\|_\infty.
\]
\end{proof}

Each function $F_0$ in \eqref{glob:patch-extension}, in its rotated
coordinates, is the extension of a uniformly bounded density supported
on one of the original sphere pieces. Lemma~\ref{glob:fourth-power}
therefore supplies uniform $L^4$, $L^\infty$, and Lipschitz bounds for
$F_0$.

\begin{lemma}[Initial peak count]\label{glob:initial-peaks}
For $0<\mu\le1$, the number of unit lattice cubes $Q'$ satisfying
$\sup_{Q'}|F_0|\ge\mu$ is at most $C\mu^{-7}$.
One may select a point $x_{Q'}\in Q'$ in each such cube with
$|F_0(x_{Q'})|\ge\mu/2$.
\end{lemma}

\begin{proof}
Take the unit cubes to be half-open, and select such a point in every
cube of an arbitrary finite subcollection. A uniform Lipschitz bound
shows that $|F_0|\ge\mu/4$ on the ball of radius $c\mu$ about each
selected point, where $c>0$ is fixed and small. Each ball contributes
at least $c'\mu^7$ to the integral of $|F_0|^4$. These balls have
bounded overlap, since their radii are bounded and there is one center
in each of distinct unit lattice cubes. Lemma~\ref{glob:fourth-power}
bounds the size of the finite subcollection by $C\mu^{-7}$. Since
the same bound holds for every finite subcollection, the whole set of
such cubes is finite and has the asserted size.
\end{proof}

\subsection{Sparse covering and reproduction}

The covering lemma converts the initial polynomial peak count into
sparse families whose number is an arbitrarily small power of
\(\mu^{-1}\).  Their radii remain polynomial in \(\mu^{-1}\), with an
exponent fixed once \(\alpha\) is chosen.  This is the information
needed to apply the sparse cubic estimate and then choose its loss.

\begin{lemma}[Sparse covering]\label{glob:sparse-covering}
Fix $C_0>1$ and $A>0$. For every $0<\alpha<1$ there is a constant
$C_\alpha$ with the following property. If $\mu>0$ is sufficiently
small and $X$ is a finite indexed list of at most $A\mu^{-7}$ points
in $\R^3$, then $X$ is covered by at most
$C_\alpha\mu^{-\alpha/2}$ families of balls. Each family has a common
radius $\rho$ with
\begin{equation}\label{glob:covering-radii}
           \mu^{-1}\le\rho,\qquad 2\rho\le\mu^{-C_\alpha},
\end{equation}
and its distinct centers are separated by at least
$(2\rho/\mu)^{C_0}$. Each family has at most $\#X$ balls.
\end{lemma}

\begin{proof}
Put $\beta=\alpha/4$, choose an integer $J>8/\beta$, and set
$D=2C_0+2$. For sufficiently small $\mu$, we have $\#X\le\mu^{-8}$.
Define
\[
 R_0=\mu^{-1},\qquad R_{j+1}=(R_j/\mu)^D
              \quad(0\le j<J),
 \qquad n_j(x)=\#\bigl(X\cap B(x,R_j)\bigr),
\]
where balls are open and counts include the multiplicities of the
indexed list. For every point $x$ of the list, some $j<J$ satisfies
\begin{equation}\label{glob:slow-growth}
                n_{j+1}(x)\le\mu^{-\beta}n_j(x).
\end{equation}
Otherwise $n_J(x)>\mu^{-J\beta}n_0(x)\ge\mu^{-J\beta}$ would
contradict $\#X\le\mu^{-8}$.

Assign each point one such index $j$, and group the points further
according to $2^k\le n_j(x)<2^{k+1}$. There are
$O_\alpha(\log(1/\mu))$ groups. In each group choose a maximal
disjoint subcollection of the radius-$R_j$ balls centered at its
points. The triples of the selected balls cover that group.

Join two selected centers by an edge when their distance is less than
$R_{j+1}/2$. For a selected center $a$, the radius-$R_j$ balls
centered at its neighbors are disjoint and lie in $B(a,R_{j+1})$,
since $R_j\le R_{j+1}/2$. Each such ball captures at least $2^k$
indices of the whole list, whereas \eqref{glob:slow-growth} gives
$n_{j+1}(a)<2^{k+1}\mu^{-\beta}$. The graph therefore has degree
at most $2\mu^{-\beta}$. Greedy coloring splits the selected balls
into $O(\mu^{-\beta})$ colors. Including all the groups, the number
of families is at most
\[
 C_\alpha\mu^{-\beta}\log(1/\mu)
                         \le C_\alpha\mu^{-\alpha/2}.
\]

Use radius $\rho=3R_j$ for each covering family. Centers in one
color have distance at least $R_{j+1}/2$, and, for sufficiently small
$\mu$,
\[
       \frac{R_{j+1}}2
       =\frac12(R_j/\mu)^{2C_0+2}
       \ge(6R_j/\mu)^{C_0}
       =(2\rho/\mu)^{C_0}.
\]
Finally, write $R_j=\mu^{-a_j}$, so that
$a_0=1$ and $a_{j+1}=D(a_j+1)$. The finite number $J$ depends only
on $\alpha$ and $C_0$. Enlarging $C_\alpha$ to at least
$a_{J-1}+1$ gives $6R_j\le\mu^{-C_\alpha}$ for small $\mu$,
which proves \eqref{glob:covering-radii}. The selected balls are
centered at points of the list, so every family has at most $\#X$
balls.
\end{proof}

\Needspace{4\baselineskip}
\begin{proposition}[Distributional bound]\label{glob:distribution}
For every $0<\alpha<1$ and all sufficiently small $\mu>0$,
\begin{equation}\label{glob:distribution-bound}
       |\{y\in\R^3:|F_0(y)|>\mu\}|
                            \le C_\alpha\mu^{-3-\alpha}.
\end{equation}
\end{proposition}

\begin{proof}
Let $X=(x_{Q'})$ be the finite list supplied by
Lemma~\ref{glob:initial-peaks}. It suffices to prove
$\#X\le C_\alpha\mu^{-3-\alpha}$, since the level set in
\eqref{glob:distribution-bound} is contained in the corresponding
unit cubes.

The Fourier transform of $F_0$, as a tempered distribution, is
supported in a fixed compact set. Choose a Schwartz function
$\psi$ whose Fourier transform is $1$ on a neighborhood of that
set. Then $F_0=\psi*F_0$; the convolution is absolutely convergent
and the identity holds pointwise because $F_0$ is bounded and
continuous. Hölder's inequality yields
\begin{equation}\label{glob:reproduction}
 |F_0(x)|^3
 \le\|\psi\|_1^2
        \int_{\R^3}|\psi(x-y)|\,|F_0(y)|^3\,dy.
\end{equation}
Apply Lemma~\ref{glob:sparse-covering}, with the separation exponent
from Lemma~\ref{glob:sparse-cubic}, and assign every index of $X$
to one ball in one covering family. Suppose the assigned covering
ball has radius $\rho$. Truncating the integral in
\eqref{glob:reproduction} to $|x-y|\le\rho$ costs at most
\[
       C\|F_0\|_\infty^3
                \int_{|z|>\rho}|\psi(z)|\,dz
                    \le C\rho^{-4}\le C\mu^4.
\]
Since $|F_0(x)|\ge\mu/2$, this error is absorbed into the
left-hand side for sufficiently small $\mu$. The truncated region
lies in the doubled assigned covering ball.

There is a uniform bound
\begin{equation}\label{glob:kernel-overlap}
           \sup_{y\in\R^3}\sum_{x\in X}|\psi(x-y)|\le C_\psi.
\end{equation}
Indeed, at most one selected point lies in each unit lattice cube,
and Schwartz decay bounds the sum by the convergent lattice sum of
a sufficiently high inverse power of distance. In particular no
pairwise separation assumption on the selected points is needed.

Let $X_{\mathcal F}$ denote the indices assigned to one covering
family $\mathcal F$, of common radius $\rho$. Summing the truncated
form of \eqref{glob:reproduction} and using
\eqref{glob:kernel-overlap}, we obtain
\[
       \mu^3\#X_{\mathcal F}
       \le C\sum_{B\in\mathcal F}\int_{2B}|F_0(y)|^3\,dy
       \le C_\epsilon(2\rho)^\epsilon.
\]
For the last inequality, the doubled balls satisfy all the
hypotheses of Lemma~\ref{glob:sparse-cubic}: their radius is at
least $\mu^{-1}$, their separation is the required one, and their
number is at most $\#X\le\mu^{-8}$ after reducing the threshold
for $\mu$.

Fix $\alpha$ first, and then choose $\epsilon>0$ so small that
$C_\alpha\epsilon\le\alpha/2$, using the exponent in
\eqref{glob:covering-radii}. Thus
\[
       \mu^3\#X_{\mathcal F}\le C_\alpha\mu^{-\alpha/2}.
\]
There are at most $C_\alpha\mu^{-\alpha/2}$ families. Each index
was assigned exactly once, and summing proves
$\#X\le C_\alpha\mu^{-3-\alpha}$, as required.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Fix $p>3$ and choose $0<\alpha<\min\{1,p-3\}$.
Let $\mu_*>0$ be smaller than the thresholds required above for this
choice of $\alpha$.
By the layer-cake identity and Proposition~\ref{glob:distribution},
the contribution of sufficiently small thresholds is bounded by
\[
 p\int_0^{\mu_*}\mu^{p-1}
      |\{|F_0|>\mu\}|\,d\mu
 \le C_{p,\alpha}\int_0^{\mu_*}\mu^{p-4-\alpha}\,d\mu
 <\infty.
\]
For larger thresholds, the uniform $L^4$ estimate gives
$|\{|F_0|>\mu\}|\le C\mu^{-4}$, and the level set is empty
above the uniform $L^\infty$ bound. Their contribution is therefore
finite as well. We conclude that $\|F_0\|_p\le C_p$.

There are only finitely many graph pieces in
\eqref{glob:patch-extension}. Rotations preserve Lebesgue measure,
so the $L^p$ triangle inequality gives $\|Eg\|_p\le C_p$ under
the normalization $\|g\|_\infty=1$. Multiplication by
$\|g\|_\infty$ proves the result for arbitrary nonzero bounded
measurable complex data. When the essential supremum is zero, the
extension is identically zero. All constructions and estimates
depend only on the almost-everywhere class of the data, which
completes the assertion of the theorem.
\end{proof}
==== END Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//07-global.tex ====

==== BEGIN Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//08-consequences.tex ====
\section{Spherical consequences}\label{sec:consequences}

The global sphere estimate has now been proved. We give the complete
transfer to the translated-tube maximal operator defined in the
introduction, retaining the weighted tube estimate needed for its
strong operator norm. This uses sphere factorization in addition to
Theorem~\ref{thm:main}.

\subsection{The Kakeya maximal transfer}

\begin{proof}[Proof of Corollary~\ref{cor:sphere-kakeya-maximal}]
Sphere factorization, due to Bourgain~\cite{Bourgain1991Besicovitch},
upgrades Theorem~\ref{thm:main} to
\(\|Eg\|_{L^q(\R^3)}\lesssim_q\|g\|_{L^q(S^2)}\) for every \(q>3\);
see~\cite[Section~1.1, (1.1)--(1.2)]{Buschenhenke2024} for the
weak-type factorization and interpolation in the sphere case.
Fix \(3<q<4\) and put $r=q/2$. Consider unit tubes
$T_j=T_\delta(x_j,\omega_j)$ in $\delta$-separated directions.
Choose smooth nonnegative bumps $b_j$, bounded by $1$, supported on
disjoint caps of radius $c\delta$ about $\omega_j$, and equal to $1$
on the concentric caps of radius $c\delta/2$. Here $c>0$ is a fixed
small constant. The modulation
\[
 h_j(\zeta)=e^{-2\pi i\delta^{-2}x_j\cdot\zeta}b_j(\zeta)
\]
translates the extension to the center of $\delta^{-2}T_j$.
We have $\|h_j\|_q^q\lesssim\delta^2$ and
$|Eh_j|\gtrsim\delta^2$ on that dilated tube. To see the lower bound,
write $y-\delta^{-2}x_j=s\omega_j+v$, where
$|s|\le\delta^{-2}/2$, $v\perp\omega_j$, and $|v|\le\delta^{-1}$.
On the cap, the phase after removing its value at $\omega_j$ has
magnitude at most $C(c+c^2)$. Taking $c$ sufficiently small makes
the real part of the remaining integrand a fixed positive fraction
of $b_j$, whose integral is comparable to $\delta^2$.

For arbitrary weights $a_j\ge0$ and independent random signs
$\varepsilon_j$, Khintchine's inequality and the diagonal estimate give
\[
 \delta^{2q-6}\Big\|\sum_j a_j1_{T_j}\Big\|_r^r
 \lesssim_q
 \mathbb E\Big\|E\Big(\sum_j\varepsilon_j a_j^{1/2}h_j\Big)\Big\|_q^q
 \lesssim_q\delta^2\sum_j a_j^r.
\]
The factor $\delta^{-6}$ comes from $y=\delta^{-2}x$, while the
last inequality uses the disjoint cap supports. Taking the $r$th
root gives the weighted tube estimate
\[
 \Big\|\sum_j a_j1_{T_j}\Big\|_{L^r(\R^3)}
 \lesssim_q \delta^{12/q-4}
       \Big(\delta^2\sum_j a_j^r\Big)^{1/r}.
\]

Set $p=r'=q/(q-2)$. For nonnegative $f\in L^p(\R^3)$ and $c_j\ge0$,
the normalization $|T_j|=\pi\delta^2$ gives
\[
 \delta^2\sum_j c_j\frac1{|T_j|}\int_{T_j}f
 =\frac1\pi\int f\sum_jc_j1_{T_j}
 \lesssim_q\delta^{12/q-4}\|f\|_p
                \Big(\delta^2\sum_jc_j^r\Big)^{1/r}.
\]
Duality for the discrete measure assigning mass $\delta^2$ to each
direction bounds the $p$-norm of these tube averages. The bound is
uniform in all tube locations, so each $T_j$ may independently
approximate the supremum defining $K_\delta f(\omega_j)$.

Choose a maximal $\delta$-separated net on $S^2$, with angular cells
of area comparable to $\delta^2$. Every direction in a cell is within
$\delta$ of its center $\omega_j$. For $\delta<1/4$, the corresponding
tubes satisfy
\[
 T_\delta(a,\omega)\subset
 T_{2\delta}(a-\omega_j/4,\omega_j)
 \cup T_{2\delta}(a+\omega_j/4,\omega_j).
\]
Indeed, the transverse displacement is at most $3\delta/2$ and the
axial displacement at most $1/2+\delta^2<3/4$. Thus
$K_\delta f(\omega)\le8K_{2\delta}f(\omega_j)$.
Sphere packing splits the net into a fixed number of
$2\delta$-separated families. Apply the discrete bound at width
$2\delta$ to each family and sum over the angular cells. This gives
Bourgain's strong maximal transfer, in the form stated in
\cite[Theorem~3.3, pp.~10--11]{MattilaFourierNotes}:
\[
 \|K_\delta f\|_{L^p(S^2)}
 \lesssim_q\delta^{12/q-4}\|f\|_{L^p(\R^3)},
 \qquad p=\frac{q}{q-2}.
\]
For $\delta\ge1/4$, the same bound follows directly from H\"older's
inequality. Interpolating with
\(\|K_\delta f\|_\infty\le\|f\|_\infty\), at parameter \(p/3\), yields
\[
 \|K_\delta f\|_{L^3(S^2)}
 \lesssim_q\delta^{-4(q-3)/(3(q-2))}\|f\|_{L^3(\R^3)}.
\]
Choosing \(q>3\) sufficiently close to \(3\) proves the claim.
\end{proof}

This deduction starts from the spherical conclusion of
Theorem~\ref{thm:main} and uses sphere-specific factorization; it does
not invoke the general-surface diagonal companion or the independent
maximal theorem.


\subsection{The open mixed-norm range on the sphere}

The diagonal sphere estimates also give the strict mixed-norm range.
For $1<a\le\infty$ and $3<b<\infty$, write $a'=a/(a-1)$ when
$a<\infty$ and $a'=1$ when $a=\infty$. Then
\begin{equation}\label{eq:sphere-mixed-norm}
 \|Eg\|_{L^b(\R^3)}\le C_{a,b}\|g\|_{L^a(S^2)},
 \qquad b>2a',
\end{equation}
for every complex $g\in L^a(S^2)$. The defining integral for $E$
remains meaningful because $\sigma(S^2)<\infty$ implies
$L^a(S^2)\subset L^1(S^2)$. Thus the conclusion covers a larger
class of input densities, with an output exponent depending on
their integrability.

Indeed, for $1<a\le b$, set $r=1+b/a'>3$ and $\vartheta=r/b\le1$.
Interpolate the diagonal $L^r(S^2)\to L^r(\R^3)$ estimate from
sphere factorization with the elementary $L^1(S^2)\to L^\infty(\R^3)$
estimate. The identities
\[
 \frac1b=\frac{\vartheta}{r},\qquad
 \frac1a=1-\vartheta+\frac{\vartheta}{r}
\]
give \eqref{eq:sphere-mixed-norm}; when $a=b$, the diagonal estimate
itself suffices. For $b<a<\infty$, use the diagonal estimate at $b$
and H\"older's inequality on $S^2$. The case $a=\infty$ is
Theorem~\ref{thm:main}. This argument uses the strict inequality
$b>2a'$ to choose $r>3$ and makes no assertion on the scaling line
$b=2a'$.
==== END Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//08-consequences.tex ====

==== BEGIN Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//adjacency-radii.tex ====
\begin{figure}[tb]
\centering
\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,font=\small]
  \draw[gray!65,->] (-.4,0)--(10.7,0) node[right] {time};
  \draw[gray!65,->] (0,-1.55)--(0,1.7) node[above] {position};
  \fill[linkblue!4] (0,-1.2)--(10,-.2)--(10,2.2)--(0,1.2)--cycle;
  \fill[linkblue!17] (0,-.4)--(10,.6)--(10,1.4)--(0,.4)--cycle;
  \draw[linkblue,thick] (0,0)--(10,1);
  \draw[linkblue!50,dashed] (0,-1.2)--(10,-.2);
  \draw[linkblue!50,dashed] (0,1.2)--(10,2.2);
  \draw[linkblue!80] (0,-.4)--(10,.6);
  \draw[linkblue!80] (0,.4)--(10,1.4);
  \draw[densely dotted] (5,-1.15)--(5,1.8);
  \fill (5,.66) circle (2pt)
    node[right=4pt,above=1pt] {\(b\)};
  \fill (5,.30) circle (2pt)
    node[left=4pt,below=1pt] {\(b'\)};
  \node[anchor=west,fill=white,inner sep=2pt] at (7.5,1.75)
    {parent high};
  \node[anchor=west,fill=white,inner sep=2pt] at (7.8,1.02)
    {parent low};
  \node[anchor=north] at (5,-1.28) {one child time bin};
  \node[anchor=south west] at (.6,.15) {parent centerline};

  \node at (5,-2.10) {Position cross-section at the marked bin};
  \begin{scope}[shift={(5,-3.55)},x=1.3cm]
    \filldraw[fill=linkblue!4,draw=linkblue!50,dashed]
      (-3,-.42) rectangle (3,.42);
    \filldraw[fill=linkblue!10,draw=linkblue!65]
      (-1.6,-.30) rectangle (2.4,.30);
    \filldraw[fill=linkblue!17,draw=linkblue!80]
      (-1,-.18) rectangle (1,.18);
    \draw[gray!65,densely dotted] (0,-.90)--(0,.50);
    \draw[linkblue!65,densely dotted] (.4,-.30)--(.4,.30);
    \fill (.4,0) circle (2pt);
    \fill (-.5,0) circle (2pt);
    \node[anchor=north] at (.4,-.48) {\(b\)};
    \node[anchor=north] at (-.5,-.48) {\(b'\)};
    \node[anchor=north,font=\scriptsize,text=gray!80] at (0,-.96)
      {parent center};
    \node at (-2.25,.80) {parent high};
    \draw[linkblue!50] (-2.25,.59)--(-2.55,.42);
    \node at (1.10,.80) {intermediate about \(b\)};
    \draw[linkblue!65] (1.10,.59)--(1.65,.30);
    \node at (-2.20,-.72) {parent low};
    \draw[linkblue!80] (-1.60,-.62)--(-1,-.18);
  \end{scope}
\end{tikzpicture}
\caption{Schematic localization neighborhoods in one position coordinate.
At the marked child time bin, the intermediate neighborhood centered at the
selected low child \(b\) contains every low child of the parent in that bin
and is contained in the parent's high neighborhood. The angular bounds in
the actual neighborhoods are not depicted. These inclusions are used for
counting paths; the analytic evolution uses the exact coefficient maps with
vertex masks.}
\label{fig:adjacency-radii}
\end{figure}
==== END Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//adjacency-radii.tex ====

==== BEGIN Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//capacity-tests.tex ====
\begin{figure}[tb]
\centering
\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,
  axes/.style={gray!65,thin},
  capacityellipse/.style={draw=linkblue,fill=linkblue!8,thick}]
  \foreach \x/\name in {0/U,4.2/V,8.4/X_v} {
    \begin{scope}[shift={(\x,0)}]
      \draw[axes,->] (-1.25,0)--(1.35,0);
      \draw[axes,->] (0,-.85)--(0,1.15);
      \node at (0,-1.15) {\(\name\)-plane};
    \end{scope}
  }
  \begin{scope}[shift={(.12,.20)},rotate=25]
    \draw[capacityellipse] (0,0) ellipse (1.00 and .37);
    \draw[linkblue,dashed] (-1,0)--(1,0);
  \end{scope}
  \begin{scope}[shift={(4.05,.26)},rotate=25]
    \draw[capacityellipse] (0,0) ellipse (1.00 and .37);
    \draw[linkblue,dashed] (-1,0)--(1,0);
  \end{scope}
  \begin{scope}[shift={(8.57,.21)},rotate=25]
    \draw[capacityellipse] (0,0) ellipse (.50 and .185);
    \draw[linkblue,dashed] (-.5,0)--(.5,0);
  \end{scope}
  \node at (.05,1.4) {\(E+u_0\)};
  \node at (4.15,1.4) {\(E+v_0\)};
  \node at (8.45,1.4) {\(M^{-1}E+x_0\)};
  \draw[decorate,decoration={brace,mirror,amplitude=4pt}]
    (-1.1,-1.53)--(5.3,-1.53);
  \node at (2.1,-1.98) {initial test};
  \draw[->,thick] (5.8,.3)--(6.8,.3)
    node[midway,above] {\(V+vU\)};
\end{tikzpicture}
\caption{An elliptic capacity test uses the same shape and orientation
in the two spatial coordinates, with independently chosen centers.
The terminal test keeps the \(U\)-ellipse and contracts the
position ellipse by \(M^{-1}\). The picture depicts testing sets,
not a product assumption on the law; the displayed contraction is
schematic.}
\label{fig:capacity-tests}
\end{figure}
==== END Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//capacity-tests.tex ====

==== BEGIN Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//packet-proof-map.tex ====
\begin{figure}[tb]
\centering
\begin{tikzpicture}[
  >=Stealth,
  proofbox/.style={draw=linkblue!70,fill=linkblue!4,rounded corners=2pt,
    align=center,text width=4.6cm,minimum height=1.05cm,
    inner sep=5pt,font=\small},
  proofarrow/.style={->,thick,draw=linkblue!80}]
  \node[proofbox] (time) at (-3.15,0)
    {First extremum: \(s\)\\\(\gamma_\sigma<\gamma\) for \(\sigma<s\)};
  \node[proofbox] (paths) at (3.15,0)
    {Second extremum: paths per root\\retain paths under later masks};
  \node[proofbox] (law) at (-3.15,-1.75)
    {Control conditional time-bin\\counts and probabilities};
  \node[proofbox] (leaves) at (3.15,-1.75)
    {Select terminal coefficients\\control incoming path counts};
  \node[proofbox] (capacity) at (-3.15,-3.5)
    {Geometric propagation\\bound output capacity};
  \node[proofbox] (mass) at (3.15,-3.5)
    {Exact maps and refined decoupling\\second and fourth moments};
  \node[proofbox,text width=5.2cm] (end) at (0,-5.3)
    {Capacity--mass cancellation\\\(\gamma\le10\kappa\)};
  \draw[proofarrow] (time)--(paths);
  \draw[proofarrow] (paths)--(leaves);
  \draw[proofarrow] (leaves)--(law);
  \draw[proofarrow] (law)--(capacity);
  \draw[proofarrow] (leaves)--(mass);
  \draw[proofarrow] (capacity.south)--(end.north west);
  \draw[proofarrow] (mass.south)--(end.north east);
\end{tikzpicture}
\caption{The two extrema in the packet proof. Here \(s\) is the infimum
of the terminal time-support exponents permitting amplification approaching
\(\gamma\). Graph paths run from initial packet indices (roots) to
terminal indices (leaves). Minimizing the limiting exponent of the
path count per original root
forces later coordinate selections to retain enough paths whenever
they preserve nearly maximal amplification. Uniform counting on a
selected path set supplies the time law needed for geometric
propagation. The second- and fourth-moment estimates concern the
exact masked coefficient maps.}
\label{fig:packet-proof-map}
\end{figure}
==== END Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//packet-proof-map.tex ====

==== BEGIN Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//paper.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{Elliptic capacity propagation and Fourier restriction to the sphere}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\input{sections/00-introduction}
\tableofcontents
\input{sections/01-geometric}
\input{sections/02-entropy}
\input{sections/03-projection}
\input{sections/04-aspect}
\input{sections/05-packets}
\input{sections/06-extremal}
\input{sections/07-global}
\input{sections/08-consequences}
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//paper.tex ====

==== BEGIN Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//preamble.tex ====
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\newtheorem{lemma}[theorem]{Lemma}
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==== END Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//preamble.tex ====

==== BEGIN Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//references.bib ====

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}

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@misc{Sawyer2026TestingComparison,
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@article{Shmerkin2022Packing,
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  year = {2022},
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}

@misc{SinghSingh2026Refined,
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  archivePrefix = {arXiv},
  primaryClass = {math.CA},
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@incollection{Stein1979,
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@article{Tao2003,
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}

@article{Tao2010Entropy,
  author = {Tao, Terence},
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@book{Tao2011Measure,
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}

@article{TaoVargasVega1998,
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}

@article{Tomas1975,
  author = {Tomas, Peter A.},
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@incollection{Tomas1979,
  author = {Tomas, Peter A.},
  title = {Restriction theorems for the {Fourier} transform},
  booktitle = {Harmonic Analysis in Euclidean Spaces, Part 1},
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  url = {https://www.ams.org/books/pspum/035.1/},
  note = {\href{https://www.ams.org/books/pspum/035.1/}{Publisher information}}
}

@article{Wang2022Brooms,
  author = {Wang, Hong},
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  volume = {171}, number = {8}, pages = {1749--1822}, year = {2022},
  eprint = {1802.04312v2}, archivePrefix = {arXiv},
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  note = {Preprint version: \href{https://arxiv.org/abs/1802.04312v2}{arXiv:1802.04312v2}}
}

@misc{WangWu2022,
  author = {Wang, Hong and Wu, Shukun},
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  year = {2022}, eprint = {2210.03878v2}, archivePrefix = {arXiv},
  primaryClass = {math.CA},
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  note = {Preprint version: \href{https://arxiv.org/abs/2210.03878v2}{arXiv:2210.03878v2}}
}

@misc{WangWu2024,
  author = {Wang, Hong and Wu, Shukun},
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  year = {2024}, eprint = {2411.08871v3}, archivePrefix = {arXiv},
  primaryClass = {math.CA},
  url = {https://arxiv.org/abs/2411.08871v3},
  note = {Preprint version: \href{https://arxiv.org/abs/2411.08871v3}{arXiv:2411.08871v3}}
}

@misc{WangZahl2025Convex,
  author        = {Wang, Hong and Zahl, Joshua},
  title         = {Volume estimates for unions of convex sets, and the {Kakeya} set conjecture in three dimensions},
  year          = {2025},
  eprint = {2502.17655v1},
  archivePrefix = {arXiv},
  primaryClass  = {math.CA},
  doi           = {10.48550/arXiv.2502.17655},
  url           = {https://arxiv.org/abs/2502.17655v1},
  note = {Preprint version: \href{https://arxiv.org/abs/2502.17655v1}{arXiv:2502.17655v1}}
}

@article{Wolff1995Kakeya,
  author = {Wolff, Thomas H.},
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}

@article{Wolff2001,
  author = {Wolff, Thomas},
  title = {A sharp bilinear cone restriction estimate},
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  volume = {153}, number = {3}, pages = {661--698}, year = {2001},
  eprint = {math/0105266v1}, archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/math/0105266v1},
  note = {Preprint version: \href{https://arxiv.org/abs/math/0105266v1}{arXiv:math/0105266v1}}
}
==== END Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere//references.bib ====