==== TABLE OF CONTENTS ====
A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula//00-introduction.tex
A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula//01-inputs.tex
A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula//02-annuli.tex
A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula//03-spectrum.tex
A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula//04-shape.tex
A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula//05-transfer.tex
A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula//06-amplitude.tex
A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula//lobes.tex
A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula//paper.tex
A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula//references.tex
===========================

==== BEGIN A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula//00-introduction.tex ====
\section{Introduction}\label{sec:introduction}

Let $\eta_\eps$ be a homogeneous Poisson process in $\R^2$ of intensity
$\eps^{-2}$. Color its Voronoi cells independently black or white, each
with probability $1/2$. For a bounded quadrilateral $Q$, a black crossing
means a connected black-cell path inside the closed quadrilateral joining
its two specified opposite sides. The cells are those of the full-plane
tessellation: the process is not resampled at the boundary, and no
boundary sites are wired together.

A nucleus is \emph{color-pivotal} for this event if changing only its cell's
color changes whether the crossing occurs. Write $N_\eps(Q)$ for the
number of such nuclei, including nuclei outside $Q$ when their cells
meet $Q$, and set $N_\eps=N_\eps([0,1]^2)$ for a left-to-right crossing.
All expectations below average both the Poisson geometry and the colors.

\begin{assumption}[Scalar Cardy limit]\label{ass:cardy}
For every bounded Jordan quadrilateral $Q$ with four distinct marked
boundary points, the critical crossing probability converges, as
$\eps\downarrow0$, to Cardy's conformally invariant value.
\end{assumption}

Here a Jordan quadrilateral is a Jordan domain with four marked boundary
points in cyclic order and a specified pair of opposite boundary arcs.
In the upper half-plane normalization with marked points $0,x,1,\infty$,
$00$, and constants $C,c>0$ such
that, for $b_*\le r0$, in the mesh-first limit the probability of
four bulk arms with any specified color order is at most
$C(r/s)^{1+c_0}$, and that of six bulk arms with any specified color
order is at most $C(r/s)^{2+c_0}$. Three free arms at a regular
boundary, measured in a conformal chart, have an upper bound with
exponent strictly larger than one.
\end{enumerate}
\end{proposition}

For precise locators, part~(i) uses Proposition~2.4, Lemma~2.5, and
Proposition~4.5 of~\cite{V}; part~(ii) uses Proposition~3.1 and
Lemmas~3.4--3.7; part~(iii) uses Lemmas~2.7, 2.8, and~2.11.
The free boundary-arm estimate in Lemma~2.11 has exponent $4/3$.
``Free'' means that the arms use the independent fair interior sites,
not the sites in the prescribed base arc.

We emphasize two domain qualifications. First, a boundary chart must
converge uniformly on its closure, have no constant boundary subarc,
and project injectively on the compact bank interval where the
estimate is used. Separate prime-end banks need not have disjoint
physical projections. The local estimate applies in an annulus by
choosing a small graph disk patch with one side on that bank and the
remaining sides in the interior. These are Jordan approximations;
reflection in the parameter coordinate gives bounded distortion on
smaller patches. Second, product colors are conditional on the graph.
After an exploration, its remaining colors are fresh, but a
remaining-domain theorem can be used only after its geometric
hypotheses have been established. We never infer uniform convergence
after conditioning on arbitrary atypical explored colors. The uniform
operator estimate in Section~\ref{sec:transfer} will instead use
independent core variables and bounds on the signed weights.

We use the site triangulation of the Voronoi tiling from~\cite{V}.
Its faces are triangles, drawn through the tiling rather than required
to have straight edges. An interface strand, or \emph{wall}, passes
through a triangle between its two heterochromatic edges. Inside a
screened collar, simple site cycles approximate circles in curve
order. They may be constructed by following narrow strips and
retaining simple cuts; same-boundary chords on the domain-facing side
can be shortcut while retaining the specified inner region. Auxiliary
uniform angular choices, if needed, preserve rotation invariance.

All transfer cuts in this paper lie in the bulk. Their incident cells
and stars stay away from the quad boundary, where site connectivity
and cell connectivity could otherwise require additional care.
At the actual quad boundary, we retain the original closed-cell event
and use the necessary-arm estimates just stated. Null nongeneric
geometries, and null sets of insertion locations in integrals, are
discarded throughout.

\subsection{Calibration and the preliminary exponent}
Let $S_\eps$ be the probability that a site inserted at the center of
$Q_0=[-1,1]\times[-1/2,1/2]$ is color-pivotal for its left-to-right
crossing in intensity $\eps^{-2}$. Equivalently, $S_\eps$ is the
expectation of the black-forced crossing indicator minus the
white-forced crossing indicator for that site. Set
\[
 m_\eps=\eps^{-2},\qquad
 q_\eps=\bigl(\E N_\eps(Q_0)\bigr)^{-1},\qquad
 f(R)=S_{1/R}.
\]
The last equality agrees by Euclidean scaling with the unit-intensity
definition in the introduction.

\begin{proposition}[Calibration]\label{in:calibration}
There are constants $c,B\in(0,\infty)$ such that
\begin{equation}\label{in:calibration-eq}
 m_\eps q_\eps S_\eps\longrightarrow c,
 \qquad q_\eps\E N_\eps\longrightarrow B.
\end{equation}
Moreover,
\begin{equation}\label{in:comparison}
 f(R)\asymp A(b_*,R).
\end{equation}
\end{proposition}

The limits are precisely the single-time Poisson specialization of
Lemmas~7.1 and~7.2 of~\cite{V}, called ``Common calibration'' and
``The prescribed normalization.'' The signed color weight at the
inserted site is $\delta_1-\delta_0$; crossing monotonicity identifies
its expectation with $S_\eps$. The comparison follows from the local
pivotal bounds and the square-to-site calibration in Sections~2 and~6
of~\cite{V}. These inputs do not assert that
$R^{5/4}f(R)$ converges.

\begin{lemma}[Exponent without an amplitude]\label{in:exponent}
Under Assumption~\ref{ass:cardy},
\begin{equation}\label{in:exponent-eq}
 f(R)=R^{-5/4+o(1)}\qquad(R\to\infty).
\end{equation}
\end{lemma}
\begin{proof}
Force all cells meeting a central square of radius $h$ in $Q_0$
first black and then white, and denote the difference of crossing
probabilities by $D_\eps(h)$. Proposition~4.5 of~\cite{V} gives,
for each fixed small $h>0$, a common limit of this quantity in the
Voronoi and triangular models. The upper arm bound and
\cite[Lemma~6.1]{V}, ``A square can transmit a pivotal change,''
compare $D_\eps(h)$ with the four-arm probability from radius $h$
to radius $1$, with constants uniform in small $h$, after taking
the mesh limit. The corresponding triangular four-arm probabilities
have exponent $\lambda=5/4$ by~\cite{SW}.

Fix $\eta>0$. For an arbitrarily large but fixed ratio $L=1/h$,
these comparisons and qualitative convergence imply, for all
sufficiently large $r$ depending on $L$,
\[
 c_\eta L^{-\lambda-\eta/3}
 \le A(r,Lr)\le C_\eta L^{-\lambda+\eta/3}.
\]
Choose $L$ so large that both these constants and the constants in
\eqref{in:quasimult} are absorbed by an additional factor
$L^{\eta/3}$. Only after fixing this $L$ choose a starting radius
above the convergence threshold. Iterating over radii $r,Lr,L^2r,
\ldots$ gives four-arm bounds with exponents between
$\lambda-\eta$ and $\lambda+\eta$. The bounded initial range and
the final incomplete ratio change only the multiplicative constants,
by \eqref{in:power} and \eqref{in:quasimult}. Equation~\eqref{in:comparison}
then yields
$C_\eta^{-1}R^{-\lambda-\eta}\le f(R)\le C_\eta R^{-\lambda+\eta}$.
Letting $\eta\downarrow0$ proves the lemma.
\end{proof}

We will henceforth write $\lambda=5/4$. The remaining task is to
replace the $o(1)$ in this exponent by convergence of the multiplicative
coefficient. The next two analytic sections identify which annular
symmetries make that improvement possible.
==== END A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula//01-inputs.tex ====

==== BEGIN A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula//02-annuli.tex ====
\section{Four-interface transmission in an annulus}
\label{sec:annuli}

The continuum quantity used below is the probability that four interfaces
cross an annulus from one boundary component to the other.  Its existence
does not follow by applying the scalar Cardy hypothesis directly to a
multiply connected domain.  We establish it from the Jordan-domain inputs
of \cite{V}.  We also allow the boundary banks produced by stopped
explorations: a bank is viewed with its access from the domain, so distinct
banks may touch in the plane without being identified.

\subsection{Connection states and two caps}

Use the site triangulation drawn through the Voronoi tiling, with interface
strands drawn across the heterochromatic edges of its triangular faces.
These strands will be called \emph{walls}.  On a simple site cycle with
known colors, a \emph{run} is a maximal consecutive set of sites of one
color, and a \emph{switch} is the edge between consecutive runs.  Filling
the disk bounded by the cycle determines a noncrossing pairing of its switches by
walls.  The pairing, together with the run colors, determines connectivity
between boundary sites.  Closed walls need not be recorded.  We call this
pairing the connection state on the cycle.

Suppose each boundary component of a graph annulus has four switches and
four alternating prescribed runs.  All remaining sites have independent
fair colors.  Let $\mathcal T$ be the event that all four outer wall ends
reach the inner boundary.  There are two possible connection states in a
disk filling the inner cycle: the cap joining its two black runs, denoted
$\mathsf b$, and the cap joining its two white runs, denoted $\mathsf w$.
These are formal planar caps; no random colors are sampled in them.  Write
$H_{\mathsf b}$ and $H_{\mathsf w}$ for the indicators that the two outer
black runs are connected after the corresponding cap is attached.
Figure~\ref{fig:annular-caps} shows the two cap states when all four
walls are transmitted.

\begin{lemma}[The cap identity]\label{an:caps}
For every coloring of the annulus,
\begin{equation}\label{an:cap-equation}
 H_{\mathsf b}-H_{\mathsf w}=\mathbf 1_{\mathcal T}.
\end{equation}
Consequently the signed difference between the two inner cap states is
transferred to the same signed difference at the outer cycle, multiplied
by the indicator of four-interface transmission.  The identity remains
valid under composition across several cycles.
\end{lemma}

\begin{proof}
Walls in a triangulation neither branch nor cross.  If fewer than four
inner ends are connected to outer ends, the wall pairing visible from the
outer cycle is independent of the inner cap: there is no choice of pairing
on zero or two surviving ends.  Thus the left side of
\eqref{an:cap-equation} is zero.  If all four are transmitted, their cyclic
order is preserved, with any possible winding retained.  The intervening
monochromatic bands carry black connectivity to black connectivity and
white connectivity to white connectivity.  Hence $H_{\mathsf b}=1$ and
$H_{\mathsf w}=0$.  This is also the usual Hex duality proof after each run
is contracted to a vertex.  Composing the planar pairings proves the last
assertion; any closed walls created in the composition have weight one.
\end{proof}

\begin{figure}[tb]
\centering
\begin{tikzpicture}[scale=0.87,line cap=round]
\foreach \a/\txt in {0/{Black cap: $H_{\mathsf b}=1$},6/{White cap: $H_{\mathsf w}=0$}} {
  \begin{scope}[xshift=\a cm]
    \draw[gray] (0,0) circle (1.85);
    \draw[gray] (0,0) circle (.76);
    \draw[very thick] (45:1.85) arc (45:135:1.85);
    \draw[very thick] (225:1.85) arc (225:315:1.85);
    \draw[very thick] (45:.76) arc (45:135:.76);
    \draw[very thick] (225:.76) arc (225:315:.76);
    \foreach \t in {45,135,225,315} {
      \draw[blue!65!black,thick] (\t:.76)--(\t:1.85);
      \fill (\t:.76) circle (1.4pt);
      \fill (\t:1.85) circle (1.4pt);
    }
    \node at (0,2.12) {$B$};
    \node at (0,-2.12) {$B$};
    \node at (-2.12,0) {$W$};
    \node at (2.12,0) {$W$};
    \node[align=center] at (0,-2.65) {\txt};
  \end{scope}
}
\draw[blue!65!black,thick] (45:.76) .. controls (.20,.25) and (.20,-.25) .. (315:.76);
\draw[blue!65!black,thick] (135:.76) .. controls (-.20,.25) and (-.20,-.25) .. (225:.76);
\begin{scope}[xshift=6cm]
\draw[blue!65!black,thick] (45:.76) .. controls (.25,.20) and (-.25,.20) .. (135:.76);
\draw[blue!65!black,thick] (225:.76) .. controls (-.25,-.20) and (.25,-.20) .. (315:.76);
\end{scope}
\end{tikzpicture}
\caption{The two cap pairings when all four walls are transmitted.
Blue curves are walls; thick black boundary arcs are black runs, and the
other runs are white.  The cap changes the outer black connection bit by
one.  The straight, unwound walls are schematic: the identity permits
winding and does not prescribe a pairing of numbered inner and outer
switches.}\label{fig:annular-caps}
\end{figure}

\subsection{The boundary convergence used below}

For $p>0$ let
\[
 \mathbb A_p=\{e^{-p}<|z|<1\}.
\]
Thus $p=\log(R/r)$ for a round annulus with radii $r0$, and a free arm from a
shrinking neighborhood to a fixed scale has probability tending to zero.
The word \emph{free} excludes prescribed sites and cap wires.  The local
bank versions, including arms approaching the bank arbitrarily closely,
are those of \cite[Section 2, ``Cardy tests in a boundary chart'' and
``Free boundary-arm bounds'']{V}.

\begin{lemma}[Continuity under boundary motion]\label{an:motion}
For a sequence with admissible boundary convergence, move either cycle
slightly into the annulus in round coordinates, and move its switches
slightly with it.  Couple all unchanged free colors.  The probability
that either capped connection test changes tends to zero as the size of
the motion tends to zero, after taking the mesh limit.  The assertion also
holds for small piecewise smooth motions in spare space around an inset
annulus.  It is uniform along any convergent sequence of admissible data
with distinct limiting switches.  More precisely, if $H_n$ and
$H_{n,\delta}$ are either capped indicators before and after a motion of
size at most $\delta$, preserving the run order and its labels away from
the moving switches, then
\begin{equation}\label{an:motion-limit}
 \lim_{\delta\downarrow0}\limsup_{n\to\infty}
 \Pp(H_n\ne H_{n,\delta})=0.
\end{equation}
\end{lemma}

\begin{proof}
First the mesh in annular coordinates tends to zero.  Otherwise a
connected cell or triangle portion of vanishing physical diameter would
have preimages with a subsequential nontrivial continuum limit on which
$\phi$ is constant.  Such a continuum cannot meet the interior, where
$\phi$ is one-to-one, and cannot lie on the boundary, whose finitely many
simple arcs contain no constant interval.  Here graph portions, rather
than whole cells crossing the domain boundary, are used.

Remove small coordinate neighborhoods of the switches and of the
finitely many joins of simple bank arcs.  On each remaining compact bank
interval, use disk patches whose boundary consists of that bank and cuts
strictly inside the annulus.  Their physical boundaries are Jordan and
their graph approximations converge in curve order.  Their uniformizing
coordinates agree, with fixed-scale margins, with the annular coordinates;
reflection in the limiting round bank controls distortion on smaller
patches.  Thus the local boundary-arm estimates of \cite{V} apply on these
patches even when another prime-end bank has the same physical image.

Cover these intervals by $O(s^{-1})$ coordinate disks of radius $s$.
Implement a motion much smaller than $s$ by successively replacing bank
intervals, with tapered ends.  At graph level choose short transverse
joins in disjoint narrow strips from the old cycle to its replacement.
Substitute the intervals in cyclic order, retaining the old cycle off the
interval currently being changed and erasing the joins once the whole
cycle is replaced.  The graph approximation in curve order ensures that
these are simple cuts.  Each join has the same constant color as its run.
Moves containing switches or bank joins are instead included in the
exceptional neighborhoods.  Two nearby nonnested contours can be
compared through a common slightly inset contour.

Consider one ordinary move on a black bank.  If a capped connection test
changes, choose a winning black connection in one test and a winning
dual white connection in the other.  Contract the prescribed runs and
use the same cap in both tests.  Purely wired connections do not change,
so a free segment of a winning path must meet the support of the move.
The test neighborhood avoids every other run and mark.  The white path
has two free branches leaving that neighborhood.  At least one black
branch must also leave freely: a black branch already reaching the same
prescribed run outside the moved interval could be joined there without
the local change.  Taking simple paths after contraction gives disjoint
white branches, and the opposite-colored black branch is disjoint from
them in the common fair region.  This gives three free arms in a slightly
shifted half-patch, with fixed-factor margins.  The same argument applies
with the colors interchanged.  Forced sites on intermediate cuts are not
counted as free arms.

For any fixed larger chart scale, the mesh-first bound for a single disk
is $O(s^{1+\eta})$.  The union over $O(s^{-1})$ disks therefore costs
$O(s^\eta)$.  The number of exceptional neighborhoods is fixed.  A change
there requires a fair arm approaching from positive parameter distance:
connections already provided by the same run or by the cap do not
change.  This arm has positive physical diameter, uniformly after fixing
its outer parameter scale, by the no-continuum-fiber observation above.
The ordinary physical one-arm bound consequently makes the exceptional
cost tend to zero, including at a contact between different accesses.
One may move each exceptional neighborhood as a single piece, so this
cost is not multiplied by the number of ordinary intervals.

First take the mesh limit, then make the ordinary motions small compared
with $s$, then let $s$ tend to zero, and finally shrink the exceptional
neighborhoods.  This proves the assertion.  The same proof with moving
marks, or with a moving round radius, proves the stated continuity.
\end{proof}

After applying Lemma~\ref{an:motion}, all prescribed banks can be placed
on smooth contours strictly inside the parameter annulus, with spare
space on both sides.  There are still two separate issues: joint ordinary
crossings must have a conformally invariant annular law, and those
crossings must determine the capped tests.  We address them in that order.

\begin{lemma}[Ordinary crossings in an annular open set]
\label{an:ordinary}
In an annular open set compactly inside the parameter annulus, the joint
limiting law of any countable collection of regular ordinary quad tests
is conformally invariant.  The tests may be chosen from a countable dense
family of polygon or piecewise smooth quads with positive margins.
\end{lemma}

\begin{proof}
On every strict Jordan disk patch the joint law is the critical triangular
percolation law by \cite[Sections 3--4]{V}.  This law is conformally
invariant; see \cite[Theorems 1 and 7]{CN} and the quad-crossing
formulation in \cite[Section 2.3]{GPS}.  Scalar Cardy sandwiches identify smooth images of polygon
tests and nearby graph crossing conventions with the same limit.

Choose a finite-length smooth cut opening the annulus.  Any finite
collection of tests compactly contained off the cut fits in a strict
Jordan disk, so its joint law agrees in the physical domain and in the
conformal chart.  To recover tests meeting the cut, use the
Schramm--Smirnov gluing theorem
\cite[Theorem 1.19 and Proposition 4.1]{SS}, only in strict Jordan disk
patches.  For the critical triangular crossing law, the crossing of a
regular quad in such a patch is measurable, modulo null sets, with
respect to countably many tests avoiding the finite smooth cut in that
patch.  Its hypotheses hold here: independence in the discrete model,
RSW, one-arm decay, and a four-arm exponent strictly greater than one.
Regular continuity tests may be used throughout, by Cardy sandwiches.

Each required quad meeting the annular cut has a slightly larger Jordan
disk neighborhood inside the annulus.  The recovering measurable rule
in that neighborhood is the same in the chart and in its image, since
their whole disk laws already agree.  All recovering tests belong to the
common countable off-cut family.  Hence the recovered crossings have the
same joint law as well.  This proves the assertion.  White crossings are
included by Hex duality and continuity.  No gluing assertion along a rough
slit is used.
\end{proof}

\begin{proposition}[The annular transmission limit]\label{an:limit}
For every sequence with admissible boundary convergence, the probability
of $\mathcal T$ has a limit depending only on the round data $(p,x,y)$ and
the choice of inner colors.  Denote these limits by
\[
 T^+(p,x,y),\qquad T^-(p,x,y).
\]
They are continuous for $p>0$ and distinct switches.  The same convergence
holds for each capped connection probability.  In particular, it applies
to random remaining domains whenever, conditional on their geometry and
explored boundary, their interior colors are fresh and their marked
domains have admissible convergence almost surely in a subsequential
coupling.  In that case the difference between the conditional
transmission probability and $T^\pm$ evaluated at the actual annular
conformal data tends to zero in mean absolute value.
\end{proposition}

\begin{proof}
By Lemma~\ref{an:motion}, it suffices first to consider smooth inset
annuli with spare space.  We show that ordinary quad crossings determine
the limiting value of each capped black connection test.  Sample fair
colors throughout a slightly larger annular neighborhood in that spare
space, overriding them on prescribed runs only when evaluating a capped
test.  Ordinary tube tests use the fair colors.  Compare two
candidate sequences, one of which may be in round coordinates.  Couple
their countable ordinary crossing vectors through the common limiting
law of Lemma~\ref{an:ordinary}.  Along further subsequences include the
cap indicators in the coupling; the curve tightness proved below also
allows the chosen winning paths to be included.  The two limiting
collections are coupled to have the same ordinary crossing vector.

Fix a small \emph{favorable} perturbation of the black test in the second
sequence.  Move black banks into the annular fluid and extend their runs
slightly past the old switches; retreat white banks from the old fluid.
Use positive slacks and taper the motions within the old white
intervals, putting each new switch on the retreated part.  Thus every old
black attachment lies behind a new black bank, the new white banks are
positively separated from the old fluid, and any removed fluid is in
separate collars of the expanded black runs.  All these choices take
place in the spare space.

On a positive configuration in the first sequence, contract each
prescribed run and attach the cap edge when it is black.  Take an
outermost simple winning path between the two outer black terminals.
For example attach an external terminal edge and take the appropriate
boundary of the union of cycles containing it.  Contracting connected
boundary runs and attaching the cap preserves the planar embedding.
The resulting path has no bypass on its chosen exterior side.  Since it
visits each contracted run at most once, it decomposes into a bounded
number of free black segments joining different runs, together with
wired or cap joins.  Crop their endpoints by a vanishing graph distance
if they end adjacent to a wire.

We recall why these free segments admit simple limiting curves.  Many
oscillations across an annulus supply many disjoint free arms; BK and
one-arm decay bound their probabilities.  Successive-excursion
compactness therefore gives tightness in the curve metric.  A bulk
self-contact with two positive excursions would give four black
traversals.  In two gaps on the no-bypass side, intermediate transverse
cuts and Hex duality give two additional white traversals: a black
transverse crossing would instead be a bypass.  The six arms are
disjoint after fixed-factor crops, also for backtracking pieces.
The mesh-first six-arm exponent greater than two excludes such contacts
by a covering argument.  At a smooth boundary bank, a repeated visit to
the same point gives at least three free traversals, including when one
of the visits is an endpoint; the boundary exponent greater than one
excludes these repeated visits.  A single tangential bank contact need
not be excluded: the favorable perturbation accommodates it.  At the finitely
many switches one free arm suffices.  These are precisely the local
no-bypass and free-segment arguments of \cite[Section 4, ``Free segments
of a terminal path'']{V}.  Contracted runs do not change the bulk argument,
whose working annulus avoids them.  Constant pauses may be removed.

In a subsequential coupled representation, fix the favorable displacement
and then crop each limiting free segment a small positive amount at its
ends, keeping the new endpoints strictly behind the expanded black
banks.  A cropped endpoint may still lie on an old bank; the ambient
tube may straddle that bank within the spare space.  This positive crop is distinct from the vanishing
graph crop used to remove incidences to a wire.  Surround the cropped
simple segment by a thin Jordan tube, with end caps in those endpoint
neighborhoods.  The approximating free graph segments supply ordinary
fair-color crossings of this ambient tube.  The tube can be approximated with positive margins by a quad
from the countable ordinary test family.  Its crossing occurs in the
first sequence, and hence in the second for all sufficiently large
indices in the coupling.  The witnesses may be chosen from the countable
family after taking the subsequential limit; coordinatewise convergence
still applies to each of the finitely many chosen witnesses.

Clip these transferred crossings when they hit the new black banks.
Excursions outside the new fluid can only lie in the collars now wired
black.  They can be shorted through the corresponding run, and collars
of different black banks are separated.  This operation also handles
arbitrarily many backtracks: at graph level every passage between a
removed collar and the fluid meets its boundary cycle.  The transferred
tubes, together with the unchanged cap or wire joins, therefore give a
winning black connection in the favorably perturbed second test.

Apply this comparison to subsequences realizing the upper and lower
limits of the unperturbed cap probabilities.  The upper limit of the
first is bounded by the lower limit of the second plus its boundary-motion
error.  Remove the favorable perturbations using Lemma~\ref{an:motion}
and reverse the roles of the sequences.  Their upper and lower limits
coincide.  This proves convergence and conformal invariance of each cap
test on inset annuli.  Taking the inset size to zero proves the same
assertions under Definition~\ref{an:admissible}.  The cap identity gives
the transmission limit.  Allowing the marks and round radii to move in
the same comparison proves continuity in $(p,x,y)$.

Finally consider random remaining domains under the stated coupling.
For almost every convergent domain realization, the deterministic
statement applies to its fresh product-color law; continuity identifies
the same limit for $T^\pm$ at the actual conformal data.  All probabilities
are bounded by one, so bounded convergence gives convergence in mean
absolute value.  This conclusion does not assert a uniform conditional
estimate over arbitrary atypical explored boundaries.
\end{proof}

\begin{lemma}[Collisions and symmetries]\label{an:collision}
The functions $T^\pm$ extend continuously by zero when switches collide
on either boundary, uniformly for $p$ in compact subsets of $(0,\infty)$.
They are invariant under simultaneous rotation of $x$ and $y$ and under
global color reversal.  Cyclic relabeling by one switch in either tuple
interchanges $T^+$ and $T^-$.
\end{lemma}

\begin{proof}
\leavevmode\clubpenalty=10000
If two consecutive switches coalesce, four-interface transmission
requires a fair arm from their intervening shrinking run to the other
boundary.  This arm cannot use another prescribed run: it lies between
the two corresponding transmitted walls.  A mesh-first one-arm bound,
with a fixed outer scale and then a shrinking inner scale, proves
vanishing.  The bound does not depend on the positions of the other
switches and therefore also covers simultaneous collisions.
Conformal invariance gives simultaneous rotational invariance, and
fair colors give global color symmetry.  Moving the first switch to the
next switch reverses the alternating run labels on that component.
By the defining convention, this interchanges the two inner prescriptions.
\end{proof}
==== END A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula//02-annuli.tex ====

==== BEGIN A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula//03-spectrum.tex ====
\section{The spectrum of four-interface transmission}
\label{sp:section}
\label{sec:spectrum}

We now estimate the limiting transmission probabilities
$T^\pm(p,x,y)$ of Section~\ref{sec:annuli} when the modulus $p$ is large.
There are two different conclusions.  The leading term separates into an
inner factor and an outer factor.  The signed difference $T^+-T^-$ has
additional decay, and averaging the rotation of one end improves that
decay beyond exponent $2$.  This last improvement will make the response
to a fair Poisson insertion integrable over the plane.

\subsection{The modulus equation}

The space of four cyclically ordered, labeled angles is
\[
 \mathcal C=
 \{w\in\R^4:w_10$ and every fixed finite derivative order $k$,
the vector field $B_p$ extends smoothly across angle collisions and
\[
 \|B_p\|_{C^k}\le C_{k,p_0}e^{-p}\qquad(p\ge p_0).
\]
It is equivariant under permutations within either tuple, common
rotations, and cyclic relabelings.  In particular, its normal component
at every collision face is zero.
\end{proposition}

\begin{proof}
We first derive a local equation with just one switch active.  Write the
annulus as the cylinder $0<\operatorname{Im}z0$.  For either transmission function, or its sum or difference,
the transformed solution satisfies
\begin{equation}\label{sp:uniform-smoothing}
 \|W(p)\|_\infty+\operatorname{Lip}(W(p))\le C_{p_0}
 \qquad(p\ge p_0+2).
\end{equation}
The semigroup $P_t$ of $D$ is symmetric on $L^2(\mu)$ and, for every
$q>1$, maps $L^q(\mu)$ to bounded functions at every fixed positive
time.  At unit time the evolution in \eqref{sp:transformed-pde} maps
bounded functions to Lipschitz functions with a constant uniform in
the starting modulus $p\ge p_0$.
\end{lemma}

\begin{proof}
We first justify using the diffusion all the way to the chamber
boundary.  The drift in one tuple is
$\frac14(\sum_{j\ne i}\cot((w_i-w_j)/2))_{i=1}^4$.
On the lifted convex chamber, use $-\log m$, shifted to be nonnegative,
as a Lyapunov function.  If $\alpha=4/3$ is the density exponent, the
cotangent identity shows that its generator has a singular contribution
proportional to $(\alpha-\alpha^2)\sum_{i