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\title{A construction of the sphere-packing magic functions in dimensions 8 and 24}
\author{}
\date{}
\begin{document}
\maketitle
\begin{abstract}
We develop an alternative route to the solutions of the sphere-packing
problem in dimensions \(8\) and \(24\), through the auxiliary functions
in the Cohn--Elkies Fourier bound. In each dimension we construct a
radial Schwartz function with the prescribed lattice zeros by solving
a countable system of linear equations in absolutely summable
coefficients. We prove existence and uniqueness after normalization,
without using the known magic functions or Fourier interpolation.
Through a Cauchy boundary problem, solutions of the homogeneous
equations are identified with global sections of a rank-six
holomorphic vector bundle over the compactified modular curve
of the full modular group, with its orbifold structure retained.
Riemann--Roch shows that this bundle has Euler characteristic
one, forcing a nonzero global section and hence a nontrivial
solution of the equations. A separate argument proves uniqueness
after normalization.
An equivalent positive operator equation has a bounded inverse.
The construction is motivated by the coefficient method for planar
circle packing. A subsequent comparison identifies our solutions
with the functions of Viazovska and of Cohn, Kumar, Miller,
Radchenko, and Viazovska. The route to sphere packing is not yet
independent in its entirety: we do not prove the required global
sign inequalities from the coefficient equations. Those inequalities
are inherited from the known functions after identification.
\end{abstract}

\section{Introduction}

The Fourier bound of Cohn and Elkies \cite{CE} converts sphere
packing into a problem about a function and its Fourier transform.
A suitable auxiliary function gives an upper bound on the density
of every packing. In dimensions \(8\) and \(24\), specially chosen
functions make this bound equal to the densities of the \(E_8\)
and Leech lattices, respectively, and hence prove their optimality.
Such sharp auxiliary functions are often called \emph{magic functions}.
This paper gives a construction of those functions from countably
many linear equations.

There are two distinct tasks in a sharp Fourier bound: constructing
the function with the required zeros and normalization, and proving
its signs between those zeros. We give existence and uniqueness
proofs for the coefficient systems in both dimensions. We then
identify the resulting functions with the known magic functions.
We do not give independent proofs of the global sign inequalities.
Thus the construction part is independent of the earlier magic
functions, but a complete sphere-packing proof still uses their
established signs. Section~\ref{sec:signs} describes this remaining
task and the methods used to address it in the planar case.

\subsection{The functions to be constructed}

Fix \(d\in\{8,24\}\). Let \(\mathbf y,\boldsymbol{\xi}\in\mathbb R^d\) denote a
spatial vector and a frequency vector, respectively. For a Schwartz
function \(f:\mathbb R^d\to\mathbb C\), define
\begin{equation}
 \widehat f(\boldsymbol{\xi})=\int_{\mathbb R^d}
                   f(\mathbf y)e^{-2\pi i \mathbf y\cdot\boldsymbol{\xi}}\,d\mathbf y.
 \label{eq:Fourier-convention}
\end{equation}
Here \(\mathbf y\cdot\boldsymbol{\xi}\) is the Euclidean inner product. A Schwartz
function is smooth and, together with each derivative, decays
faster than every inverse power of \(|\mathbf y|\).
We consider radial functions: \(f(\mathbf y)\) depends only on \(|\mathbf y|\).
Their Fourier transforms are radial as well.

For a radial \(f\), define \(\Phi,G:[0,\infty)\to\mathbb C\) by
\begin{equation}
 f(\mathbf y)=\Phi(|\mathbf y|^2),\qquad \widehat f(\boldsymbol{\xi})=G(|\boldsymbol{\xi}|^2).
 \label{eq:radial-profiles}
\end{equation}
The letters \(u,v\) denote independent real arguments in
\([0,\infty)\) of \(\Phi,G\), respectively. In the displayed
identities we evaluate these functions at \(|\mathbf y|^2,|\boldsymbol{\xi}|^2\).
Primes denote ordinary derivatives of the one-variable functions;
in particular \(\frac{d}{dr}\Phi(r^2)=2r\Phi'(r^2)\).

Put \(n_8=1\) and \(n_{24}=2\). In the normalization where
each lattice has covolume one, its first nonzero squared length
is \(2n_d\). The Cohn--Elkies bound asks for a real radial
Schwartz function satisfying
\begin{equation}
 \Phi(0)=G(0)=1,\qquad
 \Phi(u)\leq0\quad(u\geq2n_d),\qquad
 G(v)\geq0\quad(v\geq0).
 \label{eq:sign-goal}
\end{equation}
It then bounds the packing density by the volume of a ball of
radius \(\sqrt{2n_d}/2\), which is precisely the density of
the corresponding lattice packing.

The lattices are even, so their nonzero squared lengths belong
to \(2\mathbb Z\), starting at \(2n_d\); see
\cite[Section 1]{V} and \cite[Section 1]{CKMRV24}.
Equality in the Fourier bound forces both profiles to vanish
at the lattice lengths. We impose these conditions at every
node \(2n\), \(n\geq n_d\). The sign inequalities also force
their derivatives to vanish, except possibly at the first zero
of \(\Phi\), the endpoint of its required sign interval.
The equalities to be imposed are therefore
\begin{align}
 \Phi(2n)&=0 &&(n\geq n_d),&
 \Phi'(2n)&=0 &&(n>n_d),\nonumber\\
 G(2n)&=0 &&(n\geq n_d),&
 G'(2n)&=0 &&(n\geq n_d).
 \label{eq:desired-zeros}
\end{align}
Our coefficient theorems construct normalized functions satisfying
these equalities. From this point on, our proofs concern their
construction and uniqueness; we do not return to an independent
proof of the more general inequalities \eqref{eq:sign-goal}.
The later comparisons with the known magic functions transfer
the established signs to our solutions.

There are well-documented methods for proving the required
inequalities. Viazovska's original dimension-eight proof combines
interval arithmetic with asymptotic expansions and rigorous
remainder bounds near the two ends of the integral parameter
range \cite[Section 5]{V}. In dimension \(24\), Cohn, Kumar,
Miller, Radchenko, and Viazovska bound the errors in truncated
Fourier series of modular forms and use Sturm's theorem for
real polynomials, with exact rational arithmetic, to certify
the remaining signs \cite[Appendix A]{CKMRV24}. Subsequent
proofs replace these numerical certifications by structural
arguments: Dan Romik proves the dimension-eight modular-form
inequalities directly using modular identities, coefficient
positivity, and special values \cite{Romik}; Seewoo Lee gives
algebraic proofs in both dimensions using differential identities,
positivity under Serre differentiation, and monotonicity of
ratios of modular forms \cite[Sections 3, 5, and 6]{Lee}.
The recent planar construction instead uses certified polynomial
approximations in the Bernstein basis, together with coefficient
error bounds and estimates for the unbounded region
\cite[Section 4 and Appendices A--B]{OpenAI}.
Section~\ref{sec:signs} describes that Bernstein approach and
what would be needed to apply it to the present coefficient
construction.

\subsection{The planar construction as the starting point}

The guiding method comes from the planar circle-packing preprint
of OpenAI \cite[Sections 2, 3, and 5]{OpenAI}. That paper starts
with a trigonometric polynomial having double zeros at a periodic
set of nodes. Dividing by linear and quadratic factors permits
changes to a value or first derivative at a chosen node. Gaussian
damping turns these expressions into radial Schwartz functions.
Their coefficients are determined by imposing Fourier conditions.
Estimates on the entire coefficient space control the infinite
system, and Poisson summation determines the normalization.

We adopt this sequence of ideas: choose a family with the required
zeros, express the Fourier conditions as linear equations, control
the infinite coefficient space, and recover normalization by
Poisson summation. In both dimensions studied here, the evenly
spaced nodes allow the common factor
\[
 P(u)=\sin^2(\pi u/2).
\]
We combine it with integrals of Gaussians over \(t\geq1\) and
\(0<t\leq1\), where \(t\) is the Gaussian parameter. Additional
functions permit the values and derivatives that must remain
unconstrained at the first few nodes. There are five such
correction functions in dimension \(24\) and three in dimension
\(8\). Sections~\ref{sec:system} and \ref{sec:eight} specify
the families, including the parity restrictions on their
coefficients. These choices are part of the construction; they
are not asserted to follow from the zero set alone.

Our method for solving the resulting system differs from that of
the planar paper. Fourier samples become moments of two measures
on a fixed contour. Their Cauchy integrals give a boundary problem,
from which we derive translation and inversion laws. These laws
lead to an index calculation and a vanishing argument. The planar
paper thus supplies the starting construction strategy; the
boundary and index analysis proves existence and uniqueness for
the systems studied here.

\subsection{Earlier constructions and the scope of the result}

Viazovska constructed the eight-dimensional magic function from
modular and quasimodular forms \cite{V}. Cohn, Kumar, Miller,
Radchenko, and Viazovska adapted this method to dimension \(24\)
\cite{CKMRV24}. Earlier computations of Cohn and Miller \cite{CM}
had revealed striking special values and suggested convergence
of finite approximations.

Two subsequent constructions are especially close to the present
work. Feigenbaum, Grabner, and Hardin \cite{FGH} start with a
holomorphic Laplace input satisfying specified endpoint conditions.
Their Proposition 2.4 characterizes Fourier eigenfunctions by
functional equations for that input. Proposition 3.1 and Section 4
identify the resulting quasimodular and level-two structures;
Sections 3.1 and 4.1 use dimension counts, and Section 5 gives
recurrences. Thus deriving modular structure from Fourier conditions
is already part of that work. Rolen and Wagner
\cite[Theorems 1.1--1.2]{RW}, with Theorem 1.2 read in its
corrected form \cite{RWcorr}, construct families of radial Schwartz functions from modular and
quasimodular forms, control their prescribed zeros, and recover the
known functions in dimensions \(8\) and \(24\).

Our starting data are different: square-summable coefficients in
an explicit elementary family, with discrete Fourier values and
derivatives required to vanish. We do not assume a holomorphic
Laplace input on the whole upper half-plane or an eigenfunction
identity. The contribution proved here is the equivalence of this
entire coefficient kernel with a space of global sections, including
endpoint continuation and geometric coefficient decay. Once that
equivalence is established, the familiar quasimodular and level-two
structures reappear in its two blocks. The rank-six bundle and its
index organize the resulting finite-dimensional problem; the novelty
claimed is not a new source of modular structure or a replacement
for the earlier modular classifications. The compact regularizer
adds an explicitly specified bounded operation on arbitrary
coefficient vectors, rather than merely an abstract projection onto
the finite-dimensional solution space.

The subsequent Fourier interpolation theorem \cite{CKMRVinterp}
determines a radial Schwartz function from its values and radial
derivatives, and those of its Fourier transform, at the relevant
lattice radii. Its proof develops the modular integral machinery
of the sphere-packing constructions. The interpolation paper first
appeared in 2019 and was published in 2022; the sphere-packing
constructions first appeared in 2016.

Our existence and uniqueness proofs in the chosen coefficient
spaces use neither the known magic functions nor the interpolation
theorem. They do use modular geometry: the boundary equations
produce transformation laws, Riemann--Roch gives existence, and
low-weight consequences of the valence formula give uniqueness.
The known functions enter only in the later comparisons in
Sections~\ref{sec:comparison} and \ref{sec:eight-comparison}.
Those comparisons transfer their signs to our functions.
Interpolation also gives uniqueness in the larger class of all
radial Schwartz functions with the prescribed data, whereas our
independent uniqueness statements concern the specified coefficient
families.

\subsection{Strategy of the proof}\label{sec:strategy}

The problem is to solve a countable system of linear equations:
choose coefficients so that the Fourier values and derivatives in
\eqref{eq:desired-zeros} vanish, and impose one scalar normalization.
The zeros of \(\Phi\) are already built into the chosen functions.
Solving successively larger finite systems does not by itself
produce a solution of all the equations, nor prove uniqueness.
We control the entire coefficient family by interpreting the
equations as an analytic continuation problem.

\paragraph{1. Build in the zeros of the function.}
Section~\ref{sec:system} gives the explicit functions for dimension
\(24\) and assigns a coefficient to each. The trigonometric
factor supplies the zeros, and five correction functions permit
the unconstrained values and derivatives. We seek absolutely
summable coefficients but initially work in the larger Hilbert
space of square-summable families. Section~\ref{sec:analytic}
proves that each family defines a radial Schwartz function.
Poisson summation relates its normalization to the derivative
at the first lattice radius. Dimension \(8\) uses the same
steps with three correction functions.

\paragraph{2. Replace Fourier samples by moments.}
Lemma~\ref{lem:samples} expresses \(G(2n)\) and \(G'(2n)\)
as integrals of \(w^{n-1}\) against two complex measures depending
linearly on the coefficients. Their supports are fixed. In
dimension \(24\), where \(n\geq2\), the equations prescribe
the moments of positive degree. The constant moments remain
unrestricted: setting them to zero would add equations at
\(v=2\). In dimension \(8\), the samples start at \(n=1\),
so the constant moments are prescribed as well. This distinction
will change the allowed poles in the global problem.

\paragraph{3. Turn vanishing moments into a boundary problem.}
A Cauchy integral expanded at infinity reads off the moments as
Laurent coefficients. Subtracting the contribution of any
unprescribed constant moment gives two functions that vanish
outside the contour. Inside it they are holomorphic except
along a real segment. Their boundary values and their jumps
across that segment are explicit expressions in four generating
series of coefficients. We prove both implications: coefficient
solutions give this boundary problem, and sufficiently regular
solutions of the boundary problem give the original equations.

\paragraph{4. Recover the generating series and continue them.}
Four boundary values determine the four generating series through
an invertible matrix. Away from two endpoints this gives
holomorphic continuation immediately. Section~\ref{sec:regularity}
removes the endpoint singularities using area estimates and
square summability. Continuation beyond the unit circle gives
geometric decay of the coefficients. Thus every solution in the
Hilbert space belongs to the required absolutely summable space.

\paragraph{5. Count solutions of the global continuation problem.}
Section~\ref{sec:bundle} joins the continued functions over a
fundamental domain. The four generating series and two Cauchy
integrals give six holomorphic functions with translation and
inversion laws. We prove equivalence with this global problem;
the laws are consequences of the boundary equations, not extra
constraints. In each dimension they define a rank-six holomorphic
vector bundle over the compactified modular curve of the full
modular group, retaining its orbifold structure. In particular,
the stabilizers at the elliptic points enter the index calculation.
The cusp conditions specify how the bundle extends across the
added cusp. Solutions of the homogeneous coefficient equations
correspond exactly to global holomorphic sections of this bundle.
If \(h^0\) is the dimension of its space of global sections and
\(h^1\) is the dimension of its first sheaf cohomology group,
Riemann--Roch gives
\[
 h^0-h^1=1.
\]
Since \(h^1\geq0\), a nonzero section exists, and the
correspondence gives a nontrivial coefficient solution. A separate
vanishing argument shows that a solution with zero normalization
is zero. The solution space is therefore one-dimensional and has
a unique normalized element. Both dimensions have Euler
characteristic one, although their scalar weights and cusp
conditions differ. The index is that of the bundle's cohomology,
not an index of the compact moment operator.

\paragraph{6. Obtain a stable equation and identify the function.}
The moment operator is compact, so a small output alone cannot
control the coefficient norm. The continuation estimates supply
a second compact operator that fixes every exact solution.
Section~\ref{sec:operator} combines the operators and the
normalization into a positive equation with a bounded inverse.
Its exact Galerkin approximations converge; a certified numerical
implementation would require additional quantitative estimates.
Only after these
proofs do we compare the coefficients with the known magic
functions and inherit their sign inequalities.

We give the detailed argument in dimension \(24\) in
Sections~\ref{sec:system}--\ref{sec:operator}.
Section~\ref{sec:eight} then proves the dimension-eight theorem:
it specifies the changed coefficient system, checks the boundary
and global equivalences, recomputes the index, and supplies its
vanishing proof. Section~\ref{sec:signs} returns to the missing
independent sign estimates in both dimensions.

The variables used along the way belong to different spaces.
The real arguments \(u,v\) describe \(\Phi,G\). A complex
parameter \(z\) describes Gaussians. A complex variable \(q\)
is the argument of a generating series, while \(w\) is a
coordinate in the contour plane. Maps between these planes have
separate names, and each integral specifies which parameters
are held fixed. Later \(\tau\in\mathbb H\) is the upper-half-plane
coordinate and \(\mathfrak q=e^{2\pi i\tau}\) is the cusp coordinate.
For reference, the coordinate maps are
\[
 \begin{array}{c|l}
 u,v&\text{independent real arguments of the two radial profiles}\\
 z&\text{complex Gaussian parameter}\\
 q=Q(z)=e^{-\pi(z-1)}&\text{argument of the coefficient series}\\
 w=\omega(z)=e^{-2\pi/z}&\text{coordinate in the contour plane}\\
 \tau=iz,\quad\cuspq=e^{2\pi i\tau}
       &\text{upper-half-plane and modular cusp coordinates}.
 \end{array}
\]
In particular, \(q=e^\pi\cuspq^{1/2}\) under these substitutions.
It is not an identification of the two independent series variables.

Bold lowercase letters denote vectors; sans serif uppercase
letters denote linear operators and finite matrices. Calligraphic
letters denote the principal indexing sets and geometric objects.
Scalar functions use ordinary or Greek letters, retaining the
standard notation for Eisenstein series and the elementary
families \(P,U,V\). A superscript \(T\) denotes transpose,
\(*\) denotes an adjoint, and a subscript \(*\) denotes a
normalized solution. Hats denote Fourier transforms and primes
denote derivatives. Tildes mark candidate functions; the
superscript \(\sharp\), used only in Appendix~\ref{app:K},
marks a holomorphic average. Auxiliary matrices have descriptive
subscripts: \(\mathrm{pair}\) for paired traces, \(\mathrm{frm}\)
for the six-component frame, \(\mathrm{cell}\) for recovery of four
components, and \(\mathrm{rec},\mathrm{match}\) for recovery and
matching of six components. A parenthesized dimension distinguishes
the latter matrices when their powers depend on \(d\). The permanent
gluing matrices are \(\mathsf J_{\mathrm{tr}},\mathsf J_{\mathrm{inv}}\).

\section{The coefficient system in dimension 24}\label{sec:system}

From here through Section~\ref{sec:operator}, the dimension is fixed at
\(24\). Section~\ref{sec:eight} gives the dimension-eight version.

\subsection{Elementary functions and the zeros they enforce}

We now choose the functions whose coefficients will be the
unknowns. Throughout, \(\mathbb N=\{1,2,\ldots\}\)
and \(\mathbb N_0=\{0,1,2,\ldots\}\).
For \(u\geq0\), \(m\in\mathbb N\), and \(k=0,1,2\), define
\begin{align}
 U_{m,k}(u)&=e^{\pi m}P(u)\int_1^\infty
                 t^k e^{-\pi(m+u)t}\,dt,\label{eq:U}\\
 V_m(u)&=e^{\pi m}P(u)\int_0^1
                 t^{10}e^{-\pi(m/t+ut)}\,dt.\label{eq:V}
\end{align}
Here \(P(u)=\sin^2(\pi u/2)\) as above. The index \(m\) labels
the exponential factor in the integral; it does not label a
point where a zero is prescribed. We reserve \(n\) for the
sample points \(2n\).
The factors \(e^{\pi m}\) rewrite the coefficient damping as
\(e^{-\pi m(t-1)}\) in the first integral and
\(e^{-\pi m(1/t-1)}\) in the second. This will allow us to
collect coefficients into power series on a unit disk.

There is no convergence exception at \(u=0\) in these definitions:
\(m\geq1\), so \(\int_1^\infty t^k e^{-\pi mt}\,dt\) is finite.
The reciprocal integral also converges there, since \(e^{-\pi m/t}\)
controls its integrand as \(t\downarrow0\). Both integrals are
analytic in a neighborhood of each \(u=2n\), including \(n=0\).
Multiplication by \(P\) therefore gives a zero of order at least
two at every nonnegative even integer. The zero index \(n=0\)
does not introduce a mode \(m=0\). We use the
following combinations:
\[
 U_{m,0}-2V_m\quad(m\text{ odd}),\qquad
 U_{m,0},\ U_{m,1},\ U_{m,2}+V_m\quad(m\text{ even}).
\]
These are the functions to which we assign independent
coefficients. In particular, one coefficient multiplies the
whole expression \(U_{m,0}-2V_m\) when \(m\) is odd,
and one multiplies \(U_{m,2}+V_m\) when \(m\) is even.
Appendix~\ref{app:motivation} explains the choice: Gaussian
inversion suggests the two integration ranges, and cancellation
of the unbounded contour integrals at the Fourier sample points
gives exactly these parity restrictions and combinations.
Section~\ref{sec:comparison} subsequently compares the family
with the known integral formula.

The common factor \(P\) imposes more zeros than we want:
\(\Phi(0)\) must be nonzero, neither \(\Phi(2)\) nor
\(\Phi'(2)\) is prescribed, and \(\Phi'(4)\) need not vanish.
To permit precisely these changes, together with an unrestricted
derivative at \(0\), define five correction functions:
\begin{align}
 R_{\ell,0}(u)&=\frac{P(u)e^{-\pi(u-\ell)}}{\pi(u-\ell)},
                  &&\ell=0,2,4,\label{eq:Hfinite}\\
 R_{\ell,1}(u)&=P(u)e^{-\pi(u-\ell)}
       \left(\frac1{\pi(u-\ell)}+\frac1{\pi^2(u-\ell)^2}\right),
                  &&\ell=0,2.\nonumber
\end{align}
These are definitions by explicit formulas and removable
continuation, not by possibly divergent integrals. For \(u>\ell\)
they agree with the expressions obtained formally from
\eqref{eq:U} at \(m=-\ell\), after the corresponding exponential
rescaling. At \(u=\ell\) their singularities are removable,
since \(P\) has a double zero. In particular,
\[
 \lim_{u\to0}\frac{R_{0,0}(u)}u=\frac{\pi}{4},
 \qquad R_{0,1}(0)=\frac14.
\]
At the respective removable points one also has
\[
 R_{\ell,0}(\ell)=0,\qquad R_{\ell,0}'(\ell)=\frac{\pi}{4},
 \qquad R_{\ell,1}(\ell)=\frac14,\qquad R_{\ell,1}'(\ell)=0.
\]
Thus the correction functions deliberately change the zero order;
they are not included in the double-zero assertion for \(m\geq1\). A simple denominator permits a nonzero
first derivative; a double denominator permits a nonzero
value. These five functions therefore allow a value and a
derivative at \(0\), a value and a derivative at \(2\), and
a derivative at \(4\). All other zeros required of \(\Phi\)
in \eqref{eq:desired-zeros} remain built into the family.
The point \(2\) is allowed because the Leech lattice has no
vectors of squared length \(2\).

\subsection{Unknown coefficients and their norms}

The coefficients are indexed by pairs \((j,m)\). The first
subscript specifies a generating series and the second its
exponent. Define the common set of positive-mode indices by
\[
 \mathcal I_+
 =\bigl(\{0,1,2\}\times2\mathbb N\bigr)
   \cup\bigl(\{3\}\times(2\mathbb N-1)\bigr),
\]
and the dimension-24 indexing set by
\[
 \mathcal I_{24}
 =\{(0,-4),(0,-2),(0,0),(1,-2),(1,0)\}\cup\mathcal I_+.
\]
A coefficient vector is a family
\(\mathbf c=(c_{j,m})_{(j,m)\in\mathcal I_{24}}\) of complex
numbers. Its five nonpositive modes multiply the correction
functions. Its positive modes give three sequences for even
\(m\) and one for odd \(m\). This indexing will also be used
in the generating series of Section~\ref{sec:boundary}.

For \(p=1,2\), define the norm by
\begin{align}
 \|\mathbf c\|_p^p={}&
 |c_{0,-4}|^p+|c_{0,-2}|^p+|c_{0,0}|^p
          +|c_{1,-2}|^p+|c_{1,0}|^p\nonumber\\
 &+\sum_{m\in2\mathbb N}\sum_{k=0}^2|c_{k,m}|^p
       +\sum_{m\in2\mathbb N-1}|c_{3,m}|^p. \label{eq:coefficient-norm}
\end{align}
For \(p=1,2\), write \(\ell^p(\mathcal I_{24})\) for the space of these
coefficient families with finite norm \(\|\mathbf c\|_p\).
Thus
\begin{align*}
 \ell^1(\mathcal I_{24})&=\mathbb C^5\oplus\ell^1(2\mathbb N)^3
                        \oplus\ell^1(2\mathbb N-1),\\
 \ell^2(\mathcal I_{24})&=\mathbb C^5\oplus\ell^2(2\mathbb N)^3
                        \oplus\ell^2(2\mathbb N-1).
\end{align*}
We equip \(\ell^2(\mathcal I_{24})\) with its standard coefficient inner product
associated with the sum in \eqref{eq:coefficient-norm}.
Every sequence space below displays its indexing set.
Lebesgue spaces of functions, used later for boundary values,
are denoted by \(L^p\).
The construction seeks a vector in \(\ell^1(\mathcal I_{24})\). Working
temporarily in the larger Hilbert space \(\ell^2(\mathcal I_{24})\) permits the
use of adjoints and compact operators. We will prove that
every exact solution in \(\ell^2(\mathcal I_{24})\) already belongs to \(\ell^1(\mathcal I_{24})\).

For \(\mathbf c\in\ell^2(\mathcal I_{24})\), define a function
\(\Phi_{\mathbf c}:[0,\infty)\to\mathbb C\) by the following
series, with independent argument \(u\geq0\):
\begin{align}
 \Phi_{\mathbf c}(u)={}&
    c_{1,0}R_{0,1}(u)+c_{0,0}R_{0,0}(u)\nonumber\\
 &+e^{-2\pi}c_{1,-2}R_{2,1}(u)
    +e^{-2\pi}c_{0,-2}R_{2,0}(u)
    +e^{-4\pi}c_{0,-4}R_{4,0}(u)\nonumber\\
 &+\sum_{m\in2\mathbb N-1}c_{3,m}\bigl(U_{m,0}(u)-2V_m(u)\bigr)
      \nonumber\\
 &+\sum_{m\in2\mathbb N}
       \bigl(c_{0,m}U_{m,0}(u)+c_{1,m}U_{m,1}(u)
                       +c_{2,m}(U_{m,2}(u)+V_m(u))\bigr).
 \label{eq:ansatz}
\end{align}
Lemma~\ref{lem:Schwartz} proves that this series defines a
radial Schwartz function
\[
 f_{\mathbf c}(\mathbf y)=\Phi_{\mathbf c}(|\mathbf y|^2),\qquad \mathbf y\in\mathbb R^{24}.
\]
Using \eqref{eq:Fourier-convention}, let \(G_{\mathbf c}\) be the
unique function on \([0,\infty)\) satisfying
\[
 \widehat f_{\mathbf c}(\boldsymbol{\xi})=G_{\mathbf c}(|\boldsymbol{\xi}|^2),
                  \qquad \boldsymbol{\xi}\in\mathbb R^{24}.
\]
Thus \(\Phi_{\mathbf c}\) is specified by the coefficients,
whereas \(G_{\mathbf c}\) is obtained from it by Fourier
transformation. Both depend linearly on \(\mathbf c\).

Only \(R_{0,1}\) in \eqref{eq:ansatz} is nonzero at \(u=0\),
and \(R_{0,1}(0)=1/4\). It is therefore convenient to name the
normalizing linear functional
\begin{equation}
 \lambda(\mathbf c)=c_{1,0}/4=\Phi_{\mathbf c}(0).
 \label{eq:lambda}
\end{equation}

\subsection{The equations and the main theorem}

The required zeros of \(\Phi_{\mathbf c}\) are automatic from
the choice of functions. The unknown coefficients must make
the Fourier profile satisfy
\begin{equation}
 G_{\mathbf c}(2n)=G_{\mathbf c}'(2n)=0\quad(n\geq2),
       \qquad \lambda(\mathbf c)=1. \label{eq:system}
\end{equation}
Here is the linear system explicitly. In this paragraph only,
enumerate the entries of \(\mathbf c\) by \(j\in\mathbb N\),
write \(c_j\) for the corresponding scalar, and let
\(\mathbf e_j\) be the coefficient family with that entry
equal to one and all other entries zero. Put
\(g_j=G_{\mathbf e_j}\); these are fixed functions, obtained
by Fourier transforming the individual terms of
\eqref{eq:ansatz}. The equations are
\begin{equation}
 \begin{gathered}
 \sum_{j\geq1}g_j(2n)c_j=0,\qquad
 \sum_{j\geq1}g_j'(2n)c_j=0\quad(n\geq2),\\
 \sum_{j\geq1}\lambda(\mathbf e_j)c_j=1.
 \end{gathered}
 \label{eq:matrix-system}
\end{equation}
Thus \(g_j(2n)\) and \(g_j'(2n)\) are matrix entries,
not unknowns. Only the \(c_j\) are unknown. The last row
has a single nonzero entry, \(1/4\), in the position of
\(c_{1,0}\). Lemma~\ref{lem:Schwartz} justifies the
series and evaluation term by term. Lemma~\ref{lem:samples}
will express the two types of Fourier rows as contour integrals.

\begin{theorem}[Coefficient construction]\label{thm:main}
The system \eqref{eq:system} has exactly one solution
\(\mathbf c_*\) in \(\ell^2(\mathcal I_{24})\). Its entries are real, and it belongs
to \(\ell^1(\mathcal I_{24})\). More precisely, for every \(1<r<e^\pi\),
\[
 \sum_{m\in2\mathbb N}
       (|c_{0,m}|+|c_{1,m}|+|c_{2,m}|)r^m
       +\sum_{m\in2\mathbb N-1}|c_{3,m}|r^m<\infty.
\]
The associated profiles satisfy
\begin{align*}
 \Phi_{\mathbf c_*}(0)&=G_{\mathbf c_*}(0)=1,&
 \Phi_{\mathbf c_*}(2n)&=0 &&(n\geq2),\\
 \Phi_{\mathbf c_*}'(4)&=-\frac1{65520},&
 \Phi_{\mathbf c_*}'(2n)&=0 &&(n\geq3).
\end{align*}
Before imposing \(\lambda(\mathbf c)=1\), the Fourier equations
have a one-dimensional solution space in \(\ell^2(\mathcal I_{24})\).
\end{theorem}

Poisson summation will force
\(c_{0,-4}=-e^{4\pi}/(16380\pi)\) for the normalized solution.
Its correction part can therefore be written
\[
 4R_{0,1}-\frac{R_{4,0}}{16380\pi}
 +c_{0,0}R_{0,0}
 +e^{-2\pi}c_{1,-2}R_{2,1}
 +e^{-2\pi}c_{0,-2}R_{2,0}.
\]
Only the three coefficients \(c_{0,0},c_{1,-2},c_{0,-2}\)
and the four positive-mode sequences remain unknown.

Section~\ref{sec:analytic} proves the regularity of
\(f_{\mathbf c}\) and the Poisson identities.
Sections~\ref{sec:boundary}--\ref{sec:vanishing} prove the
coefficient theorem. Section~\ref{sec:operator} gives an
equivalent equation with a bounded inverse. The comparison
with the known magic function follows in Section~\ref{sec:comparison};
Section~\ref{sec:eight} then proves the dimension-eight theorem.

\section{The coefficient map and Poisson normalization}\label{sec:analytic}

Before manipulating Fourier samples, we justify the series defining
\(f_{\mathbf c}\). We also show that distinct coefficient vectors
give distinct functions. The second part of this section explains
why the two normalizations at the origin and the first derivative
at \(u=4\) are tied together.

\begin{lemma}\label{lem:Schwartz}
The map \(\mathbf c\mapsto f_{\mathbf c}\) defined by \eqref{eq:ansatz} is a continuous
injective linear map from \(\ell^2(\mathcal I_{24})\) to the radial Schwartz space on
\(\mathbb R^{24}\). For every \(\mathbf c\in\ell^2(\mathcal I_{24})\),
\begin{align}
 \Phi_{\mathbf c}(0)&=\lambda(\mathbf c),&
 \Phi_{\mathbf c}(2n)&=0\quad(n\geq2),\nonumber\\
 \Phi_{\mathbf c}'(4)&=\frac{\pi}{4}e^{-4\pi}c_{0,-4},&
 \Phi_{\mathbf c}'(2n)&=0\quad(n\geq3).\label{eq:physical-jets}
\end{align}
\end{lemma}
\begin{proof}
For \(k=0,1,2\), integration gives
\[
 U_{m,k}(u)=P(u)e^{-\pi u}
       \sum_{j=0}^k\frac{k!}{(k-j)!}
                         \frac1{[\pi(m+u)]^{j+1}}.
\]
For \(u\geq0\), each derivative of the factor after \(P\) is bounded
by a constant times \(e^{-\pi u}/m\). Cauchy--Schwarz and
\(\sum_{m\geq1}m^{-2}<\infty\) control all corresponding series.

Write \(d_m=c_{2,m}\) for even \(m\) and \(d_m=-2c_{3,m}\) for odd
\(m\). The remaining sum is
\[
 P(u)\int_0^1t^{10}e^{-\pi ut}
          \sum_{m\geq1}d_me^{-\pi m(1/t-1)}\,dt.
\]
The inner sum is bounded by \(C\|\mathbf c\|_2 e^{-\pi/t}\) for
\(0<t\leq1/2\), and by \(C\|\mathbf c\|_2(1-t)^{-1/2}\) for
\(1/2<t<1\). Differentiation in \(u\) adds only powers of \(t\)
and bounded derivatives of \(P\). The inequality
\(t^{-1}+ut\geq2\sqrt u\) then bounds each derivative, for \(u\geq1\),
by
\[
 C_j\|\mathbf c\|_2\bigl(e^{-2\pi\sqrt u}+e^{-\pi u/2}\bigr).
\]
The five functions \(R_{\ell,k}\) in the finite sum have removable
singularities and Gaussian decay.
Each spatial derivative of \(\Phi_{\mathbf c}(|\mathbf y|^2)\)
is a finite sum of a polynomial in \(\mathbf y\) times a derivative
of \(\Phi_{\mathbf c}\). The displayed decay absorbs all such
polynomials and all polynomial weights in the Schwartz seminorms.
These estimates therefore prove continuity into Schwartz space. The double
zeros of \(P\), together with
\(R_{0,1}(0)=1/4\) and
\(R_{4,0}'(4)=\pi/4\), prove
\eqref{eq:physical-jets}.

For injectivity, continue the profile meromorphically in \(u\).
For each compact set \(K\subset\mathbb C\), the inequality
\(1/t-1\geq1-t\), \(0<t\leq1\), gives
\[
 \sup_{u\in K}|V_m(u)|\leq C_K
       \int_0^1e^{-\pi m(1-t)}\,dt\leq C_K/(\pi m).
\]
Thus the \(V_m\) series converges normally by Cauchy--Schwarz
and is entire. The rational formula for \(U_{m,k}\) gives the
same \(C_K/m\) bound on compact sets avoiding its poles, with
finitely many small \(m\) treated separately. Its series therefore
converges normally away from the negative integers.
If the profile vanishes, its residues at \(u=-m\), for odd and
even positive \(m\), respectively, give
\[
 \frac{e^{\pi m}}{\pi}c_{3,m}=0,\qquad
 \frac{e^{\pi m}}{2\pi}c_{2,m}=0.
\]
No \(V_m\) terms remain. Divide by \(P(u)e^{-\pi u}\).
The double and simple principal parts at each negative even integer
give \(c_{1,m}=c_{0,m}=0\); the principal parts at \(0,2,4\)
give the five coefficients of the correction functions.
Thus \(\mathbf c=0\).
\end{proof}

Let \(\Lambda\subset\mathbb R^{24}\) be the Leech lattice, scaled
so that a fundamental domain has volume one. For \(n\in\mathbb N\),
let \(N_n=\#\{\mathbf y\in\Lambda:|\mathbf y|^2=2n\}\).
Its dual lattice is
\(\Lambda^*=\{\boldsymbol{\xi}\in\mathbb R^{24}:\boldsymbol{\xi}\cdot \mathbf y\in\mathbb Z
\text{ for all }\mathbf y\in\Lambda\}\).
We use \(\Lambda^*=\Lambda\), the fact that all squared norms
are even integers, and
\(N_1=0,\ N_2=196560\); see \cite[Appendix B.1]{CK}.

\begin{lemma}[Poisson normalization]\label{lem:Poisson}
For every \(\mathbf c\in\ell^2(\mathcal I_{24})\),
\begin{align}
 G_{\mathbf c}(0)+\sum_{n\geq2}N_nG_{\mathbf c}(2n)&=\lambda(\mathbf c),\label{eq:Poisson1}\\
 \pi N_2 e^{-4\pi}c_{0,-4}+12\lambda(\mathbf c)
       &=-\sum_{n\geq2}2nN_nG_{\mathbf c}'(2n).\label{eq:Poisson2}
\end{align}
Consequently every solution of the homogeneous sample equations
satisfies
\begin{equation}
 G_{\mathbf c}(0)=\lambda(\mathbf c),\qquad
 c_{0,-4}=-\frac{e^{4\pi}}{16380\pi}\lambda(\mathbf c).
 \label{eq:normalizations}
\end{equation}
\end{lemma}
\begin{proof}
For a real dilation parameter \(t>0\), apply Poisson summation
to \(\mathbf y\mapsto f_{\mathbf c}(\sqrt t\,\mathbf y)\):
\[
 \Phi_{\mathbf c}(0)+\sum_{n\geq2}N_n\Phi_{\mathbf c}(2nt)
 =t^{-12}\left(G_{\mathbf c}(0)+\sum_{n\geq2}N_nG_{\mathbf c}(2n/t)\right).
\]
Schwartz regularity justifies differentiation locally uniformly in
\(t>0\). Evaluate at \(t=1\), and then differentiate and evaluate
there, using \eqref{eq:physical-jets}. The identities follow;
\(N_2/12=16380\) gives \eqref{eq:normalizations}.
\end{proof}

\section{From Fourier samples to a boundary problem}\label{sec:boundary}

This section replaces the Fourier equations by a problem in one
complex variable. Its first objective is the following identity.
The measures in the statement are defined explicitly in
\eqref{eq:measures}; their supports are independent of the
coefficient vector. Their dependence on that vector is linear.

\begin{lemma}[Fourier samples as contour integrals]\label{lem:samples}
For each \(\mathbf c\in\ell^2(\mathcal I_{24})\), let \(\mu_0,\mu_1\) be the
finite complex measures constructed in \eqref{eq:measures}.
For every integer \(n\geq0\),
\begin{equation}
 G_{\mathbf c}(2n)=\frac1{8\pi}\int_{\mathbb C}w^{n-1}\,d\mu_0(w),\qquad
 G_{\mathbf c}'(2n)=-\frac18\int_{\mathbb C}w^{n-1}\,d\mu_1(w).
 \label{eq:samples}
\end{equation}
The integrals converge absolutely, including when \(n=0\).
\end{lemma}

Thus the equations for \(n\geq2\) are exactly the vanishing
of the moments of positive degree of these measures. Once the
lemma has been proved, we package the moments as an operator
and use Cauchy integrals to express their vanishing as boundary
conditions for holomorphic functions.

\subsection{Gaussian parameters and generating series}

For each fixed \(z\in\mathbb C\) with \(\Re z>0\), define
\(g_z(\mathbf y)=e^{-\pi z|\mathbf y|^2}\), \(\mathbf y\in\mathbb R^{24}\).
Gaussian Fourier inversion gives, for every fixed
\(\boldsymbol{\xi}\in\mathbb R^{24}\),
\begin{equation}
 \widehat g_z(\boldsymbol{\xi})
 =\int_{\mathbb R^{24}}e^{-\pi z|\mathbf y|^2}e^{-2\pi i \mathbf y\cdot\boldsymbol{\xi}}\,d\mathbf y
 =z^{-12}e^{-\pi|\boldsymbol{\xi}|^2/z}.
 \label{eq:Gaussian}
\end{equation}
Only \(\mathbf y\) is integrated in this formula. Equivalently, the
one-variable function giving its radial Fourier values is
\(v\mapsto z^{-12}e^{-\pi v/z}\), for an independent real
argument \(v\geq0\). In the contour calculations below we
hold \(v\) fixed and integrate in \(z\). We then evaluate at
\(v=2n\), or differentiate with respect to \(v\) and evaluate
there. No frequency vector \(\boldsymbol{\xi}\) occurs in those calculations.

Define two maps, with different roles:
\begin{equation}
 Q(z)=e^{-\pi(z-1)}\quad(z\in\mathbb C),\qquad
 \omega(z)=e^{-2\pi/z}\quad(z\in\mathbb C\setminus\{0\}).
 \label{eq:parameter-maps}
\end{equation}
The map \(Q\) supplies arguments for generating series; the
map \(\omega\) supplies points of the contour plane. In
particular \(e^{-\pi(2n)/z}=\omega(z)^n\), which is the source
of the powers in \eqref{eq:samples}. We reserve \(q\) for an
independent complex argument of a generating series and \(w\)
for an independent complex coordinate in the contour plane.
Neither letter denotes a map.

Fix \(\mathbf c\in\ell^2(\mathcal I_{24})\) for the definitions that follow.
All generating series, densities, and measures depend on this
fixed vector; we omit that dependence from their subscripts.
For \(0<|q|<1\), define
\begin{align}
 A_0(q)&=c_{0,-4}q^{-4}+c_{0,-2}q^{-2}+c_{0,0}
       +\sum_{m\in2\mathbb N}c_{0,m}q^m,\nonumber\\
 A_1(q)&=c_{1,-2}q^{-2}+c_{1,0}
       +\sum_{m\in2\mathbb N}c_{1,m}q^m,\nonumber\\
 A_2(q)&=\sum_{m\in2\mathbb N}c_{2,m}q^m,\nonumber\\
 A_3(q)&=\sum_{m\in2\mathbb N-1}c_{3,m}q^m.
 \label{eq:series}
\end{align}
These series already contain four of the six functions that will
represent a global section of the bundle. More precisely, for a
complex variable \(\tau\) with \(\Im\tau>1\), substitute
\(q=Q(-i\tau)=e^\pi e^{\pi i\tau}\) and form
\[
 \bigl(A_0(q),-iA_1(q),-A_2(q),A_3(q)\bigr).
\]
When the Fourier equations are satisfied, these four functions
continue holomorphically to the upper half-plane and become the
first four components of a section of the rank-six bundle over
the compactified modular curve, with the orbifold and cusp
conditions described in Section~\ref{sec:bundle}. The other two
components come from the Cauchy integrals. This assertion is
proved in Proposition~\ref{prop:equivalence}; it is not an
additional assumption on the series. The components are
coordinate functions of a bundle section, rather than four
separately invariant functions on the modular curve.

Set \(A_-(q)=A_2(q)-2A_3(q)\) and
\(A_+(q)=A_2(q)+2A_3(q)\).
The first three series are even functions of \(q\); the fourth
is odd. Define, for \(\Re z>1\),
\begin{align}
 b_0(z)&=A_0(Q(z))+zA_1(Q(z))+z^2A_2(Q(z))+A_3(Q(z)),\nonumber\\
 b_1(z)&=A_1(Q(z))+2zA_2(Q(z)).
 \label{eq:BL}
\end{align}
These evaluations are within the convergence disk because
\(|Q(z)|<1\) precisely when \(\Re z>1\).

We also need values on the line \(\Re z=1\). They are initially
boundary limits, not values of a continuation already known to
exist. Indeed, \(Q(\sigma+it)=e^{-\pi(\sigma-1)}e^{-\pi it}\).
As \(\sigma\downarrow1\), square summability and Parseval's
identity give convergence in \(L^2\) on bounded \(t\)-intervals
of each series and of \(b_0,b_1\). These \(L^2\) limits are called
\emph{boundary traces}. Evaluations of these functions on
\(\Re z=1\) below mean these traces, defined almost everywhere.

\subsection{The parametrized contours and the measures}

Define paths in the Gaussian-parameter plane by
\[
 z_+(t)=1+it,\qquad z_-(t)=1-i+it,\qquad 0\leq t\leq1,
\]
and paths in the contour plane by
\[
 \gamma_+(t)=\omega(z_+(t)),\qquad
 \gamma_-(t)=\omega(z_-(t)),\qquad 0\leq t\leq1.
\]
Put \(w_0=e^{-2\pi}\) and \(w_c=-e^{-\pi}\).
As \(t\) increases, \(\gamma_+\) runs from \(w_0\) to
\(w_c\) in the upper half-plane, and \(\gamma_-\) runs from
\(w_c\) to \(w_0\) in the lower half-plane. Explicitly,
\(\gamma_-(t)=\overline{\gamma_+(1-t)}\); this formula
specifies the reversal of direction in the conjugate path.
Let \(\Gamma_\pm=\gamma_\pm([0,1])\) be their images, and
let \(\Gamma=\Gamma_+\cup\Gamma_-\) be the resulting set.

The set \(\Gamma\) is a Jordan curve. For example, along
\(\gamma_+\) the argument \(2\pi t/(1+t^2)\) increases from
\(0\) to \(\pi\), which proves simplicity on that half.
Let \(\Omega\) be the bounded component of
\(\mathbb C\setminus\Gamma\). It contains \(0\).
Traversing \(\gamma_+\) followed by \(\gamma_-\) gives the
positive orientation of its boundary. Whenever we integrate
over \(\Gamma\), we mean the sum of the integrals along
these two parametrized paths.

The two analytic arcs meet tangentially at \(w_c\), with the
direction of traversal reversing along their common tangent.
This is the nonsmooth point of the Jordan curve, called its
\emph{contour cusp} below. The curve is smooth at \(w_0\),
but an additional segment will meet it there. Set
\(I=(0,w_0)\subset\mathbb R\). Integrals along this segment
use the path \(s\mapsto s\), \(0<s<w_0\), directed from
\(0\) towards \(w_0\). Thus \(ds\) in those integrals is
ordinary real integration with the displayed limits.

Figure~\ref{fig:contour} shows the two arcs and the segment,
with arrows indicating these directions of integration.
\begin{figure}[htbp]
 \centering
 \begin{tikzpicture}[x=11cm,y=11cm,>=latex,font=\small]
  % A uniform enlargement by exp(pi) preserves the contour's shape.
  \coordinate (wc) at (-1,0);
  \coordinate (wzero) at ({exp(-pi)},0);
  \coordinate (origin) at (0,0);
  \draw[gray!55,thin,->] (-1.065,0)--(.235,0)
                node[right,black] {$\Re w$};
  \draw[gray!55,thin,->] (0,-.135)--(0,.155)
                node[above,black] {$\Im w$};
  \draw[thick,postaction={decorate},
        decoration={markings,mark=at position .55 with {\arrow{>}}}]
   plot[domain=0:1,samples=201,variable=\t]
    ({exp(pi-2*pi/(1+\t*\t))*cos(360*\t/(1+\t*\t))},
     {exp(pi-2*pi/(1+\t*\t))*sin(360*\t/(1+\t*\t))});
  \draw[thick,postaction={decorate},
        decoration={markings,mark=at position .45 with {\arrow{>}}}]
   plot[domain=1:0,samples=201,variable=\t]
    ({exp(pi-2*pi/(1+\t*\t))*cos(360*\t/(1+\t*\t))},
     {-exp(pi-2*pi/(1+\t*\t))*sin(360*\t/(1+\t*\t))});
  \draw[very thick,->] (origin)--(wzero);
  \node at (-.16,.116) {$\Gamma_+$};
  \node at (-.16,-.116) {$\Gamma_-$};
  \node at (-.36,0) [fill=white,inner sep=2pt] {$\Omega$};
  \fill (wc) circle[radius=1.1pt];
  \fill (origin) circle[radius=1.1pt];
  \fill (wzero) circle[radius=1.1pt];
  \node[below=5pt] at (wc) {$w_c=-e^{-\pi}$};
  \node[below left=3pt] at (origin) {$0$};
  \draw[thin] (wzero)--(.09,.065)
                       node[right] {$w_0=e^{-2\pi}$};
  \draw[thin] ({exp(-pi)/2},0)--(.09,-.065)
                       node[right] {$I=(0,w_0)$};
 \end{tikzpicture}
 \caption{The contour \(\Gamma=\Gamma_+\cup\Gamma_-\) and the
 segment \(I\), drawn with a uniform enlargement. The upper arc
 runs from \(w_0\) to \(w_c\), the lower arc returns to \(w_0\),
 and \(I\) is directed from \(0\) towards \(w_0\).
 The arcs meet tangentially at the cusp \(w_c\).}
 \label{fig:contour}
\end{figure}

Define the boundary densities \(\rho_{0,\Gamma},\rho_{1,\Gamma}\) almost everywhere
on \(\Gamma\) by the following four formulas. Every occurrence
of \(b_0,b_1,A_j,A_\pm\) has its argument displayed:
\begin{align}
 \rho_{0,\Gamma}(\gamma_+(t))
   &=z_+(t)^{-10}\bigl[b_0(z_+(t))-ib_1(z_+(t))\bigr],\nonumber\\
 \rho_{1,\Gamma}(\gamma_+(t))
   &=z_+(t)^{-11}\bigl[b_0(z_+(t))-ib_1(z_+(t))
           +iz_+(t)A_+(Q(z_+(t)))\bigr],\nonumber\\
 \rho_{0,\Gamma}(\gamma_-(t))
   &=-z_-(t)^{-10}\bigl[b_0(z_-(t))+ib_1(z_-(t))\bigr],\nonumber\\
 \rho_{1,\Gamma}(\gamma_-(t))
   &=-z_-(t)^{-11}\bigl[b_0(z_-(t))+ib_1(z_-(t))
           -iz_-(t)A_+(Q(z_-(t)))\bigr].
 \label{eq:densities}
\end{align}
Here \(0<t<1\), and all boundary evaluations have the trace
meaning specified above. Define densities on the real segment by
\[
 \rho_{0,I}(s)=2A_-(e^\pi\sqrt s),\qquad
 \rho_{1,I}(s)=-\pi^{-1}(\log s)\,A_-(e^\pi\sqrt s),\qquad 0<s<w_0,
\]
using the positive square root and the real logarithm.

We define the complex Borel measures \(\mu_\nu\), \(\nu=0,1\),
by specifying their integrals against bounded Borel functions
\(\varphi:\mathbb C\to\mathbb C\):
\begin{equation}
\begin{split}
 \int_{\mathbb C}\varphi(w)\,d\mu_\nu(w)
  ={}&\sum_{\sigma\in\{+,-\}}\int_0^1
    \varphi(\gamma_\sigma(t))\rho_{\nu,\Gamma}(\gamma_\sigma(t))
                              \gamma_\sigma'(t)\,dt\\
    &+\int_0^{w_0}\varphi(s)\rho_{\nu,I}(s)\,ds.
\end{split}
 \label{eq:measures}
\end{equation}
In the left integral \(w\) is a dummy complex variable;
on the right \(t,s\) are real integration variables.
This definition includes all orientation signs and makes the
meaning of the contour integrals precise. The measures have
no atoms and are supported in the compact set
\(\Gamma\cup[0,w_0]\).

The traces on the arcs are square integrable. On the segment,
Cauchy--Schwarz gives
\[
 |A_-(x)|\leq2\|\mathbf c\|_2\frac{x}{\sqrt{1-x^2}}
 \qquad(0<x<1).
\]
These estimates prove that the measures are finite and, for a
constant \(C_\mu\) independent of \(\mathbf c\),
\begin{equation}
 \|\mu_0\|_{\rm TV}+\|\mu_1\|_{\rm TV}
       \leq C_\mu\|\mathbf c\|_2.
 \label{eq:TV}
\end{equation}
The symbol \(\|\mu\|_{\rm TV}\) denotes the total variation
norm of a complex measure. Near \(s=0\), the two segment
densities are bounded by a constant times \(\sqrt s\) and
\(\sqrt s\,|\log s|\), respectively. Their products with
\(s^{-1}\) are integrable, which explains the convergence
at \(n=0\) in Lemma~\ref{lem:samples}.

\subsection{Proof of the sample identity}

\begin{proof}[Proof of Lemma~\ref{lem:samples}]
Use the identity
\(P(u)=(2-e^{\pi iu}-e^{-\pi iu})/4\) in the defining
integrals of the elementary functions.
Apply \eqref{eq:Gaussian} to the three Gaussian terms in
each elementary function and shift their integration paths
by \(i\) and \(-i\). The argument \(v\geq0\) remains fixed
during these operations. To record the result, define
\(J_{m,k}(v)\) for integers \(m\), \(k=0,1,2\), and
\(W_m(v)\) for positive integers \(m\), by
\[
 J_{m,k}(v)=\int_1^{1+i}(z-i)^kz^{-12}
               e^{-\pi m(z-1)-\pi v/z}\,dz,\qquad
 W_m(v)=\frac{e^{-\pi v}}{2\pi(m+v)}.
\]
The first integral follows the straight path \(z_+\) defined
above; \(z\) is its integration variable. For an elementary
function \(h(u)\) in the left column below, Fourier transform
the radial function \(\mathbf y\mapsto h(|\mathbf y|^2)\). Its value at any
frequency vector of squared length \(2n\) is the right column:
\[
\begin{array}{c|c}
 h(u)&\text{Fourier value at }v=2n\\ \hline
 U_{m,0}-2V_m\quad(m\text{ odd})
     &\tfrac12\Re J_{m,0}(2n)-2W_m(2n)\\
 U_{m,0}\quad(m\text{ even})&\tfrac12\Re J_{m,0}(2n)\\
 U_{m,1}\quad(m\text{ even})&\tfrac12\Re J_{m,1}(2n)\\
 U_{m,2}+V_m\quad(m\text{ even})
     &\tfrac12\Re\bigl(J_{m,2}(2n)+J_{m,0}(2n)\bigr)+W_m(2n)
\end{array}
\]
For the Fourier derivative, first replace each \(2n\) in
the right column by the independent variable \(v\),
differentiate that expression, and then set \(v=2n\).
Add \(\pi\Im J_{m,0}(2n)\) in the first row and
\(\frac\pi2\Im J_{m,0}(2n)\) in the last row.
These extra terms come from differentiating the phase factors
before evaluating at an even integer; the other rows need no
correction. They account for \(A_+\) in \eqref{eq:densities}.
The identities before specialization to \(v=2n\), including
the phases that produce these corrections, are
\eqref{eq:app-U-transform}--\eqref{eq:app-V-transform} in
Appendix~\ref{app:motivation}.
Real and imaginary parts are taken for the individual real
elementary functions before multiplying by complex coefficients.

For the five correction functions, direct integration gives
\[
 R_{\ell,k}(u)=\frac12\Re\int_1^{1+i}
                   (z-i)^ke^{-\pi(u-\ell)z}\,dz.
\]
Here \(u\) is fixed and \(z\) is again integrated along
\(z_+\). Their Fourier values are
\(e^{\pi\ell}\Re J_{-\ell,k}(2n)/2\); differentiating the
same Fourier integral gives their Fourier derivatives.

Now substitute \(w=\omega(z)\) on each path.
The differential identity
\(\omega'(z)=2\pi\omega(z)z^{-2}\) turns these path
integrals into the arc terms of \eqref{eq:measures}.
For the remaining terms set \(d_m=c_{2,m}\) when \(m\)
is even and \(d_m=-2c_{3,m}\) when \(m\) is odd. Then
\[
 \sum_{m\geq1}d_mW_m(2n)
    =\frac1{4\pi}\int_0^{w_0}s^{n-1}A_-(e^\pi\sqrt s)\,ds.
\]
Before evaluation at \(v=2n\), the power is \(s^{v/2-1}\).
Differentiation with respect to \(v\) contributes
\((\log s)/2\), giving the second segment density.
This proves the lemma for coefficient families with finite
support. Continuity into Schwartz space, \eqref{eq:TV},
and the integrable bounds near \(s=0\) extend it to every
\(\mathbf c\in\ell^2(\mathcal I_{24})\).
In particular, for \(n=0\) the segment contribution is bounded
by \(C\|\mathbf c\|_2\int_0^{w_0/2}
s^{-1/2}(1+|\log s|)\,ds\) near zero, with a similar uniform
bound on the remaining segment. Thus this limiting step is
uniform in the coefficient norm even for the negative power
\(w^{-1}\) in that sample.
\end{proof}

\subsection{The moment operator and the unprescribed moments}

Allow \(\mathbf c\) to vary in the preceding construction.
Set \(\mathcal J_{24}=\{0,1\}\times\mathbb N\). The first
index records which measure is used and the second its moment
degree. Thus \(k\in\mathbb N\) means exactly \(k\geq1\);
the two moments of degree zero are omitted.
Define the moment operator
\(\Mom:\ell^2(\mathcal I_{24})\to\ell^2(\mathcal J_{24})\) by
\begin{equation}
 (\Mom\mathbf c)_{\nu,k}=\int_{\mathbb C}w^k\,d\mu_\nu(w),
 \qquad \nu=0,1,\quad k\geq1.
 \label{eq:A}
\end{equation}
Lemma~\ref{lem:samples} shows that \(\Mom\mathbf c=0\)
is equivalent to the Fourier equations in \eqref{eq:system},
without normalization: \(k=n-1\). The zeroth moments
\[
 m_{\nu,0}=\int_{\mathbb C}d\mu_\nu(w),\qquad \nu=0,1,
\]
are not among the equations. More specifically,
\(m_{0,0}=8\pi G_{\mathbf c}(2)\) and
\(m_{1,0}=-8G_{\mathbf c}'(2)\). They are determined by
\(\mathbf c\), but the system does not prescribe them as
additional data. This is the meaning of leaving them unrestricted.

The claimed target space follows from the total variation bound.
Both measures are supported in \(\{|w|\leq\rho\}\), where
\(\rho=e^{-\pi}<1\), and hence
\[
 \left|(\Mom\mathbf c)_{\nu,k}\right|
       \leq \rho^k\|\mu_\nu\|_{\mathrm{TV}}.
\]
Consequently
\[
 \|\Mom\mathbf c\|_{\ell^2(\mathcal J_{24})}^2
 \leq \frac{\rho^2}{1-\rho^2}
       \bigl(\|\mu_0\|_{\mathrm{TV}}^2+\|\mu_1\|_{\mathrm{TV}}^2\bigr)
 \leq \frac{C_\mu^2\rho^2}{1-\rho^2}
       \|\mathbf c\|_{\ell^2(\mathcal I_{24})}^2.
\]
This proves both square summability of the moments and
boundedness of the linear operator.

Write \(\Id_{\mathcal J_{24}}\) for the identity on
\(\ell^2(\mathcal J_{24})\).
For an integer \(J\geq1\), define the orthogonal projection
\(\mathsf Q_J\) on the moment sequence space by
\[
 (\mathsf Q_J\mathbf b)_{\nu,k}=
 \begin{cases}b_{\nu,k},&1\leq k\leq J,\\0,&k>J.
 \end{cases}
\]
Here \(\mathbf b=(b_{\nu,k})\in\ell^2(\mathcal J_{24})\).
This projection has rank \(2J\). The same geometric estimate gives
\begin{equation}
 \|(\Id_{\mathcal J_{24}}-\mathsf Q_J)\Mom\|\leq
       C_\mu\frac{\rho^{J+1}}{\sqrt{1-\rho^2}}.
 \label{eq:Acompact}
\end{equation}
Thus \(\Mom\) is a compact linear operator: the finite-rank
operators \(\mathsf Q_J\Mom\) converge to it in operator norm.
This compactness is useful, but is not an existence theorem
for the normalized equations. The continuation and index
arguments below supply existence.

\subsection{Cauchy functions, traces, and jumps}\label{sec:traces}

Fix \(\mathbf c\in\ell^2(\mathcal I_{24})\), and form its measures and their
zeroth moments as above. For an evaluation point
\(\zeta\in\mathbb C\setminus(\Gamma\cup[0,w_0])\), define
\begin{equation}
 h_\nu(\zeta)=\frac1{2\pi i}
       \left(\int_{\mathbb C}\frac{d\mu_\nu(w)}{w-\zeta}
                         +\frac{m_{\nu,0}}{\zeta}\right),\qquad \nu=0,1.
 \label{eq:Cauchy}
\end{equation}
We call these the \emph{corrected Cauchy transforms}, or
\emph{Cauchy functions}, of the coefficient vector. In this
integral \(\zeta\) is fixed and \(w\) is integrated. The correction
term removes the zeroth moment from the expansion at infinity.
Write \(\mathbf h=(h_0,h_1)^T\).

If \(\mathbf c\in\ker\Mom\), then for \(|\zeta|>\rho\)
the convergent expansion of \((w-\zeta)^{-1}\) gives
\[
 \int_{\mathbb C}\frac{d\mu_\nu(w)}{w-\zeta}
       =-\sum_{k\geq0}\zeta^{-k-1}
                    \int_{\mathbb C}w^k\,d\mu_\nu(w)
       =-m_{\nu,0}/\zeta.
\]
Hence \(h_\nu=0\) on the unbounded component of
\(\mathbb C\setminus\Gamma\), by analytic continuation.
Inside \(\Omega\), it is holomorphic away from the closed
segment \([0,w_0]\). Removing that segment gives a
\emph{slit domain}. At a point \(s\in I\) there are two
boundary approaches within this domain: from above and from
below the real axis. We denote their traces by
\(h_{\nu,+}(s)\) and \(h_{\nu,-}(s)\).
On \(\Gamma\), the interior trace is the limit approached
from \(\Omega\); the exterior trace is approached from the
unbounded component. All assertions about these limits are
local on open arcs away from \(0,w_0,w_c\). More precisely,
after straightening an arc by a holomorphic coordinate, the
restrictions to parallel lines converge in \(L^2\) on every
smaller closed interval. We use this meaning of local \(L^2\)
trace convergence throughout. The limits also exist almost
everywhere in the nontangential sense.
A nontangential approach stays inside a cone that makes a
positive angle with the boundary tangent.

The Cauchy jump formula then gives
\begin{align}
 h_{\nu,\mathrm{int}}(w)&=\rho_{\nu,\Gamma}(w)
                       &&(w\in\Gamma\setminus\{w_0,w_c\}),\nonumber\\
 h_{0,+}(s)-h_{0,-}(s)
       &=2A_-(e^\pi\sqrt s) &&(s\in I),\nonumber\\
 h_{1,+}(s)-h_{1,-}(s)
       &=-\pi^{-1}(\log s)\,A_-(e^\pi\sqrt s) &&(s\in I).
 \label{eq:jumps}
\end{align}
The first line uses the zero exterior trace. The next two
give the jump, meaning the upper trace minus the lower trace,
across the segment oriented towards \(w_0\).

For clarity, we recall the analytic facts used here. On the
real line directed from left to right, the Cauchy integral of
\(g\in L^2(\mathbb R)\) is
\[
 (\mathsf Cg)(\zeta)=\frac1{2\pi i}\int_{\mathbb R}
                              \frac{g(x)}{x-\zeta}\,dx,
                   \qquad \Im\zeta\ne0.
\]
Its upper and lower traces are
\(\mathsf C_+g=\mathsf P_+g\) and
\(\mathsf C_-g=-\mathsf P_-g\), where \(\mathsf P_\pm\)
are the orthogonal projections onto the subspaces with Fourier
transform supported in the positive and negative half-lines.
They are called \emph{Hardy projections}. Their ranges are
the boundary values of the Hardy spaces: holomorphic functions
on the respective half-planes whose squared integrals along
horizontal lines are uniformly bounded. Thus
\(\mathsf C_+-\mathsf C_-=\mathsf P_++\mathsf P_-=\Id_{L^2(\mathbb R)}\).
See \cite[Sections 8.2--8.3 and 8.7]{Srivastava} for the
boundary-value description and Cauchy representation.

For an open analytic arc, choose a smaller subarc, cut off
the density outside a slightly larger subarc, and straighten
the arc by a holomorphic coordinate. The transformed Cauchy
kernel differs from the real-line kernel by a bounded smooth
kernel; the omitted part of the contour contributes a
holomorphic function near the smaller subarc. The real-line
result therefore proves the local \(L^2\) trace and jump
statements used in \eqref{eq:jumps}. This is the local form
of the Plemelj formula; see \cite[Section 3.5, especially
(3.5.3)--(3.5.5)]{Trogdon}. No trace theorem at the nonsmooth
point \(w_c\) is being invoked.

\begin{lemma}[Gluing across an open arc]\label{lem:trace-gluing}
Let an analytic arc divide a neighborhood into two sides, and
let \(h_+,h_-\) be holomorphic on those sides. Suppose their
traces converge locally in \(L^2\), in the sense just specified,
to the same function. Then they are restrictions of one
holomorphic function on a neighborhood of the open arc.
\end{lemma}
\begin{proof}
Straighten the arc and restrict to a smaller rectangle.
Trace convergence bounds the \(L^2\) norms on parallel intervals
uniformly near the real axis. Cauchy--Schwarz followed by
integration in the perpendicular coordinate gives local area
integrability of the function \(h_{\mathrm{glue}}\) defined by
the two pieces. For a smooth compactly supported test function,
integrate by parts above the line of height \(r\) and below
the line of height \(-r\). Holomorphy leaves only the two boundary
terms. Their difference tends to zero by \(L^2\) trace convergence,
so \(\bar\partial h_{\mathrm{glue}}=0\) in distributions.
Local convolution with a smooth approximate identity gives
holomorphic functions converging to \(h_{\mathrm{glue}}\) in
local \(L^1\). The area mean inequality makes this convergence
uniform on smaller disks. Its limit is the required extension.
\end{proof}

For continuous matching values this is also the consequence
of Morera's theorem obtained by splitting triangles along the
arc; see \cite[Section 1]{Zeff}. In our applications the original
coefficient series have the asserted trace convergence by the
Hardy-space theorem, and the Cauchy functions have it by the
localized Plemelj formula above. We require no trace theorem at
a contour endpoint.

\subsection{Reconstructing the equations from boundary data}

We have proved that a coefficient solution gives holomorphic
Cauchy functions with the boundary conditions \eqref{eq:jumps}.
The reverse implication, needed for the existence proof, is
the following statement. Its endpoint assumptions prevent
additional singular terms that the jumps on open arcs cannot detect.
For \(\zeta_0\in\mathbb C\) and \(r>0\), write
\(D(\zeta_0,r)=\{w:|w-\zeta_0|<r\}\).

\begin{lemma}[Reconstruction from boundary values]\label{lem:reconstruction}
Fix \(\mathbf c\in\ell^2(\mathcal I_{24})\), with densities and measures
as in \eqref{eq:densities}--\eqref{eq:measures}.
Suppose \(\widetilde h_0,\widetilde h_1\) are holomorphic on
\(\Omega\setminus[0,w_0]\), have boundary limits converging
in \(L^2\) on every compact subarc away from the three endpoints,
and satisfy the outer trace and segment jump conditions
\eqref{eq:jumps}, with \(\widetilde h_\nu\) in place of \(h_\nu\).
Suppose there are constants \(\kappa_0,\kappa_1\) such that,
for \(\nu=0,1\),
\begin{align*}
 \lim_{r\downarrow0}
 \int_{\partial D(\zeta_0,r)\cap(\Omega\setminus[0,w_0])}
       |\widetilde h_\nu(w)|\,|dw|&=0 &&(\zeta_0=w_0,w_c),\\
 \lim_{r\downarrow0}
 \int_{\partial D(0,r)\setminus[0,w_0]}
       \left|\widetilde h_\nu(w)-\frac{\kappa_\nu}{2\pi iw}\right|\,|dw|&=0.
\end{align*}
Then \(\kappa_\nu=\int d\mu_\nu\), \(\mathbf c\in\ker\Mom\),
and \(\widetilde h_\nu\) equals the Cauchy function \eqref{eq:Cauchy}
on the slit domain.
\end{lemma}
\begin{proof}
Fix an evaluation point \(\zeta\) away from the contour and
the slit. First fix small disjoint disks about \(0,w_0,w_c\)
that avoid \(\zeta\), choosing their boundaries transverse to
the incident arcs. Exhaust the remaining slit domain from
inside by curves parallel to its smooth boundary pieces.
For fixed disk radii, Cauchy's formula passes to the boundary
by local \(L^2\), hence local \(L^1\), trace convergence.
The two banks of the slit are combined at this stage: their
difference is \(\rho_{\nu,I}\). There is no need to integrate
either bank separately all the way to an endpoint.

Now let the disk radii decrease to zero. Finite total variation
controls the omitted pieces of the arc and slit integrals.
The small-circle integrals at \(w_0,w_c\) tend to zero by
hypothesis, since \((w-\zeta)^{-1}\) is bounded there. At zero,
subtract the indicated pole before taking the limit. The
remainder has a vanishing circle integral, and the pole contributes
\(\kappa_\nu/(2\pi i\zeta)\).
Cauchy's formula has value zero at an exterior evaluation point
and value \(\widetilde h_\nu(\zeta)\) at an interior one.
It therefore gives
\[
 \frac1{2\pi i}\left(
       \int_{\mathbb C}\frac{d\mu_\nu(w)}{w-\zeta}
                       +\frac{\kappa_\nu}{\zeta}\right)
  =\begin{cases}
      \widetilde h_\nu(\zeta),&\zeta\in\Omega\setminus[0,w_0],\\
      0,&\zeta\in\mathbb C\setminus\overline\Omega.
    \end{cases}
\]
Expansion of the exterior expression at infinity first gives
\(\kappa_\nu=\int d\mu_\nu\) and then gives every positive
moment equal to zero. The interior expression is therefore
\eqref{eq:Cauchy}. The argument uses Cauchy's formula with
boundary limits, justified by the same local \(L^2\)
approximation used for the jump formula.
\end{proof}

We have reached the promised boundary formulation. To use it
for existence, we still need endpoint regularity, a global
continuation problem, and a proof that this global problem has
a nonzero solution. The next three sections address those tasks.

\section{Regularity at the boundary endpoints}\label{sec:regularity}

We will prove that a solution in \(\ell^2(\mathcal I_{24})\) has generating
functions that extend beyond the unit circle. Cauchy's
coefficient estimates will then give absolute summability.
The open arcs permit continuation by an invertible matrix.
What remains is to justify passage through the contour endpoints
\(w_0,w_c\). Most of this section concerns those two points.
The estimates will also supply the compact regularizing operator
used in Section~\ref{sec:operator}.

Fix \(\mathbf c\in\ker\Mom\), and use its Cauchy functions
\(\mathbf h\) from \eqref{eq:Cauchy}. We now work on open
sets of the Gaussian-parameter plane, not just along the paths
\(z_+,z_-\). The letter \(z\) continues to denote an independent
complex variable; write \(\sigma=\Re z\), \(t=\Im z\).
Define the open set
\[
 \mathcal D=\{\sigma+it:0<\sigma<1,\
             |t|<1-\sqrt{1-\sigma^2}\}.
\]
The composite Cauchy functions will be needed on a larger set than
\(\mathcal D\). Define
\[
 \mathcal W=\{z\in\mathbb C\setminus\{0\}:
                    \omega(z)\in\Omega\setminus[0,w_0]\}.
\]
On this entire open set define
\[
 \mathbf h_{\mathrm{par}}(z)=
 \begin{pmatrix}h_{0,\mathrm{par}}(z)\\h_{1,\mathrm{par}}(z)\end{pmatrix}
       =z^{10}\mathbf h(\omega(z)).
\]
The upper and lower halves of \(\mathcal D\) lie in
\(\mathcal W\) and map to the upper and lower halves of \(\Omega\)
under \(\omega\). We initially use these portions of the domain;
later occurrences of \(\mathbf h_{\mathrm{par}}(1/z)\) use the
same composite on \(\mathcal W\), with a separate domain check.
Thus the arguments of the Cauchy functions lie in their
holomorphic domains. Boundary values of \(\mathbf h_{\mathrm{par}}\) mean
traces from these halves. For each fixed \(t\in(0,1)\),
evaluate at \(z=z_+(t)\) and \(z_1=z_-(t)=z-i\).
Then \(Q(z_1)=-Q(z)\). At this point it is convenient to collect
the four series into the column vector
\[
 \mathbf a_{\mathrm{ser}}(q)
       =(A_0(q),A_1(q),A_2(q),A_3(q))^T.
\]
Using their parity to pair the outer traces at \(z\) and \(z_1\)
gives
\begin{equation}
 \begin{gathered}
 \begin{pmatrix}h_{0,\mathrm{par}}(z)\\h_{1,\mathrm{par}}(z)\\h_{0,\mathrm{par}}(z_1)\\h_{1,\mathrm{par}}(z_1)\end{pmatrix}
   =\mathsf S_{\mathrm{pair}}(z)\mathbf a_{\mathrm{ser}}(Q(z)),\\[4pt]
 \mathsf S_{\mathrm{pair}}(z)=
 \begin{pmatrix}
 1&z_1&z^2-2iz&1\\
 z^{-1}&z_1/z&z_1&z^{-1}+2i\\
 -1&-z&-z^2-1&1\\
 -z_1^{-1}&-z/z_1&-z&z_1^{-1}-2i
 \end{pmatrix},\qquad \det\mathsf S_{\mathrm{pair}}=4i.
 \end{gathered}
 \label{eq:paired-transfer}
\end{equation}
In this matrix \(z_1\) always means \(z-i\), including when
\(z\) varies off the boundary. The determinant is nonzero,
so its inverse recovers all four generating functions from
the Cauchy values. In a neighborhood to the left of each open
boundary point, those values are holomorphic. They have the
same traces as the original series on the right. The gluing
lemma \ref{lem:trace-gluing} therefore gives
continuation across \(z_+((0,1))\). Parity gives the corresponding
continuation across its translates by integer multiples of \(i\).
The points at the ends of these segments require separate work.

We also need two matching identities for \(\mathbf h_{\mathrm{par}}\).
In the first line below, \(z\) ranges over the open upper curved boundary
of \(\mathcal D\); the point \(z/(1+iz)\) is on the lower
curved boundary and has the same image in the \(w\)-plane.
In the second line, \(+\) and \(-\) denote limits from above
and below the real interval:
\begin{align}
 \mathbf h_{\mathrm{par}}(z)&=(1+iz)^{10}\mathbf h_{\mathrm{par}}\left(\frac{z}{1+iz}\right)
       &&(|z-i|=1),\label{eq:circle-transfer}\\
 \mathbf h_{\mathrm{par},+}(z)-\mathbf h_{\mathrm{par},-}(z)
    &=2z^{10}\Aminus(Q(1/z))
                       \begin{pmatrix}1\\z^{-1}\end{pmatrix}
       &&(0<z<1).\label{eq:real-transfer}
\end{align}
The first identity follows because \(\mathbf h\) is
holomorphic across the negative real axis in \(\Omega\);
it also identifies neighborhoods of the two open circle arcs.
The second is the segment jump \eqref{eq:jumps}. Its argument
\(1/z>1\) lies in the half-plane where the coefficient series
already converge.

\subsection{Area bounds and removal of a puncture}

We record the estimates before constructing any continuation.
They apply to arbitrary coefficient vectors; matching traces will
be used only when constructing a solution's holomorphic extension.
Throughout this section \(dA\) denotes ordinary planar area.

\begin{lemma}[Local area estimates]\label{lem:local-area}
For a bounded measurable set \(E\) whose closure avoids zero,
\[
 \int_E|h_\nu(\zeta)|\,dA(\zeta)
 \leq C_E\|\mathbf c\|_2,\qquad \nu=0,1.
\]
The same assertion holds for the compositions used below on fixed
coordinate neighborhoods, provided the coordinate maps are
one-to-one with nonzero derivatives and have images avoiding zero.
For each fixed \(0<r<1/2\), the original coefficient functions
also satisfy
\[
 \int_{\substack{|z-1|<r\\\Re z>1}}
       \sum_{j=0}^3|A_j(Q(z))|\,dA(z)\leq C_r\|\mathbf c\|_2 .
\]
\end{lemma}
\begin{proof}
For a finite measure \(\mu\), Tonelli's theorem and the layer-cake
formula for the radially decreasing kernel give
\[
 \int_E\left|\int\frac{d\mu(w)}{w-\zeta}\right|dA(\zeta)
 \leq\|\mu\|_{\mathrm{TV}}
       \sup_w\int_E|w-\zeta|^{-1}dA(\zeta)
 \leq2\sqrt{\pi\,\operatorname{area}(E)}\,\|\mu\|_{\mathrm{TV}}.
\]
The correction term \(m_{\nu,0}/\zeta\) has the required bound
because \(E\) avoids zero. Use \eqref{eq:TV} to obtain the first
assertion. Changes of variables with bounded Jacobian and
bounded multipliers give the assertion about compositions.
One may split a fixed neighborhood into finitely many coordinate
patches when necessary.

Writing \(x=\Re z-1>0\), Cauchy--Schwarz gives
\begin{equation}
 \left|\sum_{m\geq1}\beta_m e^{-\pi m(z-1)}\right|
 \leq\frac{\|\boldsymbol\beta\|_2}{\sqrt{e^{2\pi x}-1}}
 \leq\frac{\|\boldsymbol\beta\|_2}{\sqrt{2\pi x}}.
 \label{eq:radial-series-bound}
\end{equation}
The finitely many nonpositive modes are uniformly bounded by
\(C_r\|\mathbf c\|_2\) on this disk. Since \(x^{-1/2}\)
is area-integrable on its right half, the last assertion follows.
\end{proof}

\begin{lemma}[Removal using one radius]\label{lem:area-removal}
Let \(h\) be holomorphic and area-integrable on
\(0<|z-z_0|<r_0\). Then \(h\) has at most a simple pole at \(z_0\).
If \(t\,h(z_0+te^{i\theta_0})\to0\) on one radius as
\(t\downarrow0\), the singularity is removable. After removal,
an area-norm bound on the disk gives a uniform bound on every
fixed smaller disk.
\end{lemma}
\begin{proof}
Translate \(z_0\) to zero and let
\(m(r)=(2\pi)^{-1}\int_0^{2\pi}|h(re^{i\theta})|\,d\theta\).
Area integrability implies that some \(r_j\downarrow0\) has
\(r_j^2m(r_j)\to0\); otherwise \(\int_0^{r_0}r m(r)\,dr\)
would diverge. The Laurent coefficient of \(z^{-k}\) has
absolute value at most \(r_j^k m(r_j)\). All coefficients
with \(k\geq2\) therefore vanish. The condition on one radius
excludes the remaining coefficient of \(z^{-1}\). The final
claim is the area mean inequality applied to the resulting
holomorphic function on disks of a fixed smaller radius.
\end{proof}

\subsection{Continuation through the two endpoints}

\begin{proposition}[Uniform continuation]\label{prop:regularity}
There are constants \(R>1\) and \(C<\infty\), independent of \(\mathbf c\),
such that every \(\mathbf c\in\ker_{\ell^2(\mathcal I_{24})}\Mom\) has \(q^4\mathbf a_{\mathrm{ser}}(q)\)
holomorphic on a neighborhood of \(|q|\leq R\), and
\begin{equation}
 \sup_{|q|=R}\|q^4\mathbf a_{\mathrm{ser}}(q)\|_1\leq C\|\mathbf c\|_2.
 \label{eq:regularity}
\end{equation}
In particular \(\ker_{\ell^2(\mathcal I_{24})}\Mom\subset\ell^1(\mathcal I_{24})\).
\end{proposition}
\begin{proof}
\textit{The contour cusp.}\quad
Use a local parameter \(z\) near \(1\); the arguments \(z+i,z-i\)
approach the two ends of the contour at \(w_c=-e^{-\pi}\).
Write
\[
 D_r=D(1,r),\qquad
 D_r^{\mathrm{out}}=D_r\cap\{|z|>1\},\qquad
 D_r^{\mathrm{in}}=D_r\cap\{|z|<1\}.
\]
Initially, on \(\Re z>1\), put
\begin{align}
 b_{\mathrm{end}}(z)&=A_0(Q(z))+zA_1(Q(z))
                         +(z^2+1)A_2(Q(z))-A_3(Q(z)),\nonumber\\
 b_-(z)&=\Aminus(Q(z)). \label{eq:localHX}
\end{align}
On the two portions of \(D_{1/4}^{\mathrm{out}}\) with
\(\Re z<1\), define these two functions by solving
\begin{align}
 \mathbf h_{\mathrm{par}}(z+i)
    &=\begin{pmatrix}b_{\mathrm{end}}(z)\\
           b_{\mathrm{end}}(z)/(z+i)+ib_-(z)\end{pmatrix}
                                      &&(\Im z<0),\nonumber\\
 \mathbf h_{\mathrm{par}}(z-i)
    &=\begin{pmatrix}-b_{\mathrm{end}}(z)\\
           -b_{\mathrm{end}}(z)/(z-i)+ib_-(z)\end{pmatrix}
                                      &&(\Im z>0).
 \label{eq:local-lenses}
\end{align}
These are legitimate Cauchy evaluations. In the lower portion,
\(|z|>1\) implies
\(\Im z<-\sqrt{1-(\Re z)^2}\), so \(z+i\) lies in the
upper half of \(\mathcal D\); the upper portion is analogous.
The two equations are triangular and uniquely determine the
two functions. On each open segment \(\Re z=1\), substitution
of the outer densities, using the parity of \(A_j\), gives
exactly the traces of \eqref{eq:localHX}. Lemma~\ref{lem:trace-gluing}
therefore joins the three definitions into holomorphic functions
on \(D_{1/4}^{\mathrm{out}}\). Lemma~\ref{lem:local-area}
bounds their area norms there by \(C\|\mathbf c\|_2\):
it covers both the original-series portion and the two Cauchy
portions. All denominators \(z\pm i\) stay away from zero.

On the two open arcs \(|z|=1\), \(z\ne1\), the circle matching
law \eqref{eq:circle-transfer} yields
\begin{equation}
 b_{\mathrm{end}}(z)=z^{10}b_{\mathrm{end}}(1/z),\qquad
 b_-(z)+z^{10}b_-(1/z)=b_{\mathrm{end}}(z).
 \label{eq:local-involution}
\end{equation}
Here is the algebra at the lower arc. The two arguments in
\eqref{eq:local-lenses} are \(z+i\) and \(1/z-i\).
Their circle-transfer multiplier is \((iz)^{10}=-z^{10}\).
The first component gives the first identity, and the second
gives the second, using
\((1/z-i)^{-1}-(z+i)^{-1}=i\).
The upper arc gives the inverse identities. These identities
hold for the analytic germs at the open arcs, because their
Cauchy arguments lie on the interior negative real interval
of \(\Omega\), across which \(\mathbf h\) is holomorphic.

For \(z\in D_{1/8}^{\mathrm{in}}\) define
\[
 b_{\mathrm{end}}(z)=z^{10}b_{\mathrm{end}}(1/z),\qquad
 b_-(z)=b_{\mathrm{end}}(z)-z^{10}b_-(1/z).
\]
The right sides have already been defined: \(|1/z|>1\) and
\[
 |1/z-1|\leq\frac{|z-1|}{1-|z-1|}<\frac17<\frac14.
\]
The transformation on the ordered pair is \(z^{10}\) times
\(\left(\begin{smallmatrix}1&0\\1&-1\end{smallmatrix}\right)\);
applying it at \(z\) and then \(1/z\) is the identity.
The arc identities just proved match these inner pieces to
the outer pieces. We thus have a single holomorphic pair on
the whole punctured disk \(D_{1/8}\setminus\{1\}\).
This specifies both the cover and all its overlaps.

Inversion has bounded Jacobian on these disks, and \(z^{10}\)
is bounded there. Lemma~\ref{lem:local-area} consequently bounds
the area norm of the whole pair, including the inner pieces.
On the radius \(z=1+t\), \eqref{eq:radial-series-bound} bounds
each component by \(C\|\mathbf c\|_2(1+t^{-1/2})\).
Lemma~\ref{lem:area-removal} removes both punctures and gives
uniform bounds on \(D_{1/16}\). The involution identities
continue wherever both sides are defined. In particular,
\(b_-\) continues \(\Aminus(Q(z))\) through \(z=1\).

\textit{The real endpoint and the remaining components.}\quad
At the other contour endpoint \(w_0=e^{-2\pi}\), which corresponds
to \(z=1\), define on the left of \(z=1\)
\begin{equation}
 b_+(z)=-i\bigl(h_{1,\mathrm{par}}(z)-h_{0,\mathrm{par}}(z)/z\bigr). \label{eq:localXplus}
\end{equation}
Use this formula separately on the upper and lower left
half-disks of \(D_{1/8}\), both contained in the corresponding
halves of \(\mathcal D\). Its real-interval jump is zero:
the jump vector in \eqref{eq:real-transfer} has second component
equal to its first divided by \(z\). On the right half-disk
use \(b_+(z)=\Aplus(Q(z))\). The outer trace formulas give this
same value on the two open vertical segments. Applying
Lemma~\ref{lem:trace-gluing} to the real interval and the vertical
segments yields a holomorphic function on
\(D_{1/8}\setminus\{1\}\).
Its left pieces have the Cauchy area bound and its right piece
has the series area bound of Lemma~\ref{lem:local-area}.
On \(z=1+t\) it satisfies \eqref{eq:radial-series-bound}.
Lemma~\ref{lem:area-removal} removes the puncture and bounds
the extension on \(D_{1/16}\).
Consequently
\[
 A_2(Q(z))=(b_+(z)+b_-(z))/2,\qquad
 A_3(Q(z))=(b_+(z)-b_-(z))/4
\]
are holomorphic near that point.

It remains to recover \(A_1,A_0\). Work on \(D_{1/32}\),
where all three functions just constructed are bounded by
\(C\|\mathbf c\|_2\). In the upper and lower left half-disks, the subscripts \(+\) and \(-\) on \(h_{0,\mathrm{par}}\)
designate its holomorphic restrictions to those half-disks;
on the intervening real interval they designate the corresponding
traces. Use, respectively,
\begin{align}
 A_1(Q(z))&=i\bigl[h_{0,\mathrm{par},+}(z)-b_{\mathrm{end}}(z)
                    +(1+2iz)A_2(Q(z))-2A_3(Q(z))\bigr],\nonumber\\
 A_1(Q(z))&=i\bigl[h_{0,\mathrm{par},-}(z)+b_{\mathrm{end}}(z)
                    -(1-2iz)A_2(Q(z))+2A_3(Q(z))\bigr].
 \label{eq:localA1}
\end{align}
Their difference is
\(i[h_{0,\mathrm{par},+}(z)-h_{0,\mathrm{par},-}(z)-2b_{\mathrm{end}}(z)+2b_-(z)]\).
On the real interval, \eqref{eq:real-transfer} identifies the
jump of \(h_{0,\mathrm{par}}\) with \(2z^{10}b_-(1/z)\).
The second identity of \eqref{eq:local-involution} identifies
this with \(2b_{\mathrm{end}}(z)-2b_-(z)\). The difference
is therefore zero. On \(\Re z=1\), direct substitution of the
outer traces gives \(A_1(Q(z))\) from either formula.
Glue the two left pieces to that original right-hand piece by
Lemma~\ref{lem:trace-gluing}. Their area norms are bounded:
use Lemma~\ref{lem:local-area} for the Cauchy terms and the
original series, and the bounds on \(D_{1/32}\) for the other
terms. The right-hand radius again satisfies
\eqref{eq:radial-series-bound}. Lemma~\ref{lem:area-removal}
now removes the puncture and bounds \(A_1(Q(z))\) on
\(D_{1/64}\). Finally,
\begin{equation}
 A_0(Q(z))=b_{\mathrm{end}}(z)-zA_1(Q(z))-(z^2+1)A_2(Q(z))+A_3(Q(z)).
 \label{eq:localA0}
\end{equation}
This formula agrees with the original \(A_0(Q(z))\) on the
right half-disk by its definition in \eqref{eq:localHX};
it is holomorphic throughout \(D_{1/32}\) and bounded on
\(D_{1/64}\). All four functions now extend across \(q=1\), and parity gives
the same conclusion at \(q=-1\).

\textit{A common annulus and its norm bound.}\quad
For an explicit common neighborhood, put
\(\delta_{\mathrm{strip}}=2^{-18}\) and \(\delta_{\mathrm{end}}=2^{-7}\).
If \(1-\delta_{\mathrm{strip}}<\Re z<1\) and
\(\delta_{\mathrm{end}}\leq\Im z\leq1-\delta_{\mathrm{end}}\), both \(z,z-i\) lie in the required
halves of \(\mathcal D\), since
\((1-\delta_{\mathrm{strip}})^2>1-\delta_{\mathrm{end}}^2\).
Use \(\mathsf S_{\mathrm{pair}}(z)^{-1}\) there and the endpoint disks elsewhere.
Parity covers a full period. Hence \(q^4\mathbf a_{\mathrm{ser}}\) is holomorphic
for \(|q|<e^{\pi\delta_{\mathrm{strip}}}\). On the circle
\[
 R=e^{\pi\delta_{\mathrm{strip}}/2},
\]
the middle-arc Cauchy evaluations range over fixed compact sets
away from the contours, and the endpoint evaluations have the
area-mean bounds already proved. This proves \eqref{eq:regularity}.
Cauchy's coefficient estimate gives geometric coefficient tails,
and hence absolute summability.
\end{proof}


\section{The global holomorphic problem}\label{sec:bundle}

The preceding section gives continuation a little beyond the
unit circle. We now describe a global continuation problem.
It involves six scalar holomorphic functions: four encode the
coefficients, and two encode the Cauchy integrals after their
explicit jumps have been subtracted.
Proposition~\ref{prop:equivalence} will prove equivalence in
both directions. In particular, none of the global continuation
conditions is being assumed for a coefficient solution.

\subsection{Six functions and their gluing matrices}

Let \(\mathbb H=\{\tau\in\mathbb C:\Im\tau>0\}\).
The linear change of coordinate \(z\mapsto\tau=iz\)
identifies the right half-plane in the \(z\)-plane with
\(\mathbb H\). In this section \(\tau\) is an independent
variable in \(\mathbb H\), and the corresponding Gaussian
parameter is \(-i\tau\). The map \(Q\) satisfies
\begin{equation}
 Q(-i\tau)=e^\pi e^{\pi i\tau}.
 \label{eq:coordinates}
\end{equation}
For \(\Im\tau>1\), define
\begin{align*}
 a_0(\tau)&=A_0(Q(-i\tau)),&a_1(\tau)&=-iA_1(Q(-i\tau)),\\
 a_2(\tau)&=-A_2(Q(-i\tau)),&a_3(\tau)&=A_3(Q(-i\tau)).
\end{align*}
The first three are periodic under \(\tau\mapsto\tau+1\);
the fourth changes sign. The global problem seeks continuations
of these four functions to all of \(\mathbb H\), together with
two more holomorphic functions \(a_4,a_5\) that are periodic.

To explain the additional functions, first suppose such global
functions are given. Define the branch
\[
 \log_{\mathrm{cut}}w=\log|w|+i\arg w,\qquad 0<\arg w<2\pi,
\]
on \(\mathbb C\setminus[0,\infty)\). Its restriction to
\(\Omega\setminus[0,w_0]\) is used throughout this construction.
For \(w\) in that slit domain, put
\(\tau_w=\log_{\mathrm{cut}}w/(2\pi i)\).
The upper and lower traces on the positive real segment use
arguments \(0\) and \(2\pi\), respectively.
Define candidate Cauchy functions by
\begin{align}
 \widetilde{h}_0(w)
    &=2\tau_w a_2(\tau_w)-2a_3(\tau_w)+a_4(\tau_w),\label{eq:index-H0}\\
 \widetilde{h}_1(w)
    &=i\bigl[(-\tau_w^2+\tau_w)a_2(\tau_w)
                    +(2\tau_w-1)a_3(\tau_w)+a_5(\tau_w)\bigr].
 \label{eq:index-H1}
\end{align}
Indeed, the lower bank uses \(\tau_w+1\); subtracting its value
from the upper bank gives exactly \eqref{eq:jumps}.
Conversely, subtracting these explicit jumping terms from
Cauchy functions with \eqref{eq:jumps} leaves periodic
remainders \(a_4,a_5\). Their definition for an actual
coefficient solution is carried out on a fundamental domain
in Lemma~\ref{lem:coefficients-to-section}.

Let \(I_r\) denote the \(r\)-by-\(r\) identity matrix.
Collect the six functions as
\(\mathbf a=(a_0,a_1,a_2,a_3,a_4,a_5)^T\).
The translation matrix is \(\mathsf J_{\mathrm{tr}}\).
To specify the inversion matrix, define the frame matrix
\(\mathsf S_{\mathrm{frm}}(\tau)\) and the matrix
\(\mathsf S_{\mathrm{swap}}\) that interchanges two triples:
\[
 \mathsf J_{\mathrm{tr}}=\operatorname{diag}(1,1,1,-1,1,1),\qquad
 \mathsf S_{\mathrm{swap}}=\begin{pmatrix}0&I_3\\I_3&0\end{pmatrix},
\]
\begin{equation}
 \mathsf S_{\mathrm{frm}}(\tau)=
 \begin{pmatrix}
 1&\tau&\tau^2&1&0&0\\
 0&1&2\tau&0&0&0\\
 0&\tau^{-1}&1&2&0&0\\
 0&0&1&2&0&0\\
 0&0&-1-2\tau&0&-1&0\\
 0&0&\tau^2&1&0&-1
 \end{pmatrix},\qquad
 \mathsf J_{\mathrm{inv}}(\tau)=\mathsf S_{\mathrm{frm}}(-1/\tau)^{-1}\mathsf S_{\mathrm{swap}}\mathsf S_{\mathrm{frm}}(\tau).
 \label{eq:J}
\end{equation}
The gluing laws are
\begin{equation}
 \mathbf a(\tau+1)=\mathsf J_{\mathrm{tr}}\mathbf a(\tau),\qquad
 \mathbf a(-1/\tau)=\tau^{-10}\mathsf J_{\mathrm{inv}}(\tau)\mathbf a(\tau).
 \label{eq:automorphy}
\end{equation}
All matrices are holomorphic and invertible in the upper half-plane:
\(\det\mathsf S_{\mathrm{frm}}=-4\) and \(\det\mathsf J_{\mathrm{inv}}=-1\).
Writing \(S\tau=-1/\tau\), \(T\tau=\tau+1\), \(R_3=ST\), and
\(\mathsf J_{ST}(\tau)=\mathsf J_{\mathrm{inv}}(\tau+1)\mathsf J_{\mathrm{tr}}\), direct
multiplication gives
\begin{equation}
 \mathsf J_{\mathrm{inv}}(S\tau)\mathsf J_{\mathrm{inv}}(\tau)=I_6,\qquad
 \mathsf J_{ST}(R_3^2\tau)\mathsf J_{ST}(R_3\tau)
                        \mathsf J_{ST}(\tau)=I_6. \label{eq:group-relations}
\end{equation}
The scalar weight factors also have product \(1\).
The transformations \(S,T\) generate the modular group
\(\PSL=\SL/\{\pm I_2\}\), acting on \(\mathbb H\) by
fractional linear transformations.
The group relations say that applying
\eqref{eq:automorphy} along two words representing the same
transformation gives the same result. This consistency is
what is meant here by a holomorphic automorphy factor.

The inversion matrix \(\mathsf J_{\mathrm{inv}}(\tau)\) in \eqref{eq:J}
can be recovered directly from the boundary data.
The first three entries of \(\mathsf S_{\mathrm{frm}}(\tau)\mathbf a(\tau)\)
are the following expressions, in which all six scalar functions
are evaluated at \(\tau\):
\[
 a_0+\tau a_1+\tau^2a_2+a_3,\quad
 a_1+2\tau a_2,\quad a_1/\tau+a_2+2a_3.
\]
The last three are
\[
 a_2+2a_3,\quad -(1+2\tau)a_2-a_4,\quad
 \tau^2a_2+a_3-a_5.
\]
Substituting \eqref{eq:index-H0}--\eqref{eq:index-H1} in both
outer traces pairs the two triples by inversion. More explicitly,
write \(\widetilde{\mathbf h}
=(\widetilde{h}_0,\widetilde{h}_1)^T\).
For this check, the four expressions \(A_j(Q(z))\) are
obtained from \(a_j(iz)\) by reversing their definitions,
and \(b_0(z),b_1(z)\) are formed using \eqref{eq:BL}.
For \(z\in z_+((0,1))\) or \(z\in z_-((0,1))\),
evaluate the candidate Cauchy functions at \(\omega(z)\).
Equation~\eqref{eq:automorphy} gives the first identity below
on the first path and the second identity on the second path:
\begin{align}
 z^{10}\widetilde{\mathbf h}(\omega(z))
   &=\begin{pmatrix}b_0(z)-ib_1(z)\\z^{-1}(b_0(z)-ib_1(z))+i\Aplus(Q(z))\end{pmatrix}
                         ,\nonumber\\
 z^{10}\widetilde{\mathbf h}(\omega(z))
   &=\begin{pmatrix}-b_0(z)-ib_1(z)\\-z^{-1}(b_0(z)+ib_1(z))+i\Aplus(Q(z))\end{pmatrix}
                         .
 \label{eq:outer-check}
\end{align}
For these evaluations, put \(\tau=iz\). The logarithmic
parameter of \(\omega(z)\) is \(-1/\tau\) on the first
arc and \(-1/\tau+1\) on the second. These are algebraic row identities in
\(\mathsf J_{\mathrm{inv}}\) and \(\mathsf J_{\mathrm{tr}}\).
To show that every coefficient solution produces these
transformation laws, we must extend its functions over a full
fundamental domain. Lemma~\ref{lem:coefficients-to-section}
establishes this direction.

\subsection{Cusp conditions and equivalence}

The end \(\Im\tau\to\infty\) is called the modular cusp.
Let \(\cuspq\) be a local complex coordinate at this cusp:
substitution of \(\cuspq=e^{2\pi i\tau}\) converts Laurent
series in \(\cuspq\) into periodic functions of \(\tau\).
The half-power \(\cuspq^{1/2}\) means \(e^{\pi i\tau}\)
when pulled back to the upper half-plane. Thus the cusp
corresponds to \(\cuspq=0\). It is distinct from the
contour cusp \(w_c\) treated in Section~\ref{sec:regularity}.
At this modular cusp, prescribe the following bounds:
\begin{equation}
 \begin{array}{c|rrrrrr}
 \text{function}&a_0&a_1&a_2&a_3&a_4&a_5\\ \hline
 \text{lowest allowed exponent of }\cuspq
       &-2&-1&1&\tfrac12&-1&-1.
 \end{array} \label{eq:cusp-orders}
\end{equation}
Each function is \(\cuspq\) to the displayed exponent times a
convergent power series in \(\cuspq\).
In particular, \(a_4,a_5\) may each have a simple pole.
Write their principal parts as
\(\kappa_0\cuspq^{-1}/(2\pi i)\) and
\(-\kappa_1\cuspq^{-1}/(2\pi)\), respectively. The constants
\(\kappa_0,\kappa_1\) are not prescribed in this global
problem. Lemma~\ref{lem:reconstruction} will identify them
with the zeroth moments \(m_{0,0},m_{1,0}\) of the recovered coefficient
vector. The constant terms of \(a_4,a_5\) are also unrestricted.
Neither \(\lambda=1\) nor
\(G_{\mathbf c}(0)=1\) has been imposed here.

Here is the geometric meaning of this problem.
Form the quotient of \(\mathbb H\) by \(\SL\), retaining
the stabilizers, and add the cusp. In this quotient the central
element \(-I_2\) is retained as a stabilizer even though it
acts trivially on \(\mathbb H\); the stabilizers are larger
at the two elliptic points. This is the compactified modular orbifold
\(\overline{\mathcal X}\), identified with the weighted
projective line \(\mathbb P(4,6)\) in \cite{CF}.
The six-dimensional vectors are identified over equivalent
points by \eqref{eq:automorphy}. At the cusp, divide each
component by its displayed power in \eqref{eq:cusp-orders};
these quotients are the coordinates required to be holomorphic
there. Explicitly, write \(\kappa_{\mathrm{cusp},j}\), \(0\leq j\leq5\),
for the six exponents in \eqref{eq:cusp-orders}, and put
\[
 \mathsf S_{\mathrm{cusp}}(\tau)
   =\operatorname{diag}\bigl(e^{2\pi i\kappa_{\mathrm{cusp},j}\tau}\bigr)_{j=0}^5.
\]
Then \(\mathsf S_{\mathrm{cusp}}(\tau+1)
=\mathsf J_{\mathrm{tr}}\mathsf S_{\mathrm{cusp}}(\tau)\).
Thus \(\mathsf S_{\mathrm{cusp}}^{-1}\mathbf a\) is a
single-valued vector function of \(\cuspq\), and requiring it
to be holomorphic at zero defines the cusp trivialization.
In particular the half-integer exponent cancels the fourth
component's antiperiodicity. These rules define a rank-six
holomorphic vector bundle \(\mathcal E\).

For the arguments below, a global section of \(\mathcal E\)
means exactly a vector of six holomorphic functions on
\(\mathbb H\) satisfying \eqref{eq:automorphy} and
\eqref{eq:cusp-orders}. We denote the vector space of these
sections by \(H^0(\overline{\mathcal X},\mathcal E)\), or simply
\(H^0(\mathcal E)\).
The geometric description allows us to apply Riemann--Roch
in Section~\ref{sec:index}.

\subsection{From a section to the coefficient equations}

\begin{lemma}\label{lem:section-to-coefficients}
Taking the Laurent coefficients of the first four components
defines an injective linear map
\[
 H^0(\overline{\mathcal X},\mathcal E)
       \longrightarrow\ker_{\ell^2(\mathcal I_{24})}\Mom.
\]
Every coefficient vector in its image is absolutely summable
with weight \(r^m\) for every \(1<r<e^\pi\). Its Cauchy
functions are \eqref{eq:index-H0}--\eqref{eq:index-H1}.
\end{lemma}
\begin{proof}
Let \(\mathbf a\) be a section. Recover
\(A_0,A_1,A_2,A_3\) by reversing the four definitions above:
for \(q\ne0\), \(|q|<e^\pi\), choose \(\tau\in\mathbb H\)
with \(Q(-i\tau)=q\), and set
\[
 A_0(q)=a_0(\tau),\quad A_1(q)=ia_1(\tau),\quad
 A_2(q)=-a_2(\tau),\quad A_3(q)=a_3(\tau).
\]
Any two choices of \(\tau\) differ by an even integer, so
periodicity makes these definitions independent of the choice.
The translation law and \eqref{eq:cusp-orders} give exactly
the allowed exponents in \eqref{eq:series}. These four functions are holomorphic for
\(|q|<e^\pi\), apart from the specified poles at \(0\).
For a given \(r<e^\pi\), choose \(r_1\) with \(r<r_1<e^\pi\).
Cauchy's coefficient estimates on \(|q|=r_1\) bound each
positive coefficient by a constant times \(r_1^{-m}\);
summing \((r/r_1)^m\) gives the stated weighted absolute
summability and, in particular, membership in \(\ell^2\).

Define the candidate functions \(\widetilde{h}_\nu\) by
\eqref{eq:index-H0}--\eqref{eq:index-H1}.
Their cut jumps and outer traces are \eqref{eq:jumps} and
\eqref{eq:outer-check}. At \(w_0,w_c\), their arguments have finite
limits in the upper half-plane, so the functions are bounded on
each incident side. At \(0\), the terms
\(a_2(\tau_w)\) and \(a_3(\tau_w)\) are bounded by constant
multiples of \(|w|\) and \(|w|^{1/2}\), respectively. Even
after multiplication by the displayed logarithmic factors,
they tend to zero. Only the specified simple poles remain.
The endpoint hypotheses of Lemma~\ref{lem:reconstruction}
therefore hold. That lemma gives all the moment equations. It also identifies these candidate
functions with the actual Cauchy integrals \eqref{eq:Cauchy}.
This map is injective: if the four \(a_j\) vanish, the first triple
in the six expressions following \eqref{eq:group-relations} vanishes;
inversion forces the last
triple to vanish, and hence \(a_4=a_5=0\).

\end{proof}

\subsection{From the coefficient equations to a section}

We will need one elementary endpoint estimate for Cauchy integrals.
It is separate from the endpoint continuation theorem: that theorem
will make the densities analytic, and the following lemma will then
control the two remaining components at the elliptic vertices.

\begin{lemma}[Analytic densities at an endpoint]\label{lem:endpoint-log}
Suppose finitely many finite analytic arcs end at \(w_*\),
with integrable densities. Suppose the density with respect to
the directed differential \(dw\) on each arc extends
holomorphically to a neighborhood of \(w_*\).
The sum of their Cauchy integrals is bounded, on each incident
sector near \(w_*\), by
\[
 C\bigl(1+|\log|w-w_*||\bigr).
\]
Adding a function holomorphic near \(w_*\) preserves this bound.
\end{lemma}
\begin{proof}
Restrict an arc to a small neighborhood where its density \(g\)
is holomorphic. For an evaluation point \(\zeta\) there, write
\[
 \frac{g(w)}{w-\zeta}
   =\frac{g(\zeta)}{w-\zeta}
           +\frac{g(w)-g(\zeta)}{w-\zeta}.
\]
The second quotient is uniformly bounded, including its removable
diagonal value. The integral of the first term is \(g(\zeta)\)
times the difference of logarithms at the two arc endpoints.
On each incident sector their arguments can be chosen in a fixed
bounded interval. The endpoint different from \(w_*\) stays
away from \(\zeta\), and the endpoint \(w_*\) gives the asserted
logarithm. The portions of all arcs outside the neighborhood
give holomorphic bounded terms. Sum over the finitely many arcs.
\end{proof}

\begin{lemma}\label{lem:coefficients-to-section}
Every \(\mathbf c\in\ker_{\ell^2(\mathcal I_{24})}\Mom\)
determines a holomorphic section of \(\mathcal E\) whose first
four components have the prescribed coefficient series.
\end{lemma}
\begin{proof}
Fix \(\mathbf c\in\ker\Mom\). Proposition~\ref{prop:regularity}
gives its local holomorphic continuation. We construct the
section on a fundamental cell, check every side identification,
remove the vertices in local orbifold charts, and then verify
the cusp conditions.

\textit{The cell and its analytic domains.}\quad

Use the Gaussian-parameter
cell
\[
 \mathcal R=\{z\in\mathbb C:0<\Im z<1,\ \Re z>0,
                                      \ |z|>1,\ |z-i|>1\}.
\]
Its image under $\tau=iz$ is the fundamental domain
\[
 -1<\Re\tau<0,\qquad |\tau|>1,\qquad|\tau+1|>1,
\]
with its two halves related to the usual fundamental domain by a
translation. For the following domain checks write $\sigma=\Re z$ and
$t=\Im z$. On the part $\sigma>1$ the four coefficient functions
already exist. On the cap $\sigma<1$, both $z$ and $z-i$ lie in the
two respective halves of $\mathcal D$. Indeed,
\[
 \sqrt{1-\sigma^2}<t<1-\sqrt{1-\sigma^2}.
\]
Consequently the inverse of the matrix in
\eqref{eq:paired-transfer} defines all four coefficient functions
on the cap. Its determinant is $4i$. The original boundary
identities glue this definition to the original functions across
$\sigma=1$. Thus all four $a_j$ are holomorphic on this entire cell.

The two other functions can also be defined there. For each
\(z\in\mathcal R\), evaluate the Cauchy functions at
\(e^{-2\pi z}=\omega(1/z)\). This point lies in
\(\Omega\setminus[0,w_0]\). To check the only
nontrivial case, put $r=\min(t,1-t)\le1/2$. The upper boundary
height of the logarithmic image of $\Omega$ is
\[
 h_\Omega(r)=\frac{1+\sqrt{1-4r^2}}2,
\]
whereas the cell has \(\sigma>\sqrt{1-r^2}\ge h_\Omega(r)\).
For the last inequality put \(x=\sqrt{1-r^2}\).
Then \(2x-1\geq0\), and
\((2x-1)^2-(1-4r^2)=4(1-x)\geq0\).
This also proves \(1/z\in\mathcal W\), the domain on which
\(\mathbf h_{\mathrm{par}}(1/z)\) was defined.
Since $\tau=iz$ has $-1<\Re\tau<0$, the chosen logarithm of \(e^{-2\pi z}\)
corresponds to $\tau+1$. Therefore set
\begin{align}
 a_4(\tau)&=h_0(e^{2\pi i\tau})
                      -2(\tau+1)a_2(\tau)-2a_3(\tau),\\
 a_5(\tau)&=h_1(e^{2\pi i\tau})/i
                      +(\tau^2+\tau)a_2(\tau)
                      +(2\tau+1)a_3(\tau).
 \label{eq:converse-p-fields}
\end{align}
\textit{The translation sides.}\quad
The sides \(t=0,1\) of \(\mathcal R\) have \(\sigma>1\).
There the original series converge normally. Their parity
gives the translation law for the first four components.
The two traces of \(h_\nu(e^{2\pi i\tau})\) differ by
\eqref{eq:jumps}; subtracting the polynomial terms displayed
in \eqref{eq:converse-p-fields} cancels that difference.
For example, across the positive slit the logarithmic parameter
changes from \(\tau\) to \(\tau+1\);
\(a_2(\tau+1)=a_2(\tau)\) and \(a_3(\tau+1)=-a_3(\tau)\).
The difference of the terms \(2\tau a_2-2a_3\) is
\(-2a_2-4a_3=2A_-\), exactly the first jump. Subtracting the
terms in \eqref{eq:index-H1} similarly gives
\(2i\tau(a_2+2a_3)=-\pi^{-1}(\log s)A_-\).
The remaining components \(a_4,a_5\) consequently match
periodically. Lemma~\ref{lem:trace-gluing} glues the functions
across these open sides, giving the full translation law.

\textit{The curved sides.}\quad
We specify the ordered Cauchy inputs and the recovery matrices
to check these identifications without assuming any continuation
law. Write
\[
 \mathbf h_{\mathrm{cell}}(z)=\begin{pmatrix}h_{0,\mathrm{par}}(z)\\h_{1,\mathrm{par}}(z)/i\end{pmatrix}.
\]
On the cap the four coefficient functions are recovered from
\begin{equation}
 \begin{pmatrix}\mathbf h_{\mathrm{cell}}(z)\\\mathbf h_{\mathrm{cell}}(z-i)\end{pmatrix}
 =\mathsf S_{\mathrm{cell}}(\tau)
      \begin{pmatrix}a_0\\a_1\\a_2\\a_3\end{pmatrix},
 \qquad \tau=iz,
 \label{eq:cell-recovery}
\end{equation}
where
\[
 \mathsf S_{\mathrm{cell}}(\tau)=
 \begin{pmatrix}
 1&\tau+1&\tau^2+2\tau&1\\
 \tau^{-1}&1+\tau^{-1}&\tau+1&2+\tau^{-1}\\
 -1&-\tau&1-\tau^2&1\\
 -(\tau+1)^{-1}&-\tau/(\tau+1)&-\tau&-(2\tau+1)/(\tau+1)
 \end{pmatrix},
 \qquad \det\mathsf S_{\mathrm{cell}}=-4.
\]
The relation to the earlier paired-trace matrix is
\[
 \mathsf S_{\mathrm{cell}}(\tau)
   =\operatorname{diag}(1,-i,1,-i)\,
      \mathsf S_{\mathrm{pair}}(-i\tau)\,
        \operatorname{diag}(1,i,-1,1).
\]
The two diagonal factors simply implement the changes from
\(\mathbf h_{\mathrm{par}}\) to \(\mathbf h_{\mathrm{cell}}\)
and from \(\mathbf a_{\mathrm{ser}}\) to \((a_0,a_1,a_2,a_3)^T\).
Define a six-by-six rational recovery matrix
$\mathsf S_{\mathrm{rec}}^{(24)}(\tau)$ by the following recipe on three
two-component inputs $\mathbf b_1,\mathbf b_2,\mathbf b_3$,
where $\mathbf b_j=(b_{j,0},b_{j,1})^T$:
\begin{align*}
 (a_0,a_1,a_2,a_3)^T
   &=\mathsf S_{\mathrm{cell}}(\tau)^{-1}
                     (\mathbf b_1^T,\mathbf b_2^T)^T,\\
 a_4&=-\tau^{10}b_{3,0}-2(\tau+1)a_2-2a_3,\\
 a_5&=-\tau^{10}b_{3,1}+(\tau^2+\tau)a_2+(2\tau+1)a_3.
\end{align*}
The actual six functions on the cap are thus
\[
 \mathbf a(\tau)=\mathsf S_{\mathrm{rec}}^{(24)}(\tau)
  \begin{pmatrix}\mathbf h_{\mathrm{cell}}(z)\\\mathbf h_{\mathrm{cell}}(z-i)\\\mathbf h_{\mathrm{cell}}(1/z)\end{pmatrix}.
\]
For the third input, $\mathbf h_{\mathrm{par}}(1/z)=z^{-10}
\mathbf h(e^{-2\pi z})$, so this expression agrees with
\eqref{eq:converse-p-fields}.

Consider the curved side $|z|=1$, $0<\Im z<1/2$, and put
$z_1=z-i$, $z_2=1/z+i$. Its partner is $z_2$ on the other curved side.
The elementary identities
\[
 \frac1{z_2}=\frac1{z_1}-i,\qquad
 e^{-2\pi/(1/z_2)}=e^{-2\pi/z}
\]
and the single-valuedness of $\mathbf h$ across the negative
real axis give
\begin{equation}
 \mathbf h_{\mathrm{cell}}(z_2)=\tau^{-10}\mathbf h_{\mathrm{cell}}(z_1),\qquad
 \mathbf h_{\mathrm{cell}}(z_2-i)=\mathbf h_{\mathrm{cell}}(1/z),\qquad
 \mathbf h_{\mathrm{cell}}(1/z_2)=(1+\tau)^{-10}\mathbf h_{\mathrm{cell}}(z).
 \label{eq:cell-cauchy-matching}
\end{equation}
These identities hold in a neighborhood of each open side point.
For example, on that side $e^{-2\pi/z_1}$ and $e^{-2\pi/z_2}$ are
the same negative real point strictly inside $\Omega$, so this
matching does not cross the cut or appeal to an unknown continuation.

Let $\mathsf S_{\mathrm{match}}^{(24)}(\tau)$ denote the block matrix
\[
 \begin{pmatrix}
 0&\tau^{-10}I_2&0\\
 0&0&I_2\\
 (1+\tau)^{-10}I_2&0&0
 \end{pmatrix}.
\]
Substituting the displayed recovery recipe gives the rational identity
\begin{equation}
 \mathsf J_{\mathrm{tr}}\mathsf S_{\mathrm{rec}}^{(24)}(-1/\tau-1)
       \mathsf S_{\mathrm{match}}^{(24)}(\tau)
   =\tau^{-10}\mathsf J_{\mathrm{inv}}(\tau)\mathsf S_{\mathrm{rec}}^{(24)}(\tau).
 \label{eq:cell-matrix-matching}
\end{equation}
All entries are specified above, so this can be checked by direct
multiplication; no identities of special modular forms enter this multiplication.
Together with \eqref{eq:cell-cauchy-matching}, it proves precisely
the inversion law \eqref{eq:automorphy} across the paired
curved sides. This verifies the gluing for the actual solution,
rather than imposing that law on it.

\textit{The elliptic vertices and extension to \(\mathbb H\).}\quad
At \(z=e^{i\pi/6}\), all three Cauchy arguments in the recovery
formula equal \(-e^{-\pi\sqrt3}\), an interior point of the
negative real interval in \(\Omega\).
The matrices and coordinate maps are nonsingular there.
The six recovered functions therefore extend holomorphically
to a full neighborhood of this vertex in the parameter plane.

At \(z=1\) and \(1+i\), Proposition~\ref{prop:regularity}
makes the four coefficient functions holomorphic on neighborhoods
of the vertices. The outer densities near \(w_0\) are consequently
holomorphic functions on each incident arc: the map \(\omega\)
has nonzero derivative at \(z=1\), and all the factors in
\eqref{eq:densities} are holomorphic there. The segment density
is also holomorphic near \(w_0>0\), using the local square root
and real logarithm in \eqref{eq:densities}.
Lemma~\ref{lem:endpoint-log}, with the correction \(1/w\)
holomorphic near \(w_0\), gives
\[
 |h_\nu(w)|\leq C\bigl(1+|\log|w-w_0||\bigr)
\]
in each incident sector. Since
\(e^{-2\pi z}-w_0\) has a simple zero at \(z=1\), the definitions
\eqref{eq:converse-p-fields} give the same logarithmic bound
for \(a_4,a_5\) at the corresponding vertex.

For completeness, the gluing at a vertex takes place on the
upper-half-plane cover, not merely on the quotient. Take a
sufficiently small disk about the elliptic fixed point. Finitely
many translates of the incident cell sectors fill the punctured
disk. At the point of order two, translation first joins the
sectors represented by \(z=1\) and \(1+i\); inversion supplies
the remaining sectors. At the point of order three use the
three sectors related by the order-three transformation.
Assign functions on successive sectors by the stated side laws.
The open-side identities already proved glue adjacent assignments
holomorphically. A complete turn composes either the order-two
relation or the order-three relation in \eqref{eq:group-relations},
including the scalar factors; the product is the identity.
Thus the assignment is single-valued on this punctured disk.

All transition matrices and their inverses are holomorphic and
bounded on a smaller disk. The sector bounds therefore give
a common bound \(C(1+|\log|\tau-\tau_*||)\) in all directions.
For every \(k\geq1\), the coefficient of
\((\tau-\tau_*)^{-k}\) has absolute value at most
\(Cr^k(1+|\log r|)\), which tends to zero.
Every negative Laurent coefficient vanishes. The six functions
extend holomorphically across the fixed point, and equivariance
there follows by continuity. This is precisely holomorphy in
the local orbifold chart.

Now extend over all translates of the cell. Across an open
side the assignments agree by the identities already checked;
at vertices they agree by the local argument above. The
relations of \(\PSL\) ensure independence of the word used to
reach a translated cell. We obtain a holomorphic vector on
\(\mathbb H\) satisfying \eqref{eq:automorphy}. The retained
central element \(-I_2\) acts trivially, so this vector defines
a section on the modular orbifold away from the cusp.

\textit{The modular cusp.}\quad
The first four components still have their original series
for \(\Im\tau>1\), and hence the first four exponents of
\eqref{eq:cusp-orders}. The periodic components \(a_4,a_5\)
are single-valued holomorphic functions of \(\cuspq\) on a
punctured disk. On a slit version of that disk their expressions
are the Cauchy functions minus the explicit jumping terms.
The Cauchy kernel area estimate in Lemma~\ref{lem:local-area}
holds also on a disk containing zero; the separate correction
\(1/w\) is itself area-integrable there. The subtracted terms
are bounded by constants times
\(|w|\,|\log w|^2+|w|^{1/2}(1+|\log w|)\).
Thus \(a_4,a_5\), as functions of \(\cuspq\), are area-integrable.
Lemma~\ref{lem:area-removal} leaves at most a simple pole in each.

We compute those poles without any unproved boundary estimate
at zero. Split the measure in \eqref{eq:Cauchy} into its arc
and segment parts, and take \(w=-r<0\), \(r\downarrow0\).
The arc stays a positive distance from zero, so multiplying
its Cauchy integral by \(w\) makes it tend to zero. On the
segment, with variable \(s>0\),
\[
 \left|\frac{w}{s-w}\right|=\frac{r}{s+r}\leq1,
 \qquad \frac{w}{s-w}\longrightarrow0 .
\]
Finite total variation and the absence of an atom at zero
give the same conclusion by dominated convergence.
The explicit correction contributes exactly
\(m_{\nu,0}/(2\pi i)\) to \(\lim_{w\to0}w h_\nu(w)\).
The jumping terms contribute zero to that limit. Hence the
principal parts of \(a_4,a_5\) are, respectively,
\[
 \frac{m_{0,0}}{2\pi i}\cuspq^{-1},
       \qquad -\frac{m_{1,0}}{2\pi}\cuspq^{-1}.
\]
No restriction is imposed on their constant terms.
These are exactly the last two conditions in
\eqref{eq:cusp-orders}, so the section extends across the cusp
in the prescribed frame and recovers the original coefficients.

\end{proof}



\begin{proposition}[Boundary--bundle correspondence]\label{prop:equivalence}
The coefficient map is an isomorphism
\[
 H^0(\overline{\mathcal X},\mathcal E)
       \simeq\ker_{\ell^2(\mathcal I_{24})}\Mom.
\]
For every vector in this kernel, the infinite coefficient
sequences are absolutely summable with weight \(r^m\),
for each \(1<r<e^\pi\).
\end{proposition}
\begin{proof}
Lemma~\ref{lem:section-to-coefficients} gives an injective map
to the coefficient kernel and its weighted summability.
Lemma~\ref{lem:coefficients-to-section} gives a section over every
kernel vector, so that map is surjective. Because the first
map is injective, the recovered section is unique and the two
constructions are inverse. No known magic function or Fourier
interpolation theorem enters either construction.
\end{proof}


\section{The index and existence}\label{sec:index}

The boundary problem has now become a problem about holomorphic
sections of \(\mathcal E\). Write \(h^j(\mathcal E)\) for the
dimension of its sheaf cohomology group \(H^j(\mathcal E)\),
for \(j=0,1\). In particular, \(h^0\) counts the independent
holomorphic sections. Riemann--Roch computes their difference,
the Euler characteristic
\(\chi(\mathcal E)=h^0(\mathcal E)-h^1(\mathcal E)\), from
the transition laws. A positive value will force a nonzero section
and hence a nonzero coefficient solution. This is the index we
compute; it is not an index of the compact moment map \(\Mom\).

\begin{proposition}\label{prop:index}
The bundle \(\mathcal E\) has Euler characteristic \(1\).
In particular, \(\ker_{\ell^2(\mathcal I_{24})}\Mom\ne\{0\}\).
\end{proposition}
\begin{proof}
The determinant automorphy factor has scalar weight \(-60\)
and character
\[
 \varepsilon_{\mathrm{det}}(S)=\varepsilon_{\mathrm{det}}(T)=-1,\qquad \varepsilon_{\mathrm{det}}(ST)=1.
\]
The sum of the cusp exponents in \eqref{eq:cusp-orders} is
\(-7/2\). Let \(\mathcal L_k\) denote the line bundle whose
sections are scalar modular forms of weight \(k\), holomorphic
at the cusp. The determinant bundle \(\det\mathcal E\) is a
line bundle: its transition functions are the determinants of
those of \(\mathcal E\). The line-bundle extension formula
\cite[Theorem 3.4]{CF} gives
\begin{equation}
 \det\mathcal E\simeq\mathcal L_{-60-12(-7/2)}
                 =\mathcal L_{-18}. \label{eq:detbundle}
\end{equation}
For the character convention, Dedekind's eta function is
\(\eta(\tau)=\mathfrak q^{1/24}\prod_{n\geq1}(1-\mathfrak q^n)\),
where \(\mathfrak q^{1/24}=e^{2\pi i\tau/24}\).
The character \(\varepsilon_{\mathrm{det}}\) is the sixth power of the
character of \(\eta^2\). Its allowed exponent
is \(1/2-4\). Thus \eqref{eq:detbundle} includes both the
character and the integer shift of the cusp frame.
One can see the isomorphism directly: divide a determinant
section by \(\eta^{12}\), of weight \(6\), character
\(\varepsilon_{\mathrm{det}}\), and leading exponent \(1/2\).
The result has trivial character, weight \(-66\), and lowest
allowed exponent \(-4\). Set
\(\Delta_{\mathrm{mod}}=\eta^{24}\), the modular discriminant.
Multiplication by \(\Delta_{\mathrm{mod}}^4\) gives weight
\(-18\) and a holomorphic cusp frame. These multipliers have no zeros in
\(\mathbb H\), and their leading cusp factors are nonzero
after the specified powers have been removed; hence this is
an isomorphism of the extended line bundles.

It remains to account for the elliptic fibers. Put
\(\rho_3=e^{2\pi i/3}\).
The elliptic points are fixed by \(S\) and \(ST\), respectively.
The actions of these stabilizers on the fibers, including the
scalar weight factor, are the following two matrices, denoted by
\[
 \mathsf S_{\mathrm{ell},2}^{(24)}=-\mathsf J_{\mathrm{inv}}(i),\qquad
 \mathsf S_{\mathrm{ell},3}^{(24)}=\rho_3\mathsf J_{\mathrm{inv}}(\rho_3+1)\mathsf J_{\mathrm{tr}}.
\]
Using \eqref{eq:J}, with an independent polynomial variable \(x\), gives
\[
 \det(xI_6-\mathsf S_{\mathrm{ell},2}^{(24)})=(x-1)^3(x+1)^3,\qquad
 \det(xI_6-\mathsf S_{\mathrm{ell},3}^{(24)})=(x-1)^2(x-\rho_3)^2(x-\rho_3^2)^2.
\]
These identities can be verified by reducing the rational
matrices modulo \(\tau^2+1\) and \(\tau^2+\tau+1\),
respectively, and using \eqref{eq:group-relations}.
All nontrivial elliptic traces are zero, and the central element
\(-I_2\) acts trivially.

The Riemann--Roch formula for a general holomorphic bundle on
\(\mathbb P(4,6)\), in this case, is
\[
 \chi(\mathcal E)=\frac{5\,\operatorname{rank}\mathcal E+
                         \deg_{\mathcal L_1}(\det\mathcal E)}{12}.
\]
Here the degree is normalized by
\(\deg_{\mathcal L_1}(\mathcal L_k)=k\).
The elliptic correction terms vanish because of the traces just
computed. The identity and the central stabilizer contribute equal
terms, since the latter acts trivially; together they give the
denominator \(12\) displayed here. This is the general bundle
formula in \cite[Theorem 6.1]{CF};
no constant matrix representation is required.
It gives \(\chi(\mathcal E)=(30-18)/12=1\).
Since \(h^0-h^1=1\), a nonzero holomorphic section exists.
Proposition~\ref{prop:equivalence} sends it to a nonzero
coefficient solution.
\end{proof}

The cusp count has a useful independent check. With the baseline
exponents \((0,0,0,1/2,0,0)\), the determinant is
\(\mathcal L_{-66}\) and the index is \(-3\).
Allowing the poles in \(a_0,a_1,a_4,a_5\) adds \(2+1+1+1=5\);
removing the constant term of \(a_2\) subtracts \(1\).
The resulting index is again \(-3+5-1=1\).

\section{Vanishing and uniqueness}\label{sec:vanishing}

The index proves existence but does not by itself determine the
dimension. To prove uniqueness we show that normalization is
injective on the kernel. The following constant change of
variables separates the system of six functions into two blocks
of three. We analyze the blocks separately.

\begin{align*}
 a_0^+&=(a_0+a_5)/2,&
 a_1^+&=(a_1+a_2+a_4)/2,&
 a_2^+&=a_2,\\
 a_0^-&=(a_0-a_5)/2,&
 a_1^-&=(a_1-a_2-a_4)/2,&
 a_2^-&=a_3.
\end{align*}
Here all functions on the right are evaluated at the same
argument \(\tau\) as the function on the left.
The superscripts \(+\) and \(-\) distinguish the two invariant
blocks of the gluing laws. They do not assert that these scalar
functions are Fourier eigenfunctions. This is an invertible
constant change of variables. Direct
substitution in $\mathsf J_{\mathrm{inv}}$ gives two separate systems:
\begin{align}
 a_0^+(-1/\tau)&=\tau^{-12}a_0^+(\tau),\\
 a_1^+(-1/\tau)&=\tau^{-10}a_1^+(\tau)
                          +2\tau^{-11}a_0^+(\tau),\\
 a_2^+(-1/\tau)&=\tau^{-8}a_2^+(\tau)
                   +\tau^{-9}a_1^+(\tau)
                   +\tau^{-10}a_0^+(\tau),
 \label{eq:triangular-splitting}\\
 \begin{pmatrix}a_0^-\\a_1^-\\a_2^-\end{pmatrix}(-1/\tau)
 &=\tau^{-10}
 \begin{pmatrix}
 -1/2&\tau^{-1}-\tau/2&3/2\\
 0&1&0\\
 1/2&\tau/2&1/2
 \end{pmatrix}
 \begin{pmatrix}a_0^-\\a_1^-\\a_2^-\end{pmatrix}(\tau).
 \label{eq:second-splitting}
\end{align}
All these functions except $a_2^-$ are periodic; $a_2^-$ is
antiperiodic. These identities follow from the derived matrix;
none of the functions is selected from a known magic-function formula.

\begin{proposition}[Vanishing at zero normalization]
\label{prop:vanishing}
If $\mathbf c\in\ker\Mom$ and $\lambda(\mathbf c)=0$, then $\mathbf c=0$.
\end{proposition}

\begin{proof}
By Proposition~\ref{prop:equivalence}, $\mathbf c$ gives global
holomorphic functions with the transformation laws above.
We use \(h\ll g\), with \(g\geq0\), to mean
\(|h|\leq Cg\) near the indicated limit for a fixed constant
\(C\). The Poisson identity \eqref{eq:normalizations} gives
$c_{0,-4}=0$. Consequently the largest pole in $a_0$ drops
from order two to order one in $\mathfrak q$. The cusp bounds are
now, as \(\cuspq\to0\),
\begin{equation}
 \begin{gathered}
 |a_0^+|+|a_1^+|+|a_0^-|+|a_1^-|\ll|\cuspq|^{-1},\\
 |a_2^+|\ll|\cuspq|,\qquad |a_2^-|\ll|\cuspq|^{1/2}.
 \end{gathered}
 \label{eq:zero-normalization-orders}
\end{equation}

Recall the product expansion of the modular discriminant:
\[
 \Delta_{\rm mod}(\tau)
   =\mathfrak q\prod_{n\ge1}(1-\mathfrak q^n)^{24}
   =\mathfrak q-24\mathfrak q^2+\sum_{n\geq3}\delta_n\mathfrak q^n.
\]
The coefficients \(\delta_n\) are defined by the product;
the displayed series converges for \(|\cuspq|<1\).
This is a weight-$12$ cusp form with no zeros in the upper half-plane.
Let $\mathrm{Mod}_k$ be the space of scalar modular forms of
weight $k$ for $\mathrm{SL}_2(\mathbb Z)$, holomorphic on the
upper half-plane and at the cusp. Let
$E_4=1+240\sum_{n\geq1}(\sum_{\ell\mid n}\ell^3)\mathfrak q^n$
be the normalized Eisenstein series of weight $4$.
We use only $\mathrm{Mod}_0=\mathbb C$, $\mathrm{Mod}_2=\{0\}$,
$\mathrm{Mod}_4=\mathbb C E_4$, and the
weight-$12$ bound $\operatorname{ord}_\infty f\le1$ for a nonzero
holomorphic form. These are immediate consequences of the usual
valence formula. See \cite[Examples 4.13--4.15 and Proposition 4.16]{Milne}; his weight indexing
is half the weight used here.

Since $a_0^+$ has weight $-12$ and at most a simple cusp pole,
$\Delta_{\rm mod}a_0^+$ is a holomorphic weight-zero form.
Thus
\[
 a_0^+=\frac{c}{\Delta_{\rm mod}}
 \quad\text{for a constant }c.
\]
Write $\partial_\tau=d/d\tau$. Differentiating its weight-$-12$
law gives
\begin{align*}
 (\partial_\tau a_0^+)(-1/\tau)
   &=\tau^{-10}\partial_\tau a_0^+
                         -12\tau^{-11}a_0^+,\\
 (\partial_\tau^2a_0^+)(-1/\tau)
   &=\tau^{-8}\partial_\tau^2a_0^+
       -22\tau^{-9}\partial_\tau a_0^+
       +132\tau^{-10}a_0^+.
\end{align*}
It follows that
$a_1^++\tfrac16\partial_\tau a_0^+$ is a weight-$-10$
modular form with at most a simple cusp pole. Multiplication by
$\Delta_{\rm mod}$ puts it in $\mathrm{Mod}_2=\{0\}$, so
\[
 a_1^+=-\tfrac16\partial_\tau a_0^+.
\]
Using this in the third law in \eqref{eq:triangular-splitting}
shows that
$a_2^+-\tfrac1{132}\partial_\tau^2a_0^+$ is a
weight-$-8$ modular form with at most a simple cusp pole.
Multiplying by $\Delta_{\rm mod}$ now puts it in $\mathrm{Mod}_4$:
\begin{equation}
 a_2^+=
  \frac{c}{132}\partial_\tau^2
                       \left(\frac1{\Delta_{\rm mod}}\right)
       +b\,\frac{E_4}{\Delta_{\rm mod}}
 \label{eq:triangular-two-constants}
\end{equation}
for a constant $b$.

Only two elementary coefficients are needed to finish this block.
The coefficient of \(\cuspq\) in \(E_4\) is \(240\),
and its constant coefficient is \(1\). Using the product for
\(\Delta_{\rm mod}\), we obtain, as \(\cuspq\to0\),
\[
 \partial_\tau^2(1/\Delta_{\rm mod})-(2\pi i)^2\cuspq^{-1}
      \ll |\cuspq|,\qquad
 \frac{E_4}{\Delta_{\rm mod}}-\cuspq^{-1}-264\ll|\cuspq|.
\]
The absence of both a pole and a constant term in
\(a_2^+\), by \eqref{eq:zero-normalization-orders}, gives
\[
 \frac{(2\pi i)^2}{132}c+b=0,\qquad 264b=0.
\]
Therefore $b=c=0$ and all three $a_j^+$ vanish.

For the other block, $a_1^-$ is a weight-$-10$ modular form
with at most a simple cusp pole. Again
$\Delta_{\rm mod}a_1^-\in\mathrm{Mod}_2=\{0\}$, so $a_1^-=0$.
For this part of the proof, define the two-by-two matrices
\(\mathsf S_{\mathrm{lev},S},\mathsf S_{\mathrm{lev},T}\) in the following display.
The remaining two functions obey
\[
 \begin{pmatrix}a_0^-\\a_2^-\end{pmatrix}(-1/\tau)
 =\tau^{-10}\mathsf S_{\mathrm{lev},S}
       \begin{pmatrix}a_0^-\\a_2^-\end{pmatrix}(\tau),
 \quad
 \mathsf S_{\mathrm{lev},S}=\tfrac12\begin{pmatrix}-1&3\\1&1\end{pmatrix},
 \quad
 \mathsf S_{\mathrm{lev},T}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
\]
The matrices satisfy
$\mathsf S_{\mathrm{lev},S}^2=\mathsf S_{\mathrm{lev},T}^2=(\mathsf S_{\mathrm{lev},S}\mathsf S_{\mathrm{lev},T})^3=I_2$;
their representation factors through
$\mathrm{PSL}_2(\mathbb F_2)$.
Thus
\[
 h(\tau)=\Delta_{\rm mod}(\tau)a_2^-(\tau)
\]
is a weight-$2$ scalar form for the principal congruence subgroup
$\Gamma(2)=\{\gamma\in\mathrm{SL}_2(\mathbb Z):
\gamma\equiv I_2\pmod 2\}$.
Write \(\gamma_{jk}\) for the entries of
\(\gamma\in\SL\). The weight-\(k\) slash operator is
\[
 (h|_k\gamma)(\tau)=(\gamma_{21}\tau+\gamma_{22})^{-k}
           h\left(\frac{\gamma_{11}\tau+\gamma_{12}}
                        {\gamma_{21}\tau+\gamma_{22}}\right).
\]
All its cusp transforms are bounded: every weight-$2$ slash
transform of $h$ is a constant linear combination of
\(\Delta_{\rm mod}a_0^-\), which is bounded, and
\(\Delta_{\rm mod}a_2^-\), whose absolute value is at most
a constant times \(|\cuspq|^{3/2}\).

Here is a reduction to the level-one valence bound, avoiding any
dimension formula for level two. If $h\ne0$, form the product
\[
 p_h=\prod_{\gamma\in\Gamma(2)\backslash
                                      \mathrm{SL}_2(\mathbb Z)}
                    (h|_2\gamma).
\]
It is a nonzero holomorphic weight-$12$ form for the full modular
group. Among the six cosets choose $I_2$ and $T$; their factors
are \(h\) and \(-h\). Since \(|h|\ll|\cuspq|^{3/2}\)
and the other factors are bounded, \(|p_h|\ll|\cuspq|^3\).
Thus its order of vanishing at the cusp is at least three.
This contradicts the weight-$12$ valence bound.
Therefore $a_2^-=0$. The inversion formula then gives
$a_0^-=0$ also.

All six functions vanish. Inverting their constant change of variables
gives $a_0=a_1=a_2=a_3=0$, hence all original coefficients vanish.
\end{proof}



\begin{proof}[Proof of Theorem~\ref{thm:main}]
Proposition~\ref{prop:index} gives a nonzero homogeneous solution,
and Proposition~\ref{prop:vanishing} makes \(\lambda\) injective
on the homogeneous kernel. The kernel is therefore one-dimensional,
and its unique vector with \(\lambda=1\) is \(\mathbf c_*\).
Complex conjugation preserves the real coefficient equations and
the normalization, so \(\mathbf c_*\) is real.
Proposition~\ref{prop:equivalence} proves the weighted absolute
summability. Lemmas~\ref{lem:Schwartz} and \ref{lem:Poisson}
give the claimed values and derivatives.
\end{proof}

The proof also shows \(h^0(\mathcal E)=1\) and \(h^1(\mathcal E)=0\).
No explicit coefficient of the known magic function has entered
the existence or uniqueness argument.

\section{A boundedly invertible coefficient equation}\label{sec:operator}

The sample equations already specify the solution uniquely.
For approximation it is useful to replace them by an equivalent
equation whose inverse is bounded on the full Hilbert space.
Write \(\Id_{\mathcal I_{24}}\) for the identity on
\(\ell^2(\mathcal I_{24})\). The extra ingredient is a compact
linear operator that fixes every exact solution. Appendix~\ref{app:K} defines such an operator
\(\mathsf K:\ell^2(\mathcal I_{24})\to\ell^2(\mathcal I_{24})\) by fixed Cauchy integrals, local disk
averages, and coefficient extraction. It uses no unknown kernel
vector and satisfies
\begin{equation}
 \mathsf K\mathbf c=\mathbf c\quad(\mathbf c\in\ker\Mom),\qquad
 \|(\Id_{\mathcal I_{24}}-\mathsf P_J)\mathsf K\|\leq C_K R^{-J}, \label{eq:Kproperties}
\end{equation}
where \(R>1\) is fixed and \(J\in\mathbb N\). The orthogonal projection
\(\mathsf P_J:\ell^2(\mathcal I_{24})\to\ell^2(\mathcal I_{24})\) is defined to be the identity on
the five entries with nonpositive exponent and to act on each infinite
sequence by setting its entries with \(m>J\) to zero.

Use the usual coefficient inner products on \(\ell^2(\mathcal I_{24})\), on the
moment space \(\ell^2(\mathcal J_{24})\), and on
\(\mathbb C\). A superscript \(*\) denotes the adjoint with
respect to these inner products. Define
\begin{equation}
 \begin{aligned}
 \mathsf L&=(\Id_{\mathcal I_{24}}-\mathsf K)^*
                  (\Id_{\mathcal I_{24}}-\mathsf K)+\Mom^*\Mom,\\
 \mathsf N&=\mathsf L+\lambda^*\lambda.
 \end{aligned}
 \label{eq:N}
\end{equation}
Both are bounded positive self-adjoint operators. They are the
identity plus compact operators, and
\[
 \ker\mathsf L=\ker\Mom,
\]
because \(\langle\mathsf L\mathbf c,\mathbf c\rangle
=\|(\Id_{\mathcal I_{24}}-\mathsf K)\mathbf c\|^2+\|\Mom \mathbf c\|^2\).

\begin{theorem}[Bounded inverse]\label{thm:inverse}
There is a constant \(c_{\mathrm{coer},24}>0\) such that, for every \(\mathbf c\in\ell^2(\mathcal I_{24})\),
\begin{equation}
 \begin{aligned}
 \langle\mathsf N\mathbf c,\mathbf c\rangle
  &=\|(\Id_{\mathcal I_{24}}-\mathsf K)\mathbf c\|_2^2
          +\|\Mom\mathbf c\|_2^2+|\lambda(\mathbf c)|^2\\
  &\geq c_{\mathrm{coer},24}\|\mathbf c\|_2^2.
 \end{aligned}
 \label{eq:coercivity}
\end{equation}
Consequently \(\mathsf N\) is boundedly invertible and
\begin{equation}
 \mathbf c_*=\mathsf N^{-1}\lambda^*(1). \label{eq:inverse-solution}
\end{equation}
\end{theorem}
\begin{proof}
If the inequality failed, there would be unit vectors \(\mathbf c_j\) with
\[
 (\Id_{\mathcal I_{24}}-\mathsf K)\mathbf c_j\to0,\qquad \Mom \mathbf c_j\to0,\qquad \lambda(\mathbf c_j)\to0.
\]
Compactness of \(\mathsf K\) gives a convergent subsequence of
\(\mathsf K\mathbf c_j\); the same subsequence of \(\mathbf c_j\) converges to a
unit vector \(\mathbf c\). Then \(\Mom \mathbf c=0\) and \(\lambda(\mathbf c)=0\),
contradicting Proposition~\ref{prop:vanishing}.
Coercivity gives closed range and zero kernel. Self-adjointness
then gives dense range, hence surjectivity.

Let \(\mathbf b\ne0\) be a vector in \(\ker\Mom\). For
\(\mathbf c=\mathsf N^{-1}\lambda^*(1)\), pair the equation with \(\mathbf b\).
Since \(\lambda(\mathbf b)\ne0\) and \(\mathsf L\mathbf b=0\), this gives
\(\lambda(\mathbf c)=1\). The equation becomes \(\mathsf L\mathbf c=0\).
Thus \(\mathbf c\) is the normalized coefficient solution.
\end{proof}

This also gives convergence of the exact Galerkin approximations.
On \(\mathsf P_J\ell^2(\mathcal I_{24})\) solve
\[
 \mathsf P_J\mathsf N \mathsf P_J \mathbf c_J=\mathsf P_J\lambda^*(1).
\]
Every compression is coercive with the same constant \(c_{\mathrm{coer},24}\).
Galerkin orthogonality and \eqref{eq:coercivity} give
\[
 \|\mathbf c_J-\mathbf c_*\|_2\leq
       \frac{\|\mathsf N\|}{c_{\mathrm{coer},24}}\|(\Id_{\mathcal I_{24}}-\mathsf P_J)\mathbf c_*\|_2
       \longrightarrow0.
\]
The right side has a geometric tail by Theorem~\ref{thm:main}.
This is a convergence statement for the positive equation
\eqref{eq:N}; it does not assert stability of arbitrary finite
truncations of the original compact moment map \(\Mom\).
The proof establishes \(c_{\mathrm{coer},24}>0\) without supplying a numerical
value. The compressed matrix still contains integrals and infinite
inner products, through \(\mathsf K^*\mathsf K\) and \(\Mom^*\Mom\).
A certified numerical implementation would additionally require
error bounds for their evaluation, a computable coercivity bound,
and control of the resulting perturbation of the finite equations.
No such implementation or numerical efficiency claim is needed
for the existence theorem.


\section{Comparison with the known function in dimension 24}\label{sec:comparison}

Existence and uniqueness in the coefficient space have now been
proved. We next identify the answer. This comparison is not used
in either dimension's existence or uniqueness proof.

\subsection{Coefficient identification}

Use \(\cuspq=e^{2\pi i\tau}\), and normalize
\[
 E_2=1-24\sum_{n\geq1}\sigma_1(n)\cuspq^n,\quad
 E_4=1+240\sum_{n\geq1}\sigma_3(n)\cuspq^n,\quad
 E_6=1-504\sum_{n\geq1}\sigma_5(n)\cuspq^n,
\]
where \(\sigma_j(n)=\sum_{\ell\mid n}\ell^j\).
Here \(\Delta_{\rm mod}=(E_4^3-E_6^2)/1728\).
For the comparison only, set
\begin{align*}
 r_0&=-49E_4^3+25E_6^2,&
 r_1&=48E_6E_4^2,&
 r_2&=25E_4^4-49E_4E_6^2,\\
 \varphi_{24,2}&=\frac{r_2+r_1E_2+r_0E_2^2}{\Delta_{\rm mod}^2},&
 \varphi_{24,1}&=\frac{-6r_1-12E_2r_0}{\Delta_{\rm mod}^2},&
 \varphi_{24,0}&=\frac{-36r_0}{\Delta_{\rm mod}^2}.
\end{align*}
Let
\[
 \theta_2(\tau)=\sum_{n\in\mathbb Z}e^{\pi i(n+1/2)^2\tau},
 \qquad
 \theta_4(\tau)=\sum_{n\in\mathbb Z}(-1)^ne^{\pi in^2\tau}.
\]
Writing \(x=\theta_4^4,\ y=\theta_2^4\), define
\[
 \psi_{24,I}=\frac{7x^5y^2+7x^6y+2x^7}{\Delta_{\rm mod}^2},\qquad
 \psi_{24,S}=-\frac{7y^5x^2+7y^6x+2y^7}{\Delta_{\rm mod}^2}.
\]
These are the ingredients of \cite[Sections 2--4]{CKMRV24}:
\(\varphi_{24,2}\) is a weakly holomorphic quasimodular form of
weight \(-8\) and depth \(2\), and \(\psi_{24,I}\) has weight
\(-10\) for \(\Gamma(2)\).
Let \(f_{\rm known}\) be the normalized magic function
constructed in \cite{CKMRV24}, and define
\(\Phi_{\rm known}:[0,\infty)\to\mathbb R\) by
\(f_{\rm known}(\mathbf y)=\Phi_{\rm known}(|\mathbf y|^2)\).
For \(u>4\), it is given by
\begin{equation}
 \Phi_{\rm known}(u)=P(u)\int_0^\infty
 \left(\frac{\pi}{K_{24}^{\varphi}}t^{10}\varphi_{24,2}(i/t)
          -\frac1{K_{24}^{\psi}\pi}\psi_{24,I}(it)\right)e^{-\pi ut}\,dt,
 \label{eq:known-integral}
\end{equation}
Here \(K_{24}^{\varphi}=28304640\) and \(K_{24}^{\psi}=65520\).
The profile extends to all \(u\geq0\).
The required transformation identities are
\begin{align}
 t^{10}\varphi_{24,2}(i/t)
    &=t^2\varphi_{24,2}(it)+\frac{t}{\pi}\varphi_{24,1}(it)
                                      -\frac1{\pi^2}\varphi_{24,0}(it),
 \nonumber\\
 \psi_{24,I}(it)&=-t^{10}\psi_{24,S}(i/t),&
 \psi_{24,S}(\tau)&=\psi_{24,I}(\tau)-\psi_{24,I}(\tau+1).
 \label{eq:known-transformations}
\end{align}

Here is an explicit translation into our coefficients.
Use the cusp coordinate \(\mathfrak q=e^{2\pi i\tau}\), with
\(\mathfrak q^{1/2}=e^{\pi i\tau}\). Define the scalar coefficients by
\begin{align*}
 \varphi_{24,j}(\tau)&=\sum_m\varphi_{24,j,m}\mathfrak q^{m/2}
                                      &&(j=0,1,2),\\
 \psi_{24,I}(\tau)&=\sum_m\psi_{24,I,m}\mathfrak q^{m/2},&
 \psi_{24,S}(\tau)&=\sum_m\psi_{24,S,m}\mathfrak q^{m/2}.
\end{align*}
The indices \(m\) are integers and only finitely many negative
powers occur. For example,
\(\varphi_{24,2,m}=[\mathfrak q^{m/2}]\varphi_{24,2}\)
denotes the scalar coefficient of \(\mathfrak q^{m/2}\) in this
expansion. The brackets denote coefficient extraction.
The relation to our generating-series variable is
\(q=e^\pi\mathfrak q^{1/2}\). The second subscript \(j\)
in \(\varphi_{24,j}\) is chosen so that its weight is
\(2j-12\), matching the three polynomial degrees in the
large-parameter integrals.

Split \eqref{eq:known-integral} at \(t=1\), and use
\eqref{eq:known-transformations} on the appropriate halves.
The large-parameter coefficients before damping are
\begin{align}
 \gamma_{24,0,m}&=-\frac{\varphi_{24,0,m}+432\psi_{24,I,m}}{K_{24}^{\varphi}\pi},\nonumber\\
 \gamma_{24,1,m}&=\frac{\varphi_{24,1,m}}{K_{24}^{\varphi}},\nonumber\\
 \gamma_{24,2,m}&=\frac{\pi \varphi_{24,2,m}}{K_{24}^{\varphi}}, \label{eq:known-c}
\end{align}
and the small-parameter coefficient is
\[
 \beta_{24,m}=\frac{\pi \varphi_{24,2,m}}{K_{24}^{\varphi}}
                            +\frac{\psi_{24,S,m}}{K_{24}^{\psi}\pi}.
\]
Each of \(\varphi_{24,0},\varphi_{24,1},\varphi_{24,2}\) has only even powers of \(\cuspq^{1/2}\), and
\(\psi_{24,S,m}=(1-(-1)^m)\psi_{24,I,m}\).
Thus \(\beta_{24,m}=\gamma_{24,2,m}\) for even \(m\),
and \(\beta_{24,m}=-2\gamma_{24,0,m}\) for odd \(m\).
These are exactly the parity relations in \eqref{eq:ansatz}.
Because the elementary profiles \(U_{m,k}\) and \(V_m\)
include the factor \(e^{\pi m}\), the corresponding sequence
coefficients in \eqref{eq:ansatz} are
\[
 c_{j,m}=e^{-\pi m}\gamma_{24,j,m}\quad(m\ {\rm even}),\qquad
 c_{3,m}=e^{-\pi m}\gamma_{24,0,m}\quad(m\ {\rm odd}).
\]
In particular the two functions from the splitting in
Section~\ref{sec:vanishing} satisfy
\[
 a_2^+=-\frac{\pi}{K_{24}^{\varphi}}\varphi_{24,2},\qquad
 a_2^-=-\frac1{2K_{24}^{\psi}\pi}\psi_{24,S}.
\]
The quasimodular and level-two structures thus agree with the
structures derived from the boundary equations.
The five coefficients of the correction functions are
\(c_{j,m}=e^{-\pi m}\gamma_{24,j,m}\) at
\((j,m)=(0,-4),(0,-2),(0,0),(1,-2),(1,0)\).
There are no other nonpositive modes. In particular the resulting
finite part is
\begin{equation}
 4R_{0,1}-\frac{R_{4,0}}{16380\pi}
 -\frac{14347}{1365\pi}R_{0,0}
 +\frac1{39}R_{2,1}-\frac{292}{4095\pi}R_{2,0}.
 \label{eq:finite-known}
\end{equation}
These identities follow by multiplication of the displayed series;
they also check the normalization of the coefficient coordinates.
Convergence of these expansions for \(|\mathfrak q^{1/2}|<1\),
with only the stated poles at \(0\),
gives weighted absolute summability after the damping factor.

The split integral first identifies the two profiles for \(u>4\).
Here is the analytic domain needed to extend this identity.
Subtract the finitely many nonpositive exponential modes from
the large-\(t\) integrand and integrate them explicitly, giving
the entire correction functions in \eqref{eq:finite-known}.
The remaining large-\(t\) integrand decays at least as a
polynomial times \(e^{-\pi t}\), and the small-\(t\) integrand
decays exponentially in \(1/t\). The resulting integrals
converge normally for complex \(u\) on compact subsets of
\(\Re u>-1\). They give the analytic continuation used in
\cite[Sections 2--4]{CKMRV24}. Our coefficient profile is
holomorphic on this same half-plane by the normal-convergence
estimates in Lemma~\ref{lem:Schwartz}; its possible negative
integer poles lie outside it. The identity theorem gives
equality throughout \(\Re u>-1\), and in particular at all
nonnegative real \(u\).
The known Fourier zeros and normalization now show that the
coefficient vector solves \eqref{eq:system}.
Theorem~\ref{thm:main} identifies it with \(\mathbf c_*\).
Consequently the sign inequalities proved in \cite{CKMRV24}
hold for \(f_{\mathbf c_*}\). In this sense the coefficient construction
recovers the Leech lattice magic function.

\subsection{The alternative uniqueness argument from interpolation}

The interpolation theorem of \cite[Theorems 1.7 and 1.9]{CKMRVinterp}
states that a radial Schwartz function on \(\mathbb R^{24}\) is
determined by the values and radial derivatives of itself and its
Fourier transform at \(\sqrt{2n}\), \(n\geq2\).
For our normalized function, all these data vanish except
the radial derivative
\[
 \left.\frac{d}{dr}\Phi_{\mathbf c_*}(r^2)\right|_{r=2}
       =4\Phi_{\mathbf c_*}'(4)=-\frac1{16380}.
\]
In the interpolation basis of that paper this says
\[
 f_{\mathbf c_*}=-\frac1{16380}b_2,
\]
where \(b_2\) has radial derivative \(1\) at \(2\) and all
other interpolation data zero.
It follows at once that the function is unique in the full radial
Schwartz class with these data. Injectivity in
Lemma~\ref{lem:Schwartz} then gives coefficient uniqueness as well.
This is an alternative argument, not a premise of our proof.
Our independent uniqueness statement is within the coefficient
class \eqref{eq:ansatz}; the assertion about all radial Schwartz
functions uses the interpolation theorem.

\section{The construction in dimension 8}\label{sec:eight}
\label{sec:dimension}

We now prove the analogous theorem for \(E_8\). The first
nonzero squared length is \(2\), so the Fourier equations start
one node earlier. This removes two correction functions and
forces the constant moments to vanish. Gaussian inversion changes
the scalar power from \(10\) to \(2\). We give the resulting
system explicitly, check the continuation problem and its index,
and prove uniqueness. None of these steps uses the comparison
with the known function in Section~\ref{sec:eight-comparison}.

\subsection{The coefficient system and its theorem}

Keep \(P,U_{m,k},R_{\ell,k}\) as defined in
\eqref{eq:U} and \eqref{eq:Hfinite}, and define
\begin{equation}
 V_m^{(8)}(u)=e^{\pi m}P(u)\int_0^1
             t^2e^{-\pi(m/t+ut)}\,dt,
 \qquad u\geq0,\quad m\in\mathbb N.
 \label{eq:eight-V}
\end{equation}
The power is \(d/2-2=2\): Gaussian inversion contributes
\(t^{-d/2}\), and the substitution \(t\mapsto1/t\) contributes
its Jacobian. The large-parameter integrals \(U_{m,k}\) do
not change.

Define
\[
 \mathcal I_8=\{(0,-2),(0,0),(1,0)\}\cup\mathcal I_+.
\]
An eight-dimensional coefficient vector
\(\mathbf c=(c_{j,m})_{(j,m)\in\mathcal I_8}\) consists
of the three numbers \(c_{0,-2},c_{0,0},c_{1,0}\),
three sequences \((c_{k,m})_{m\in2\mathbb N}\), \(k=0,1,2\),
and one sequence \((c_{3,m})_{m\in2\mathbb N-1}\).
For \(p=1,2\), define
\begin{equation}
 \|\mathbf c\|_{p,8}^p=
 |c_{0,-2}|^p+|c_{0,0}|^p+|c_{1,0}|^p
 +\sum_{m\in2\mathbb N}\sum_{k=0}^2|c_{k,m}|^p
 +\sum_{m\in2\mathbb N-1}|c_{3,m}|^p.
 \label{eq:eight-norm}
\end{equation}
Write \(\ell^p(\mathcal I_8)\) for the space of these families with finite
norm. Thus \(\ell^p(\mathcal I_8)=\mathbb C^3\oplus\ell^p(2\mathbb N)^3
\oplus\ell^p(2\mathbb N-1)\). These are new coefficient
vectors; they are not the dimension-24 solutions with two
entries removed.

For \(\mathbf c\in\ell^2(\mathcal I_8)\), set
\begin{align}
 \Phi^{(8)}_{\mathbf c}(u)={}&
 c_{1,0}R_{0,1}(u)+c_{0,0}R_{0,0}(u)
        +e^{-2\pi}c_{0,-2}R_{2,0}(u)\nonumber\\
 &+\sum_{m\in2\mathbb N-1}c_{3,m}
                  \bigl(U_{m,0}(u)-2V_m^{(8)}(u)\bigr)\nonumber\\
 &+\sum_{m\in2\mathbb N}
   \bigl(c_{0,m}U_{m,0}(u)+c_{1,m}U_{m,1}(u)
                +c_{2,m}(U_{m,2}(u)+V_m^{(8)}(u))\bigr).
 \label{eq:eight-family}
\end{align}
Define \(f^{(8)}_{\mathbf c}(\mathbf y)=\Phi^{(8)}_{\mathbf c}(|\mathbf y|^2)\)
for \(\mathbf y\in\mathbb R^8\), and define \(G^{(8)}_{\mathbf c}\)
on \([0,\infty)\) by
\(\widehat f^{(8)}_{\mathbf c}(\boldsymbol{\xi})=G^{(8)}_{\mathbf c}(|\boldsymbol{\xi}|^2)\).
The Fourier transform here is taken in \(\mathbb R^8\).
Finally put \(\lambda_8(\mathbf c)=c_{1,0}/4\).

\begin{theorem}[Eight-dimensional coefficient construction]
\label{thm:eight}
The countable system of linear equations
\begin{equation}
 G^{(8)}_{\mathbf c}(2n)=(G^{(8)}_{\mathbf c})'(2n)=0
       \quad(n\geq1),\qquad \lambda_8(\mathbf c)=1
 \label{eq:eight-system}
\end{equation}
has exactly one solution \(\mathbf c_{*,8}\in\ell^2(\mathcal I_8)\).
Its entries are real, and for every \(1<r<e^\pi\),
\[
 \sum_{m\in2\mathbb N}\sum_{k=0}^2|c_{k,m}|r^m
       +\sum_{m\in2\mathbb N-1}|c_{3,m}|r^m<\infty.
\]
In particular it belongs to \(\ell^1(\mathcal I_8)\). Its radial Schwartz
function satisfies
\begin{align*}
 \Phi^{(8)}_{\mathbf c_{*,8}}(0)&=G^{(8)}_{\mathbf c_{*,8}}(0)=1,&
 \Phi^{(8)}_{\mathbf c_{*,8}}(2n)&=0 &&(n\geq1),\\
 (\Phi^{(8)}_{\mathbf c_{*,8}})'(2)&=-\frac1{120},&
 (\Phi^{(8)}_{\mathbf c_{*,8}})'(2n)&=0 &&(n\geq2).
\end{align*}
The homogeneous Fourier equations have a one-dimensional solution
space in \(\ell^2(\mathcal I_8)\).
\end{theorem}

As in \eqref{eq:matrix-system}, the matrix entries are the
Fourier values and derivatives of the explicitly specified
individual functions in \eqref{eq:eight-family}; only their
coefficients are unknown. The following subsections prove the
theorem.

\subsection{Regularity and Poisson normalization}

The proof of Lemma~\ref{lem:Schwartz} applies to
\eqref{eq:eight-family}. To check the change at small \(t\),
the relevant estimate is now
\[
 t^2\left|\sum_{m\geq1}d_m e^{-\pi m(1/t-1)}\right|
 \leq C\|\mathbf c\|_{2,8}
 \begin{cases}
 t^2e^{-\pi/t},&0<t\leq1/2,\\
 (1-t)^{-1/2},&1/2<t<1,
 \end{cases}
\]
where \(d_m=c_{2,m}\) for even \(m\) and \(d_m=-2c_{3,m}\)
for odd \(m\). The exponential at zero still controls every
power introduced by differentiation. The same estimate
\(t^{-1}+ut\geq2\sqrt u\) proves continuity into the radial
Schwartz space. The meromorphic argument in
Lemma~\ref{lem:Schwartz} proves injectivity: the small-parameter
part is entire, and the large-parameter rational functions are
unchanged. The three correction functions are linearly independent
by their values and derivatives at \(0,2\).

The built-in values are
\begin{equation}
 \Phi^{(8)}_{\mathbf c}(0)=\lambda_8(\mathbf c),\qquad
 (\Phi^{(8)}_{\mathbf c})'(2)
       =\frac\pi4e^{-2\pi}c_{0,-2}.
 \label{eq:eight-jets}
\end{equation}
All values at \(2n\), \(n\geq1\), and derivatives there
for \(n\geq2\), vanish.

For Poisson summation use the even self-dual lattice
\[
 E_8=\{\mathbf y\in\mathbb Z^8:\textstyle\sum_jy_j\in2\mathbb Z\}
 \ \cup\
 \{\mathbf y\in(\mathbb Z+1/2)^8:\textstyle\sum_jy_j\in2\mathbb Z\}.
\]
It has covolume one and \(240\) vectors of squared length \(2\).
Indeed, \(112=4\binom82\) have two integer coordinates equal
to \(\pm1\), and \(128\) have all coordinates equal to
\(\pm1/2\) with an even number of minus signs. See also
\cite[Section 1]{V} for these lattice conventions.

If \(\mathbf c\) solves the homogeneous Fourier equations,
Poisson summation and its differentiated dilation identity,
as in Lemma~\ref{lem:Poisson}, give
\[
 G^{(8)}_{\mathbf c}(0)=\lambda_8(\mathbf c),\qquad
 2\cdot240\,(\Phi^{(8)}_{\mathbf c})'(2)
                              =-4\lambda_8(\mathbf c).
\]
The factor \(4\) is \(d/2\). Thus
\begin{equation}
 (\Phi^{(8)}_{\mathbf c})'(2)=-\frac{\lambda_8(\mathbf c)}{120},
 \qquad
 c_{0,-2}=-\frac{e^{2\pi}}{30\pi}\lambda_8(\mathbf c).
 \label{eq:eight-Poisson}
\end{equation}
After normalization, the correction terms are consequently
\[
 4R_{0,1}-\frac{R_{2,0}}{30\pi}+c_{0,0}R_{0,0}.
\]
Only \(c_{0,0}\) and the four positive-mode sequences remain unknown.

\subsection{The boundary problem and continuation}

Fix \(\mathbf c\in\ell^2(\mathcal I_8)\). In this section the symbols
\(A_j,A_\pm,b_0,b_1\) refer to the series and functions formed
from this eight-dimensional vector. Specifically, replace
\eqref{eq:series} by
\begin{align}
 A_0(q)&=c_{0,-2}q^{-2}+c_{0,0}
                       +\sum_{m\in2\mathbb N}c_{0,m}q^m,\nonumber\\
 A_1(q)&=c_{1,0}+\sum_{m\in2\mathbb N}c_{1,m}q^m,\nonumber\\
 A_2(q)&=\sum_{m\in2\mathbb N}c_{2,m}q^m,\nonumber\\
 A_3(q)&=\sum_{m\in2\mathbb N-1}c_{3,m}q^m,
 \qquad 0<|q|<1.
 \label{eq:eight-series}
\end{align}
The definitions of \(A_\pm\), \(b_0,b_1\) in \eqref{eq:BL},
the maps \(Q,\omega\), and all the parametrized contours of
Section~\ref{sec:boundary} remain the same.

For each fixed \(z\) with \(\Re z>0\), Gaussian inversion
in \(\mathbb R^8\) is
\[
 \widehat{\bigl(\mathbf y\mapsto e^{-\pi z|\mathbf y|^2}\bigr)}(\boldsymbol{\xi})
       =z^{-4}e^{-\pi|\boldsymbol{\xi}|^2/z}.
\]
Consequently define \(\rho_{0,\Gamma}^{(8)},\rho_{1,\Gamma}^{(8)}\) by the four
formulas in \eqref{eq:densities}, replacing the powers
\(z_\pm(t)^{-10},z_\pm(t)^{-11}\) by
\(z_\pm(t)^{-2},z_\pm(t)^{-3}\), respectively. The segment
densities are unchanged:
\[
 \rho_{0,I}^{(8)}(s)=2A_-(e^\pi\sqrt s),\qquad
 \rho_{1,I}^{(8)}(s)=-\pi^{-1}(\log s)\,A_-(e^\pi\sqrt s)
 \quad(0<s<w_0).
\]
Equation~\eqref{eq:measures}, with these densities, defines
\(\mu_0^{(8)},\mu_1^{(8)}\). All arguments and orientations
in that definition are retained.

The calculation proving Lemma~\ref{lem:samples} now gives
\begin{equation}
 G^{(8)}_{\mathbf c}(2n)=\frac1{8\pi}
                  \int_{\mathbb C}w^{n-1}\,d\mu_0^{(8)}(w),\qquad
 (G^{(8)}_{\mathbf c})'(2n)=-\frac18
                  \int_{\mathbb C}w^{n-1}\,d\mu_1^{(8)}(w).
 \label{eq:eight-samples}
\end{equation}
This holds for every integer \(n\geq0\). In that calculation
the path integral \(J_{m,k}\) uses \(z^{-4}\) in place of
\(z^{-12}\). The rational term \(W_m\), the phase-derivative
corrections, and the substitution
\(\omega'(z)=2\pi\omega(z)z^{-2}\) do not change. They give
exactly the densities just specified. The same total variation
bound and integrability at \(s=0\) extend the calculation from
finite coefficient families to all of \(\ell^2(\mathcal I_8)\).

Set \(\mathcal J_8=\{0,1\}\times\mathbb N_0\), so that both
moments of degree zero are now included. Define the moment operator by
\begin{equation}
 (\Mom_8\mathbf c)_{\nu,k}
      =\int_{\mathbb C}w^k\,d\mu_\nu^{(8)}(w),
 \quad \nu=0,1,\quad k=0,1,2,\ldots.
 \label{eq:eight-A}
\end{equation}
Its domain is \(\ell^2(\mathcal I_8)\) and its target is
\(\ell^2(\mathcal J_8)\).
The total variation estimate gives
\(\|\Mom_8\mathbf c\|_{\ell^2(\mathcal J_8)}
\leq C_{\mu,8}(1-\rho^2)^{-1/2}\|\mathbf c\|_{\ell^2(\mathcal I_8)}\),
where \(C_{\mu,8}\) is the dimension-eight total variation constant.
It is a compact linear operator by the geometric moment estimate
\eqref{eq:Acompact}; the two additional constant-moment rows
have finite rank. By \eqref{eq:eight-samples}, its kernel
is exactly the homogeneous Fourier system in
\eqref{eq:eight-system}.

For \(\mathbf c\in\ker\Mom_8\), both zeroth moments are zero.
The Cauchy functions therefore have no explicit pole correction:
\begin{equation}
 h^{(8)}_\nu(\zeta)=\frac1{2\pi i}
       \int_{\mathbb C}\frac{d\mu_\nu^{(8)}(w)}{w-\zeta},
 \qquad
 \mathbf h_{\mathrm{par}}^{(8)}(z)=z^2\mathbf h^{(8)}(\omega(z)).
 \label{eq:eight-Cauchy}
\end{equation}
Here \(\mathbf h^{(8)}=(h_0^{(8)},h_1^{(8)})^T\).
The first formula is defined off \(\Gamma\cup[0,w_0]\);
the second is used wherever its argument lies in that set's
complement. The exterior Cauchy functions vanish, and their
interior traces and segment jumps obey the same formulas as
before with the new densities. The reconstruction statement
of Lemma~\ref{lem:reconstruction} applies with both pole
constants equal to zero. It then gives the vanishing of
\emph{all} nonnegative moments, including the two new rows.

The local continuation proof also applies. Here are the changes
needed to check this assertion. The paired matrix
\(\mathsf S_{\mathrm{pair}}(z)\) in \eqref{eq:paired-transfer} is unchanged,
because rescaling by \(z^2\) cancels the new density power.
Its determinant is still \(4i\). The circle transfer factor
is \((1+iz)^2\), and the real-interval jump is
\[
 2z^2A_-(Q(1/z))\begin{pmatrix}1\\z^{-1}\end{pmatrix}.
\]
For the functions \(b_{\mathrm{end}},b_-\) defined by \eqref{eq:localHX},
with the present coefficient series, the endpoint involutions are
\[
 b_{\mathrm{end}}(z)=z^2b_{\mathrm{end}}(1/z),\qquad b_-(z)+z^2b_-(1/z)=b_{\mathrm{end}}(z).
\]
Every other recovery identity in the proof of
Proposition~\ref{prop:regularity} is unchanged. In particular,
the punctured-disk cover, the bound
\(|1/z-1|<1/7\), and every trace-overlap check remain valid.
The lower circle multiplier is now \((iz)^2=-z^2\), so the
same two-by-two involution still squares to the identity.
Lemmas~\ref{lem:local-area} and \ref{lem:area-removal} apply
with the dimension-eight total variation bound. Powers of \(z\)
are bounded, with bounded inverses, on the
fixed endpoint neighborhoods used there. The Cauchy area
estimates and the coefficient bound by a constant times
the inverse square root of radial distance therefore give
the same removable-singularity argument. In particular there
are constants \(R>1\) and \(C_8<\infty\), independent of
the solution, such that the four functions continue to
\(0<|q|<R\) and
\begin{equation}
 \sup_{|q|=R}\sum_{j=0}^3|A_j(q)|
                         \leq C_8\|\mathbf c\|_{2,8}.
 \label{eq:eight-continuation}
\end{equation}
At \(q=0\), multiplying by \(q^2\) removes the sole allowed
pole. This proves absolute summability already at the local stage.

\subsection{The global problem and its index}

Define \(a_0,a_1,a_2,a_3\) from \eqref{eq:eight-series}
exactly as in Section~\ref{sec:bundle}, using
\(Q(-i\tau)=e^\pi e^{\pi i\tau}\). Define \(a_4,a_5\)
on the fundamental cell by \eqref{eq:converse-p-fields},
with \(h_\nu^{(8)}\) in place of \(h_\nu\).
For \(\mathbf a=(a_0,a_1,a_2,a_3,a_4,a_5)^T\), the global
laws are
\begin{equation}
 \mathbf a(\tau+1)=\mathsf J_{\mathrm{tr}}\mathbf a(\tau),\qquad
 \mathbf a(-1/\tau)=\tau^{-2}\mathsf J_{\mathrm{inv}}(\tau)\mathbf a(\tau).
 \label{eq:eight-automorphy}
\end{equation}
Here \(\mathsf J_{\mathrm{tr}},\mathsf J_{\mathrm{inv}}\) are the same
permanent gluing matrices as in \eqref{eq:J}; they have no
dimension parameter. The allowed cusp exponents are now
\begin{equation}
 \begin{array}{c|rrrrrr}
 \text{function}&a_0&a_1&a_2&a_3&a_4&a_5\\ \hline
 \text{lowest allowed exponent of }\cuspq
        &-1&0&1&\tfrac12&0&0.
 \end{array}
 \label{eq:eight-cusp}
\end{equation}
As before, each component divided by its displayed power is
holomorphic at \(\cuspq=0\). Denote this holomorphic bundle
on \(\overline{\mathcal X}\) by \(\mathcal E_8\).

\begin{lemma}\label{lem:eight-equivalence}
The coefficient map gives an isomorphism
\[
 H^0(\overline{\mathcal X},\mathcal E_8)
                      \simeq\ker_{\ell^2(\mathcal I_8)}\Mom_8.
\]
Every vector in this kernel has the weighted absolute summability
asserted in Theorem~\ref{thm:eight}.
\end{lemma}
\begin{proof}
We apply the two directions proved in
Lemmas~\ref{lem:section-to-coefficients} and
\ref{lem:coefficients-to-section}, checking each change
with the dimension. The open cell and its
Cauchy evaluation points are unchanged. The four coefficient
functions are recovered by the four-by-four matrix
\(\mathsf S_{\mathrm{cell}}\) in \eqref{eq:cell-recovery}, whose determinant
is \(-4\). Write
\(\mathsf S_{\mathrm{rec}}^{(8)},\mathsf S_{\mathrm{match}}^{(8)}\)
for the dimension-eight recovery and matching matrices.
In the six-by-six recovery recipe, replace the two factors
\(-\tau^{10}\) by \(-\tau^2\); call the resulting matrix
\(\mathsf S_{\mathrm{rec}}^{(8)}(\tau)\). Its last two output coordinates are
explicitly
\[
 a_4=-\tau^2b_{3,0}-2(\tau+1)a_2-2a_3,\qquad
 a_5=-\tau^2b_{3,1}+(\tau^2+\tau)a_2+(2\tau+1)a_3.
\]
The inputs \(\mathbf b_1,\mathbf b_2,\mathbf b_3\) are the
three two-component Cauchy values used in
\eqref{eq:cell-recovery}, now formed from \(\mathbf h_{\mathrm{par}}^{(8)}\).
Curved-side matching is expressed by
\[
 \mathsf S_{\mathrm{match}}^{(8)}(\tau)=
 \begin{pmatrix}
 0&\tau^{-2}I_2&0\\0&0&I_2\\(1+\tau)^{-2}I_2&0&0
 \end{pmatrix}.
\]
The same direct multiplication, with the new powers, gives
\begin{equation}
 \mathsf J_{\mathrm{tr}}\mathsf S_{\mathrm{rec}}^{(8)}(-1/\tau-1)\mathsf S_{\mathrm{match}}^{(8)}(\tau)
       =\tau^{-2}\mathsf J_{\mathrm{inv}}(\tau)\mathsf S_{\mathrm{rec}}^{(8)}(\tau).
 \label{eq:eight-cell}
\end{equation}
The boundary trace identities in \eqref{eq:outer-check} use
\(z^2\) on the left. Their row identities in
\(\mathsf J_{\mathrm{inv}},\mathsf J_{\mathrm{tr}}\) are unchanged, since
\(i^2=i^{10}=-1\). Equations~\eqref{eq:eight-cell} and
\eqref{eq:group-relations} prove the side identifications
\eqref{eq:eight-automorphy}. For the open sides the domain checks and the trace-convergence
hypotheses are identical to those in
Lemma~\ref{lem:coefficients-to-section}. At the order-three
vertex the three Cauchy arguments are still
\(-e^{-\pi\sqrt3}\), and the matrices are regular there.
At the order-two vertex the endpoint continuation makes the
incident densities analytic, so Lemma~\ref{lem:endpoint-log}
gives the logarithmic bound. The finite-sector construction
on the upper-half-plane cover applies because the scalar
weight \(-2\), like weight \(-10\), has product one in both
group relations. Its Laurent coefficient estimate removes
the vertices in all sectors. These arguments establish
holomorphy on \(\mathbb H\), with the required side laws.

At the cusp, the first four exponents follow directly from
\eqref{eq:eight-series}. To check the last two, the subtracted
cut-jump terms are bounded by constants times
\(|w|\,|\log w|^2\) and \(|w|^{1/2}|\log w|\).
After these terms have been removed, \(a_4,a_5\) are
single-valued and area-integrable near zero. They can therefore
have at most simple poles. The residue calculation in
Lemma~\ref{lem:coefficients-to-section} identifies these residues
with the two zeroth moments, which now vanish. Thus the two
singularities are removable, giving the last two exponents
in \eqref{eq:eight-cusp}.

Conversely, a section gives four generating functions on
\(0<|q|<e^\pi\) with exactly the powers in
\eqref{eq:eight-series}. Define candidate Cauchy functions
by \eqref{eq:index-H0}--\eqref{eq:index-H1}.
They have the required jumps and outer traces. The cusp
conditions give no pole at zero, and their endpoint values
are bounded on each incident side. Lemma~\ref{lem:reconstruction}
with zero pole constants proves all nonnegative moment
equations, including the two zeroth moments. It also identifies
the candidates with \eqref{eq:eight-Cauchy}. Injectivity
follows by the same triple-pairing argument in
the six expressions following \eqref{eq:group-relations}.
Finally, holomorphy of the recovered
generating functions up to \(|q|<e^\pi\), apart from their
specified pole at zero, gives weighted absolute summability
for every \(r<e^\pi\) by Cauchy's estimates.
\end{proof}

\begin{lemma}\label{lem:eight-index}
The Euler characteristic of \(\mathcal E_8\) is \(1\).
In particular \(\ker\Mom_8\ne\{0\}\).
\end{lemma}
\begin{proof}
The determinant factor in \eqref{eq:eight-automorphy} has
scalar weight \(-12\) and the same character
\(\varepsilon_{\mathrm{det}}(S)=\varepsilon_{\mathrm{det}}(T)=-1\) as before.
The sum of the exponents in \eqref{eq:eight-cusp} is \(1/2\).
Thus the line-bundle extension formula used in
\eqref{eq:detbundle} gives
\[
 \det\mathcal E_8\simeq\mathcal L_{-12-12(1/2)}
                         =\mathcal L_{-18}.
\]
The character's exponent here is \(1/2\), without the
integer shift \(-4\) required in dimension \(24\).
Equivalently, division of a determinant section by
\(\eta^{12}\) already gives weight \(-18\), trivial character,
and a holomorphic cusp frame, so no power of
\(\Delta_{\mathrm{mod}}\) is needed.

At the elliptic fixed points the fiber actions are
\[
 -\mathsf J_{\mathrm{inv}}(i),\qquad
 \rho_3^2\mathsf J_{\mathrm{inv}}(\rho_3+1)\mathsf J_{\mathrm{tr}},
 \qquad \rho_3=e^{2\pi i/3}.
\]
The first is unchanged. The second is \(\rho_3\) times
the dimension-24 action, so its three eigenvalues are cyclically
permuted; each still has multiplicity two. The eigenvalue
multiplicities are therefore \((3,3)\) and \((2,2,2)\).
All nontrivial elliptic traces vanish, and the central element
acts trivially. The Riemann--Roch formula used in
Proposition~\ref{prop:index} now gives
\[
 \chi(\mathcal E_8)=\frac{5\cdot6-18}{12}=1.
\]
Hence a nonzero section exists, and
Lemma~\ref{lem:eight-equivalence} gives a nonzero coefficient
solution.
\end{proof}

The equality of the two indices has a simple accounting explanation.
Changing the scalar factor from weight \(-10\) to weight \(-2\)
raises the determinant weight by \(48\). Removing the two
coefficient poles and the two Cauchy poles raises the sum of
cusp exponents by \(4\), subtracting exactly \(48\) from the
extended determinant weight. Both determinant bundles are
therefore \(\mathcal L_{-18}\).

\subsection{Vanishing, uniqueness, and the bounded inverse}

\begin{lemma}\label{lem:eight-vanishing}
If \(\mathbf c\in\ker\Mom_8\) and
\(\lambda_8(\mathbf c)=0\), then \(\mathbf c=0\).
\end{lemma}
\begin{proof}
Poisson summation in \eqref{eq:eight-Poisson} gives
\(c_{0,-2}=0\). Consequently every component in
\eqref{eq:eight-cusp} is bounded at the cusp; moreover
\(a_2\) vanishes there and \(a_3\) is bounded by a constant
times \(|\cuspq|^{1/2}\).

Apply the same constant splitting as in
Section~\ref{sec:vanishing}:
\begin{align*}
 a_0^+&=(a_0+a_5)/2,&
 a_1^+&=(a_1+a_2+a_4)/2,&a_2^+&=a_2,\\
 a_0^-&=(a_0-a_5)/2,&
 a_1^-&=(a_1-a_2-a_4)/2,&a_2^-&=a_3.
\end{align*}
All functions are holomorphic on \(\mathbb H\). All except
\(a_2^-\) are periodic under \(\tau\mapsto\tau+1\);
\(a_2^-\) is antiperiodic. The first block obeys
\begin{align*}
 a_0^+(-1/\tau)&=\tau^{-4}a_0^+(\tau),\\
 a_1^+(-1/\tau)&=\tau^{-2}a_1^+(\tau)
                                      +2\tau^{-3}a_0^+(\tau),\\
 a_2^+(-1/\tau)&=a_2^+(\tau)
                      +\tau^{-1}a_1^+(\tau)+\tau^{-2}a_0^+(\tau).
\end{align*}
These identities follow by multiplying the splitting matrix
against \(\mathsf J_{\mathrm{inv}}\), as in \eqref{eq:triangular-splitting}.
The valence formula admits no nonzero holomorphic full-group
modular form of negative weight. It first gives
\(a_0^+=0\), at weight \(-4\), and then
\(a_1^+=0\), at weight \(-2\).
Now \(a_2^+\) is a holomorphic form of weight zero,
hence constant. Its vanishing at the cusp makes it zero.

For the second block, \(a_1^-\) is a holomorphic full-group
form of weight \(-2\), so it too is zero. The remaining vector satisfies the following laws, using the
two-by-two matrices \(\mathsf S_{\mathrm{lev},S},\mathsf S_{\mathrm{lev},T}\)
already used in Section~\ref{sec:vanishing}:
\[
 \begin{pmatrix}a_0^-\\a_2^-\end{pmatrix}(-1/\tau)
   =\tau^{-2}\mathsf S_{\mathrm{lev},S}
         \begin{pmatrix}a_0^-\\a_2^-\end{pmatrix}(\tau),
 \qquad
 \begin{pmatrix}a_0^-\\a_2^-\end{pmatrix}(\tau+1)
   =\mathsf S_{\mathrm{lev},T}\begin{pmatrix}a_0^-\\a_2^-\end{pmatrix}(\tau),
\]
where
\[
 \mathsf S_{\mathrm{lev},S}=\tfrac12\begin{pmatrix}-1&3\\1&1\end{pmatrix},
 \qquad \mathsf S_{\mathrm{lev},T}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.
\]
Their representation factors through the group modulo \(2\).
Thus \(h=a_2^-\) is a weight-\(-2\) form for \(\Gamma(2)\),
holomorphic at every cusp: each slash transform is a constant
linear combination of the bounded functions \(a_0^-,a_2^-\).
If \(h\ne0\), the product over the six cosets,
\[
 \prod_{\gamma\in\Gamma(2)\backslash\SL}(h|_{-2}\gamma),
\]
would be a nonzero holomorphic full-group form of weight \(-12\).
The valence formula forbids it. Hence \(a_2^-=0\), and its
inversion law gives \(a_0^-=0\). All six split components
vanish. Inverting the splitting and recovering the coefficients
gives \(\mathbf c=0\).
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:eight}]
Lemma~\ref{lem:eight-index} gives a nonzero element of
\(\ker\Mom_8\), and Lemma~\ref{lem:eight-vanishing} makes
\(\lambda_8\) injective on that kernel. The kernel is therefore
one-dimensional, with a unique vector of normalization one.
Complex conjugation of the real coefficient equations shows
that its entries are real. Lemma~\ref{lem:eight-equivalence}
gives the weighted absolute summability, and
\eqref{eq:eight-jets}--\eqref{eq:eight-Poisson} give all
the asserted values and derivatives.
\end{proof}

There is also a boundedly invertible equation, with the same
purpose as in Section~\ref{sec:operator}. Define
\(\mathsf K_8\) by the recipe in Appendix~\ref{app:K}, using
the eight-dimensional measures, replacing the powers \(10\)
by \(2\), and extracting only the powers allowed in
\eqref{eq:eight-series}. For arbitrary coefficient vectors
use the corrected Cauchy formula \eqref{eq:Cauchy} with their
actual zeroth moments; on \(\ker\Mom_8\) those moments are zero.
The disk averaging and coefficient extraction then give a
bounded compact operator satisfying
\[
 \mathsf K_8\mathbf c=\mathbf c\quad(\mathbf c\in\ker\Mom_8).
\]
Indeed the estimates for arbitrary raw assignments in the
appendix use only the total variation and area bounds, which
remain valid, and reproduction on solutions follows from
\eqref{eq:eight-continuation}.

With adjoints in the coefficient and moment Hilbert spaces,
put
\begin{equation}
 \mathsf N_8=(\Id_{\mathcal I_8}-\mathsf K_8)^*(\Id_{\mathcal I_8}-\mathsf K_8)
                      +\Mom_8^*\Mom_8+\lambda_8^*\lambda_8.
 \label{eq:eight-N}
\end{equation}
Here \(\Id_{\mathcal I_8}\) is the identity on \(\ell^2(\mathcal I_8)\).
The compactness argument in Theorem~\ref{thm:inverse}, using
Lemma~\ref{lem:eight-vanishing}, proves
\[
 \langle\mathsf N_8\mathbf c,\mathbf c\rangle
           \geq c_{\mathrm{coer},8}\|\mathbf c\|_{2,8}^2
 \quad\text{for some }c_{\mathrm{coer},8}>0.
\]
Thus \(\mathsf N_8\) has a bounded inverse and
\(\mathbf c_{*,8}=\mathsf N_8^{-1}\lambda_8^*(1)\).
The exact finite-dimensional positive compressions converge by the
same Galerkin argument, subject to the distinction between exact
entries and certified numerical evaluation made in
Section~\ref{sec:operator}. This completes the coefficient construction
before any comparison with Viazovska's function.

\subsection{Identification with Viazovska's function}
\label{sec:eight-comparison}

We now identify the answer, using \cite[Sections 4--5]{V}.
This comparison is not part of the preceding existence or
uniqueness proof. Use the Eisenstein series, discriminant,
and theta functions from Section~\ref{sec:comparison}, and
also define
\(\theta_3(\tau)=\sum_{n\in\mathbb Z}e^{\pi in^2\tau}\).
Set
\[
 \varphi_{8,0}=\frac{E_4^2}{\Delta_{\rm mod}},\qquad
 \varphi_{8,1}=\frac{E_4^2E_2-E_4E_6}{\Delta_{\rm mod}},\qquad
 \varphi_{8,2}=\frac{(E_4E_2-E_6)^2}{\Delta_{\rm mod}}.
\]
\begin{align*}
 \psi_{8,I}&=128\left(
       \frac{\theta_3^4+\theta_4^4}{\theta_2^8}
                +\frac{\theta_4^4-\theta_2^4}{\theta_3^8}\right),\\
 \psi_{8,S}&=-128\left(
       \frac{\theta_3^4+\theta_2^4}{\theta_4^8}
                +\frac{\theta_2^4-\theta_4^4}{\theta_3^8}\right).
\end{align*}
All functions on the right are evaluated at
\(\tau\in\mathbb H\). The second subscript \(j=0,1,2\) in
\(\varphi_{8,j}\) has the same role as in dimension \(24\):
the weight is \(2j-d/2\), now \(-4,-2,0\).
The first subscript specifies the dimension.

Let \(\Phi^{(8)}_{\rm known}\) be the profile of Viazovska's
normalized function. For \(u>2\), its integral formula is
\begin{equation}
 \Phi^{(8)}_{\rm known}(u)=P(u)\int_0^\infty
 \left(-\frac\pi{2160}t^2\varphi_{8,2}(i/t)
       -\frac1{60\pi}\psi_{8,I}(it)\right)e^{-\pi ut}\,dt.
 \label{eq:eight-known}
\end{equation}
The transformation identities needed to split this integral are
\begin{align}
 t^2\varphi_{8,2}(i/t)&=t^2\varphi_{8,2}(it)
         -\frac{12t}{\pi}\varphi_{8,1}(it)
         +\frac{36}{\pi^2}\varphi_{8,0}(it),\nonumber\\
 \psi_{8,I}(it)&=-t^2\psi_{8,S}(i/t),\nonumber\\
 \psi_{8,S}(\tau)&=\psi_{8,I}(\tau)-\psi_{8,I}(\tau+1).
 \label{eq:eight-known-laws}
\end{align}

Define the scalar coefficients of the cusp expansions by
\begin{align*}
 \varphi_{8,j}(\tau)&=\sum_m\varphi_{8,j,m}\mathfrak q^{m/2}
                                    &&(j=0,1,2),\\
 \psi_{8,I}(\tau)&=\sum_m\psi_{8,I,m}\mathfrak q^{m/2},&
 \psi_{8,S}(\tau)&=\sum_m\psi_{8,S,m}\mathfrak q^{m/2}.
\end{align*}
Here again \(\mathfrak q^{1/2}=e^{\pi i\tau}\), and the
coefficients are numbers, defined by coefficient extraction.
Splitting \eqref{eq:eight-known}
at \(t=1\) and applying \eqref{eq:eight-known-laws} gives
the undamped large-parameter coefficients
\begin{align}
 \gamma_{8,0,m}&=-\frac{\varphi_{8,0,m}+\psi_{8,I,m}}{60\pi},\nonumber\\
 \gamma_{8,1,m}&=\frac{\varphi_{8,1,m}}{180},\nonumber\\
 \gamma_{8,2,m}&=-\frac{\pi \varphi_{8,2,m}}{2160},
 \label{eq:eight-known-coefficients}
\end{align}
and the small-parameter coefficients
\[
 \beta_{8,m}=-\frac{\pi \varphi_{8,2,m}}{2160}
                                  +\frac{\psi_{8,S,m}}{60\pi}.
\]
All three \(\varphi_{8,j}\) have only even powers, and
\(\psi_{8,S,m}=(1-(-1)^m)\psi_{8,I,m}\). Hence
\(\beta_{8,m}=\gamma_{8,2,m}\) for even \(m\),
and \(\beta_{8,m}=-2\gamma_{8,0,m}\) for odd \(m\).
The coefficients with positive exponent \(m\) in
\eqref{eq:eight-family} are
\[
 c_{k,m}=e^{-\pi m}\gamma_{8,k,m}\quad(m\in2\mathbb N),
 \qquad c_{3,m}=e^{-\pi m}\gamma_{8,0,m}\quad(m\in2\mathbb N-1).
\]
The permitted coefficients with exponent \(m\leq0\) are obtained by the
same rule \(c_{k,m}=e^{-\pi m}\gamma_{8,k,m}\).

For clarity, the coefficients at powers \(\mathfrak q^{-1},\mathfrak q^0\)
of \((\varphi_{8,2},\varphi_{8,1},\varphi_{8,0},
\psi_{8,I})\) are respectively
\[
 (0,0,1,1),\qquad(0,720,504,144).
\]
There are no other nonpositive terms. Thus the correction part
is exactly
\begin{equation}
 4R_{0,1}-\frac{R_{2,0}}{30\pi}
                         -\frac{54}{5\pi}R_{0,0}.
 \label{eq:eight-known-finite}
\end{equation}
In particular its normalization is one. These Laurent series,
viewed in the variable \(\mathfrak q^{1/2}\), converge for
\(|\mathfrak q^{1/2}|<1\), apart from the stated poles at zero.
Cauchy's estimates give weighted absolute summability after
the factor \(e^{-\pi m}\).
The split integral gives equality of the profiles for \(u>2\).
Subtracting its nonpositive large-\(t\) modes gives the entire
correction functions in \eqref{eq:eight-known-finite}.
The remaining large-\(t\) integrand decays at least as a
polynomial times \(e^{-\pi t}\), and the small-\(t\) part
decays exponentially in \(1/t\).
As in the preceding comparison, these expressions define
holomorphic profiles on the common connected domain
\(\Re u>-1\); this is the continuation of the original contour
formula in \cite[Sections 4--5]{V}.
The identity theorem therefore gives equality for every
nonnegative real \(u\). The known Fourier zeros and
normalization make the coefficient vector a solution of
\eqref{eq:eight-system}, and Theorem~\ref{thm:eight} identifies
it with \(\mathbf c_{*,8}\). The signs proved in
\cite[Theorem 4]{V} therefore hold for our function.

Fourier interpolation gives the parallel uniqueness statement
in the full radial Schwartz class. By
\cite[Theorems 1.7 and 1.9]{CKMRVinterp}, the data at
\(\sqrt{2n}\), \(n\geq1\), determine such a function in
\(\mathbb R^8\). All our interpolation data are zero except
\[
 \left.\frac{d}{dr}\Phi^{(8)}_{\mathbf c_{*,8}}(r^2)
             \right|_{r=\sqrt2}=-\frac1{30\sqrt2}.
\]
If \(b_1^{\rm int}\) denotes the interpolation basis function
whose radial derivative at \(\sqrt2\) is one and whose other
data vanish, then
\(f^{(8)}_{\mathbf c_{*,8}}=-(30\sqrt2)^{-1}b_1^{\rm int}\).
The superscript marks an interpolation basis function. This
identification is an additional consequence, not an input to
the construction.

\section{The remaining sign problem}\label{sec:signs}

Theorems~\ref{thm:main} and \ref{thm:eight} solve the coefficient
systems: they produce real radial Schwartz functions with the
required zeros, normalization, and first derivative. To complete
an independent Cohn--Elkies proof, one must still establish
\[
 \Phi^{(8)}_{\mathbf c_{*,8}}(u)\leq0\ (u\geq2),\qquad
 G^{(8)}_{\mathbf c_{*,8}}(v)\geq0\ (v\geq0),
\]
and
\[
 \Phi_{\mathbf c_*}(u)\leq0\ (u\geq4),\qquad
 G_{\mathbf c_*}(v)\geq0\ (v\geq0).
\]
Prescribing values and first derivatives at a discrete set does
not establish these inequalities on the intervals between nodes.
Nor does an index or a uniqueness argument establish a sign.
Our comparisons prove that the inequalities hold by transferring
the earlier sign theorems. Without that comparison, the global
sign proof in both dimensions remains unfinished. Thus we have
an independent construction of the functions, not yet a complete
independent proof of the sphere-packing bounds.

It is useful to distinguish this situation from the planar paper,
which proves its signs within the coefficient construction
\cite[Section 4 and Appendices A--B]{OpenAI}. Its finite
coefficient approximations come with rigorous bounds for the
error in each \(\ell^1\) coefficient list. On bounded intervals,
the exact prescribed zeros are factored out. Near a zero, the
lower Taylor terms of the approximation error are subtracted
before division by the vanishing factor, so the error bound
does not become singular. Polynomial approximations in the
Bernstein basis then certify signs on entire intervals through
positive Bernstein coefficients, with rigorous interval enclosures
and quadrature remainder bounds.

The unbounded region is controlled separately by an explicit
barrier built from the trigonometric product, together with
second-derivative and tail estimates. These analytic estimates
are what let finitely many certified computations prove signs
for the infinite coefficient series on the entire required
half-lines. They are not a numerical sampling argument.
Adapting that program here would require effective coefficient
error bounds and sign margins suited to the dimension-eight
and dimension-24 families. Our bounded-inverse theorems provide
convergence, but do not supply those quantitative certificates.

\appendix
\section{Why these elementary functions?}\label{app:motivation}

The factor \(P\) builds in the desired zeros of the physical
profile. It does not determine the remaining factors in
\(U_{m,k}\) and \(V_m\), nor does it explain their parity
restrictions. This appendix motivates those choices by Gaussian
inversion and an explicit cancellation in the Fourier samples.
The calculation applies to both \(d=8\) and \(d=24\).
It uses no modular form and does not assume that a nonzero
coefficient solution exists.

\subsection{Gaussian inversion and the two integration ranges}

For a radial profile \(h\) whose associated function is Schwartz,
write \(\mathcal F_d h\) for its radial Fourier profile:
\[
 \widehat{(\mathbf y\mapsto h(|\mathbf y|^2))}(\boldsymbol{\xi})
       =(\mathcal F_d h)(|\boldsymbol{\xi}|^2).
\]
This is the Fourier transform in \(\mathbb R^d\), expressed in
one-variable profiles; it is not the one-dimensional Fourier
transform of \(h\). For \(m\geq1\), start with
\[
 g_m(u)=e^{\pi m}\int_1^\infty e^{-\pi mt}e^{-\pi tu}\,dt.
\]
Gaussian inversion and the substitution \(t=1/s\) give
\begin{align}
 (\mathcal F_d g_m)(v)
 &=e^{\pi m}\int_1^\infty
                t^{-d/2}e^{-\pi(mt+v/t)}\,dt \nonumber\\
 &=e^{\pi m}\int_0^1
                s^{d/2-2}e^{-\pi(m/s+vs)}\,ds .
 \label{eq:app-reciprocal}
\end{align}
Consequently
\[
 U_{m,0}=P g_m,\qquad V_m^{(d)}=P\,\mathcal F_d g_m,
\]
where \(V_m^{(24)}=V_m\) and \(V_m^{(8)}\) is
\eqref{eq:eight-V}. The power \(d/2-2\) is forced by
Gaussian inversion and the Jacobian. The split at \(1\) is
natural because \(1\) is fixed by inversion of the Gaussian
parameter. Multiplication by \(P\) does not commute with
\(\mathcal F_d\), so these identities alone do not determine
the Fourier samples of \(U_{m,0}\) or \(V_m^{(d)}\).
That is the purpose of the next calculation.

The factor \(e^{\pi m}\) normalizes the exponential mode at
the splitting point:
\[
 e^{\pi m}e^{-\pi mt}=e^{-\pi m(t-1)}.
\]
It therefore makes the coefficient series converge in the
variable \(q=Q(t)=e^{-\pi(t-1)}\) on the unit disk.
For each fixed \(m\), it is a scalar normalization of the profile;
the coefficient norm is then prescribed in these coordinates.
Finally, polynomial weights in \(t\)
give elementary rational factors:
\begin{equation}
 U_{m,k}(u)=P(u)e^{-\pi u}
       \sum_{j=0}^k\frac{k!}{(k-j)![\pi(m+u)]^{j+1}}.
 \label{eq:app-rational}
\end{equation}
Thus \(k=0,1,2\) permits successively higher principal parts
at the negative modes while preserving all the physical zeros.

\subsection{The parity restrictions and the coupled terms}

Let \(p\) be a polynomial with real coefficients and define
\[
 U_m[p](u)=e^{\pi m}P(u)
               \int_1^\infty p(t)e^{-\pi(m+u)t}\,dt.
\]
For \(p(t)=t^k\) this is \(U_{m,k}\).
Put \(\sigma_m=(-1)^m\) and define the polynomial
\begin{equation}
 (\delta_m p)(t)=2p(t)-\sigma_m\bigl(p(t+i)+p(t-i)\bigr).
 \label{eq:app-difference}
\end{equation}
The sign \(\sigma_m\) comes from shifting the exponential
mode by \(i\) or \(-i\). To see this precisely, use
\[
 P(u)e^{-\pi tu}
 =\frac14\bigl(2e^{-\pi tu}
              -e^{-\pi(t-i)u}-e^{-\pi(t+i)u}\bigr).
\]
Hold \(v\geq0\) fixed, apply Gaussian inversion, and shift
the two displaced horizontal paths back to the positive real
axis. For \(m\geq1\), the integrands decay exponentially
at the right ends, so the connecting integrals there tend
to zero. Define the remaining integral along the straight
path from \(1\) to \(1+i\) by
\[
 J_m[p](v)=e^{\pi m}\int_1^{1+i}
              p(z-i)z^{-d/2}e^{-\pi(mz+v/z)}\,dz.
\]
The lower path contributes its complex conjugate. The result is
\begin{equation}
 \begin{split}
 (\mathcal F_d U_m[p])(v)
 ={}&\frac{\sigma_m}{2}\Re J_m[p](v)\\
 &+\frac{e^{\pi m}}4\int_1^\infty
       (\delta_m p)(t)t^{-d/2}e^{-\pi(mt+v/t)}\,dt.
 \end{split}
 \label{eq:app-U-transform}
\end{equation}
The notation \(J_m[1]\) means that the polynomial is the
constant \(1\), not the monomial of degree one.
For \(p(t)=t^k\), this path integral is \(J_{m,k}\) from the
sample calculation, with the exponent appropriate to \(d\).

There is a companion identity for the reciprocal family.
Let \(W_m(v)=e^{-\pi v}/(2\pi(m+v))\), as in the sample
calculation. Then
\begin{equation}
 \begin{split}
 (\mathcal F_d V_m^{(d)})(v)
 ={}&W_m(v)
   +\frac{\sigma_m}{2}\Re\bigl(e^{-\pi iv}J_m[1](v)\bigr)\\
 &-\frac{\sigma_m}{2}\cos(\pi v)(\mathcal F_d g_m)(v).
 \end{split}
 \label{eq:app-V-transform}
\end{equation}
For completeness, the unshifted Gaussian term in this
calculation is
\[
 \frac{e^{\pi m}}2\int_0^1
                   t^{-2}e^{-\pi(m+v)/t}\,dt=W_m(v).
\]
In the other two terms, first substitute \(s=1/t\).
The transformed Gaussian parameters involve \(s+i\) and
\(s-i\); the identity
\(s/(1+is)=-i+1/(s-i)\) and its conjugate separate the
factors \(e^{\pi iv}\) and \(e^{-\pi iv}\).
Shifting the resulting paths back to the real axis gives
the last two terms of \eqref{eq:app-V-transform}.
The constant phase \(i^{-d/2}\) equals \(1\) in both
dimensions under consideration.
All these identities can be differentiated in \(v\) on compact
real intervals by dominated convergence.

Suppose now that \(\delta_m p\) is a constant \(b\).
For the combination
\[
 U_m[p]+\frac{b}{2\sigma_m}V_m^{(d)},
\]
the sum of the unbounded real-path terms in
\eqref{eq:app-U-transform}--\eqref{eq:app-V-transform} is
\[
 \frac b4(1-\cos(\pi v))(\mathcal F_d g_m)(v)
       =\frac b2 P(v)(\mathcal F_d g_m)(v).
\]
This has a double zero at every \(v=2n\), including \(n=0\).
Its contribution to both Fourier sample equations is therefore
zero. Only the fixed finite path integrals and \(W_m\)
remain. The latter also becomes a bounded-interval moment:
\[
 W_m(2n)=\frac1{4\pi}\int_0^{w_0}
                  s^{n-1}e^{\pi m}s^{m/2}\,ds,
 \qquad w_0=e^{-2\pi}.
\]

The elementary polynomial differences are
\[
 \begin{array}{c|c|c|c}
 \text{parity of }m&p(t)&\delta_m p&
       U_m[p]+\dfrac{\delta_m p}{2\sigma_m}V_m^{(d)}
       \\ \hline
 \text{even}&1&0&U_{m,0}\\
 \text{even}&t&0&U_{m,1}\\
 \text{even}&t^2&2&U_{m,2}+V_m^{(d)}\\
 \text{odd}&1&4&U_{m,0}-2V_m^{(d)}
 \end{array}
\]
These are precisely the four coefficient families used in the
paper. Among polynomial weights, the requirement that
\(\delta_m p\) be constant permits degree at most two for
even \(m\), and degree zero for odd \(m\). For even \(m\),
the difference kills constants and linear polynomials, and
sends degree \(k\geq2\) to degree \(k-2\); for odd \(m\),
it preserves the leading degree. Higher-degree
weights could be considered, but their real-path terms would
require additional reciprocal families for this cancellation.

\subsection{What the calculation explains}

The construction now has a sequence of reasons: the lattice
nodes suggest \(P\); inversion of Gaussians pairs the two
integration ranges; exponential modes organize the unknowns
into series; and the polynomial difference calculation leaves
three even series and one odd series. The resulting Fourier
values and derivatives become two moment families on fixed
supports. Their two Cauchy transforms, together with the four
generating series, account for the six functions in the
boundary and bundle problems.

These observations motivate a tractable coefficient space.
They neither characterize every family with the prescribed
zeros nor prove that the chosen system has a nonzero solution.
Existence and uniqueness require the continuation, index,
and vanishing arguments in the body of the paper.
The later comparison with the known magic functions confirms
which function this construction produces; it is not used
to justify the choices in this appendix.

\section{A compact regularizing operator}\label{app:K}

For completeness we specify the operator used in
Section~\ref{sec:operator}. The construction turns the local
continuation proof into a bounded linear operation on arbitrary
coefficient vectors, including vectors that do not solve the
equations.

For arbitrary \(\mathbf c\in\ell^2(\mathcal I_{24})\), use \eqref{eq:measures} and
\eqref{eq:Cauchy} to define \(\mathbf h\) and \(\mathbf h_{\mathrm{par}}\)
on the same interior domains. No vanishing of the exterior
Cauchy function is needed for these definitions.
For a center \(\zeta_0\in\mathbb C\), a radius \(r>0\), and an
area-integrable function \(h\) on the disk \(D(\zeta_0,r)\),
define its holomorphic average by
\begin{equation}
 (\mathsf B_{\zeta_0,r}h)(z)=\frac1{\pi r^2}
       \int_{D(\zeta_0,r)}
       \frac{h(\zeta)}
       {(1-(z-\zeta_0)\overline{(\zeta-\zeta_0)}/r^2)^2}\,dA(\zeta).
 \label{eq:Bergman}
\end{equation}
For \(0<\sigma<1\),
\[
 \sup_{|z-\zeta_0|\leq\sigma r}|\mathsf B_{\zeta_0,r}h(z)|
       \leq\frac{\|h\|_{L^1(D(\zeta_0,r))}}{\pi r^2(1-\sigma)^2}.
\]
Expanding the kernel and integrating angularly proves that
\(\mathsf B_{\zeta_0,r}\) reproduces every holomorphic area-integrable
function on the disk.

The continuation formulas will first assign functions separately
on open pieces. For arbitrary coefficient vectors those assignments
need not agree across the separating curves. We denote them by
the subscript \(\mathrm{raw}\), and use a superscript \(\sharp\)
for the holomorphic functions obtained by applying \(\mathsf B\).
Values on the separating curves do not affect the area integrals.

\begin{enumerate}
\item On \(\{|z-1|<1/4,\ |z|>1\}\), assign functions
\(b_{\mathrm{end},\mathrm{raw}},b_{-,\mathrm{raw}}\) from \eqref{eq:localHX} where \(\Re z>1\), from the
first identity of \eqref{eq:local-lenses} in the lower left
lens, and from the second in the upper left lens.
Inside \(|z|<1,\ |z-1|<1/8\), assign
\[
 b_{\mathrm{end},\mathrm{raw}}(z)=z^{10}b_{\mathrm{end},\mathrm{raw}}(1/z),\qquad
 b_{-,\mathrm{raw}}(z)=b_{\mathrm{end},\mathrm{raw}}(z)-z^{10}b_{-,\mathrm{raw}}(1/z).
\]
Apply \(\mathsf B_{1,1/8}\) to get \(b_{\mathrm{end}}^\sharp,b_-^\sharp\).

\item For \(z\in D(1,1/8)\) with \(\Re z>1\), set
\(b_{+,\mathrm{raw}}(z)=\Aplus(Q(z))\). For \(\Re z<1\),
use the expression in \eqref{eq:localXplus}.
Apply \(\mathsf B_{1,1/8}\) to obtain \(b_+^\sharp\), and define
\[
 a_{2,\mathrm{par}}^\sharp(z)=(b_+^\sharp(z)+b_-^\sharp(z))/2,\qquad
 a_{3,\mathrm{par}}^\sharp(z)=(b_+^\sharp(z)-b_-^\sharp(z))/4.
\]
These averaged functions have argument \(z\), whereas the
original \(A_j\) have argument \(q\). For an exact coefficient
solution their relation is \(a_{j,\mathrm{par}}^\sharp(z)=A_j(Q(z))\).

\item For \(z\in D(1,1/32)\) with \(\Re z>1\), set
\(a_{1,\mathrm{par},\mathrm{raw}}(z)=A_1(Q(z))\). On the two left half-disks use
\eqref{eq:localA1}, with the sharp functions in place of
\(b_{\mathrm{end}},A_2,A_3\). Apply \(\mathsf B_{1,1/32}\) to get
\(a_{1,\mathrm{par}}^\sharp\), and use \eqref{eq:localA0} to define
\(a_{0,\mathrm{par}}^\sharp\). Write
\(\mathbf a_{\mathrm{par}}^\sharp=(a_{0,\mathrm{par}}^\sharp,a_{1,\mathrm{par}}^\sharp,a_{2,\mathrm{par}}^\sharp,a_{3,\mathrm{par}}^\sharp)^T\).
\end{enumerate}

\paragraph{Bounds for arbitrary coefficients.}
These estimates use no matching identity. In the first two stages,
Lemma~\ref{lem:local-area} bounds the Cauchy pieces, the original
series pieces, and their pullbacks by inversion. In particular
\[
 \|b_{\mathrm{end},\mathrm{raw}}\|_{L^1(D(1,1/8))}
 +\|b_{-,\mathrm{raw}}\|_{L^1(D(1,1/8))}
 +\|b_{+,\mathrm{raw}}\|_{L^1(D(1,1/8))}
       \leq C\|\mathbf c\|_2.
\]
The averaging estimate bounds the three sharp functions, and
hence \(a_{2,\mathrm{par}}^\sharp,a_{3,\mathrm{par}}^\sharp\),
on \(D(1,1/16)\) by \(C\|\mathbf c\|_2\).
In the third stage these functions are consequently bounded
on all of \(D(1,1/32)\). The remaining terms in
\eqref{eq:localA1} are Cauchy pieces on the left and the
original series on the right. Lemma~\ref{lem:local-area}
therefore gives
\[
 \|a_{1,\mathrm{par},\mathrm{raw}}\|_{L^1(D(1,1/32))}
        \leq C\|\mathbf c\|_2.
\]
A second application of the averaging estimate, followed by
\eqref{eq:localA0}, yields
\[
 \sup_{|z-1|\leq1/64}\|\mathbf a_{\mathrm{par}}^\sharp(z)\|_1
            \leq C_0\|\mathbf c\|_2.
\]
These are bounds for the piecewise raw assignments of an arbitrary
vector, not conclusions about its analytic continuation.

\paragraph{Reproduction for exact solutions.}
If \(\mathbf c\in\ker\Mom\), the overlap checks and removals
in Proposition~\ref{prop:regularity} show that each raw assignment
is the restriction of the actual holomorphic continuation.
Disk averaging reproduces it. The first two stages therefore
recover \(b_{\mathrm{end}},b_\pm\) and \(A_2(Q(z)),A_3(Q(z))\);
the third recovers \(A_1(Q(z))\) and then \(A_0(Q(z))\).
Thus \(\mathbf a_{\mathrm{par}}^\sharp(z)=
\mathbf a_{\mathrm{ser}}(Q(z))\) for an exact solution.
This reproduction statement is separate from the arbitrary-input
bounds above.

\paragraph{Coefficient extraction.}

Put \(R=e^{\pi2^{-19}}\), \(\sigma_0=1-2^{-19}\),
\(\delta_{\mathrm{end}}=2^{-7}\), and
\(\mathsf S_{\mathrm{par}}=\operatorname{diag}(1,1,1,-1)\).
Here \(\mathsf S_{\mathrm{pair}}\) is the paired-trace matrix
of \eqref{eq:paired-transfer}, and \(\mathsf S_{\mathrm{par}}\)
records the parity of the four series. Define paths
\(z_R(t)=\sigma_0+it\) and
\(\gamma_R(t)=Re^{-\pi it}\), \(0\leq t\leq1\).
Then \(Q(z_R(t))=\gamma_R(t)\), and \(\gamma_R\)
parametrizes the lower semicircle in the \(q\)-plane from
\(R\) to \(-R\). Define a vector-valued function on this
semicircle by
\[
 \widetilde{\mathbf a}_{\mathrm{ser},\mathbf c}(\gamma_R(t))=
 \begin{cases}
 \mathbf a_{\mathrm{par}}^\sharp(z_R(t)),&0\leq t<\delta_{\mathrm{end}},\\
 \mathsf S_{\mathrm{pair}}(z_R(t))^{-1}
       \begin{pmatrix}\mathbf h_{\mathrm{par}}(z_R(t))\\\mathbf h_{\mathrm{par}}(z_R(t)-i)\end{pmatrix},
                  &\delta_{\mathrm{end}}\leq t\leq1-\delta_{\mathrm{end}},\\
 \mathsf S_{\mathrm{par}}\mathbf a_{\mathrm{par}}^\sharp(z_R(t)-i),&1-\delta_{\mathrm{end}}<t\leq1.
 \end{cases}
\]
Define its values on the upper semicircle by
\(\widetilde{\mathbf a}_{\mathrm{ser},\mathbf c}(-q)=\mathsf S_{\mathrm{par}}\widetilde{\mathbf a}_{\mathrm{ser},\mathbf c}(q)\)
for \(q\) on the lower semicircle. The endpoint definitions
agree because \(\mathsf S_{\mathrm{par}}^2=I_4\).
The middle Cauchy arguments lie in fixed compact sets off the
contours. It follows that
\begin{equation}
 \sup_{|q|=R}\|\widetilde{\mathbf a}_{\mathrm{ser},\mathbf c}(q)\|_1
                      \leq C_{\mathrm{patch}}\|\mathbf c\|_2. \label{eq:patch}
\end{equation}
Write
\(\widetilde{\mathbf a}_{\mathrm{ser},\mathbf c}
 =(\widetilde A_{\mathbf c,0},\widetilde A_{\mathbf c,1},
   \widetilde A_{\mathbf c,2},\widetilde A_{\mathbf c,3})^T\).
The four components are ordered as \(A_0,A_1,A_2,A_3\)
and are numbered \(j=0,1,2,3\). The allowed powers are precisely those in
\eqref{eq:series}: the five specified nonpositive powers in
the first two components, positive even powers in the first
three, and positive odd powers in the fourth. Extract their
coefficients by
\begin{equation}
 (\mathsf K\mathbf c)_{j,m}
    =\frac{R^{-m}}{2\pi}\int_0^{2\pi}
       \widetilde A_{\mathbf c,j}(Re^{i\theta})e^{-im\theta}\,d\theta.
 \label{eq:K}
\end{equation}
The output index \((j,m)\) ranges over \(\mathcal I_{24}\).
Thus \eqref{eq:K} produces the coefficient vector in the same
coordinates as \(\mathbf c\); coefficients at all other powers
are discarded. This defines a linear map from
\(\ell^2(\mathcal I_{24})\) to coefficient families on
\(\mathcal I_{24}\).
For \(m\geq1\), let \((\mathsf K\mathbf c)_m\) be the tuple
of entries at exponent \(m\): three entries if \(m\) is even,
and one if \(m\) is odd. By \eqref{eq:patch},
\[
 \|(\mathsf K\mathbf c)_m\|_1\leq C_{\mathrm{patch}}R^{-m}\|\mathbf c\|_2
 \quad(m\geq1),
\]
so
\[
 \|(\Id_{\mathcal I_{24}}-\mathsf P_J)\mathsf K\|
   \leq\frac{C_{\mathrm{patch}}R^{-J-1}}{\sqrt{1-R^{-2}}}.
\]
The five remaining coefficients are bounded linear functionals
by the same integral estimate. Hence \(\mathsf K:\ell^2(\mathcal I_{24})\to\ell^2(\mathcal I_{24})\)
is bounded, and the tail estimate proves compactness and
\eqref{eq:Kproperties}.
For \(\mathbf c\in\ker\Mom\), the boundary assignments coincide with the
actual continuation, and Cauchy's coefficient formula gives
\(\mathsf K\mathbf c=\mathbf c\). Every operation defining \(\mathsf K\)
has been specified before any coefficient of \(\mathbf c_*\) is known.

\bigskip
\noindent
This document was generated by ChatGPT-6 Astra with prompting from Thomas Hales.

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