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% Front matter
\title[Lattice kissing numbers and average contacts]
  {A Fourier consequence for lattice kissing numbers and average contacts}

\author{Scott Duke Kominers}

\date{First working version: September 26, 2026.
  This version: October 2, 2026.}

\subjclass[2020]{Primary 11H31; Secondary 52C17, 42B10}
\keywords{lattice kissing number, sphere packing, average contact degree,
  contact number, Fourier linear programming bound, auxiliary function}

\thanks{%
  I used LLMs to assist with analysis and synthesis in the preparation of
  this article, particularly GPT-6 Astra and Claude Fable 5.1 (accessed in
  part via Poe with the support of Quora, where I am an advisor).
  In particular, the bound I present was developed in dialogue with
  GPT-6 Astra.
  I greatly appreciate helpful comments from Henry Cohn and Thijs
  Laarhoven.
  The problem, methods, and final written form are my own; and of course
  any errors remain my responsibility. This work was conducted while I
  was visiting the Technological Innovation, Entrepreneurship, and
  Strategic Management (TIES) Group at the MIT Sloan School of
  Management.%
}

\begin{document}

\begin{abstract}
We deduce a new asymptotic upper bound
$\Klat(n)\leq\left(\sqrt{2e/\pi}+o(1)\right)^n$
on the maximal lattice kissing number $\Klat(n)$ in dimension $n$, using
the explicit auxiliary functions constructed in OpenAI's ``Ten Advances''
preprint~\cite{OAI}.
The same bound holds for the average contact degree of a finite packing
of congruent balls. The bound's base-$2$ exponential rate rounds to
$0.3956$, matching the value extrapolated empirically by Afkhami-Jeddi,
Cohn, Hartman, de Laat, and Tajdini~\cite{AJCHLT}.
\end{abstract}

\maketitle

For a rank-$n$ Euclidean lattice $L$, we write
$\min L=\min_{0\ne x\in L}\langle x,x\rangle$ for the minimum squared
length of nonzero vectors in $L$; the \emph{kissing number}
\[
  \tau(L)=\bigl|\{x\in L:\langle x,x\rangle=\min L\}\bigr|
\]
counts the number of such vectors, including both signs. The
\emph{maximal lattice kissing number in dimension $n$} is
\[
  \Klat(n)=\max_{\operatorname{rank}L=n}\tau(L).
\]

In this note, we obtain a new asymptotic upper bound for $\Klat(n)$,
using explicit auxiliary functions constructed in OpenAI's
preprint~\cite{OAI}.

\begin{theorem}
\label{thm:main}
As $n\to\infty$,
\[
  \Klat(n)\leq\left(\sqrt{\frac{2e}{\pi}}+o(1)\right)^n
  =2^{\left(\frac{1}{2}\log_2(2e/\pi)+o(1)\right)n}.
\]
Here $\frac{1}{2}\log_2(2e/\pi)=0.395599455708322\ldots$.
\end{theorem}

For a packing of congruent balls, the \emph{contact degree} of a ball is
the number of other balls tangent to it; the \emph{average contact
degree} of a nonempty finite packing is the arithmetic mean of these
degrees. After scaling such balls to radius $1/2$, their centers form a
finite set $C\subset\R^n$ with minimum separation at least $1$, and the
average contact degree is
\[
  \overline{\tau}(C)
  =\frac{\#\{(x,y)\in C^2:\|x-y\|=1\}}{|C|}.
\]
Corollary~\ref{cor:average} gives the same upper bound as
Theorem~\ref{thm:main} for $\overline{\tau}(C)$, uniformly over all
such $C$.

To place the bound in context, let $K(n)$ be the ordinary kissing
number---i.e., the largest number of nonoverlapping unit balls that can
touch one unit ball in $\R^n$. Equivalently, $K(n)$ is the largest size
of a set of unit vectors with pairwise inner products at most $1/2$.
Normalizing the shortest vectors of a lattice gives such a set, so
$\Klat(n)\leq K(n)$. The classical spherical-code bound of Kabatiansky
and Levenshtein~\cite{KL}, from 1978, gives the exponent $0.40141\ldots$
for $K(n)$. Combining it with Sidel'nikov's spherical-cap
reduction~\cite{Sidelnikov} gives the optimized classical
bound~\cite[Theorem~4]{KL}, with exponent $0.400944\ldots$; subsequent
work of Sardari and Zargar~\cite{SZ}, and of Zargar~\cite{Zargar},
improved multiplicative factors.

Fourier methods offer a second route, through the contacts in a packing.
In 2020, Afkhami-Jeddi, Cohn, Hartman, de Laat, and
Tajdini~\cite{AJCHLT} extrapolated an implied kissing exponent of
$0.3956$ (the $d=\infty$ row of~\cite[Table~3]{AJCHLT}) from their
numerically optimized density bound. Its interpretation as an
average-contact bound relied on interpolation and sign conjectures.
Their Theorem~5.3 gives an unconditional inequality for admissible
auxiliary functions; we apply this result to the explicit functions
of~\cite{OAI}, obtaining an exponent that agrees with their extrapolation
to the displayed precision, without relying on the intermediate
conjectures.

The cap-optimized spherical-code hierarchy of~\cite{OAI} gives
\[
  K(n)\leq 2^{(0.39661+o(1))n}
\]
at level $2$; see~\cite[Chapter~2, Remark~1.7]{OAI}.
Our Theorem~\ref{thm:main} and Corollary~\ref{cor:average} give a smaller
exponent for lattice kissing numbers and average contact degree,
respectively. In fact, Lemma~\ref{lem:hierarchy-gap} in
Appendix~\ref{app:hierarchy-gap} shows that the infimum of the
cap-optimized fixed-level asymptotic kissing exponents of the hierarchy
in~\cite{OAI} remains strictly above ours.
To our knowledge, Theorem~\ref{thm:main} gives the first proven
asymptotic upper bound for lattice kissing numbers whose exponential
rate is strictly smaller than the best known upper-bound rate for the
unrestricted kissing number.\footnote{%
  We thank Thijs Laarhoven for pointing out this possibility.
  Earlier lattice-specific bounds include Skoruppa's Mellin-transform
  bound~\cite[Theorem~2]{Skoruppa}; its proof gives the exponent
  $\log_2(e/2)=0.442695\ldots$, which is weaker than the classical general
  bound. Meanwhile, stronger bounds in individual dimensions have long
  been known for lattices and antipodal codes (see, e.g.,
  \cite{Watson,Anstreicher}).%
}

Meanwhile, there remains a large gap between the upper and lower bounds
for lattice kissing numbers.
The Barnes--Wall lattices~\cite{BarnesWall} in dimensions $n=2^m$ have
kissing numbers
\[
  \prod_{i=1}^{m}(2^i+2)
  =\exp\bigl(\Theta((\log n)^2)\bigr);
\]
see Nebe~\cite[Lemma~3.4]{Nebe}.
Recent work of Laarhoven and the present
author~\cite[Theorem~1.1]{LK} improves this to
\[
  \Klat(n)\geq
  \left(\frac{1}{2\pi e^2}+o(1)\right)\sqrt{n}\,e^{2\sqrt{n}}
\]
as $n\to\infty$ through prime powers.\footnote{%
  Vl\u{a}du\cb{t}~\cite{Vladut} attempted to obtain lattices with
  exponentially many shortest vectors, but Bennett, Golovnev, and
  Stephens-Davidowitz~\cite{BGS} later identified errors in his
  construction.%
}

Our analytic input comes from Chapter~1 of~\cite{OAI}; all references to
that preprint use the version updated August~6, 2026.
That chapter determines the exponential strength of the Fourier linear
program for sphere-packing density, developed by
Gorbachev~\cite{Gorbachev} and Cohn--Elkies~\cite{CE}.
Writing $\LP_n$ for the infimum of the density upper bounds supplied by
this program, \cite[Chapter~1, Theorem~1.1]{OAI} states
\[
  \lim_{n\to\infty}\LP_n^{1/n}
  =\sqrt{\frac{e}{2\pi}}.
\]
Consequently, the maximal packing density $\Delta_n$ satisfies
$\Delta_n\leq 2^{-(0.604400544\ldots+o(1))n}$; the equality above
determines the rate of the linear program itself.
A comparison theorem of Cohn and Zhao~\cite[Theorem~3.4]{CZ} then
implies that the ordinary spherical linear program for $K(n)$ cannot
attain an exponent smaller than $\frac{1}{2}\log_2(2e/\pi)$.

For our bounds, we use the explicit auxiliary functions $f$ constructed
in~\cite{OAI} to establish the asymptotic density upper bound, rather
than the density-rate theorem alone.
The density bound depends on $f(0)/\widehat{f}(0)$ together with a volume
factor, whereas the kissing and contact bounds depend on $f(0)/[-f(R)]$,
where $R$ is the center-to-center distance at contact. We evaluate the
latter ratio using the saddle-point estimate at the target parameter
$u_0$ in~\cite[Chapter~1, \S4]{OAI}.
Lemma~\ref{lem:input} below collects the required properties of these
functions and the resulting contact-value estimate, with precise
references to the global sign and saddle-point estimates on which our
argument relies.

All logarithms without a subscript are natural.

\section*{The Fourier contact bound}

The following bound for average contact degree appears
in~\cite[Theorem~5.3]{AJCHLT}. Its finite-packing assertion also follows
from the positive-type bound of Dostert, Kolpakov, and de Oliveira
Filho~\cite[Theorem~4.1]{DKOF}, specialized to congruent balls.
As noted in~\cite{AJCHLT}, it is the Euclidean analogue of Fortier Bourque
and Petri's trace-formula bound~\cite[Proposition~4.1]{FBP}.

We use the normalization
$\widehat{f}(\xi)=\int_{\R^n}f(x)e^{-2\pi i\langle x,\xi\rangle}\,dx$.
For radial $f$, write $f(r)$ for its value at a vector of norm $r$.

\begin{lemma}[{\cite[Theorem~5.3]{AJCHLT}; \cite[Theorem~4.1]{DKOF}}]
\label{lem:contact}
Let $f$ be a real radial Schwartz function on $\R^n$, and suppose that,
for some $R>0$,
\[
  \widehat{f}\geq0,
  \qquad
  f(r)\leq0\quad(r\geq R),
  \qquad
  f(R)<0.
\]
Then every rank-$n$ lattice satisfies $\tau(L)\leq f(0)/[-f(R)]$.
The same ratio bounds the average contact degree of every nonempty
finite packing of balls of radius $R/2$, with contacts at center
distance $R$.
\end{lemma}

\begin{proof}
We scale $L$ so that its shortest nonzero vectors have length $R$,
equivalently $\min L=R^2$. Writing $L^*$ for the dual lattice, Poisson
summation gives
\[
  0\leq
  \frac{1}{\operatorname{covol}(L)}
  \sum_{\xi\in L^*}\widehat{f}(\xi)
  =\sum_{x\in L}f(x)
  \leq f(0)+\tau(L)f(R);
\]
since $f(R)<0$, this proves the lattice bound.

For a nonempty finite set $C$ of minimum separation at least $R$,
Fourier inversion gives
\[
  0\leq\int_{\R^n}\widehat{f}(\xi)
  \left|\sum_{x\in C}e^{2\pi i\langle x,\xi\rangle}\right|^2\,d\xi
  =\sum_{x,y\in C}f(x-y).
\]
If $M$ counts the ordered pairs at distance $R$, then the last sum is
at most $|C|f(0)+Mf(R)$. Thus $M/|C|\leq f(0)/[-f(R)]$.
\end{proof}

\section*{The auxiliary functions}

Fix sufficiently small $\eps>0$, and put
\[
  \lambda=\frac{n}{2},
  \qquad
  u=1+\frac{\eps}{4},
  \qquad
  m=\frac{\lambda(1+u)}{2},
  \qquad
  \psi=\frac{\Gamma'}{\Gamma}.
\]
Let $w_\eps$ be the compactly supported signed density
in~\cite[Chapter~1, \S4]{OAI},\footnote{%
  Explicitly,
  \[
    \begin{aligned}
      w_\eps(a)
      &=-\frac{(1-2\eps(1+a))e^{-2a}}{2a^2\cosh a}
        \mathbf{1}_{[\eps^2,\,\log(1/\eps)]}(a)\\
      &\quad+\frac{e^{-3/(8\eps^2)}}{\cosh a}
        \mathbf{1}_{[\eps^{-3},\,\eps^{-3}+1]}(a),
    \end{aligned}
  \]
  where $\mathbf{1}_I$ denotes the indicator of $I$;
  see~\cite[Chapter~1, (34)--(35)]{OAI}.%
}
and define the real function
\[
  g_\eps(t)=\int_0^\infty w_\eps(a)(\cosh(at)-1)\,da.
\]
Thus $g_\eps(t)=h_\eps(it)$ in the notation of~\cite[Chapter~1]{OAI}.
We collect the analytic input, including the contact value extracted
from the~\cite[Chapter~1]{OAI} saddle estimate, in the following lemma.

\begin{lemma}
\label{lem:input}
For every fixed sufficiently small $\eps>0$, there is $n_0(\eps)$ such
that, for every $n\geq n_0(\eps)$, there exist a real radial Schwartz
function $f=f_{n,\eps}$ on $\R^n$ and a radius $R=R_{n,\eps}>0$
satisfying the hypotheses of Lemma~\ref{lem:contact}, with
\begin{align}
  \log R
  &=-\frac{1}{2}\log\pi+\frac{1}{2}\psi(m)+g_\eps'(u),
  \label{eq:radius}\\
  f(0)
  &=2(u-1)\pi^{\lambda/2}e^{\lambda g_\eps(1)},
  \label{eq:origin}\\
  -f(R)
  &=A_{n,\eps}\,\pi^{-\lambda u/2}\Gamma(m)
    e^{\lambda g_\eps(u)}R^{-2m},
    \qquad A_{n,\eps}\asymp_\eps\sqrt{n},
  \label{eq:contact-value}\\
  g_\eps'(u)
  &=-\frac{1}{2}\log(\pi/2)+O(\eps).
  \label{eq:derivative}
\end{align}
In particular, $A_{n,\eps}>0$. The constants implicit in $\asymp_\eps$
may depend on $\eps$ but not on $n$.\footnote{%
  Here and below, $o_\eps(1)$ denotes a quantity tending to zero as
  $n\to\infty$ with $\eps$ fixed.%
}
More precisely, put
\[
  p_\eps=(u-1)\bigl[1-(1+u)^2\bigr]<0,
  \qquad
  V_{n,\eps}=\frac{\lambda}{4}\psi'(m)+g_\eps''(u).
\]
Then $V_{n,\eps}\asymp_\eps1$ and
\begin{equation}
  \label{eq:prefactor}
  A_{n,\eps}
  =(-p_\eps)\sqrt{\frac{\lambda}{2\pi V_{n,\eps}}}
  \,\bigl(1+o_\eps(1)\bigr).
\end{equation}
\end{lemma}

\begin{proof}
We take $f$ to be the function denoted $f_-$
in~\cite[Chapter~1, \S4]{OAI}.
The origin value~\eqref{eq:origin} and radius~\eqref{eq:radius} are given
in~\cite[Chapter~1, (42) and (44)]{OAI}, respectively.
The derivative estimate~\eqref{eq:derivative} is
\cite[Chapter~1, Lemma~4.2]{OAI}, and the global sign conditions are
established in~\cite[Chapter~1, Theorem~4.1]{OAI}.
For the contact value, the target parameter is $u=u_0$ in that
preprint's notation, and this endpoint is included in
\cite[Chapter~1, Lemma~4.8]{OAI}.
The saddle coefficient $p_\eps$ is given
in~\cite[Chapter~1, (38)--(39)]{OAI}.
In~\cite[Chapter~1, (70)--(71)]{OAI}, the inverse Mellin factor
$\lambda/(2\pi)$ and Gaussian factor combine as
\[
  \frac{\lambda}{2\pi}\sqrt{\frac{2\pi}{\lambda V_{n,\eps}}}
  =\sqrt{\frac{\lambda}{2\pi V_{n,\eps}}},
\]
giving~\eqref{eq:contact-value} and~\eqref{eq:prefactor}.
The curvature bounds in~\cite[Chapter~1, Lemma~4.5, (56)]{OAI} give
$V_{n,\eps}\asymp_\eps1$. Since $p_\eps<0$, the contact value is negative
and $A_{n,\eps}\asymp_\eps\sqrt{n}$.
\end{proof}

We also need the estimate
\begin{equation}
  \label{eq:difference}
  g_\eps(u)-g_\eps(1)=O(\eps).
\end{equation}
\nopagebreak
To verify this, let $w_{\eps,+}$ and $w_{\eps,-}$ be the positive and
negative parts of $w_\eps$, so that $w_\eps=w_{\eps,+}-w_{\eps,-}$.
The density construction and \cite[Chapter~1, Lemma~4.2]{OAI} give
\[
  w_{\eps,-}(a)\leq\frac{e^{-2a}}{2a^2\cosh a},
  \qquad
  \int_0^\infty w_{\eps,+}(a)a\sinh(ua)\,da
  =O(e^{-c/\eps^2})
\]
for an absolute $c>0$. For $1\leq t\leq u\leq5/4$, we use the
inequalities $\sinh(ta)\leq ta\cosh(ta)$ and
$\cosh(ta)\leq e^{(t-1)a}\cosh a$ to obtain
\[
  |g_\eps'(t)|
  \leq\frac{t}{2}\int_0^\infty e^{-(3-t)a}\,da
  +O(e^{-c/\eps^2})
  \leq\frac{5}{14}+O(e^{-c/\eps^2}).
\]
Integration over $[1,u]$, of length $\eps/4$,
proves~\eqref{eq:difference}.

\section*{The kissing number bound}

We now estimate the ratio $f(0)/[-f(R)]$ and apply
Lemma~\ref{lem:contact} to obtain the lattice kissing number bound.

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Fix sufficiently small $\eps>0$.
Taking logarithms in \eqref{eq:origin} and~\eqref{eq:contact-value} and
substituting~\eqref{eq:radius} gives
\begin{equation}
  \label{eq:key-log}
  \begin{aligned}
    \log\frac{f(0)}{-f(R)}
    ={}&\lambda\bigl[g_\eps(1)-g_\eps(u)
        +(1+u)g_\eps'(u)\bigr]\\
    &+m\psi(m)-\log\Gamma(m)
      +\log\frac{2(u-1)}{A_{n,\eps}}.
  \end{aligned}
\end{equation}
(The powers of $\pi$ cancel because $2m=\lambda(1+u)$.)
Stirling's formula yields
\[
  m\psi(m)-\log\Gamma(m)
  =m+\frac{1}{2}\log m+O(1).
\]
Since $\log A_{n,\eps}=\frac{1}{2}\log n+O_\eps(1)$ and
$m/n=(1+u)/4$, we have
\[
  \frac{1}{2}\log m+\log\frac{2(u-1)}{A_{n,\eps}}
  =O_\eps(1).
\]
Dividing~\eqref{eq:key-log} by $n=2\lambda$ therefore gives
\[
  \frac{1}{n}\log\frac{f(0)}{-f(R)}
  =\frac{1}{2}\left[
    g_\eps(1)-g_\eps(u)
    +(1+u)g_\eps'(u)+\frac{1+u}{2}
  \right]+O_\eps(n^{-1}).
\]
Since $u=1+O(\eps)$, equations~\eqref{eq:derivative}
and~\eqref{eq:difference} then give
\[
  \frac{1}{n}\log\frac{f(0)}{-f(R)}
  =\frac{1}{2}\log\frac{2e}{\pi}
  +O(\eps)+O_\eps(n^{-1}).
\]
Applying Lemma~\ref{lem:contact} and taking the upper limit as
$n\to\infty$ with $\eps$ fixed, we obtain
\[
  \limsup_{n\to\infty}\frac{1}{n}\log\Klat(n)
  \leq\frac{1}{2}\log\frac{2e}{\pi}+O(\eps).
\]
Letting $\eps\downarrow0$ proves the theorem.
\end{proof}

\section*{Average contact degree}

The same ratio estimate also bounds average contact degree.

\begin{corollary}
\label{cor:average}
As $n\to\infty$, uniformly over nonempty finite $C\subset\R^n$ with
minimum separation at least $1$,
\[
  \overline{\tau}(C)\leq
  \left(\sqrt{\frac{2e}{\pi}}+o(1)\right)^n.
\]
\end{corollary}

\begin{proof}
Fix sufficiently small $\eps>0$, and take $f$ and $R$ from
Lemma~\ref{lem:input}. The scaled set $RC$ has minimum separation at
least $R$, and its ordered pairs at distance $R$ correspond exactly to
the ordered pairs of $C$ at distance $1$.
The finite-packing part of Lemma~\ref{lem:contact} therefore gives
\[
  \overline{\tau}(C)\leq\frac{f(0)}{-f(R)};
\]
the estimate for this ratio in the proof of Theorem~\ref{thm:main} is
independent of $C$.
Taking $n\to\infty$ with $\eps$ fixed and then letting $\eps\downarrow0$
proves the stated uniform bound.
\end{proof}

\section*{Concluding remarks}

\subsection*{Uniformity and contact counts}

Explicitly, for every $\eta>0$ there is $n_0(\eta)$ such that, for every
$n\geq n_0(\eta)$, both $\Klat(n)$ and $\overline{\tau}(C)$ for every
eligible $C\subset\R^n$ are at most
$2^{(\frac{1}{2}\log_2(2e/\pi)+\eta)n}$.
For a packing of $N$ congruent balls, the \emph{contact number}---the
number of unordered tangent pairs---is therefore at most
\[
  \frac{N}{2}\left(\sqrt{\frac{2e}{\pi}}+o(1)\right)^n,
\]
uniformly in $N$.

Our bounds apply to lattice kissing numbers and to the average contact
degree of arbitrary finite packings of congruent balls. They do not
establish the same exponential bound for the maximum contact degree in
such a packing, and therefore do not give the same bound for the
unrestricted kissing number $K(n)$; they also do not address packings
with unequal radii. For background on the contact-number problem, see
the Bezdek and Khan~\cite{BK} survey.

\subsection*{Comparison with the spherical linear program}

The exponent in Theorem~\ref{thm:main} also gives a lower bound on the
exponential rate of the optimal ordinary kissing-number
linear-programming bound.
Let $K_n^{\LP}$ denote the infimum of the upper bounds for $K(n)$
supplied by the two-point spherical linear program of Delsarte,
Goethals, and Seidel~\cite{DGS}.
Cohn and Zhao~\cite[Theorem~3.4]{CZ}, specialized to the kissing angle
$\pi/3$, proved
\begin{equation}
  \label{eq:cz-comparison}
  \LP_n\leq 2^{-n}K_n^{\LP};
\end{equation}
their comparison applies in the same dimension $n$.\footnote{%
  The continuous compactly supported auxiliary functions constructed
  in~\cite{CZ} can be convolved with smooth compactly supported,
  nonnegative, positive-definite radial approximate identities and then
  rescaled to restore the sign radius. This gives Schwartz functions
  whose density bounds converge to the original bound, at each fixed
  $n$, so~\eqref{eq:cz-comparison} also holds for the Schwartz
  formulation used in~\cite{OAI}.%
}
Combining~\eqref{eq:cz-comparison} with
$\lim_{n\to\infty}\LP_n^{1/n}=\sqrt{e/(2\pi)}$ gives
\begin{equation}
  \label{eq:kissing-lp-floor}
  \liminf_{n\to\infty}\frac{1}{n}\log_2 K_n^{\LP}
  \geq\frac{1}{2}\log_2\frac{2e}{\pi}.
\end{equation}
Thus our bounds for lattice kissing numbers and average contact degree
attain an exponent below which the ordinary spherical linear program
cannot bound the unrestricted kissing number.\footnote{%
  We thank Henry Cohn for this observation.%
}
The inequality~\eqref{eq:kissing-lp-floor} is a lower bound on the
optimal spherical linear programming upper bound $K_n^{\LP}$, not a
lower bound on the kissing number $K(n)$ itself.
Earlier asymptotic lower bounds for the optimal spherical
linear-programming bound were obtained by
Samorodnitsky~\cite[Corollary~1.3]{Samorodnitsky}.

The same constant appears within the spherical-code hierarchy
of~\cite{OAI}, whose certificates belong to this two-point linear
program; see~\cite[Chapter~2, Theorem~4.2]{OAI}.
Specifically, write $\sigma=2\Gamma_r$ for a certificate's threshold
and $\Phi=\Phi_r$ for its code-size exponent.
Then~\cite[Chapter~2, Proposition~8.1, (101)]{OAI} gives
\[
  \Phi=\frac{1}{2}\log_2\frac{2}{1-\sigma}
  -\frac{1}{2}\log_2\frac{2\pi}{e}
  +\frac{D_{\mathrm{KL}}(\nu\Vert\rho_K)}{2\log 2},
\]
where $\nu$ and $\rho_K$ are the probability measures defined there
and $D_{\mathrm{KL}}$ denotes relative entropy.
For parameters feasible at the kissing angle, $\sigma\geq1/2$;
nonnegativity of relative entropy therefore gives
$\Phi\geq\frac{1}{2}\log_2(2e/\pi)$.
This identity makes the same lower bound explicit for those
certificates, while the Cohn--Zhao comparison applies to the entire
two-point spherical linear program.

Appendix~\ref{app:hierarchy-gap} gives a stronger limitation for the
spherical-code hierarchy of~\cite{OAI}:
Lemma~\ref{lem:hierarchy-gap} shows that, even after spherical-cap
optimization and taking the infimum over fixed levels, the hierarchy's
asymptotic kissing exponent remains strictly above ours. This statement
concerns that certificate family; it does not establish a strict gap
in~\eqref{eq:kissing-lp-floor} for the full two-point spherical linear
program.

The sharpness argument in~\cite[Chapter~2, Theorem~8.3]{OAI} sends
$\sigma\to1$, establishing the optimized sphere-packing density exponent.
This is compatible with Lemma~\ref{lem:hierarchy-gap}, which fixes the
kissing angle at $\pi/3$.

% Bibliography
\begingroup
\raggedright

\begin{thebibliography}{10}

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\bibitem{Anstreicher}
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\bibitem{BGS}
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\endgroup

% Appendix
\appendix
\numberwithin{equation}{section}

\section{Separation from the OpenAI spherical-code hierarchy}
\label{app:hierarchy-gap}

In this appendix, we show that the asymptotic kissing-number exponents
supplied by the hierarchy of~\cite[Chapter~2]{OAI} remain uniformly
separated from the exponent in Theorem~\ref{thm:main}, even after
spherical-cap optimization and taking the infimum over fixed levels.
Starting from the relative-entropy identity
of~\cite[Chapter~2, Proposition~8.1]{OAI}, we use Jensen's inequality to
bound the mass of a fixed interval and thereby obtain an explicit
positive lower bound on the relative entropy.

\begin{appendixlemma}
\label{lem:hierarchy-gap}
Put
\[
  \alpha_*=\frac{1}{2}\log_2\frac{2e}{\pi},
  \qquad
  \beta=\frac{2}{3}-\frac{2}{\pi}\arctan\frac{1}{2\sqrt{3}}.
\]
The infimum of the fixed-level asymptotic kissing-number exponents
supplied by the hierarchy of~\cite[Chapter~2, Theorem~1.2]{OAI}, over
all finite levels and the spherical-cap parameters
in~\cite[Chapter~2, (89)]{OAI}, is at least
\[
  \alpha_*-\frac{1}{4}\log_2\bigl(4\beta(1-\beta)\bigr)
  =0.395815863377\ldots>\alpha_*.
\]
\end{appendixlemma}

Here the level is fixed before the dimension tends to infinity, and the
infimum is then taken over the resulting asymptotic rates. The argument
does not address levels that vary with the dimension.

\begin{proof}[Proof of Lemma~\ref{lem:hierarchy-gap}]
For a finite-level certificate, let $x_i\geq0$ and $R_i>0$ be the
quadratic coordinates and residues
in~\cite[Chapter~2, Proposition~8.1, (102)]{OAI}, so that
$\sum_i R_i=1$. Define
\[
  w_i=\sqrt{\frac{x_i}{x_i+1/4}},
  \qquad
  \mu=\sum_i R_i\delta_{w_i},
\]
where $\delta_w$ denotes unit point mass at $w$, and put
\[
  k_w(v)=\frac{(1-v^2)(1-w^2)}{1-v^2+v^2w^2},
  \qquad
  F(v)=\int_0^1 k_w(v)\,d\mu(w)
  \qquad(0\leq v<1).
\]
In particular, $k_0(v)=1$ and $k_1(v)=0$.
Under $t=(1-v^2)/4$, the reference measures
in~\cite[Chapter~2, Proposition~8.1]{OAI} become
\[
  d\nu(v)=dv,
  \qquad
  d\rho(v)=\frac{2\,dv}{\pi\sqrt{1-v^2}}.
\]
The kernel satisfies
\[
  \int_0^1 k_w(v)\,d\rho(v)=1-w
  \qquad(0\leq w\leq1).
\]
We define the normalizing constant
\[
  Z:=\int_0^1 F(v)\,d\rho(v);
\]
the kernel identity gives
\[
  Z=1-\int_0^1 w\,d\mu(w).
\]
By~\cite[Chapter~2, (102)--(104)]{OAI}, the certificate's threshold
$\sigma$ and exponent $\Phi$ satisfy
\begin{equation}
  \label{eq:gap-normalization}
  \sigma=\int_0^1 w\,d\mu(w),
  \qquad
  Z=1-\sigma,
  \qquad
  \Phi=-\frac{1}{2\log 2}\int_0^1\log F(v)\,dv.
\end{equation}
We now enlarge the class under consideration by allowing arbitrary
probability measures $\mu$ on $[0,1]$, including endpoint atoms, rather
than only those arising from finite-level certificates. For these
measures, define $F$ and $Z$ as above and $\sigma$ and $\Phi$ by the same
formulas in~\eqref{eq:gap-normalization}. The kernel identity ensures
that $Z=1-\sigma$ continues to hold.
We set $\Phi=+\infty$ if $F\equiv0$.

We first include cap optimization and reduce to $\sigma=1/2$.
A certificate used at an inner-product threshold $0\leq s\leq1/2$ is
feasible when $\sigma\geq s$; cap reduction adds
$\tfrac{1}{2}\log_2(2(1-s))$ to its exponent.
Set $q=1/(2(1-s))$ and replace $\mu$ by $q\mu+(1-q)\delta_1$. This
replaces $F$ by $qF$, adds exactly that cap cost, and replaces the
normalizing constant by
\[
  Z'=qZ=\frac{1-\sigma}{2(1-s)}\leq\frac{1}{2};
\]
hence, the new measure has mean at least $1/2$.
Direct use at the kissing angle is the case $s=1/2$, $q=1$; the
endpoint $s=0$ is also covered by taking $\mu=\delta_0$.

We may further restrict to measures of mean $1/2$.
Indeed, for any probability measure $\mu$ with mean $\sigma\geq1/2$,
put
\[
  a=\frac{1}{2\sigma},
  \qquad
  \widetilde{\mu}=a\mu+(1-a)\delta_0.
\]
Then $\widetilde{\mu}$ has mean $1/2$. Since $k_0(v)=1$ and
$0\leq F(v)\leq1$, its associated function $\widetilde{F}$ satisfies
\[
  \widetilde{F}(v)=aF(v)+(1-a)\geq F(v).
\]
Thus its associated exponent satisfies $\widetilde{\Phi}\leq\Phi$.
A lower bound for the exponent of every probability measure of mean
$1/2$ therefore also bounds the exponent of every probability measure
of mean at least $1/2$, including the measures obtained by the cap
reduction above.

For such a measure, $P=2F\rho$ is a probability measure, and the
relative-entropy identity becomes
\begin{equation}
  \label{eq:gap-entropy}
  \Phi=\alpha_*+\frac{D_{\mathrm{KL}}(\nu\Vert P)}{2\log 2}.
\end{equation}
Indeed, $\int_0^1\log(d\nu/d\rho)\,d\nu=\log\pi-1$, which
gives~\eqref{eq:gap-entropy} directly
from~\eqref{eq:gap-normalization}. The logarithms are integrable:
$k_w(v)\geq(1-w)(1-v^2)$ gives $F(v)\geq(1-v^2)/2$.

For $0\leq v\leq1/2$, the function $w\mapsto k_w(v)$ is concave, since
\[
  \frac{\partial^2 k_w(v)}{\partial w^2}
  =-\frac{2(1-v^2)(1-v^2-3v^2w^2)}{(1-v^2+v^2w^2)^3}
  \leq0.
\]
Jensen's inequality gives
\[
  F(v)\leq k_{1/2}(v)=\frac{3(1-v^2)}{4-3v^2}
  \qquad(0\leq v\leq1/2).
\]
For $E=[0,1/2]$, we have $\nu(E)=1/2$ and consequently
\[
  P(E)\leq\frac{3}{\pi}\int_0^{1/2}
  \frac{\sqrt{1-v^2}}{1-3v^2/4}\,dv
  =\frac{2}{3}-\frac{2}{\pi}\arctan\frac{1}{2\sqrt{3}}
  =\beta<\frac{1}{2}.
\]
The integral follows by setting $v=\sin\theta$; the strict inequality
follows from the fact that
$\tan(\pi/12)=2-\sqrt{3}<1/(2\sqrt{3})$.
Applying the log-sum inequality to the partition $\{E,E^c\}$ then gives
\[
  D_{\mathrm{KL}}(\nu\Vert P)
  \geq-\frac{1}{2}\log\bigl(4P(E)(1-P(E))\bigr)
  \geq-\frac{1}{2}\log\bigl(4\beta(1-\beta)\bigr).
\]
Substitution in~\eqref{eq:gap-entropy} proves the claimed bound. (It is
independent of the level and cap parameter, so taking their infimum
preserves it.)
\end{proof}

\end{document}