\section{Approaches to Bremner's conjecture}\label{sec:bremner-conj}

Bremner's conjecture (\cite[\S7]{Bremner1981}; \S\ref{sec:conjecture}) says that every curve of arithmetic genus~$0$
defined over~$\Q$ on $V$ is obtained from one of a finite set of such curves by repeated application of automorphisms
taken from a finite set, the degree increasing at every application of an automorphism that is not a symmetry. The
Theorem~\ref{thm:conjecture} verified it through degree~$12$ with the base set $\{\Gamma_1,\dots,\Gamma_8,Q_1,\dots,Q_{12}\}$ and the
$772$ $\Q$-rational involutions of the $14$ $\Q$-rational elliptic pencils of degree $\le6$. This section
answers two questions: what the literature on automorphisms of K3 surfaces already implies, and how one could try to
prove the conjecture in full. The short answers are: (i) the conjecture, including the degree clause, follows for
\emph{every} K3 surface over \emph{every} field of characteristic $\neq2$ from Sterk's theorem in the form of
Bright--Logan--van Luijk, combined with Looijenga's Dirichlet-domain construction (Theorem~\ref{thm:Aprime}), but with
finite sets that are not explicit; (ii) making the sets explicit for $V$ is a computation which is now within reach
(Borcherds' method as implemented by Shimada and in \textsc{Oscar}, or Bremner's own cone-cutting method), and a first run
of Bremner's method on our data (\S\ref{sec:conerun}) already shows that the pencils of degree $\le6$ do not suffice for
the \emph{proof}, identifies exactly what is missing (involutions of pencils of degree~$8$, among them the negation of the
Kummer pencil), and shows that the missing automorphisms are again of Bremner's type.

Throughout, $\Sigma_k$ denotes the set of irreducible curves $\Gamma\subset V$ defined over a field $k\supset\Q$ with
$p_a(\Gamma)=0$; by adjunction these are exactly the smooth rational curves, $\Gamma^2=-2$, and we call them
$(-2)$-curves. Their classes are the $(-2)$-classes $D\in\NS(V_k)$ with $D\cdot H>0$ that are irreducible (Lemma~\ref{lem:irr}). $\deg\Gamma=\Gamma\cdot H$. $\operatorname{Aut}(V_k)$ is the group of automorphisms defined over $k$; it acts
on $\NS(V_{\bar\Q})$, on $\NS(V_k)$ and on $\Sigma_k$. The \emph{symmetries} are the elements of
$\operatorname{Aut}(V_\Q)$ fixing $H$, i.e.\ the linear automorphisms of $V\subset\PP^5$ over $\Q$; by Bremner's
Appendix~II they form the group of order $48$. We use hypothesis (H) of Section~\ref{sec:lattice}, $\rho(V_{\bar\Q})=19$
(Kuwata \cite{Kuwata2007}), so that $\NS(V_{\bar\Q})=\NS(V_{\Q(i)})=\Z\langle\Gamma_1,\dots,Q_{24}\rangle$ with the Gram
matrix of Bremner's basis (17), of determinant $48$; $c$ denotes complex conjugation acting on it.

\subsection{The conjecture in lattice terms}\label{sec:latticeterms}

Fix a finite set $\mathcal S\subset\Sigma_\Q$ and a finite set $\mathcal H\subset\operatorname{Aut}(V_\Q)$ with
$\mathcal H=\mathcal H^{-1}$. Say that $\Gamma\in\Sigma_\Q$ is \emph{monotonically generated} from $(\mathcal S,\mathcal H)$
if $\Gamma=h_m\cdots h_1 C$ with $C\in\mathcal S$, $h_j\in\mathcal H$, and $\deg(h_j\cdots h_1C)>\deg(h_{j-1}\cdots h_1C)$
for all $j$ (for $m=0$, $\Gamma=C$). Bremner's conjecture asserts the existence of $(\mathcal S,\mathcal H)$ such that
every $\Gamma\in\Sigma_\Q$ is monotonically generated, where in Bremner's formulation $\mathcal H$ may also contain the
symmetries, which are used freely and do not change the degree; replacing $\mathcal H$ by its closure under conjugation
by the symmetries and $\mathcal S$ by its orbit under them, one may equivalently drop the symmetries. Note also that
``automorphism from a finite set'' may be taken to mean ``element of a finite set $\mathcal H$ of automorphisms'', the
inverses being included, since an automorphism and its inverse play symmetric roles in a chain.

The following observation is the engine of Bremner's proof for fifth powers \cite{Bremner1981fifth} and of his
cone-cutting procedure in \cite[\S7]{Bremner1981}. For a finite set $\mathcal C$ of classes in $\NS(V_\Q)$ and a finite
$\mathcal H$ as above, let
\begin{equation}\label{eq:K}
K_{\mathcal C,\mathcal H}=\{x\in\NS(V_\Q)_\R:\ x\cdot C\ge0\ (C\in\mathcal C),\quad \deg(hx)\ge\deg x\ (h\in\mathcal H)\},
\end{equation}
a rational polyhedral cone ($\deg(hx)-\deg x=x\cdot(h^{-1}H-H)$ is linear in $x$).

\begin{lemma}[cone criterion]\label{lem:cone}
Let $\mathcal S\subset\Sigma_\Q$ be finite and let $\mathcal C$ consist of the classes of the curves in $\mathcal S$
together with classes of the form $\sum_{C\in I}C$ for some Galois orbits $I$ of $(-2)$-curves on $V_{\bar\Q}$ not defined
over~$\Q$ \textup(for $V$, conjugate pairs $C+\bar C$\textup). If every $(-2)$-class in $K_{\mathcal C,\mathcal H}$ that
is a $\Q$-rational $(-2)$-curve belongs to $\mathcal S$ --- in particular if
$K_{\mathcal C,\mathcal H}\subset\{x: x^2\ge0\}$ --- then every $\Gamma\in\Sigma_\Q$ is monotonically generated from
$(\mathcal S,\mathcal H)$.
\end{lemma}

\begin{proof}
Let $\Gamma\in\Sigma_\Q$ be of minimal degree among the curves that are not monotonically generated. Then
$\Gamma\notin\mathcal S$, so $\Gamma\cdot C\ge0$ for every $C\in\mathcal S$ (distinct irreducible curves), and
$\Gamma\cdot C\ge0$ for every $C$ in a Galois orbit $I$ not defined over $\Q$ (such $C$ are not $\Q$-rational, hence
$\neq\Gamma$). If $\deg(h\Gamma)<\deg\Gamma$ for some $h\in\mathcal H$, then $h\Gamma\in\Sigma_\Q$ is monotonically
generated by minimality, and $\Gamma=h^{-1}(h\Gamma)$ with $h^{-1}\in\mathcal H$ and $\deg\Gamma>\deg h\Gamma$ is
monotonically generated, a contradiction. Hence $\Gamma\in K_{\mathcal C,\mathcal H}$, against the hypothesis
($\Gamma^2=-2<0$).
\end{proof}

\begin{theorem}[Bremner \cite{Bremner1981fifth}, main theorem and \S\S3--5; our reading of the text]\label{thm:fifth}
Let $W\subset\PP^5$ be the K3 surface $x^5+y^5+z^5=u^5+v^5+w^5$, $x+y+z=u+v+w$, $x-y=u-v$. Then $\NS(W_\Q)$ has rank~$9$,
generated by eight rational lines $L_1,\dots,L_8$ and $H$, $W$ has $16$ symmetries, and every irreducible curve of
arithmetic genus~$0$ on $W$ defined over $\Q$ is obtained from the line $L_1$ or the conic $H-L_1-L_2$ by repeated
application of the symmetries and of two involutions $\Theta_1,\Theta_2$ coming from two elliptic pencils, the degree
increasing strictly at each application of a $\Theta_i$.
\end{theorem}

Bremner's proof is exactly Lemma~\ref{lem:cone}: the cone cut out by the $24$ inequalities ``nonnegative intersection
with the lines and conics'' and by the inequalities ``$\Theta_i$ does not decrease the degree'' is shown, by a finite
computation at its extremal rays, to lie in $\{x^2\ge0\}$ (he writes $4\,T\!\cdot\!T=(\deg T)^2-(\text{positive definite
form})$, so that $T\cdot T$ is concave on the slice $\deg T=1$ and its minimum is attained at a vertex). The surface
$W$ has $\rho(W_\Q)=9$ and the relevant polyhedron is small; for $V$ ($\rho(V_\Q)=13$) Bremner reports that his
program found about $4000$ vertices with $\mathcal S=\{$lines, conics$\}$ and $\mathcal H=\emptyset$, and that he had
``no firm guarantee'' that the cutting procedure terminates \cite[p.~577]{Bremner1981}. Bremner's model was Swinnerton-Dyer's unpublished treatment of $x^4+y^4=z^4+t^4$; what is in print is
\cite{SwinnertonDyer1968}, where the elliptic pencils on that surface are used to show that every real solution is a limit
of rational ones (compare Elkies \cite{Elkies1988} for $A^4+B^4+C^4=D^4$), and the automorphism groups of diagonal quartics
over non-closed fields are discussed in \cite[\S4.3]{BLvL2020}. We return to Bremner's procedure in
\S\ref{sec:conerun}; first we explain why it \emph{must} terminate in principle.

\subsection{What is known: Torelli, Sterk, and Bright--Logan--van Luijk}\label{sec:bc-known}

Let $X$ be a complex projective K3 surface, $\NS=\NS(X)$ (hyperbolic, even), $\mathcal C_X$ the component of
$\{x\in\NS_\R: x^2>0\}$ containing an ample class, $W(\NS)$ the group generated by the reflections $s_\delta$ in the
$(-2)$-vectors $\delta\in\NS$, $O^+(\NS)$ the isometries preserving $\mathcal C_X$, and $A_{\NS}=\NS^\vee/\NS$ with its
discriminant form $q_{\NS}$ \cite{Nikulin1980}.

\begin{theorem}[Pjateckii-Shapiro--Shafarevich \cite{PSS1971}; see {\cite[Ch.~8, 15]{Huybrechts2016}}]\label{thm:torelli}
\textup{(a)} $\operatorname{Nef}(X)\cap\mathcal C_X$ is a fundamental domain for $W(\NS)$ acting on $\mathcal C_X$; its
walls are the hyperplanes $C^\perp$, $C$ a $(-2)$-curve, and $\operatorname{Nef}(X)=\{x:\ x\cdot C\ge0$ for all
$(-2)$-curves $C\}$ when $\rho(X)\ge12$ \textup(Kov\'acs \cite{Kovacs1994}\textup).
\textup{(b)} $W(\NS)$ is normal in $O^+(\NS)$ and $\operatorname{Aut}(X)\to O^+(\NS)/W(\NS)$ has finite kernel and image
of finite index; more precisely, an isometry $g\in O^+(\NS)$ with $g(\operatorname{Nef}X)=\operatorname{Nef}X$ is induced
by an automorphism if and only if its image $\bar g\in O(q_{\NS})$ lies in the image of the group of Hodge isometries of
the transcendental lattice $T_X$ under $O(T_X)\to O(q_{T_X})\cong O(q_{\NS})$, and then the automorphism is unique.
\textup{(c)} $\operatorname{Aut}(X)$ is finite if and only if $W(\NS)$ has finite index in $O(\NS)$, i.e.\ $\NS$ is
$2$-reflective; the $2$-reflective hyperbolic lattices are classified \textup(Nikulin \cite{Nikulin1979b,Nikulin1981,Nikulin1984},
Vinberg \cite{Vinberg2007}; Kond\=o \cite{Kondo1989}\textup).
\end{theorem}

\begin{proposition}\label{prop:AutV}
\textup{(a)} $\NS(V_{\bar\Q})\cong U(2)\oplus E_8(-1)^2\oplus\langle-12\rangle$, $A_{\NS}\cong(\Z/2)^2\times\Z/12$,
$q_{\NS}\cong v(2)\oplus\langle\tfrac14\rangle\oplus\langle\tfrac23\rangle$ \textup(Proposition~\ref{prop:geo-lattices}\textup), $|O(q_{\NS})|=24$, and $T_V\cong U(2)\oplus\langle12\rangle$.
\textup{(b)} The group of Hodge isometries of $T_V$ is $\{\pm1\}$. Consequently
\[
\operatorname{Aut}(V_\C)\;\cong\;\{g\in O^+(\NS):\ g(\operatorname{Nef}V)=\operatorname{Nef}V,\ \bar g=\pm1\in O(q_{\NS})\},
\]
$\operatorname{Aut}(V_\C)\to O(\NS)$ is injective, every automorphism of $V_{\bar\Q}$ is defined over $\Q(i)$, and
$\operatorname{Aut}(V_\Q)=\{\varphi\in\operatorname{Aut}(V_{\bar\Q}):\ \varphi^*c=c\varphi^*\text{ on }\NS\}$.
\textup{(c)} $\operatorname{Aut}(V_\Q)$ is infinite, $W(\NS)$ has infinite index in $O(\NS)$, and $\operatorname{Nef}(V)$
has infinitely many walls.
\end{proposition}

\begin{proof}
(a) was computed from the Gram matrix of basis (17) (Proposition~\ref{prop:geo-lattices}; the identification with the lattice of
Artebani--Laface--Testa \cite{ALT} uses Nikulin's uniqueness theorem, and we re-verified the genus equality in
Sage). (b) Let $g$ be a Hodge isometry of $T=T_V$, of rank~$3$ and signature $(2,1)$. It has finite order, preserves
the positive $2$-plane $P$ spanned by $\operatorname{Re}\omega,\operatorname{Im}\omega$ and acts on it by a rotation,
and acts on $P^\perp$ by $\pm1$. If the rotation is not $\pm1$, its eigenvalues are primitive $m$-th roots of unity with
$\varphi(m)\le2$, $m\in\{3,4,6\}$, so $P=\ker\Phi_m(g)\otimes\R$ is defined over $\Q$ and the rank-one lattice
$P^\perp\cap T$ is orthogonal to $\omega$, i.e.\ algebraic, contradicting $T\cap\NS=0$; the same argument excludes
$g=(1_P,-1_{P^\perp})$. So $g=\pm1$. Since $-1$ acts as $-1\ne1$ on $A_T$ (which has elements of order $4$), the kernel
of $\operatorname{Aut}(V_\C)\to O(\NS)$ is trivial and the description of $\operatorname{Aut}(V_\C)$ is
Theorem~\ref{thm:torelli}(b). For $\sigma\in\Gal(\bar\Q/\Q(i))$ and $\varphi\in\operatorname{Aut}(V_{\bar\Q})$,
$\varphi^\sigma$ and $\varphi$ induce the same isometry of $\NS=\NS(V_{\Q(i)})$, so $\varphi^\sigma=\varphi$; the
statement about $\operatorname{Aut}(V_\Q)$ follows in the same way. (c) The sections of the Brudno pencil are
infinitely many $\Q$-rational $(-2)$-curves (Theorem~\ref{thm:degree}), so $\operatorname{Nef}(V)$ has infinitely
many walls and (Theorem~\ref{thm:torelli}(a),(c)) $W(\NS)$ has infinite index and $\operatorname{Aut}$ is infinite.
\end{proof}

\begin{theorem}[Sterk \cite{Sterk1985}]\label{thm:sterk}
Let $X$ be a complex algebraic K3 surface. \textup{(a)} $\operatorname{Aut}(X)$ is finitely generated. \textup{(b)} For every
even $d\ge-2$ there are finitely many $\operatorname{Aut}(X)$-orbits of complete linear systems containing irreducible
curves of self-intersection $d$; for $d=-2$: finitely many orbits of smooth rational curves, for $d=0$: of elliptic pencils,
for $d=2g-2>0$: of polarisations of degree $d$. \textup{(c)} $\operatorname{Aut}(X)$ acts on $\operatorname{Nef}_e(X)$, the
convex hull of $\operatorname{Nef}(X)\cap\NS(X)$, with a rational polyhedral fundamental domain.
\end{theorem}

Part (c), which implies (a) and (b), is the K3 case of the Morrison--Kawamata cone conjecture
\cite{Morrison1993,Kawamata1997}; see Totaro \cite{Totaro2010} for dimension~$2$ in general (using Sterk, Looijenga and
Namikawa \cite{Namikawa1985}), Lieblich--Maulik \cite{LieblichMaulik2018} for characteristic $p>2$, and Looijenga
\cite{Looijenga2014} for the general theory of ``polyhedral triples'' $(\NS_\R,\operatorname{Amp},\Gamma)$. Over a
non-closed field the relevant statements are those of Bright, Logan and van Luijk.

\begin{theorem}[Bright--Logan--van Luijk \cite{BLvL2020}]\label{thm:blvl}
Let $k$ be a field of characteristic $\neq2$ with separable closure $k^s$, $X$ a K3 surface over $k$,
$\Gamma_k=\Gal(k^s/k)$, $\operatorname{Aut}X$ the group of automorphisms defined over $k$,
$\operatorname{Nef}(X)=\operatorname{Nef}(X_{k^s})\cap\operatorname{Pic}(X)_\R$, and $R_X=W(\operatorname{Pic}X^s)^{\Gamma_k}$.
\textup{(a)} \textup{(Prop.~3.6, 3.7)} $R_X$ is a Coxeter group generated by the longest elements of the finite groups
$W_I$ attached to the Galois orbits $I$ of $(-2)$-curves consisting of pairwise disjoint curves or of disjoint pairs of
curves meeting in one point, and $\operatorname{Nef}(X)\cap\mathcal C_X$ is a fundamental domain for $R_X$ on the
positive cone of $\operatorname{Pic}(X)_\R$.
\textup{(b)} \textup{(Prop.~3.10)} The natural map $\operatorname{Aut}X\ltimes R_X\to O(\operatorname{Pic}X)$ has finite
kernel and image of finite index.
\textup{(c)} \textup{(Cor.~3.15)} $\operatorname{Aut}X$ acts on $\operatorname{Nef}_e(X)$ with a rational polyhedral
fundamental domain.
\textup{(d)} \textup{(Cor.~3.16)} There are finitely many $\operatorname{Aut}X$-orbits of $k$-rational $(-2)$-curves on $X$,
finitely many orbits of primitive classes of irreducible curves of arithmetic genus~$1$, and, for each $g\ge2$, finitely
many orbits of classes of irreducible curves of arithmetic genus~$g$.
\end{theorem}

\begin{remark}
(i) The objects in (d) are exactly Bremner's: a $k$-rational $(-2)$-curve is an irreducible curve of arithmetic genus~$0$
defined over $k$. So Theorem~\ref{thm:blvl}(d) \emph{is} the finiteness part of Bremner's conjecture for $V$ over $\Q$
(Corollary~\ref{cor:A}), with $\operatorname{Aut}X=\operatorname{Aut}(V_\Q)$, which is what Bremner needs: his automorphisms
must map $\Q$-rational curves to $\Q$-rational curves. (ii) Finite generation of $\operatorname{Aut}X$ is not stated in
\cite{BLvL2020} (we searched the text), but it follows from (b): $O(\operatorname{Pic}X)$ is an arithmetic group, hence
finitely generated, so is its finite-index subgroup, hence the finite extension $\operatorname{Aut}X\ltimes R_X$ and its
quotient $\operatorname{Aut}X$; Looijenga's Corollary~4.15 gives even finite presentation (see below). (iii) In
\cite[\S4]{BLvL2020} there are K3 surfaces $X/\Q$ with $\operatorname{Aut}X$ finite although $\operatorname{Aut}\bar X$ is
infinite; so over $\Q$ the finiteness of $\operatorname{Aut}$ is not determined by the lattice alone. For $V$ this does not
happen (Proposition~\ref{prop:AutV}(c), with $\Q$-rational automorphisms). (iv) Over $\Q$ the walls of $\operatorname{Nef}(V_\Q)$
in $\NS(V_\Q)_\R$ are of two kinds: $C^\perp$ for $\Q$-rational $(-2)$-curves $C$, and $(C+\bar C)^\perp$ for conjugate
pairs with $C\cdot\bar C\in\{0,1\}$ (classes of square $-4$, e.g.\ $\Gamma_9+\Gamma_{10}$, or of square $-2$ when the two
curves meet in a point). This is why the constraints $\mathcal C$ in Lemma~\ref{lem:cone}
must include the pair classes.
\end{remark}

\begin{corollary}\label{cor:A}
Assume \textup{(H)}. The set $\Sigma_\Q$ of smooth rational curves on $V$ defined over $\Q$ is the union of finitely many
$\operatorname{Aut}(V_\Q)$-orbits, and $\operatorname{Aut}(V_\Q)$ is finitely generated \textup(indeed finitely presented\textup).
The same holds over $\Q(i)$ and over $\bar\Q$. Neither the number of orbits nor the generators are given by the proof.
\end{corollary}

The degree clause is not addressed by \cite{BLvL2020} or \cite{Sterk1985}, but it follows from the shape of the fundamental
domain. Fix an ample class $h\in\operatorname{Pic}(X)$ (for $V$: $h=H$) and put
\begin{equation}\label{eq:dirichlet}
\sigma_h=\{x\in\operatorname{Nef}_e(X):\ x\cdot h\le x\cdot gh\ \text{for all }g\in\operatorname{Aut}X\},
\end{equation}
the Dirichlet domain of the orbit $\operatorname{Aut}(X)\cdot h$. By Looijenga \cite[Application~4.14]{Looijenga2014}, whose
hypothesis (``polyhedral type'') is supplied by Theorem~\ref{thm:blvl}(c), $\sigma_h$ is a rational polyhedral cone with
$\operatorname{Aut}(X)\cdot\sigma_h=\operatorname{Nef}_e(X)$, and its stabiliser is the finite group $\operatorname{Aut}(X)_h$
(for $V$: the $48$ symmetries). By \cite[Cor.~4.15]{Looijenga2014}, $\operatorname{Aut}X$ is finitely presented and generated by
$\operatorname{Aut}(X)_h$ together with the finitely many $g$ with $g\sigma_h\cap\sigma_h$ a facet of $\sigma_h$.

\begin{theorem}\label{thm:Aprime}
Let $k$ be a field of characteristic $\neq2$, $X$ a K3 surface over $k$, $h\in\operatorname{Pic}X$ ample, $\sigma_h$ as in
\eqref{eq:dirichlet}. Let $\mathcal S_h$ be the set of $k$-rational $(-2)$-curves $C$ such that $C^\perp\cap\sigma_h$ is a facet
of $\sigma_h$, and $\mathcal H_h$ the set of $g\in\operatorname{Aut}X$ such that $g\sigma_h\cap\sigma_h$ is a facet of
$\sigma_h$. Both are finite, $\mathcal H_h=\mathcal H_h^{-1}$, and every $k$-rational $(-2)$-curve on $X$ is monotonically
generated from $(\mathcal S_h,\mathcal H_h)$ with respect to $\deg=h\cdot(\ )$. In particular Bremner's conjecture holds for
$V$ over $\Q$, with respect to $H$, with the \textup(non-explicit\textup) finite sets $\mathcal S_H$ and
$\mathcal H_H\cup\{\text{symmetries}\}$.
\end{theorem}

\begin{proof}
Write $N=\operatorname{Pic}(X)_\R$ and $G=\operatorname{Aut}X$ acting on $N$. $\sigma_h$ is full-dimensional: for
$g\notin G_h$ one has $h\cdot h<h\cdot gh$ (reverse Cauchy--Schwarz for two non-proportional vectors of equal positive
norm in the same component of the positive cone), and only finitely many $gh$ satisfy $x\cdot gh\le x\cdot h+1$ for $x$ in
a neighbourhood of $h$, because $\{y\in\operatorname{Pic}X: y^2=h^2,\ x\cdot y\le c\}$ is finite for $x$ in the open
positive cone. Hence $\sigma_h$ is the intersection of its facet half-spaces. Let $F$ be a facet and $x$ a point of its
relative interior. Since $F$ spans a hyperplane and the isotropic cone contains no open subset of a hyperplane, we may
take $x^2>0$. Either $x\in\partial\operatorname{Nef}(X)$: then, by Theorem~\ref{thm:blvl}(a) and the local finiteness of
the $(-2)$-walls inside the open positive cone, $x\cdot C=0$ for some $(-2)$-curve $C$ on $X_{k^s}$, hence
$x\cdot\sum_{C'\in I}C'=0$ for the Galois orbit $I$ of $C$, and this hyperplane supports $\sigma_h$ at $x$, so it is the
hyperplane of $F$; the facet inequality is $x\cdot\sum_{I}C'\ge0$. Or $x$ is in the interior of $\operatorname{Nef}(X)$ and
on the boundary of the Dirichlet cell: by the finiteness just noted, $x\cdot h=x\cdot gh$ for some $g\ne1$, the hyperplane of
$F$ is the bisector $\{y\cdot h=y\cdot gh\}$, and since at most two points of the orbit of equal norm have a given bisector,
the cell on the other side is $g\sigma_h$ (translate the Dirichlet cell of $h$ by $g$); so $g\sigma_h\cap\sigma_h=F$ and
$g\in\mathcal H_h$. The facet inequality is $y\cdot gh\ge y\cdot h$, i.e.\ $\deg(g^{-1}y)\ge\deg y$. Applying $g^{-1}$ to
$g\sigma_h\cap\sigma_h=F$ shows $g^{-1}\in\mathcal H_h$. Therefore
\[
\sigma_h=\Bigl\{y:\ y\cdot\!\sum_{C'\in I}\!C'\ge0\ (I\in\mathcal I_h),\quad \deg(gy)\ge\deg y\ (g\in\mathcal H_h)\Bigr\}
=K_{\mathcal C_h,\mathcal H_h}
\]
with $\mathcal I_h$ the finite set of Galois orbits whose walls are facets of $\sigma_h$ and $\mathcal C_h$ the corresponding
classes; the $k$-rational curves in $\mathcal I_h$ form $\mathcal S_h$. Since $\sigma_h\subset\operatorname{Nef}(X)\subset
\{y^2\ge0\}$, Lemma~\ref{lem:cone} (whose proof is valid over any $k$) gives the claim.
\end{proof}

\begin{remark}
(i) The proof shows that $\sigma_h$ is Bremner's cone $K_{\mathcal C,\mathcal H}$ for the ``right'' finite sets, and that
his cutting procedure, if it terminates, computes $\sigma_h$ (or a polyhedral cone between $\sigma_h$ and the positive
cone); conversely it \emph{can} terminate, since $\sigma_h$ exists and is polyhedral. What it cannot do is certify that a
given pair $(\mathcal S,\mathcal H)$ works when $K_{\mathcal C,\mathcal H}\not\subset\{x^2\ge0\}$, as happens for the
sets of Theorem~\ref{thm:conjecture} (\S\ref{sec:conerun}). (ii) The argument is the exact analogue of the
height-reduction lemma for Coxeter groups: for a Coxeter system $(W,S)$ with simple roots $\Delta$, every positive root
$\beta\notin\Delta$ has a simple root $\alpha$ with $(\beta,\alpha)>0$, and then $s_\alpha\beta$ is a positive root of
smaller height; hence every positive root is obtained from a simple root by simple reflections increasing the height at
each step (\cite[\S\S1.3--1.5, 5.4]{Humphreys1990}, \cite{Vinberg1971} for the hyperbolic case). For $W(\NS)$ itself the
``simple roots'' are \emph{all} $(-2)$-curves (infinitely many) and the height-reduction lemma is Lemma~\ref{lem:irr}: a reducible $(-2)$-class $D$ has a $(-2)$-curve $C$ with $D\cdot C<0$ and $\deg s_C(D)<\deg D$. Bremner's conjecture
replaces this infinite Coxeter system by the finite data $(\mathcal S_h,\mathcal H_h)$ of the quotient by
$\operatorname{Aut}$. (iii) The analogous statement over $\Q(i)$ (all $(-2)$-curves) and over $\bar\Q$ holds with the same
proof, with $\operatorname{Aut}(V_{\bar\Q})$.
\end{remark}

\subsection{Explicit computation of $\operatorname{Aut}(V)$: Vinberg, Borcherds, Shimada}\label{sec:borcherds}

To make Theorem~\ref{thm:Aprime} explicit one has to compute $\operatorname{Aut}(V_\Q)$, or at least enough of it to
exhibit $\sigma_H$. Vinberg's algorithm \cite{Vinberg1972,Vinberg1971} (see Dolgachev's survey \cite{Dolgachev2008} for its use on K3
surfaces) computes a fundamental polyhedron of the reflection group of a hyperbolic lattice by adjoining walls one at a time; it terminates if and only if the polyhedron has finitely
many walls, which for $W(\NS(V))$ is not the case (Proposition~\ref{prop:AutV}(c)). Borcherds' method
\cite{Borcherds1987,Borcherds1998} circumvents this: embed $S=\NS$ primitively into the even unimodular lattice
$L=\mathrm{II}_{1,25}$, whose reflection group has the Leech lattice as its Dynkin diagram (Conway \cite{Conway1983}) and
whose chambers are explicitly known; the Conway chambers meeting $S_\R$ cut $\operatorname{Nef}(X)$ into
``induced chambers'' with finitely many walls, permuted by $\operatorname{Aut}(X)$ with finitely many orbits, and a
breadth-first search through adjacent induced chambers computes a generating set of $\operatorname{Aut}(X)$ and a
fundamental domain. This is how Kond\=o computed $\operatorname{Aut}$ of the generic Jacobian Kummer surface
\cite{Kondo1998} (a polyhedron with $32+32+60+192$ faces: nodes and tropes, $60$ Cremona involutions and Keum's $192$
automorphisms \cite{Keum1997}, the last two kinds ``behaving like reflections without being reflections''), and how
Keum and Kond\=o computed $\operatorname{Aut}\operatorname{Km}(E_1\times E_2)$ for $E_1,E_2$ non-isogenous, for $E\times E$
without CM, and for $E_\omega\times E_\omega$, $E_i\times E_i$ \cite{KeumKondo2001}; Vinberg's ``two most algebraic K3
surfaces'' $X_3,X_4$ \cite{Vinberg1983} ($\rho=20$, discriminants $3,4$) were done directly by reflection-group methods.
Our $V$ is $\operatorname{Km}(\operatorname{Jac}\mathcal C)$ with $\operatorname{Jac}\mathcal C\cong E_1\times E_2$ for a
cyclic $3$-isogeny $E_1\to E_2$ (Kuwata \cite{Kuwata2007}; Remark~\ref{rem:geo-kummer}; $T_V\cong U(2)\oplus\langle12\rangle$), a case not
covered by \cite{KeumKondo2001}; we found no computation of $\operatorname{Aut}$ for Kummer surfaces of products of
isogenous, non-isomorphic elliptic curves in the literature.

Shimada's algorithm \cite{Shimada2015} (IMRN 2015 --- not Amer.\ J.\ Math.\ as sometimes cited) makes Borcherds' method
general. Its hypotheses are: [SG1] $S$ is primitively embedded in $L=\mathrm{II}_{1,25}$; [SG2] the orthogonal complement
$R=S^\perp$ is not embeddable into the Leech lattice (this makes every induced chamber a finite polyhedron); [SG3] a
lifting condition, that the image in $O(q_S)$ of the group $G\subset O(S)$ one wants to compute is contained in the image
of $O(R)\to O(q_R)\cong O(q_S)$. Earlier applications needed $R$ to contain a root lattice of finite index; Shimada drops
this and allows several $\operatorname{Aut}$-orbits of induced chambers (his examples: $46$ chambers for a lattice of rank~$3$,
$1098$ chambers and $767$ generators for the singular K3 surface with $\operatorname{disc}T=11$; the Fermat quartic,
$\operatorname{disc}T=64$, was too heavy in 2013). The method is implemented in \textsc{Oscar}
(\texttt{K3\_surface\_automorphism\_group(S[,h])}, \texttt{borcherds\_method}) following \cite{Shimada2015,BrandhorstShimada2022},
and returns generators of the image of $\operatorname{Aut}(X)\to O(\NS)$ for a very general $S$-polarised K3 surface,
representatives of the $\operatorname{Aut}$-orbits of $(-2)$-curves, and a fundamental domain \cite{OSCAR}; for the scale of
computation it can handle see \cite{BrandhorstShimada2022b,BrandhorstMezzedimi2022}, and \cite{KondoMukai2024} for Kummer
surfaces of products of elliptic curves in characteristic~$2$ by Borcherds' method.

\begin{proposition}\label{prop:embedding}
$\NS(V_{\bar\Q})$ embeds primitively into $\mathrm{II}_{1,25}$ with orthogonal complement $R\cong A_2\oplus D_4\oplus\langle-4\rangle$
\textup(negative definite, rank~$7$, determinant $-48$, $30$ roots\textup), and $O(R)\to O(q_R)$ is surjective. Hence
\textup{[SG1]--[SG3]} hold for $V$ and every subgroup of $O(\NS)$, in particular for
$G=\{g\in O^+(\NS):\bar g=\pm1\}$, which contains the image of $\operatorname{Aut}(V_\C)$ as the stabiliser of
$\operatorname{Nef}(V)$ \textup(Proposition~\ref{prop:AutV}\textup). No embedding has a complement containing a root lattice
of finite index \textup(the only rank-$7$ root lattice of determinant $48$, $A_3\oplus A_1^2\oplus A_2$, has the wrong
discriminant form\textup), so the pre-2015 form of the method does not apply.
\end{proposition}

\begin{proof}
Existence follows from Nikulin \cite[Thm.~1.12.2]{Nikulin1980}, since $26-19=7>3=\ell(A_{\NS})$. Explicitly, in Sage:
$q_R\cong-q_{\NS}\cong v(2)\oplus\langle\tfrac74\rangle\oplus\langle\tfrac43\rangle$ for $R=A_2\oplus D_4\oplus\langle-4\rangle$ \textup(all three negative definite\textup), the
overlattice of $\NS\oplus R$ along the graph of an anti-isometry $A_{\NS}\to A_R$ is even, unimodular, of signature $(1,25)$,
hence $\mathrm{II}_{1,25}$, with $\NS$ and $R$ primitive; $|O(R)|=27648$, $|O(q_R)|=24$, and the image of $O(R)$ in
$O(q_R)$ has order $24$. (Scripts \texttt{lattice\_checks2--4.sage}.)
\end{proof}

\paragraph{The algorithm for $V$.} (1) Input the Gram matrix of basis (17), the embedding of Proposition~\ref{prop:embedding},
and the ample class $H$ (with the $56$ lines and conics as the known $(-2)$-walls). (2) Find a Weyl vector $w$ of a Conway
chamber whose induced chamber $D_0\subset\NS_\R$ contains $H$; compute its walls (finitely many, by [SG2]) --- they are
of the form $r^\perp$ with $r$ the projection to $\NS$ of a Leech root, and have norms $-2$ (then $r$ is a smooth rational
curve, if $D_0\subset\operatorname{Nef}$) or $>-2$. (3) Breadth-first search: for each wall, pass to the adjacent induced
chamber, test whether it is $G$-congruent to one already found (an isometry test in $O(\NS)$, i.e.\ a finite enumeration),
and record the congruence as a generator; stop when no new congruence classes appear. (4) Output: a generating set of the
image of $\operatorname{Aut}(V_\C)$ in $O(\NS)$ (by Proposition~\ref{prop:AutV} this is $\operatorname{Aut}(V_\C)$
itself), a finite set of induced chambers tiling a fundamental domain $F$ of $\operatorname{Nef}(V)$, and, as the
$(-2)$-walls of the chambers in $F$, representatives of all $\operatorname{Aut}(V_\C)$-orbits of smooth rational curves.
(5) Over $\Q$: $\operatorname{Aut}(V_\Q)$ is the centraliser of $c$ (Proposition~\ref{prop:AutV}(b)); since $c$ preserves
$\operatorname{Nef}$ and $H$, it permutes the induced chambers of $F$ up to congruence, and the centraliser can be read off
from the finite chamber graph (or one runs the search for $c$-stable configurations in $\NS(V_\Q)_\R$ directly; cf.\ the
reflection group $R_X$ of Theorem~\ref{thm:blvl}(a), which on $\NS(V_\Q)$ is generated by reflections in the $(-2)$-curves
and in the $(-4)$- and $(-2)$-classes $C+\bar C$). (6) From the generators, compute the Dirichlet domain $\sigma_H$ by
Bremner's cutting procedure (\S\ref{sec:conerun}), which now terminates by Theorem~\ref{thm:Aprime}, and read off
$\mathcal S_H$ and $\mathcal H_H$.

\emph{Feasibility.} The cost is governed by the number of $G$-congruence classes of induced chambers in
$\operatorname{Nef}(V)$, which we cannot predict; $V$ has rank~$19$ and discriminant~$48$, between Shimada's two
examples and the Fermat quartic. Everything needed (Gram matrix, embedding, $H$, the known curves, a working
\textsc{Oscar}/Sage/PARI environment with \texttt{qfauto}, \texttt{qfminim}, \texttt{qfisom}) is at hand, and a first
experiment with \texttt{K3\_surface\_automorphism\_group} costs a few hours of set-up. The output would settle (B) and
make Theorem~\ref{thm:Aprime} explicit for $V$; it would also decide whether the $\Q$-rational pencil involutions
generate $\operatorname{Aut}(V_\Q)$ (compare generators), which is the question behind (C).

\subsection{The degree-increasing clause: heights, Voronoi cells, and the involutions}\label{sec:degree}

The degree is a linear ``height'' $x\mapsto x\cdot H$ on $\NS$, and the clause ``the degree increases at each step'' says
that from every non-minimal class some generator strictly decreases the height. We saw in Theorem~\ref{thm:Aprime} that
this holds for the facet-pairing generators of a Dirichlet domain; here are three further comments.

\emph{Canonical heights.} Silverman \cite{Silverman1991} attached to a K3 surface with two involutions $\sigma_1,\sigma_2$
(Wehler surfaces \cite{Wehler1988}, $\rho=2$, $\operatorname{Aut}\supset\Z/2*\Z/2$) canonical heights $\hat h^\pm$ with
$\hat h^\pm\circ\sigma_i=\alpha^{\pm1}\hat h^\mp$, $\alpha=2+\sqrt3$, so that along an orbit the height grows geometrically
and ``a chain is finite iff its canonical height is $0$''; Baragar extended this to orbits of curves and to vector heights
for $\rho=2,3$ \cite{Baragar2003,Baragar2003b,Baragar2004,BaragarVanLuijk2007,Baragar2011}, showing that canonical vector
heights need not exist for $\rho\ge3$. For $V$ ($\rho=13$ over $\Q$) nothing of the sort is available; what replaces the
canonical height is the convex geometry of $\sigma_H$.

\emph{Sections of a pencil.} For the sections of the Brudno pencil the degree clause can be proved outright. Recall
(Theorem~\ref{thm:degree}, assuming Lemma~\ref{lem:generate}) $\deg C_S=2\langle S-\gamma/4,S-\gamma/4\rangle$ for
$S\in\MW=\mathcal E(\Q(\lambda))$, $\gamma=-g_1+g_2+g_3$, where $\langle\,,\rangle$ is the height pairing with Gram
matrix $\bigl(\begin{smallmatrix}1&-1/2&-1/2\\-1/2&2&1/2\\-1/2&1/2&2\end{smallmatrix}\bigr)$ on $\Z g_1\oplus\Z g_2\oplus\Z g_3$.
The translation $\tau_{S_0}\colon R\mapsto R+S_0$ by a $\Q$-rational section $S_0$ is a $\Q$-rational automorphism of $V$ (it is
$\iota_{C_1+C_2}\circ\iota_{C_3+C_4}$ for sections with $C_1+C_2-C_3-C_4=S_0$ in the group law), and $\deg\tau_{S_0}(C_S)=\deg C_{S+S_0}$.

\begin{proposition}\label{prop:voronoi}
Assume Lemma~\ref{lem:generate}. Let $\mathcal V=\{\pm g_1,\pm g_2,\pm g_3,\pm(g_2-g_3),\pm(g_1+g_2),\pm(g_1+g_3)\}$.
For every section $S$ of the Brudno pencil that is not one of the eight lines $\{kT,\,-g_1+kT\}$ there is $v\in\mathcal V$
with $\deg C_{S-v}<\deg C_S$. Hence every section is obtained from a line by the $12$ translations $\tau_v$, the degree
increasing strictly at every step.
\end{proposition}

\begin{proof}
$\mathcal V$ is the set of Voronoi-relevant vectors of the lattice $\MW/\tors$ with the height pairing (computed by
Voronoi's criterion: $\pm v$ are the only shortest vectors of $v+2\,\MW$; script \texttt{mw\_voronoi.py}). Put
$x=S-\gamma/4$. If $|x-v|\ge|x|$ for all $v\in\mathcal V$, then $x$ lies in the Voronoi cell of $0$, i.e.\ $|x|\le|x-w|$ for all
$w\in\MW$, so $S$ has minimal degree among all sections, which is $1$: $S$ is a line. Otherwise $\deg C_{S-v}=2|x-v|^2<2|x|^2=\deg C_S$.
\end{proof}

The same argument applies to every $\Q$-rational Jacobian pencil $F$ of positive rank and to the translates of any fixed
curve $D$ (not only sections): by Shioda's formulas, $\deg\tau_{S_0}(D)=\tfrac{(D\cdot F)(H\cdot F)}2\langle S_0,S_0\rangle+(\text{linear in }S_0)
+(\text{periodic in }S_0)$, a shifted positive definite quadratic form on each coset of the narrow Mordell--Weil group, so
finitely many Voronoi-relevant translations reduce the degree of all but finitely many translates. This shows that within
the orbit of a single pencil's translation group the degree clause is automatic; the whole difficulty of the conjecture is
the interaction of the $14$ (or more) pencils, which is what the Dirichlet domain encodes.

\emph{The involutions are not reflections.} An involution $\iota_{C_1+C_2}$ of a pencil acts on $\NS$ as $-1$ on the
orthogonal complement of $\langle F, C_1,C_2,\text{fibre components}\rangle$ and permutes the generators; its
$(-1)$-eigenlattice $N_-(\iota)$ is negative definite (the fixed lattice contains $F$ and $H+\iota H$). For the $532$
involutions of the Brudno pencil we computed (\texttt{involution\_structure.sage}): $\operatorname{rk}N_-\in\{7,9,10,11,13\}$,
$N_-$ contains no $(-2)$-vector, and among the vectors $v\in N_-$ with $|v^2|\le12$ at most one pair $\pm v$ (of norm $-4$)
defines an integral reflection of $\NS$. So $\iota$ is a product of reflections only over $\Q$, is not in $W(\NS)$ (no
automorphism is, by Theorem~\ref{thm:torelli}(a)), and is not a product of the few integral reflections it contains. This
matches Kond\=o's observation that the facet-pairing involutions of a Kummer surface ``behave like reflections without
being true reflections'': what they have in common with simple reflections is that they pair a facet of a fundamental
domain with itself (Theorem~\ref{thm:Aprime}). The question whether the $772$ involutions of the $14$ pencils generate a
subgroup of finite index in $\operatorname{Aut}(V_\Q)$ can be answered by (B): compare the group they generate with the
output of Shimada's algorithm (finite index can be tested on the finite chamber graph); or, without (B), by exhibiting a
polyhedral cone $\Pi$ with $\Pi\subset\operatorname{Nef}_e$ and $\langle\text{involutions}\rangle\cdot\Pi=\operatorname{Nef}_e$,
which is the content of the cone test below.

\subsection{Bremner's cone test on the Phase-A data}\label{sec:conerun}

We ran the first step of Bremner's procedure with the data of Sections~\ref{sec:sections}--\ref{sec:lattice} (scripts \texttt{build\_cone.py},
\texttt{cone\_S\_only.sage}; the cone lives in $\NS(V_\Q)_\R\cong\R^{13}$, in the basis $\Lambda_1,\dots,\Lambda_{13}$).
With $\mathcal C$ = the classes of the $56$ lines and conics ($38$ distinct functionals on $\NS(V_\Q)$, since conjugate
curves give the same functional) and $\mathcal H=\emptyset$, the cone $K_{\mathcal C,\emptyset}$ has $8266$ extremal rays
(Parma Polyhedra Library, $4$ seconds): $7782$ of negative square, $268$ isotropic, $216$ positive. (Bremner reports
about $4000$ vertices; he may have used only the $20$ rational curves or a different normalisation.) Among the degree-$5$
rays with $r^2=-2$ is the class of his curve (ii) (the class of (iii), and that of Brudno's quartic, lie in the cone but not
on extremal rays).

Next we took $\mathcal H$ = the $772$ $\Q$-rational involutions of the $14$ $\Q$-rational pencils of degree $\le6$, closed
under conjugation by the $48$ symmetries ($2730$ involutions, $2444$ distinct inequalities $\deg(\iota x)\ge\deg x$).
Of the $7782$ negative rays, $7722$ violate at least one of them; \textbf{$60$ do not}, so
$K_{\mathcal C,\mathcal H}\not\subset\{x^2\ge0\}$ and Lemma~\ref{lem:cone} does not apply to the sets of Theorem~\ref{thm:conjecture}. (The full vertex enumeration of $K_{\mathcal C,\mathcal H}$, $2482$ inequalities in $\R^{13}$, had not
finished at the time of writing; it is not needed for what follows.) The $60$ rays form three symmetry orbits:
\begin{itemize}
\item[(A)] $12$ rays with $r^2=-12$, $\deg r=24$, $\operatorname{div}(r)=6$. Each is orthogonal to $16$ pairwise disjoint
lines (for one of them $\Gamma_{5},\dots,\Gamma_8,\Gamma_{13},\dots,\Gamma_{16},\Gamma_{21},\dots,\Gamma_{24},
\Gamma_{29},\dots,\Gamma_{32}$). By Nikulin \cite{Nikulin1975} sixteen disjoint smooth rational curves on a K3 surface are
the exceptional curves of a Kummer structure $V=\operatorname{Km}(A)$, and $\langle16\text{ lines}\rangle^\perp=\pi_*\NS(A)
\cong\NS(A)(2)$; here it has rank~$3$ and determinant~$48$, so $\NS(A)\cong U\oplus\langle-6\rangle$ (the lattice of
$E_1\times E_2$ with a cyclic $3$-isogeny, Remark~\ref{rem:geo-kummer}), and $r$ is the image of a $(-6)$-vector of $\NS(A)$.
\item[(B)] $24$ rays with $r^2=-4$, $\deg r=12$, $\operatorname{div}(r)=2$; \quad (C) $24$ rays with $r^2=-12$, $\deg r=24$,
$\operatorname{div}(r)=6$.
\end{itemize}
All $60$ satisfy $r\cdot C\ge0$ for all $668\,056$ irreducible $(-2)$-curves of degree $\le12$ over $\Q(i)$; none is
effective. Since $\operatorname{Nef}(V)=\bigcap_C\{x\cdot C\ge0\}$ (Theorem~\ref{thm:torelli}(a), $\rho=19\ge12$) and $r^2<0$,
each $r$ has $r\cdot C<0$ for some smooth rational curve $C$ of degree $\ge13$: the obstructing rays point exactly at the
part of $V$ that the enumeration of Section~\ref{sec:lattice} does not see.

\begin{proposition}\label{prop:newinv}
\textup{(a)} For each of the $60$ rays the reflection $s_r$ is an integral isometry of $\NS(V_{\bar\Q})$ commuting with $c$,
and $s_r$ preserves $\operatorname{Nef}(V)$: there is no $(-2)$-class $D$ with $D\cdot H>0>D\cdot s_r(H)$.
\textup{(b)} For the rays of orbit \textup{(A)}, $\bar s_r=-1$ on $A_{\NS}$; hence $s_r$ is induced by an involution
$\varphi_r\in\operatorname{Aut}(V_\Q)$, which fixes the $16$ lines orthogonal to $r$ and satisfies $\deg\varphi_r(r)=-24<24$.
It is the fibrewise negation $[-1]=\iota_{Q+Q}$ of the $\Q$-rational elliptic pencil of degree~$8$ with four fibres of type
$\mathrm I_0^*$ spanned by $r^\perp\cap\langle16\text{ lines}\rangle^\perp$ \textup(a Kummer pencil, $Q$ one of its conic
sections\textup), i.e.\ the involution induced by $(x,y)\mapsto(x,-y)$ on $E_1\times E_2$.
\textup{(c)} For the rays of orbits \textup{(B)} and \textup{(C)}, $\bar s_r\ne\pm1$, so $s_r$ is a symmetry of
$\operatorname{Nef}(V)$ not induced by an automorphism \textup(the stabiliser of $\operatorname{Nef}(V)$ in $O^+(\NS)$ is
strictly larger than $\operatorname{Aut}(V_\C)$\textup). Nevertheless each such $r$ is orthogonal to $\Q$-rational elliptic
pencils $F$ of degree~$8$ \textup(types $\mathrm I_8\mathrm I_4\mathrm I_0^*$ and $\mathrm I_1^{*2}\mathrm I_3^2$ for \textup{(B)},
$\mathrm I_8^2\,\mathrm{III}^2$ for \textup{(C)}\textup) with $\Q$-rational sections among the known curves, and some of
Bremner's involutions $\iota_{C_1+C_2}$ of these pencils satisfy $\iota(r)=-r$.
\end{proposition}

\begin{proof}
(a) Integrality: $r^2\mid2\operatorname{div}(r)$. For the second claim we used the bound $(aH+bH')^2\ge-2$ for the component
of a separating class $D=aH+bH'+D_\perp$ in the plane $\langle H,H'\rangle$, $H'=s_r(H)$, $H\cdot H'\in\{80,104\}$,
which gives $\deg D\le31$, resp.\ $40$, and enumerated all $(-2)$-classes of these degrees with $D\cdot H'\le0$ by the
constrained Fincke--Pohst search of Section~\ref{sec:lattice} (\texttt{separating.py}; a few hundred tree nodes per degree): there are
none. A $(-2)$-curve $C$ with $C\cdot H'<0$ would be such a class; so $H'$ is nef ($H'^2=8$, $H'\cdot H>0$). Since $H'$ lies in
the open chamber $s_r(\operatorname{Amp})$ and in the closure of the chamber $\operatorname{Amp}$, the two chambers coincide,
i.e.\ $s_r(\operatorname{Nef})=\operatorname{Nef}$. (b) The action on $A_{\NS}$
was computed on the dual basis; Proposition~\ref{prop:AutV}(b) and Theorem~\ref{thm:torelli}(b) give $\varphi_r$, defined
over $\Q$ because $s_r$ commutes with $c$. The identification with $[-1]$ of the Kummer pencil: the primitive isotropic classes $F$ with
$F\cdot r=0$ of degree $\le12$ were enumerated (\texttt{pencils\_perp.py}); the two of smallest degree have degree~$8$, are
$\Q$-rational and nef (checked against the curves of degree $\le8$), have $20$ fibre components among the known curves
($4\times\mathrm I_0^*$: $16$ lines and $4$ conics) and $8$ conic sections. Building their involutions with the Phase-A code
(\texttt{cut\_by\_pencils.py}, \texttt{verify\_phi.py}) gives $\deg\iota(r)\in\{-24,24,72\}$, and the matrix of the negation
$\iota_{Q+Q}$ with respect to four of the eight conic sections $Q$ (of either pencil) \emph{equals} the matrix of $s_r$. For
such a pencil, with zero section $Q$ the image of $E_1\times\{0\}$, the negation on the fibre $\{p\}\times E_2$ is
$(p,y)\mapsto(p,-y)$. (c) As in (b).
\end{proof}

\begin{remark}
(i) Proposition~\ref{prop:newinv} carries out Bremner's own prescription for a ray $R$ with $R^2<0$ that is not
effective: add to $\mathcal H$ an automorphism built from a rational elliptic pencil $F\perp R$ and a transversal $D$ with
$D\cdot F=2$, whose involution (his Theorem~5.4) reverses $R$ \cite[pp.~576--577]{Bremner1981}. The pencils needed have degree~$8$, beyond the range of
Theorem~\ref{thm:conjecture}, which explains why the obstruction was invisible there and why the involutions of degree-$\le6$ pencils cannot
prove the conjecture by this method even though they generate all curves of degree $\le12$. (ii) No product of two of the
$2730$ involutions, the symmetries and the $12$ automorphisms $\varphi_r$ reduces the degree of a ray of orbit (B) or (C)
(\texttt{orbit\_min.py}); the degree-$8$ pencils are genuinely new generators for this purpose. (iii) Part (c) shows that
$[O^+(\NS)_{\operatorname{Nef}}:\operatorname{Aut}(V_\C)]\ge2$; this index divides $|O(q_{\NS})/\pm1|=12$ and must be built
into any computation of $\operatorname{Aut}(V)$ (Shimada's $G$).
\end{remark}

\subsection{Singular rational curves}\label{sec:singular}

Bremner excludes curves of geometric genus $0$ and arithmetic genus $g\ge1$, and says such curves exist on $V$ over $\Q$
\cite[p.~546]{Bremner1981}. An irreducible curve $C$ with $p_a(C)=g\ge1$ has $C^2=2g-2\ge0$, is nef, and moves in the
$g$-dimensional linear system $|C|$; only finitely many members of $|C|$ are rational (otherwise $V$ would be uniruled).
Sterk's Theorem~\ref{thm:sterk}(b) and Theorem~\ref{thm:blvl}(d) therefore give: for each $g\ge1$ the $\Q$-rational rational
curves of arithmetic genus $g$ on $V$ fall into finitely many $\operatorname{Aut}(V_\Q)$-orbits (first finitely many orbits
of classes, then finitely many curves per class). On the other hand there is no finiteness over all $g$: Chen, Gounelas and
Liedtke \cite{CGL2022} prove that every K3 surface over an algebraically closed field of characteristic~$0$ contains
infinitely many rational curves (for elliptic K3 surfaces this is in Bogomolov--Tschinkel \cite{BogomolovTschinkel2000};
see also \cite{BHT2011,BogomolovTschinkel2005}), and for $V$ the smooth ones alone are infinite; whether rational curves of
unbounded arithmetic genus exist on $V$, over $\bar\Q$ or over $\Q$, is a different question, which we have not settled
(for a general polarised K3 surface it holds by Chen's results, but $V$ is far from general). So the natural extension of
Bremner's conjecture to all rational curves must be formulated genus by genus, as a statement about the classes
$C\in\NS(V_\Q)$ with $C^2=2g-2$ containing a rational curve over $\Q$: for each $g$ there are finitely many
$\operatorname{Aut}(V_\Q)$-orbits, and the degree clause holds for them too, by the Dirichlet domain argument applied to
the nef classes (for $g\ge1$ these are points of $\operatorname{Nef}_e$, and $\operatorname{Aut}(V_\Q)\sigma_H\ni C$ gives a
representative in $\sigma_H$ of minimal degree in its orbit, reached by a monotone gallery; we omit the details). Which
classes actually contain rational curves is a question of a different nature (Bogomolov--Tschinkel \cite{BogomolovTschinkel2005},
Baragar--McKinnon \cite{BaragarMcKinnon2010}), on which the lattice says nothing.

\subsection{A research plan}\label{sec:plan}

We distinguish what is proved, what is a reduction, and what is speculation. Proved: Corollary~\ref{cor:A} and
Theorem~\ref{thm:Aprime} (the conjecture with non-explicit sets, for every K3 surface over every field of characteristic
$\ne2$); Propositions~\ref{prop:AutV}, \ref{prop:embedding}, \ref{prop:voronoi}, \ref{prop:newinv}. Reduction: an explicit
proof for $V$ is equivalent to exhibiting finite sets $(\mathcal C,\mathcal H)$ with $K_{\mathcal C,\mathcal H}\subset\{x^2\ge0\}$
(Lemma~\ref{lem:cone}), and Theorem~\ref{thm:Aprime} guarantees that such sets exist. Speculation: that the pencils of
degree $\le8$ suffice, and that $\mathcal S$ can be taken to be the $20$ rational lines and conics.

\begin{enumerate}[label=\textup{M\arabic*.}]
\item \emph{Pencils of degree $7$ and $8$.} Extend the Phase-A enumeration of elliptic pencils from degree $6$ to $8$
(degree~$7$: $10\,752$ pencils, $192$ $\Q$-rational, $138$ orbits under symmetries and conjugation, $4$ with $\Q$-rational
members, $3$ seconds; degree~$8$ needs the chunked version of the enumerator, the current one exceeds $8$~GB), build their
$\Q$-rational involutions from the sections of degree $\le12$, and recompute the closure of Theorem~\ref{thm:conjecture} and the cone test.
Expected outcome: the $60$ rays disappear; new, higher-degree obstructing rays may or may not appear.
\item \emph{Iterate Bremner's procedure} (lazy constraint generation: add only the inequalities violated by current rays;
when a negative ray $r$ survives, look for $\Q$-rational pencils $F\perp r$ of degree $\le12$ and their involutions, as in
Proposition~\ref{prop:newinv}, or for curves $C$ with $r\cdot C<0$). Each step is a PPL/cdd vertex enumeration in $\R^{13}$;
the first one (38 inequalities) took $4$ seconds, the one with $2482$ inequalities is slow and should be done
incrementally and with the symmetry group factored out. Termination is guaranteed in principle by Theorem~\ref{thm:Aprime}
but not along this particular path; if it terminates, the output is a complete proof of Bremner's conjecture for $V$ over
$\Q$ with explicit sets, needing no classification of curves beyond the lines and conics.
\item \emph{Borcherds--Shimada for $V$} (\S\ref{sec:borcherds}), with the embedding of Proposition~\ref{prop:embedding}, first
in \textsc{Oscar}. Output: $\operatorname{Aut}(V_\C)$ with generators, the orbits of smooth rational curves, the index of
$\operatorname{Aut}(V_\C)$ in the symmetry group of $\operatorname{Nef}(V)$ (at least $2$ by Proposition~\ref{prop:newinv}(c)),
then $\operatorname{Aut}(V_\Q)$ as the centraliser of $c$ and the Dirichlet domain $\sigma_H$ with $\mathcal S_H,\mathcal H_H$.
This is the most informative route and the only one that also answers whether the pencil involutions generate
$\operatorname{Aut}(V_\Q)$, but its running time is uncertain.
\item \emph{Partial results in the meantime.} With the existing code one can prove the conjecture in degree
$\le N$ for larger $N$ cheaply as follows: by the proof of Lemma~\ref{lem:cone}, a $\Q$-rational $(-2)$-curve of minimal degree
among those not monotonically generated lies in $K_{\mathcal C,\mathcal H}$, so the conjecture holds in degree $\le N$ as soon as
no $\Q$-rational $(-2)$-curve of degree $\le N$ outside $\mathcal S$ lies in $K_{\mathcal C,\mathcal H}$; the $(-2)$-classes of
degree $d$ in this thin cone are found by the constrained Fincke--Pohst search (thousands of tree nodes, not $10^9$ classes, as
in the proof of Proposition~\ref{prop:newinv}), and each candidate is either reducible (a known curve $C$ with $D\cdot C<0$) or
must be parametrised and tested (as in Section~\ref{sec:newcurves}). This replaces the full enumeration of irreducible curves, which stops
at degree~$12$, by a search in the obstruction cone only.
\item \emph{Structure.} Record for each generator of $\operatorname{Aut}(V_\Q)$ its geometric origin (pencil involution,
translation, Kummer automorphism from $\operatorname{Aut}(E_1\times E_2)$ --- an arithmetic group commensurable with
$\Gamma_0(3)$, since $\operatorname{End}(E_1\times E_2)$ is an Eichler order of level~$3$ --- or Cremona-type involution), and the
geometric meaning of the facets of $\sigma_H$. Proposition~\ref{prop:newinv}(b) is a first instance: a Kummer automorphism
that is at the same time a Bremner involution of a degree-$8$ pencil.
\item \emph{What a full proof is.} Either (M2) terminates, or (M3) is completed and $\sigma_H$ is computed; in both cases the
result is a theorem of the form of Theorem~\ref{thm:fifth} for $V$, with an explicit finite base set (conjecturally the
$20$ rational lines and conics) and an explicit finite set of involutions, together with the number of
$\operatorname{Aut}(V_\Q)$-orbits of $\Q$-rational smooth rational curves. Without a computation, Theorem~\ref{thm:Aprime} is
the best that the general theory gives.
\end{enumerate}