\documentclass[11pt,a4paper,reqno]{amsart} \usepackage[a4paper,margin=1.1in]{geometry} \usepackage[T1]{fontenc} \usepackage{lmodern,microtype} \usepackage{indentfirst} \setlength{\parindent}{1.6em} \setlength{\parskip}{0pt} \setlength{\emergencystretch}{2em} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{booktabs,array,longtable,enumitem} \usepackage[dvipsnames]{xcolor} \usepackage[colorlinks=true,linkcolor={red!50!black},citecolor={blue!50!black},urlcolor={blue!75!black}]{hyperref} \allowdisplaybreaks[0] \raggedbottom \newtheorem{theorem}{Theorem}[section] \newtheorem{introtheorem}{Theorem} \renewcommand{\theintrotheorem}{\Alph{introtheorem}} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{construction}[theorem]{Construction} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\C}{\mathbb C} \newcommand{\Z}{\mathbb Z} \newcommand{\N}{\mathbb Z_{\ge0}} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\Newt}{\operatorname{Newt}} \newcommand{\Supp}{\operatorname{supp}} \newcommand{\conv}{\operatorname{conv}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\ind}{\mathbf 1} \newcommand{\cI}{\mathcal I} \newcommand{\A}{\mathcal A} \newcommand{\U}{\mathcal U} \newcommand{\ubar}{\bar u} \newcommand{\vbar}{\bar v} \numberwithin{equation}{section} \hypersetup{pdftitle={Laurent phenomenon algebras on minuscule Grassmannians and their mirrors},pdfauthor={Jiarui Fei},pdfsubject={Laurent phenomenon algebras, intrinsic mirrors, and theta functions},pdfkeywords={Laurent phenomenon algebras, minuscule Grassmannians, mirror symmetry, theta functions, Newton polytopes, Coxeter boundaries}} \title[LP algebras on minuscule Grassmannians]{\small Laurent Phenomenon Algebras on Minuscule Grassmannians\\and their Mirrors} \author{Jiarui Fei} \address{School of Mathematical Sciences, Shanghai Jiao Tong University} \email{jiarui@sjtu.edu.cn} \thanks{The author was supported in part by the National Natural Science Foundation of China (Nos.~12131015 and 12571038).} \subjclass[2020]{Primary 13F60, 14J33; Secondary 14M15, 14M17, 14M25.} \keywords{Laurent phenomenon algebras, minuscule Grassmannians, mirror symmetry, theta functions, Newton polytopes, Coxeter boundaries.} \begin{document} \begin{abstract} We construct Laurent phenomenon algebra structures on the homogeneous coordinate rings of the half-spin Grassmannians and the Freudenthal variety. Their frozen variables are the linear coordinates in a Coxeter orbit, and the resulting boundary divisors are anticanonical. Together with the construction of Daisey--Ducat for the Cayley plane, these examples admit explicit mirrors described by Newton polytopes. We identify the upper Laurent phenomenon algebras of the mirror seeds with the intrinsic mirror algebras of Keel--Yu at the specialization $q^\beta=1$, including the theta functions associated with the individual boundary components. The half-spin construction is uniform in the rank. In the exceptional cases, contact calculations and Laurent divisibility determine the local expansions; the Freudenthal comparison also uses exact calculations of divisorial valuations. We obtain integral comparisons and valuative independence of the intrinsic theta basis in every reachable seed. \end{abstract} \maketitle \setcounter{tocdepth}{1} {\tableofcontents} \section*{Introduction} \phantomsection Laurent phenomenon algebras, introduced by Lam and Pylyavskyy, generalize cluster algebras by allowing irreducible exchange polynomials with more than two terms \cite{LP}. Daisey and Ducat used this flexibility to describe the homogeneous coordinate ring of the Cayley plane by a finite-type LP algebra \cite{DD}. The frozen coordinates form a Coxeter orbit. We construct analogous structures for the half-spin Grassmannians and the Freudenthal variety, and identify the intrinsic mirrors of the corresponding boundary complements. The boundary has a natural Lie-theoretic origin. If $G$ is simple and $P\subset G$ is a proper parabolic subgroup, the Fano index of $G/P$ equals the Coxeter number $h(G)$ precisely when $P$ is maximal at a minuscule node (Proposition~\ref{prop:index-minuscule}). An orbit of $h(G)$ linear sections then gives an anticanonical divisor. It is a further question whether these sections can serve as frozen variables of an LP seed. Several familiar examples answer this question. Scott's cluster algebra on $\operatorname{Gr}(k,N)$ has the $N$ cyclic interval Pl\"ucker coordinates as frozen variables \cite{Scott}. Projective space admits a rank-zero seed, and the relation $uv=\sum_{i=1}^{r-1}a_i b_i$ defines a rank-one seed on the even-dimensional quadric $D_r/P_1$, with the $a_i,b_i$ frozen. We focus on \[ X_n=\operatorname{OG}^{+}(n,2n)\quad(n\ge4),\qquad X_6^{\mathrm{exc}}=E_6/P_6,\qquad X_7^{\mathrm{exc}}=E_7/P_7. \] The polarized isomorphism $B_n/P_n\simeq D_{n+1}/P_{n+1}$ allows us to treat the type-$B$ case together with type~$D$. \subsection*{Coordinate rings and anticanonical boundaries} Let $\mathcal O_X(1)$ denote the primitive ample generator of $\operatorname{Pic}(X)$, and put \[ R_X=\bigoplus_{d\ge0}H^0(X,\mathcal O_X(d)),\qquad D=\sum_{i=1}^{h}\{a_i=0\}, \] where $a_1,\ldots,a_h$ are the chosen linear coordinates in a Coxeter orbit. We call $D$ the \emph{Coxeter boundary}, since its components come from a Coxeter orbit. The seeds are defined over $S_X=\mathbb C[a_1,\ldots,a_h]$, with these \emph{frozen variables} kept polynomial. Following Lam--Pylyavskyy, we distinguish the LP algebra $\mathcal A(\Sigma_X)$ generated by cluster variables from the \emph{upper LP algebra} $\mathcal U(\Sigma_X)$, the intersection of the seed Laurent rings over $S_X$. Daisey--Ducat use the upper algebra convention in \cite[Definition~2.6]{DD}. \begin{introtheorem}[Coordinate-ring structures]\label{thm:intro-source} For each Coxeter-boundary seed constructed below, \[ \mathcal U(\Sigma_X)=R_X. \] For the half-spin Grassmannians and the Cayley plane we also have $\mathcal A(\Sigma_X)=R_X$. The half-spin seed has mutable rank $\binom{n-2}{2}$, with interval Pfaffians as mutable variables. The Cayley-plane seed is the rank-five seed of Daisey--Ducat; the Freudenthal seed has mutable rank ten and eighteen frozen variables. The half-spin seeds have infinitely many cluster variables for $n\ge7$, as does the Freudenthal seed. \end{introtheorem} For the half-spin Grassmannians, we combine the multidimensional cube recurrence of Henriques--Speyer \cite{HS} with the wiring-diagram constructions of Lam--Pylyavskyy \cite{LP}. Signed Pfaffians satisfy the exchange relations, and two complementary wiring charts identify the upper LP algebra with $R_{X_n}$. Every spinor coordinate is a cluster variable, yielding the asserted equality with $\mathcal A(\Sigma_{X_n})$. For the Freudenthal variety, five mutations connect our seed to an auxiliary chart; Laurent expansions and prime divisors then give the upper-algebra equality. The infinite-type statements follow from rank-two subpatterns. They do not exclude a different finite-type positive seed of the kind conjectured by Daisey--Ducat \cite[Conjecture~6.1]{DD}. \subsection*{Intrinsic mirrors and boundary functions} For each $X$ above, consider the affine boundary complement $U_X=\operatorname{Spec}R_X[(a_1\cdots a_h)^{-1}]$. It is a smooth affine log Calabi--Yau variety containing the initial torus. Let $Y$ be a smooth projective compactification with simple normal-crossings boundary. Keel--Yu associate to this data an \emph{intrinsic mirror algebra} $A_Y$, defined over the original curve-class ring $\mathbb Z[\operatorname{NE}(Y)]$ and equipped with theta functions $\vartheta_m$ \cite{KY,KWY}. We study the algebra \[ A_{\mathrm{int}}(U_X)_1 =A_Y\otimes_{\mathbb Z[\operatorname{NE}(Y)],\,q^\beta\mapsto1}\mathbb C. \] It is independent of the chosen compactification. The subscript $1$ refers only to the specialization of curve-class parameters, not to the Laurent coefficient characters of the mirror. We write $\Theta_m$ for the expansion of $\vartheta_m$ in a specified Laurent chart. For each component $b$ of the chosen boundary, we identify its associated \emph{boundary function} $W_b$ on the mirror. Their sum $W=\sum_bW_b$ is the \emph{superpotential}; this is the terminology usually used for the corresponding Landau--Ginzburg function. \begin{introtheorem}[Marked intrinsic mirrors]\label{thm:intro-mirror} The Newton constructions give explicit mirror seeds $\Sigma_X^\vee$ and regular functions $W_b$ such that the intrinsic Laurent expansion induces an isomorphism \[ \Phi_X:A_{\mathrm{int}}(U_X)_1\xrightarrow{\ \sim\ } \mathcal U(\Sigma_X^\vee). \] It preserves the full lattice of coefficient characters and identifies each boundary theta function with $W_b$. These upper LP algebras are finitely generated, normal, and factorial. \end{introtheorem} We specify the mutable variables, Laurent coefficient ring, ordinary exchange polynomials, and associated exchange Laurent polynomials of each mirror seed. The half-spin formulas are uniform in $n$, while the two exceptional seeds are listed explicitly. The last column of the following table indicates where to find these descriptions. \begin{center} \begin{tabular}{@{}lrrrrl@{}} \toprule $X$ & $\dim\widehat X$ & Source rank & Mirror rank & Unit rank & Mirror seed\\ \midrule $D_n/P_n$ & $\binom n2+1$ & $\binom{n-2}{2}$ & $n(n-3)/2$ & $n+1$ & \S\ref{subsec:spinor-mirror-seed}\\ $E_6/P_6$ & $17$ & $5$ & $10$ & $7$ & \S\ref{subsubsec:e6-mirror-seed}\\ $E_7/P_7$ & $28$ & $10$ & $20$ & $8$ & \S\ref{subsubsec:e7-mirror-seed}\\ \bottomrule \end{tabular} \end{center} Here the unit rank means the rank of the unit group modulo constants; it is not the number of boundary components. For the Cayley-plane mirror, Proposition~\ref{e6mirror:thm:main} also proves that the generated LP algebra equals the upper LP algebra. The mirror comparison begins with regular functions on the source, whose tropicalizations bound the possible terms in the intrinsic Laurent expansions. Initial forms or contact calculations determine the local coefficients. We then use Laurent divisibility to recognize the remaining functions within their proved supports. Finally, the boundary functions and the logarithmic volume control the divisorial valuations of the intrinsic mirror. Normality identifies its coordinate ring with the corresponding upper LP algebra. The general steps are proved in the companion work \cite{ThetaBases}; here we verify their hypotheses in each minuscule family. In the half-spin family, the divisorial argument is uniform. With $m=n-1$, its superpotential in integral triangular coordinates is \[ W=y_{11} +\sum_{1\le a0, \] so each $x_i$ is prime. Factor a nonzero $f\in\mathcal U(t)$ in $L_0$ and normalize its irreducible factors. The result is \[ f=c\prod_i x_i^{\nu_i(f)}\prod_j(P_j^\circ)^{e_j}, \qquad c\in S^\times, \] with nonnegative exponents. This is a finite factorization into primes; no Noetherian hypothesis is required. Finally, a unit and its inverse have boundary order zero, so the mutable exponents in a Laurent unit must vanish. \end{proof} \subsection{Rank-one charts and extension across codimension two} Put $s_i=\prod_{j\ne i}x_j$ and $B_0=S[x_1,\ldots,x_r,x_1',\ldots,x_r']$. Localizing the finite intersection at $s_i$ gives \begin{equation} \label{eq:newton-rank-one} \mathcal U(t)[s_i^{-1}]=B_0[s_i^{-1}]=L_0\cap L_i =\frac{S[x_j^{\pm1}:j\ne i][x_i,x_i']} {(x_ix_i'-F_i/M_i)}. \end{equation} All other adjacent rings contain $L_0$ after this localization, and the last equality follows from \eqref{eq:newton-divisibility}. If $S$ is a finitely generated $\C$-algebra, the open set $X^\circ=\operatorname{Spec}B_0\setminus V(s_1,\ldots,s_r)$ is quasi-affine of finite type and has ring of global functions $\mathcal U(t)$. If $S$ is smooth over $\C$, these \emph{rank-one charts} are smooth whenever $F_i=0$ is smooth in $\operatorname{Spec}S[x_j^{\pm1}:j\ne i]$. For polynomial or Laurent polynomial coefficient rings, the rational logarithmic forms in the mutable and coefficient coordinates agree up to sign under mutation. \begin{proposition} \label{prop:newton-hartogs} Let $B_0\subseteq B\subseteq\mathcal U(t)$ be a finitely generated domain and let $J=(s_1,\ldots,s_r)B$. Then \[ \mathcal U(t)=\bigcap_iB[s_i^{-1}] =\bigcup_{m\ge0}(B:_{\operatorname{Frac}B}J^m). \] If $B$ is normal and $V(J)$ has codimension at least two, then $B=\mathcal U(t)$. More generally, $\operatorname{depth}_J B\ge2$ is sufficient. \end{proposition} \begin{proof} The localizations of $B_0$, $B$, and $\mathcal U(t)$ agree by \eqref{eq:newton-rank-one}. Clearing finitely many powers of the $s_i$ gives the saturation formula. A normal Noetherian domain is the intersection of its height-one localizations. If $V(J)$ has codimension at least two, all these localizations occur on the indicated open set. The depth assertion is the corresponding extension theorem. \end{proof} \begin{lemma}\label{lem:prime-divisor-upper} Let $R$ be a normal Noetherian domain containing $S$. Suppose that $x_1,\ldots,x_r$ are pairwise nonassociate prime elements of $R$, $R[(x_1\cdots x_r)^{-1}]=L_0$, and that each first mutation $x_i'$ lies in $R\setminus(x_i)$. Then $\mathcal A(t)\subseteq\mathcal U(t)\subseteq R$. If, moreover, $R\subseteq L_i$ for every $i$, then $\mathcal U(t)=R$. \end{lemma} \begin{proof} We show that every element of $L_0\cap\bigcap_iL_i$ is regular at all height-one primes of $R$. A prime containing none of the $x_i$ is covered by the initial localization. A height-one prime containing $x_i$ is $(x_i)$; the other $x_j$ and $x_i'$ are units there, so $L_i$ is contained in its local ring. Normality gives the inclusion in $R$. The Laurent phenomenon gives the first inclusion, and the starfish theorem gives equality once $R$ belongs to every adjacent ring. \end{proof} \begin{lemma}\label{lem:homogeneous-cone} Let $X=G/P$ be a positive-dimensional homogeneous variety with Picard group generated by the primitive ample line bundle $\mathcal O_X(1)$. Then its homogeneous coordinate ring $R_X$ is factorial, and every nonzero element of degree one is prime. \end{lemma} \begin{proof} The homogeneous embedding is projectively normal, and $R_X$ is generated in degree one; see \cite{Brion}. The punctured cone is $\mathcal O_X(-1)^\times$. Since the vertex has codimension at least two, localization along the zero section of the line bundle gives \[ \operatorname{Cl}(R_X)=\operatorname{Pic}(X)/\mathbb Z[\mathcal O_X(1)]=0. \] Thus $R_X$ is factorial. A nonzero degree-one element is irreducible in a positively graded domain with degree-zero part $\C$, and is therefore prime. \end{proof} \section{Half-spin Grassmannians} \label{sec:classical-spinor} Fix $n\ge4$, put $m=n-1$, and let \[ X=\operatorname{OG}^{+}(n,2n),\qquad R_X=\mathbb C[\widehat X]. \] The half-spin polarization $H$ is the primitive ample generator. Thus $\dim X=m(m+1)/2$ and $\dim R_X=m(m+1)/2+1$. We index the coordinates by subsets of $[m]$; the resulting boundary also gives the type-$B_m$ construction in Remark~\ref{rem:spinor-type-b}. \subsection{Signed principal Pfaffians and the interval seed} Let $p_I$, for even subsets $I\subseteq[m+1]$, denote the degree-one spinor coordinates. Our convention on the standard big cell is $p_I=t\operatorname{Pf}(B_I),\qquad p_{\varnothing}=t$, where $B$ is a skew-symmetric $(m+1)\times(m+1)$ matrix and principal submatrices use increasing index order. These coordinates generate $R_X$ and satisfy the quadratic \emph{Wick relations}; see \cite{Rincon}. For every $S\subseteq[m]$, define \[ S^+= \begin{cases}S,&|S|\text{ even},\\S\cup\{m+1\},&|S|\text{ odd},\end{cases} \qquad \epsilon(k)= \begin{cases}-1,&k\equiv0\pmod4,\\1,&\text{otherwise},\end{cases} \qquad q_S=\epsilon(|S|)p_{S^+}. \] This is a signed relabeling of all spinor coordinates. In particular, $q_{\varnothing}=-p_{\varnothing}$. Write \[ z=q_{\varnothing},\qquad X_{ij}=q_{[i,j]} \quad(1\leq i\leq j\leq m),\qquad X_{i,i-1}=z. \] We use $X_{i,i-1}=z$ whenever an empty interval occurs in an exchange. The frozen set and mutable set are \begin{equation} \label{eq:spin-frozen} \begin{aligned} \mathcal B_m&=\{z\}\cup\{X_{1j}:1\leq j\leq m\} \cup\{X_{im}:2\leq i\leq m\}, \\ \mathcal M_m&=\{X_{ij}:2\leq i\leq j\leq m-1\}. \end{aligned} \end{equation} We use the polynomial coefficient ring $S_m=\mathbb C[\mathcal B_m]$; the frozen variables are not inverted. Define \begin{equation} \label{eq:spin-singleton} F_{ii}=X_{i-1,i-1}X_{i,i+1}+zX_{i-1,i+1} +X_{i+1,i+1}X_{i-1,i}, \qquad 2\leq i\leq m-1, \end{equation} and, for $2\leq i7$, the same rank-two reduction is obtained by freezing. In the initial $D_n$ interval seed, $n\geq7$, keep exactly the ten mutable slots $\{X_{ij}:2\leq i\leq j\leq5\}$ and freeze all other mutable variables, whose right index is at least six. Each exchange at a variable we keep involves only intervals in $[6]$ and is exactly the corresponding $D_7$ exchange. The resulting coefficient ring is obtained from the $D_7$ polynomial coefficient ring by inverting some of its frozen coordinates and adjoining independent coefficient variables, some invertible. The initial normalizers are trivial, so no additional factor enters this identification. The six mutations \eqref{eq:spin-infinite-word} therefore give the same obstruction over this coefficient extension; monomial units do not change the two reciprocal degrees. A second application of freezing lifts the infinite rank-two path to the original $D_n$ seed. \end{proof} \section{The exceptional varieties} \label{sec:exceptional-source} The Cayley-plane and Freudenthal cones have dimensions $17$ and $28$. Their Coxeter boundaries contain $12$ and $18$ linear coordinates, respectively, so the mutable ranks are $5$ and $10$. We work over the polynomial rings generated by these frozen coordinates. \subsection{The Cayley plane} We keep the labels of Daisey--Ducat \cite[Section 5]{DD}. The $27$ degree-one coordinates split into Coxeter orbits \[ (a_i)_{i\in\mathbb Z/12},\qquad (x_i)_{i\in\mathbb Z/12},\qquad (z_i)_{i\in\mathbb Z/3}. \] Freeze the outer orbit $a_1,\ldots,a_{12}$. When a subscript on $z$ is an element of $\mathbb Z/12$, it is reduced modulo $3$. Set \[ y_i=x_i z_i-a_i x_{i-1}-a_{i+7}x_{i+1} \qquad (i\in\mathbb Z/12). \] In particular $y_3=x_3z_3-a_3x_2-a_{10}x_4$ is a degree-two coordinate function, and a convenient initial cluster is \[ \mathbf x_6=(x_1,x_2,x_3,x_4,y_3). \] Over $\mathbb C[a_1,\ldots,a_{12}]$ its ordinary exchange polynomials are \begin{equation} \label{eq:e6-seed} \begin{aligned} F_{x_1}&=y_3+a_{12}a_1,\\ F_{x_2}&=a_2x_1(y_3+a_{10}x_4) +a_9x_3(y_3+a_1a_{12}) +x_1x_3(a_7x_4+a_4a_{12}),\\ F_{x_3}&=y_3+a_3x_2+a_{10}x_4,\\ F_{x_4}&=a_{11}(y_3+a_3x_2) +x_3(a_4x_1+a_6x_2+a_1a_9),\\ F_{y_3}&=x_1x_4(a_5x_2+a_7x_3+a_2a_{10}) +a_{12}x_3(a_4x_1+a_1a_9)\\ &\hspace{8mm}+a_{12}x_2(a_6x_3+a_8x_4+a_3a_{11}). \end{aligned} \end{equation} These are the exchanges of \cite[Proposition~5.3]{DD}. Their associated Laurent polynomials satisfy \[ \widehat F_{x_2}=F_{x_2}/x_1,\qquad \widehat F_v=F_v\quad(v\ne x_2). \] Indeed, substituting $x_1=(y_3+a_1a_{12})/x_1'$ into $F_{x_2}$ introduces one common factor $y_3+a_1a_{12}$; the other normalization orders are zero. We recall the following result of Daisey and Ducat \cite[Theorem~5.4 and Corollary~5.5]{DD}. \begin{theorem} The seed \eqref{eq:e6-seed} gives the finite-type Cayley-plane LPA. It has $264$ seeds, up to the seed equivalence used in \cite{DD}, and $32$ mutable cluster variables. All exchange polynomials and all cluster-variable Laurent expansions have positive coefficients. \end{theorem} The count $32$ excludes the twelve frozen variables. Their degrees are as follows: the mutable variables consist of the fifteen linear coordinates $x_i,z_j$, twelve quadratics $y_i$, two cubics $u_1,u_2$, and three quartics $t_1,t_2,t_3$. Representatives of the last two families are \begin{align*} t_1&=x_1x_4x_7x_{10}-a_3a_6a_9a_{12},\\ u_1&=x_1x_5x_9-a_4a_5x_1-a_8a_9x_5-a_{12}a_1x_9 -a_1a_5a_9-a_4a_8a_{12}, \end{align*} with the other members obtained by the dihedral action. There are fifteen dihedral orbits of seeds: seven orbits of length $24$ and eight of length $12$. \subsection{The Freudenthal seed} \label{sec:primary-freudenthal-original} Let $R_7$ be the homogeneous coordinate ring of $E_7/P_7$, in the coordinate convention of \cite[Section~6.1]{DD}, reproduced in Appendix~\ref{app:e7-coordinate-convention}, with the upper signs in their equations (e)--(g). Subscripts of $a_i,x_i,y_i$ are read modulo $18$. We keep the Coxeter boundary $a_0,\ldots,a_{17}$ frozen over $S_7=\mathbb C[a_0,\ldots,a_{17}]$; none of these coefficients is inverted. We take the seven linear functions \[ (h,b,c,d,e,f,g)=(x_1,x_{11},x_9,x_{16},x_6,x_{14},x_4) \] and set \[ \begin{aligned} r&=x_4y_{15}-a_4x_{14},& q&=x_{11}y_0-a_{11}x_1,\\ u&=x_1x_{11}y_8-a_1x_{11}x_9-a_{11}x_1y_5-a_{12}x_1x_6. \end{aligned} \] The cluster $\mathbf x=(h,b,c,d,e,f,g,r,q,u)$ has degrees $(1,1,1,1,1,1,1,2,2,3)$. This choice gives the binomial mirror of Subsection~\ref{sec:e7-mirror}. To specify its exchanges, put \[ \begin{aligned} A_0&=g\{b(a_0f+a_1a_{14}+a_{15}h)+a_{11}a_{16}h\} +a_{17}b(a_4f+r),\\ A_1&=g\{b(a_0f+a_1a_{14}+a_{15}h)+h(a_{11}a_{16}+a_{12}d)\} +a_{17}a_4bf,\\ A_2&=a_3f+dg,\\ A_3&=a_1a_6c+a_2ce+a_4a_9h+a_5ch+a_8gh. \end{aligned} \] Define \[ F_h=a_1bc+u,\qquad F_b=a_{11}h+q,\qquad F_g=a_4f+r \] and \[ \begin{aligned} F_c&=a_9hF_g+a_8fgh+gu,\\ F_d&=A_0+a_{16}gq,\\ F_e&=a_6qF_h+h(a_6a_{11}+a_7b)(br+u) +bhq(a_4a_9+a_5c+a_8g),\\ F_f&=a_{14}gF_h+h\{cg(a_{11}a_{16}+a_{12}d+a_{15}b) +r(a_{13}c+a_{14}a_9)\},\\ F_r&=cA_2(eA_1+dgu)+dgq(fA_3+gu)+a_{16}a_3cefgq,\\ F_q&=eA_1+br(a_{17}e+dg)+dgu,\\ F_u&=F_r|_{u=0}+r\{dgh[f(a_{10}e+a_{11}a_6+a_7b) +e(a_{13}c+a_{14}a_9)]+a_{17}a_3bcef\}. \end{aligned} \] The associated Laurent exchange is $\widehat F_\xi=F_\xi/M_\xi$, where \[ \begin{array}{c|cccccccccc} \xi&h&b&c&d&e&f&g&r&q&u\\\hline M_\xi&1&1&g&bg&hb&h&1&1&g&dg. \end{array} \] \begin{theorem} \label{thm:primary-freudenthal-lpa} These functions and exchanges define a rank-ten LP seed $\Sigma_7$ over the polynomial coefficient ring $S_7$, with \[ \mathcal A(\Sigma_7)\subseteq\mathcal U(\Sigma_7)=R_7. \] The LP algebra has infinitely many cluster variables. \end{theorem} \begin{proof} Proposition~\ref{prop:primary-chart-change} identifies the seed with the auxiliary Freudenthal seed by five complete LP mutations. The seed axioms can also be checked directly: each exchange is primitive linear in a variable specified in Appendix~\ref{app:primary-comparisons}, and the normalization gives the table above, over $S_7$. For the auxiliary seed, Proposition~\ref{thm:e7-upper} computes the upper coordinate ring, and Proposition~\ref{thm:e7-infinite} gives an infinite rank-two mutation sequence. Both statements are invariant under mutation equivalence, so they hold for $\Sigma_7$. \end{proof} \begin{remark} This theorem concerns one infinite mutation class. It does not exclude a different positive finite-type Freudenthal LPA as in \cite[Conjecture~6.1]{DD}. \end{remark} %\clearpage \part{Algebraic mirrors and intrinsic comparison} \section{Newton reconstruction and intrinsic comparison} \label{sec:newton-program} For a boundary complement $U$ from Part~I, let $T$ be its initial torus, with character lattice $M_T$ and cocharacter lattice $N_T=\operatorname{Hom}(M_T,\mathbb Z)$. We write $\mathscr M=A_{\mathrm{int}}(U)_1$ and recall the results of \cite{ThetaBases} used in the mirror comparisons. All lattice changes are integral and include the coefficient directions. For a Laurent polynomial $f=\sum_a c_a z^a$, put \[ \Supp(f)=\{a:c_a\ne0\},\qquad \Newt(f)=\conv\Supp(f),\qquad \nu_w(f)=\min_{c_a\ne0}\langle a,w\rangle. \] We call $\nu_w$ the \emph{Gauss valuation} of weight $w$. A leading term has least weight in the specified chamber. The source $U$ is smooth and factorial. Indeed, the homogeneous coordinate ring is normal with class group $\operatorname{Pic}(X)/\mathbb Z[\mathcal O_X(1)]=0$, and inverting a frozen linear coordinate removes the cone vertex. Thus $\operatorname{Pic}(U)=0$. The \emph{normality theorem for the intrinsic mirror algebra} states that $\mathscr M$ is a finitely generated normal domain. It also provides a normal projective compactification $(\overline M,D_M)$ with reduced boundary, $K_{\overline M}+D_M\sim0$, and log canonical singularities. The theorem on the \emph{logarithmic volume of the intrinsic mirror} identifies the resulting rational volume, up to a nonzero scalar, with the standard logarithmic form on any algebraic torus open in $\operatorname{Spec}\mathscr M$. In particular, this form has order zero at every prime divisor of $\operatorname{Spec}\mathscr M$. The \emph{intrinsic Laurent expansion and trace theorem} constructs an injective algebra homomorphism from $\mathscr M$ to the Laurent ring of the dual torus when the local cylinder counts, boundedness of contact orders, and finiteness of the contributing curve classes satisfy its hypotheses. It identifies the images of the prescribed theta functions with their local expansions. We verify these hypotheses for each source; in particular, we obtain support bounds from regular source functions before using any proposed mirror formula. We also use \emph{restriction to a principal open subset}. If $0\ne f\in\mathcal O(U)$ and $U_f$ contains the chosen torus, the filtration \[ F_{\ge a}\mathscr M =\operatorname{Span}_{\mathbb C}\{\vartheta_m:f^{\mathrm{trop}}(m)\ge a\} \] is multiplicative, and its associated graded is $A_{\mathrm{int}}(U_f)_1$, with the corresponding theta labels unchanged. When $U_f$ is a torus $T_f$, this is the group algebra $\mathbb C[N_{T_f}]$, and the initial symbol of each theta function is its labelled monomial with coefficient one. All character changes include the frozen directions. We choose a basis of the saturated lattice generated by support differences and complete it to a basis of the source character lattice. The dual coordinates are $z_1,\ldots,z_r,\kappa_1,\ldots,\kappa_c$; the coefficient ring is $\mathscr S=\mathbb C[\kappa_1^{\pm1},\ldots,\kappa_c^{\pm1}]$. Compatibility with torus products gives \[ A_{\mathrm{int}}(V\times T)_1 \simeq A_{\mathrm{int}}(V)_1\otimes_{\mathbb C}\mathbb C[N_T], \] and sends each product-labelled theta function to the corresponding tensor product. This formula restores the torus factors after the comparison in the remaining coordinates. \subsection{Support bounds and coefficient recognition} Let $\mathsf N:\mathbb Z^s\hookrightarrow N_T$ be the contact map, with pointed correction monoid $\mathsf N\mathbb N^s$. For regular source functions $\psi_j$ with minimizing exponents $h_j$ in the chosen chamber, put \[ \delta_j(m)=\langle h_j,m\rangle- \min_{a\in\Supp\psi_j}\langle a,m\rangle. \] Writing $H$ for the matrix of the $h_j$, the regular-function slope inequalities give \[ d\ge0,\qquad H\mathsf N d\ge-\delta(m). \] A rational certificate $\lambda\ge0$ with $\lambda H\mathsf N\le-\epsilon\mathbf1$, $\epsilon>0$, gives $\sum_i d_i\le\lambda\delta(m)/\epsilon$. Conversely such a certificate exists whenever the recession cone is zero, by Farkas' lemma. We display the relevant source inequalities in each family rather than infer support bounds from candidate formulas. \begin{lemma} \label{lem:finite-curve-classes} Suppose the contact model verifies \cite[Assumption~20.27]{KY} for the pointed monoid $\mathsf N\mathbb N^s$. A finite set of allowed contact vectors involves only finitely many contributing numerical curve classes. \end{lemma} \begin{proof} Choose $k$ larger than the total degree of every allowed contact vector. All terms survive modulo the $k$th power of the monomial ideal generated by the nonzero elements of the contact monoid. Keel--Yu's Lemma~20.31 bounds the contributing curve classes in this truncation, hence in the entire expansion. \end{proof} We determine coefficients in two ways. For a mutable source coordinate $v$, let $g_v=v'\prod_{w\ne v}w$, where the product runs over the other mutable coordinates. The open set $U_{g_v}$ is the first-mutated source torus. Restriction to this principal open gives a valuation on $\mathscr M$ whose associated graded is a torus group algebra. Suppose $\Theta_m=\sum_{a\in A_m}c_a z^a$ has a proved finite support bound, and the initial symbols of the characters $z^a$ have coefficient one in that group algebra. If a unique $a_0\in A_m$ has valuation $g_v^{\mathrm{trop}}(m)$ and all other points have larger valuation, then $c_{a_0}=1$. This is the lemma on \emph{coefficient determination from an initial form} of \cite{ThetaBases}. We include the associated source normalizer when computing this valuation. The second method is \emph{finite Laurent interpolation} \cite{ThetaBases}. Once the adjacent mirror Laurent rings have been identified, write the function with a fixed monomial denominator and a numerator supported in a proved finite set. In the $i$th adjacent chart, every coefficient of $z_i^{-d}$ must be divisible by $\widehat F_i^d$ in the unchanged-variable Laurent ring. These are linear equations in the numerator coefficients. If they have a unique solution after fixing the leading coefficient, that solution is the theta function: the intrinsic expansion satisfies the same equations and normalization. We use this argument for the exceptional generators and boundary functions. Their coefficient matrices and nonzero maximal minors are included in the ancillary files. \subsection{Contact equations in toric blowup charts} \label{subsec:contact-interpolation} Let $U$ be smooth and contain a torus $T$ with character lattice $M_T$ and cocharacter lattice $N_T$. Suppose an open part of $U$ is obtained from toric ray charts by blowing up centres $F_a=0$ in their boundary divisors and removing the strict transforms of those divisors. We index the resulting coordinate divisors by $a\in I$ and write $n_a\in N_T$ for their primitive character-value vectors. Near a generic contact the chart has the form $x_aw_a=F_a$ with the other torus characters invertible and $F_a$ independent of $x_a$. Distinct divisors may have the same centre equation. These coordinate divisors lie inside $U$. Choose the compactification used for the intrinsic counts and a common open model for it and these toric blowup charts. The next proposition applies on the locus where the maps stay in this model away from their two marked ends. The applications verify the end conditions and that this locus contains the full counting fibre for generic evaluation. \begin{proposition} \label{prop:contact-interpolation} Fix two nonzero end directions, a contact vector $(b_a)_{a\in I}$, and a geometric evaluation point $c$ over a generic monomial point of $T$. Suppose the relevant counted maps have stable domains with three distinguished points, meet the coordinate divisors transversely at distinct points, and avoid the complement of the toric blowup charts away from their marked ends. Normalize the ends and evaluation to $0,\infty,1$ and write $m\in N_T$ for the incoming direction. Along each such map $f$, the characters satisfy \begin{equation} \label{eq:contact-interpolation} \chi(f(z))=\chi(c)z^{\langle\chi,m\rangle} \prod_{a\in I}\prod_{\alpha=1}^{b_a} \left(\frac{z-t_{a,\alpha}}{1-t_{a,\alpha}}\right)^{ \langle\chi,n_a\rangle},\qquad \chi\in M_T. \end{equation} The contact parameters satisfy the centre equations $F_a=0$ after substitution. For the two-ended local expansion, they lie in $0<|t_{a,\alpha}|<1$. The outgoing direction is $-m-\sum_a b_an_a$. Conversely, consider solutions of these equations with distinct contact parameters in the prescribed annuli. Assume that the interpolated maps lie in the common toric blowup model and satisfy the required open-stratum conditions at both ends and the toric-tail condition. On this locus, the incidence scheme and the geometric evaluation fibre agree. An isolated solution at which evaluation is \'etale contributes one. Parameters of the same divisor type are unordered. \end{proposition} \begin{proof} Work over a completed algebraic closure of the residue field of the chosen monomial point. Its character values have the prescribed valuations, without special algebraic cancellations. The smooth, stable-domain assertion of \cite[Lemmas 8.20--8.21]{KY} allows the normalization to a three-pointed $\mathbb P^1$ in the stated locus. The divisor of each character is determined by the two ends and the contacts. A rational function with that divisor is unique up to a constant, which evaluation at $1$ fixes. This proves \eqref{eq:contact-interpolation} and the balancing formula at infinity. Characters invertible on $U$ have no unmarked zeros or poles. At a simple contact of type $a$, the map lifts to $x_aw_a=F_a$ exactly when $F_a$ vanishes there. This is divisibility by a local parameter on the domain. The quotient is the value of $w_a$; a further condition $w_a\ne0$, when required, must be checked and is not implied by the centre equation alone. The prescribed leading coefficients at $0,\infty$ give the two marked boundary contacts. The toric-tail condition of \cite[Construction 9.3]{KY} requires the complementary tail to have no unmarked contact; for the normalized two-ended expansion it gives the stated open-unit-disk condition. Under the converse hypotheses these formulas construct a map to the chosen compactification with the prescribed contacts. The construction is reversible and works in families on this open locus, so it identifies the incidence scheme with the evaluation fibre there. A simple Jacobian over the evaluation parameters proves \'etaleness when used in the applications below. Such a point has length one. The three marked points remove automorphisms, and quotienting by permutations of contacts of the same type introduces no factorial multiplicity. A map has a uniquely determined curve class, so summing classes does not count it again. \end{proof} To apply the proposition to the full counting fibre, we must also exclude curves outside its stated locus. Once the maps are in the appropriate moduli space, \cite[Lemmas 3.9 and 8.21]{KY} allow generic evaluation to avoid fixed codimension-two loci and ensure generic transversality. Lemma~\ref{lem:finite-curve-classes} permits one generic choice for the finitely many relevant classes. We check the end-stratum and toric-tail conditions in each family. \subsection{From local expansions to the full mirror} \label{subsec:comparison-criterion} We apply the intrinsic Laurent expansion and trace theorem of \cite{ThetaBases}. After the primitive walls have been identified, consistency gives the adjacent expansions. The following elementary observation is used when a transformed expansion is initially formal. \begin{lemma}\label{lem:positive-denominator} Let $f$ be a nonnegative formal series in finitely many translates of a pointed monoid, and let $D$ be a nonnegative polynomial with positive constant term. If $Df$ is a finite Laurent polynomial, so is $f$. \end{lemma} \begin{proof} There is no cancellation, and the constant term of $D$ puts $\Supp f$ inside $\Supp(Df)$. \end{proof} We then use the \emph{intrinsic mirror comparison theorem} of \cite{ThetaBases}. For an LP seed with initial and adjacent Laurent rings $L_0,\ldots,L_r$, it is enough to establish \[ \mathscr S[x_i,x_i':1\le i\le r]\subseteq \mathscr R:=\Phi(\mathscr M) \subseteq\bigcap_{i=0}^rL_i \] and either recognize algebra generators of the intersection or prove that no height-one prime of $\mathscr R$ contains two distinct initial mutable coordinates. The Cayley-plane comparison uses generators. The half-spin and Freudenthal comparisons use divisors. For the latter method, let $w$ be the character-value vector of a normalized divisorial valuation of volume order zero, with $w\ne0$, and write $m_f(w)=\min_{a\in\Supp f}\langle a,w\rangle$. The \emph{divisorial valuation formula} of \cite{ThetaBases} gives an irreducible Laurent polynomial $q$ in the characters of value zero. More precisely, write $w=dw_0$, with $w_0$ primitive, and choose a character $t$ with $\langle t,w_0\rangle=1$. For a collected Laurent sum in $t$, \[ \nu\!\left(\sum_jt^j f_j\right) =\min_j\{dj+\operatorname{ord}_q(f_j)\}. \] Consequently, every regular $f$ with $m_f(w)<0$ satisfies $q^{-m_f(w)}\mid\operatorname{in}_w(f)$. It follows that $m_f(w)\ge-b_f$ when all face factors of $f$ have multiplicity at most $b_f$. \begin{corollary}\label{dnint:toric-face} If a negative minimizing face is irreducible up to a character and has multiplicity one, then $w$ is primitive, $m_f(w)=-1$, and $\nu(f)=0$. All such negative faces define the same spectator hypersurface. \end{corollary} \begin{proof} The divisibility forces $-m_f(w)=1$. In the exact slice formula its lowest slice has value zero, while every slice has nonnegative value. \end{proof} \section{Algebraic mirrors of the half-spin varieties} \label{sec:classical-mirror-seeds} \label{sec:classical-boundary} Fix $n\ge4$ and put \[ m=n-1,\qquad N=\binom n2+1,\qquad r_0=\binom{n-2}{2},\qquad r=\frac{n(n-3)}2. \] We use the interval seed of Section~\ref{sec:classical-spinor}, writing $h=q_\varnothing$, $X_{ij}=q_{[i,j]}$, and $X_{i,i-1}=h$. Let \[ \mathcal M=\{(i,j):2\le i\le j\le m-1\},\qquad \mathcal B_n=\{h,X_{11},\ldots,X_{1m},X_{2m},\ldots,X_{mm}\}. \] The mirror will have $r$ mutable coordinates, $n+1$ coefficient units, and $2n-2$ boundary functions indexed by $\mathcal B_n$. \subsection{An integral system of Newton coordinates} \label{dnmirror:sec:lattice} In the interval-coordinate torus define \begin{align} \chi_{B_i}&=\frac{X_{i+2,i+2}X_{i,i+1}} {X_{ii}X_{i+1,i+2}} &&(1\le i\le m-3),\label{dnmirror:eq:chiB}\\ \chi_{E_i}&=\frac{X_{i,m-2}X_{i-1,m}} {X_{i-1,m-2}X_{im}} &&(2\le i\le m-2),\notag\\ \chi_{R_{ij}}&=\frac{X_{ij}X_{i-1,j+1}X_{i+1,j+2}} {X_{i+1,j+1}X_{i-1,j}X_{i,j+2}} &&(2\le i\le j\le m-2),\notag\\ \chi_P&=\frac{hX_{m-2,m}}{X_{m-2,m-2}X_{m-1,m}}, &\chi_Q&=\frac{X_{m-2,m-1}X_{mm}} {X_{m-2,m-2}X_{m-1,m}}.\label{dnmirror:eq:chiPQ} \end{align} There are $r$ characters. Complete them by \begin{equation} \label{dnmirror:eq:gauge} \begin{split} g_0&=\begin{cases} X_{1,m-2},&n\text{ odd},\\ X_{2,m-1},&n\text{ even}, \end{cases}\\ (g_1,\ldots,g_n)&=(X_{1,m-1},X_{1m},X_{2m},\ldots,X_{mm}). \end{split} \end{equation} Let $C$ be the exponent matrix of $(\chi_\rho,g_j)$ in the original interval coordinates. The original mutable coordinates come first; the individual columns are denoted $Ce_a$, so the formulas do not depend on a further ordering convention. \begin{proposition}\label{dnmirror:prop:unimod} The characters in \eqref{dnmirror:eq:chiB}--\eqref{dnmirror:eq:gauge} form an integral basis. The $\chi_\rho$ generate the saturated support-difference lattice of the spinor-coordinate expansions. Its rank is $r=N-(n+1)$. \end{proposition} \begin{proof} We first prove the integral basis statement. Work in the quotient of the character lattice by the subgroup generated by the $\chi_\rho$. Write $x_{ij}$ for the additive class of $X_{ij}$, and put $P_j=x_{1j}$ and $S_i=x_{im}$, so $P_m=S_1$. For $m\ge4$, set \[ \Delta_{ij}=x_{ij}+x_{i-1,j+1}-x_{i-1,j}-x_{i,j+1}. \] The $R$ relations say $\Delta_{ij}=\Delta_{i+1,j+1}$, and the $E$ relations say $\Delta_{i,m-2}+\Delta_{i,m-1}=0$. Hence, with $s=\Delta_{22}$, $\Delta_{ij}=(-1)^{j-i}s$. Put $A_i=\ind_{i\,\mathrm{even}}$ and $D_j=\sum_{t=j}^{m-1}(-1)^t$. Summing the differences gives \[ x_{ij}=P_j+S_i-S_1+sA_iD_j\qquad(2\le i\le j\le m). \] The remaining $B$ relations become \[ P_{i+1}-P_i=S_{i+1}-S_{i+2}+\ind_{m\,\mathrm{even}}(-1)^i s. \] Writing $[h]$ for the class of $h$, the $Q$ and $P$ relations give \[ P_{m-1}-P_{m-2}+S_m-S_{m-1}-\ind_{m\,\mathrm{even}}s=0, \qquad [h]=P_{m-2}+S_{m-1}-S_1. \] For $m$ even, $s$ is determined integrally by $P_{m-2},P_{m-1}$ and the $S_i$. For $m$ odd, $P_{m-2}=P_{m-1}+S_m-S_{m-1}$, and $s=x_{2,m-1}-P_{m-1}-S_2+S_1$. These are exactly the complementary characters \eqref{dnmirror:eq:gauge}. These formulas express every initial character integrally in the complementary characters and impose no further relations among them. They therefore freely generate the quotient. Since the number of characters $\chi_\rho$ is the complementary rank, adjoining the complementary characters gives a unimodular basis. For $m=3$ the two characters in \eqref{dnmirror:eq:chiPQ} give the same conclusion directly. Each $\chi_\rho$ has homogeneous half-spin weight zero. The interval coordinates have weights $(1,\ind_{1\in I},\ldots,\ind_{m\in I},|I|\bmod2)$, whose rank is $m+2=n+1$. Since the lattice generated by the $\chi_\rho$ is saturated and has complementary rank, it is the kernel of this grading. Lemma~\ref{dnmirror:lem:sweep} shows that every $\chi_\rho$ occurs as a pure linear correction in a complementary-interval spinor coordinate. Consequently the spinor-coordinate support differences already generate this kernel. \end{proof} Write \[ z=(B_i,E_i,R_{ij},P,Q;\kappa_0,\ldots,\kappa_n) \] for the dual torus coordinates and put $\mathscr S_n=\mathbb C[\kappa_0^{\pm1},\ldots,\kappa_n^{\pm1}]$. The symbols $B_i,E_i,R_{ij},P,Q$ are mirror coordinates; the $\chi_\rho$ are characters on the original torus. For an initial basis vector $v$, the corresponding dual monomial is $z^{Cv}$. \subsection{The mirror seed} \label{subsec:spinor-mirror-seed} For $a=(i,j)\in\mathcal M$, set $c_a=Ce_a$ and define \begin{equation} \label{dnmirror:eq:Eab} \mathcal E_a=z^{[c_a]_+}+z^{[-c_a]_+}. \end{equation} Here $[c]_+$ denotes the coordinatewise positive part. The ordinary exchanges are \[ \begin{aligned} F_{B_i}^\vee&=\mathcal E_{i+1,i+1},& F_{R_{ij}}^\vee&=\mathcal E_{i,j+1},\\ F_{E_i}^\vee&=\mathcal E_{i,m-1},& F_P^\vee=F_Q^\vee&=\mathcal E_{m-1,m-1}. \end{aligned} \] For each $a$, let $\mathcal I_a$ be the set of positions assigned to $\mathcal E_a$. The sets with more than one element are \begin{equation} \label{dnmirror:eq:pairs} \{P,Q\},\qquad \{E_i,R_{i,m-2}\}\quad(2\le i\le m-2). \end{equation} For $Z_\rho\in\mathcal I_a$, the associated exchange and first mutation are \begin{equation} \label{dnmirror:eq:hat} \widehat F_\rho^\vee =\frac{\mathcal E_a}{\prod_{\sigma\in\mathcal I_a\setminus\{\rho\}}Z_\sigma}, \qquad Z_\rho'=Y_a:=\frac{\mathcal E_a}{\prod_{\sigma\in\mathcal I_a}Z_\sigma}. \end{equation} The $r$ mutable positions give $r_0$ distinct first-mutated functions. \begin{proposition}\label{dnmirror:thm:seed} Equations \eqref{dnmirror:eq:Eab}--\eqref{dnmirror:eq:hat} define an LP seed $\Sigma_n^\vee$ over $\mathscr S_n$. This seed is obtained by Newton reconstruction from the interval-Pfaffian seed. Its mutable rank is $r=n(n-3)/2$. \end{proposition} \begin{proof} Each $c_a$ is primitive because $C$ is unimodular. Hence $1+z^{c_a}$ is irreducible in the full Laurent ring, and clearing its coprime monomial denominator gives the irreducible polynomial $\mathcal E_a$. The assigned variable is absent since $C_{\rho,a}=0$ for $\rho\in\mathcal I_a$. Distinct columns give nonassociate binomials. In the LP normalization a different binomial cannot divide the relevant specialization: specializing an occurring variable to zero leaves a monomial, and specializing a nonoccurring variable leaves the same nonassociate binomial. For $\rho\in\mathcal I_a$, normalization removes one factor $Z_\sigma$ for each $\sigma\in\mathcal I_a\setminus\{\rho\}$. This proves \eqref{dnmirror:eq:hat}. For the reconstruction, the complementary-interval functions have mutable leading gradients $-e_a$. They give a bounded box for every correction label. At the adjacent label \begin{equation} \label{dnmirror:eq:mu} \mu_a=-\sum_{\rho\in\mathcal I_a}e_\rho+[-c_a]_+, \end{equation} the box contains only the corrections $0,e_a$; see Appendix~\ref{dnmirror:sec:reconstruction}. Thus the support is $\{\mu_a,\mu_a+c_a\}$. Laurent divisibility forces equal coefficients, which gives $Y_a=z^{\mu_a}(1+z^{c_a})$. \end{proof} For odd $n$, the exchanges contain no coefficient variables. For even $n$, only $\kappa_0$ occurs, in the exchange assigned to $(2,m-1)$. For $n=4$ this is the entire seed: \[ F_P^\vee=F_Q^\vee=1+\kappa_0,\qquad P'=Q'=\frac{1+\kappa_0}{PQ}. \] The polynomial $1+\kappa_0$ is irreducible and noninvertible in $\mathscr S_4$. For $D_5$, the five mutable variables are $B_1,E_2,R_{22},P,Q$, with \[ \begin{aligned} F_{B_1}^\vee&=E_2R_{22}+PQ,\\ F_{E_2}^\vee=F_{R_{22}}^\vee&=B_1+Q,\\ F_P^\vee=F_Q^\vee&=B_1+R_{22}. \end{aligned} \] These examples illustrate the repeated exchanges in the uniform formula. \subsection{The indexed boundary functions} Let $e_h,e_{ij}$ be the interval basis. Define \[ W_h=z^{Ce_h},\qquad W_{1m}=z^{Ce_{1m}}, \] and \begin{align} W_{1k}&=\sum_{t=0}^{m-k-1} z^{C(e_{1k}+\sum_{j=1}^{t}e_{j+1,j+k})} &&(1\le k0$. All $W_j$ belong to $\mathcal U(\Sigma_7^\vee)$. The character matrix is unimodular, and the twenty $\chi_i$ span the saturated support-difference lattice. \end{proposition} \begin{proof} The linear coordinates and the cubic mutation $f^\sharp=F_f/(hf)$ form a family of $57$ regular functions. The least monomial of \[ G=x_{15}^{\,2}x_2y_{14}x_5y_{11}x_{10}x_{13}x_0^{\,2}y_1^{\,3} \] has exponent $(-1,\ldots,-1)$ in the ten mutable coordinates. The regular-function inequalities therefore bound a contact correction $\beta\in\mathbb Z_{\ge0}^{10}$ by $\sum_i\beta_i\le\delta_G(m)$ at label $m$. Appendix~\ref{app:primary-newton} gives the finite support and adjacent Laurent-divisibility calculations. Each primitive facet has precisely the two allowed exponents occurring in its exchange. Every binomial exponent difference is primitive, and the two monomials are coprime. Hence the ordinary exchanges are irreducible. Normalization removes precisely one copy of each other variable having the same ordinary exchange: specializing a variable in a different binomial leaves either a monomial or a nonassociate binomial. This gives the stated normalizers. We determine the wall coefficients from valuations associated with source tori and the eight mixed-contact calculations in Proposition~\ref{prop:direct-facet-counts}. For every boundary label the regularity kernel is one-dimensional; normalizing the least coefficient gives exactly $W_j$. The starfish theorem \cite[Theorem~1.3]{Fei} puts these functions in the full upper LP algebra. The determinant and support-lattice calculations give the last assertion. \end{proof} \subsubsection{Intrinsic comparison} \begin{proposition}\label{prop:direct-facet-counts} The twenty primitive wall functions in the primary chamber are the binomials $E_i$ of \S\ref{sec:e7-mirror}, after removing their monomial factors. Every nonstraight term has coefficient one. \end{proposition} Six valuations from source tori determine the twelve single-contact directions. The five two-contact and three three-contact cases retain their explicit incidence equations. Both parts, including the required end conditions, are proved in Appendix~\ref{app:direct-contact-counts}. \begin{proposition} \label{prop:direct-rigidity} Suppose $w\in\mathbb Z_{\ge0}^{20}$ is the character-value vector of a normalized divisorial valuation with volume order zero, and all twenty first mutations $v_i^{\prime}$, together with the eighteen boundary functions $W_j$, are regular at that valuation. Then $w=0$ or $w=e_i$ for one $i$. \end{proposition} The finite step of the second proposition is computer-assisted. Its proof in Appendix~\ref{app:direct-global} derives rational inequalities for all eighty possible irreducible factors of negative Newton faces; these leave only zero and the coordinate vectors. \begin{theorem} \label{thm:e7-global-mirror} There is an isomorphism of $\mathscr S_7$-algebras \[ A_{\mathrm{int}}(U_{\mathrm F})_1\xrightarrow{\ \sim\ } \mathcal U(\Sigma_7^\vee) \] carrying the theta function indexed by $a_j=0$ to $W_j$ for every $0\le j\le17$. The upper LP algebra is finitely generated, normal, and factorial, and its unit group is $\mathscr S_7^\times$. Its superpotential is $W=\sum_{j=0}^{17}W_j$. \end{theorem} \begin{proof} The source theorem, full Newton marking and negative regular function of Proposition~\ref{prop:primary-newton} satisfy the Laurent-realization hypotheses of \cite[intrinsic Laurent expansion and trace theorem]{ThetaBases}. Proposition~\ref{prop:direct-facet-counts} identifies the twenty walls, and \S\ref{app:direct-intrinsic-functions} identifies their first mutations and the eighteen boundary theta functions. Thus the intrinsic image $\mathscr R_7$ satisfies \[ \mathscr S_7[v_i,v_i',W_j]\subseteq\mathscr R_7\subseteq\mathcal U(\Sigma_7^\vee). \] It is normal, and its initial torus has volume order zero, by \cite[normality and logarithmic volume of the intrinsic mirror]{ThetaBases}. Proposition~\ref{prop:direct-rigidity} shows that no prime divisor of $\mathscr R_7$ contains two initial mutable coordinates. Hence \cite[intrinsic mirror comparison theorem, codimension-one criterion]{ThetaBases} gives the asserted equality. The boundary functions were identified before this global step; finite generation, factoriality and the unit group now follow from the intrinsic and upper-algebra descriptions. \end{proof} \section{Integral forms and valuative bases} \label{sec:minuscule-theta-upgrade} We now descend the comparisons to the integral forms and apply the valuative-independence theorem to the intrinsic theta functions. \subsection{Integral descent} Choose primitive integer exchange polynomials throughout mutation and retain the independent Laurent coefficient characters. Let $\mathcal U_{X,\mathbb Z}^\vee$ be the upper LP algebra over $\mathbb Z[\kappa_j^{\pm1}]$, with $\zeta$ in place of $\kappa$ in the exceptional cases. Write $A_{\mathrm{int},\mathbb Z}(U_X)_1$ for the integral counting algebra at the specialization $q^\beta=1$. \begin{theorem} \label{thm:minuscule-integral} The isomorphisms of Theorem~\ref{thm:intro-mirror} restrict to \[ A_{\mathrm{int},\mathbb Z}(U_X)_1 \xrightarrow{\ \sim\ }\mathcal U_{X,\mathbb Z}^\vee. \] The intrinsic basis has nonnegative integral structure constants and nonnegative integral expansions in the marked initial chart. Every basis element is integrally Laurent in every LP seed. \end{theorem} \begin{proof} The finite contact bounds give finite initial expansions with integer coefficients and distinct coefficient-one least terms. Intrinsic multiplication has nonnegative integer coefficients. Apply the monic integral-descent theorem \cite[integral descent for monic Laurent expansions]{ThetaBases} to the complex comparisons of Theorems~\ref{dnint:global-comparison}, \ref{thm:e6-intrinsic}, and \ref{thm:e7-global-mirror}. Primitive exchange divisibility propagates integral Laurentness, and successive least terms recover the integral basis coefficients. This argument does not require scalar extension to commute with an infinite Laurent intersection. \end{proof} Integral Laurentness does not assert coefficientwise positivity in every seed. Nor does the theorem specialize the independent coefficient characters; only curve-class characters have been specialized. \subsection{Valuative independence of the intrinsic basis} \begin{theorem}\label{thm:minuscule-valuative-existence} \label{thm:minuscule-theta-criterion}\label{subsec:newton-minimality} In each comparison of Theorem~\ref{thm:intro-mirror}, the intrinsic theta basis is valuatively independent for every real monomial valuation in the marked initial chart and in every reachable LP seed. In particular, for every finite signed sum, \[ \Newt\!\left(\sum_m c_m\Theta_m\right) =\conv\bigcup_{c_m\ne0}\Newt(\Theta_m). \] Every regular extended cluster monomial spans an intrinsic theta line. Under the established integral and positive normalization, it equals the corresponding theta function. \end{theorem} \begin{proof} The source theorems and the regular negative-vertex functions verify the hypotheses of the valuative-independence theorem in \cite{ThetaBases}. The local realization in each comparison identifies its functions with the intrinsic theta functions; the full integral Newton marking only changes the character basis. The theorem therefore applies to these intrinsic functions. Exact Gauss transport through the exchange Laurent polynomials gives the assertion in every reachable seed. A regular seed monomial has singleton Newton support there, so valuative independence allows only one theta constituent. Integral descent and the specified positive normalization fix its scalar. \end{proof} The theorem applies to the entire mutation pattern. Positivity in every seed, and atomicity for the LP atlas, require additional arguments. \begin{corollary}\label{prop:newton-intersection} Let $P$ be a bounded rational polytope in $M_{\mathbb R}$, where $M$ is the full character lattice, and put \[ A_P=\mathcal U(\Sigma_X^\vee)\cap \operatorname{Span}_{\mathbb C}\{z^a:a\in P\cap M\}. \] Then \[ A_P=\bigoplus_{\Newt(\Theta_m)\subseteq P}\mathbb C\Theta_m. \] In particular the theta labels occurring in $A_P$ belong to the finite set $P\cap M$. Laurent-divisibility equations therefore give a finite recognition calculation once the corresponding support bounds are established. \end{corollary} \begin{proof} Apply the preceding theorem to a theta expansion of an element of $A_P$. Every nonzero constituent has Newton polytope contained in $P$. The converse inclusion is immediate; the label is a monomial in its theta function. \end{proof} For completeness, the familiar vertex ambiguity can be described without introducing a second basis. Fix the pointing order and put $P_m=\Newt(\Theta_m)$ and $P_m^-=\conv((P_m\cap M)\setminus\operatorname{Vert}(P_m))$. We set $A_{\varnothing}=0$. \begin{proposition}\label{prop:minuscule-rigid-window}\label{prop:newton-vertex} Every monic element with least label $m$ has Newton polytope containing $P_m$. The monic elements with least label $m$ and Newton polytope exactly $P_m$ form $\Theta_m+A_{P_m^-}$. Thus this representative is unique precisely when $A_{P_m^-}=0$. \end{proposition} \begin{proof} Pointing makes $\Theta_m$ the least constituent of any such expansion, with coefficient one. Newton containment proves the first assertion. If another constituent in $A_{P_m}$ had a vertex of $P_m$ in its support, a generic weight exposing that vertex would give it and $\Theta_m$ the same least exponent. Cancelling the two coefficients would contradict valuative independence. All other constituents are therefore supported in $P_m^-$. The converse follows because adding such an element changes neither the vertex coefficients nor the least label. \end{proof} \subsection{The quadric mirror} \begin{corollary}\label{cor:d4-theta-basis} The basis of Proposition~\ref{dnint:d4-model} is valuatively independent and Laurent-positive in every seed over the specified integral coefficient ring. Its elements are precisely the extended cluster monomials, and it is atomic for the LP atlas. If intrinsic scattering expansions are coefficientwise positive at every generic endpoint, it is also atomic for the intrinsic scattering atlas. \end{corollary} \begin{proof} For $m=(a,b;k)$, put $d=\max(0,-a,-b)$. The theta polytope is the segment from $m$ to $m+d\epsilon_0$, where $\epsilon_0$ is the $\kappa_0$ direction. A function in this Newton window is $\kappa^kP^aQ^bf(\kappa_0)$ with $\deg f\le d$. Adjacent Laurentness requires $(1+\kappa_0)^d\mid f$, so the window is one-dimensional. Theorem~\ref{thm:minuscule-theta-criterion} applies to every label. The clusters are $\{P,Q\}$, $\{P,R\}$, and $\{Q,R\}$. At least one exponent in $\kappa^kP^{a+d}Q^{b+d}R^d$ is zero, so every theta function is a cluster monomial. In a chart where it is monomial, that exponent cannot occur in another theta function: Newton containment transports to a chart where the latter is monomial and would force proportionality. Its coefficient therefore detects its basis coefficient. This proves LP atomicity. The same detectors work in the intrinsic charts; under the stated scattering-positivity hypothesis they also prove scattering atomicity. \end{proof} \section{Lie-theoretic structure} \label{sec:lie-theory} The mirror construction reflects the degree-and-weight lattice of the minuscule representation. In type $D$, its Newton polytope also encodes the minuscule root poset. These give two Lie-theoretic interpretations of the formulas. Neither identifies the entire mirror with a homogeneous space or an open Richardson variety. \subsection{The coefficient torus and the weight lattice} Let $G$ be simply connected, let $T_G\subset G$ be a maximal torus, and write $\ell=\operatorname{rank}G$. The degree grading and the $T_G$-action act together on $R_X$. Let $\Lambda$ be the character lattice of the initial torus of $U_X$, and let \[ \gamma_X:\Lambda\longrightarrow \mathbb Z\oplus X^*(T_G) \] send each initial coordinate to its homogeneous degree and its weight as a function. Put $\Gamma_X=\operatorname{im}(\gamma_X)$. Thus $\Gamma_X$ is the character lattice of the effective degree-and-weight torus acting on the cone. We use this image lattice, rather than the whole lattice $\mathbb Z\oplus X^*(T_G)$, to account for the finite central kernel of the action. Let $L_X\subset\Lambda$ be the saturated lattice generated by the support differences in the Newton construction. \begin{proposition}\label{prop:lie-coefficient-torus} For the three LP structures on the homogeneous coordinate rings in this paper, \[ L_X=\ker(\gamma_X),\qquad \Lambda/L_X\simeq\Gamma_X, \qquad \operatorname{rank}\Gamma_X=\ell+1. \] The mirror coefficient lattice is canonically $L_X^\perp=\operatorname{Hom}(\Gamma_X,\mathbb Z)$. Consequently the coefficient torus is dual to the effective degree-and-weight torus acting on the cone. \end{proposition} \begin{proof} The initial coordinates and the regular functions used to construct $L_X$ are homogeneous $T_G$-eigenvectors. Two monomials in the Laurent expansion of one such function have the same degree and weight. Hence $L_X\subseteq\ker(\gamma_X)$. The action of $\mathbb G_m\times T_G$ on the cone has finite kernel: a torus element acting as a scalar on the irreducible representation is central, and $G$ is simple. Since the initial coordinates generate the function field, their degree-and-weight characters have rank $\ell+1$. The Newton characters form part of an integral basis of $\Lambda$, and there are $\dim\widehat X-\ell-1$ of them; see Proposition~\ref{dnmirror:prop:unimod} and Appendices~\ref{app:e6-calculations}--\ref{app:primary-newton}. Hence $L_X$ is saturated and has the same rank as $\ker(\gamma_X)$. Their equality follows. Dualizing the resulting exact sequence gives \[ 0\longrightarrow\Gamma_X^* \longrightarrow\Lambda^* \longrightarrow L_X^*\longrightarrow0. \] The duals of the complementary characters of the initial torus are exactly the mirror coefficient characters, so they span $L_X^\perp$. \end{proof} The coefficient torus is isogenous to $\mathbb G_m\times T_G^\vee$, where $T_G^\vee$ is the dual torus. This explains the coefficient ranks $n+1$, $7$, and $8$ for $D_n$, $E_6$, and $E_7$. It also explains why the number of mirror units is not the number $h$ of chosen boundary components. The inclusion of the coefficient ring gives a morphism \[ \operatorname{Spec}\mathcal U(\Sigma_X^\vee) \longrightarrow\operatorname{Spec}\mathbb C[\Gamma_X^*]. \] The chosen lattice basis provides coordinates on its target. The character $\mathfrak q$ corresponds to cone scaling, and fixing it removes the one degree direction. These multiplicative parameters are not identified here with either the curve-class parameters of the intrinsic family or additive equivariant parameters in quantum cohomology. \subsection{The minuscule root poset in type $D$} For $X=\operatorname{OG}^+(n,2n)$, put $m=n-1$. The triangular poset \[ P_m=\{(a,b):1\le a\le b\le m\} \] with componentwise order is the minuscule poset for the half-spin representation. In the usual realization of the $D_n$ roots, its identification with the positive roots outside the Levi of $P_n$ is \[ (a,b)\longmapsto \beta_{ab}=\varepsilon_{n-b}+\varepsilon_{n-a+1}. \] The root order agrees with the poset order, and $\operatorname{ht}(\beta_{ab})=a+b-1=\mathbf h_{ab}$. Thus the horizontal and vertical ratios in the potential are indexed by covering relations in the minuscule root poset; the other two terms are indexed by its minimum and maximum. If $\Delta_m$ is the Newton polytope of the fixed-$\mathfrak q$ potential, Section~\ref{sec:classical-mirror-seeds} gives $\Delta_m^\circ=(2n-2)\mathcal O(P_m)-\mathbf h$. For the minuscule embedding, Spacek--Wang identify the order polytope with a Newton--Okounkov body and establish the corresponding Khovanskii-basis result \cite[Theorem 1.1]{SWChevalley}. In particular, \[ \#\bigl(d\mathcal O(P_m)\cap\mathbb Z^{P_m}\bigr) =\dim H^0(X,\mathcal O_X(d)). \] The polar relation therefore connects the mirror potential directly to the representation-theoretic polytope of the half-spin variety. This is a statement about the Newton polytope and a toric degeneration, not a presentation of the whole mirror as a flag variety. \subsection{Comparison with canonical homogeneous-space mirrors} Rietsch constructs Landau--Ginzburg models for $G/P$ using the Langlands dual group and an open Richardson variety \cite{Rietsch}. Spacek and Wang give canonical Pl\"ucker-coordinate models for all cominuscule spaces and identify them with Rietsch's models \cite{SWCanonical}; their exceptional and orthogonal constructions are given in \cite{SWExceptional,SWOrthogonal}. These results provide a natural comparison with our formulas. Here we study the complement of a specified Coxeter orbit, and the mirror theorem preserves its individual boundary labels. A comparison with a canonical homogeneous-space mirror must therefore identify both the affine domain and the potential. Agreement on a torus, equality of dimensions, or equality of coefficient ranks does not suffice. \begin{example}[The six-dimensional quadric]\label{ex:lie-quadric-difference} Even the underlying affine domains can differ. At $\mathfrak q=1$, the $D_4$ mirror in Section~\ref{dnint:global-completion-section} has coordinate ring \[ \frac{\mathbb C[\kappa_0^{\pm1},\kappa_1^{\pm1}, \kappa_2^{\pm1},\kappa_3^{\pm1},P,Q,R]} {(PQR-1-\kappa_0)}. \] Its units modulo constants are generated by the four $\kappa_i$. The morphism defined by these units has a nonsmooth fibre precisely over $\kappa_0=-1$, an irreducible hypersurface in the unit torus. The canonical mirror of $Q^6$ has coordinate ring \[ \frac{\mathbb C[A^{\pm1},B^{\pm1},C^{\pm1},D^{\pm1}, x,x',y,y']} {(xx'-C-D,\ yy'-AB-D)}; \] this is the specialization $m=4$, $p_0=1$ of the exchange relations in \cite[Section 5.2, equation (51)]{PRW}. Its units modulo constants are generated by $A,B,C,D$. Indeed, localizing at $x,y$ makes any unit a Laurent monomial; regularity of that unit and its inverse along $x=0$ and $y=0$ forces the exponents of $x,y$ to vanish. Its unit morphism has nonsmooth fibres over the two distinct irreducible hypersurfaces $C+D=0$ and $AB+D=0$. An isomorphism of affine varieties identifies their groups of units and hence these morphisms up to an automorphism and translation of the target torus. The two domains are therefore not isomorphic. This example does not exclude a birational comparison of potentials, but it rules out an automatic identification of their affine completions. \end{example} A uniform construction of our Coxeter-boundary mirrors from root subgroups, generalized minors, or another Lie-theoretic model remains a separate question. Proposition~\ref{prop:lie-coefficient-torus} and the root-poset description give intrinsic constraints that such a construction would have to respect. \appendix \phantomsection \addcontentsline{toc}{part}{Appendices} \section{Freudenthal coordinate ring and chart comparisons} \label{supp:aux-original} \label{app:primary-comparisons} We prove the auxiliary Freudenthal coordinate-ring statements used in Theorem~\ref{thm:primary-freudenthal-lpa}. In this appendix we suppress $\circ$ on the auxiliary variables $h,b,c,d,e,f,g,r,q,u$; in particular, $d=y_{10}$, whereas the primary seed uses $d=x_{16}$. The auxiliary seed is $\Sigma_7^\circ$. \subsection{Coordinate convention}\label{app:e7-coordinate-convention} We follow the upper-sign convention of \cite[Section~6.1]{DD}. Indices on $a_i,x_i,y_i$ are read modulo $18$; a Coxeter step shifts these indices and interchanges $z_1,z_2$. The defining ideal is generated by the cyclic translates of the eight quadrics: \begin{equation} \label{eq:e7-defining-quadrics} \begin{aligned} x_1x_2={}&a_3x_{17}+a_2y_0+a_1y_3+a_0x_4+a_{17}a_4,\\ y_{14}y_1={}&x_{15}y_4-a_0x_{10}-a_{15}x_5+x_0y_{11}+x_{13}x_2,\\ y_2y_1={}&a_2y_{12}+y_5x_0+y_{16}x_3+a_1y_9+a_5a_{16},\\ x_4y_1={}&x_2y_5+a_5x_{17}+a_4y_{16}-a_2y_8-a_1x_7,\\ x_2x_6={}&x_4y_4-a_4z_2-a_5y_0-a_3y_8+a_1a_7,\\ y_1y_3={}&x_{17}x_5+x_2z_1+a_0x_7+a_4x_{15}-a_2y_{11},\\ y_4y_{16}={}&y_1z_2-y_8x_0-y_{12}x_2-a_1x_{10}+a_5a_{15},\\ z_1z_2={}&y_3y_{12}+y_7y_{16}+y_8y_{17} +x_1x_{10}+x_3x_{12}+x_5x_{14}\\ &+a_0a_9+a_2a_{11}-a_3a_{12}+a_4a_{13}+a_6a_{15}. \end{aligned} \end{equation} The $144$ cyclic translates span a $133$-dimensional space of quadratic relations. Row reduction over $\mathbb Q$ gives this rank, which is unchanged over $\mathbb C$. The elimination data are in \nolinkurl{verification/freudenthal_quadric_rank_certificate.json}. \subsection{Coordinates and exchanges} \label{subsec:e7-source} Let $R_7=\mathbb C[\widehat{E_7/P_7}]$. Its $56$ linear coordinates are $a_i,x_i,y_i$ for $i\in\mathbb Z/18$ and $z_1,z_2$. The first three families are Coxeter orbits of length $18$, while $z_1,z_2$ form an orbit of length two. We freeze the $a_i$ and put $S_7=\mathbb C[a_0,\ldots,a_{17}]$. Define global homogeneous functions \begin{equation} \label{eq:e7-webs} \begin{aligned} r&=x_{17}y_{10}-a_{17}x_9,\\ q&=x_6y_{13}-a_6x_{14},\\ u&=x_{14}x_4y_7-a_{14}x_4x_6-a_3x_{14}x_9-a_4x_{14}y_{10}. \end{aligned} \end{equation} The initial cluster is \begin{equation} \label{eq:e7-source-cluster} (h,b,c,d,e,f,g,r,q,u) =(x_1,x_{11},x_9,y_{10},x_6,x_{14},x_4,r,q,u). \end{equation} Its degrees are $(1,1,1,1,1,1,1,2,2,3)$. \subsection{Exchange polynomials} Set \[ P=a_6a_{11}f+a_7bf+a_9a_{14}e+a_{10}ef+a_{12}de+a_{13}ce. \] The ten ordinary exchange polynomials are \begin{align*} F_h={}&efq\bigl(a_0dg+a_3r+a_{17}(a_3c+a_4d)\bigr)\\ &+df(er+u)(a_1a_6+a_2e)+a_1dq(u+a_{14}eg),\\ F_b={}&a_{11}(a_6f+q)+e(a_9a_{14}+a_{10}f+a_{12}d+a_{13}c),\\ F_c={}&fg(a_6a_{11}+a_7b+a_8d+a_{10}e) +a_9(a_4df+a_{14}eg+u),\\ F_d={}&r+a_{17}c,\\ F_e={}&q+a_6f,\\ F_f={}&u+a_{14}eg,\\ F_g={}&u+a_3cf+a_4df,\\ F_r={}&g(a_{16}d+a_{17}b)(hP+bu)+hdgq(a_{11}a_{16}+a_{15}b)\\ &+bq\bigl(dg(a_0f+a_1a_{14})+a_{17}f(a_3c+a_4d)+a_{17}u\bigr)\\ F_q={}&hP+b(er+u),\\ F_u={}&c\left.F_r\right|_{u=0} +br\left\{cf\bigl(d(a_1a_6+a_2e)+a_3q\bigr)+h\bigl(\left.F_c\right|_{u=0}+a_5cdf+a_{13}ceg\bigr)\right\}. \end{align*} In $F_u$, the notation $F_r|_{u=0}$ and $F_c|_{u=0}$ refers to the polynomials displayed above. The associated Laurent exchange polynomials are $\widehat F_v=F_v/M_v$, where \[ \begin{array}{c|cccccccccc} v&h&b&c&d&e&f&g&r&q&u\\\hline M_v&def&e&f&1&1&1&1&bg&1&bd\\ \end{array} \] Thus $v'=F_v/(vM_v)$; for example, $be\,b'=F_b$ and $rbg\,r'=F_r$. Each $F_v$ is independent of $v$ and has no mutable monomial factor. In the order $h,b,c,d,e,f,g,r,q,u$, the polynomials are primitive and linear in $a_0,a_{10},a_8,r,q,u,$ $u,a_{15},a_7,a_5$, respectively. The linear coefficients are monomials coprime to the constant terms, so Gauss's lemma proves irreducibility over $\mathbb C$. The LP normalization gives the monomials $M_v$ in the table. \subsection{The first mutations} With the same coordinate convention their formulas are \begin{align*} h'={}&x_2q-a_7(a_2x_{14}+a_1y_{13}) -(a_1a_6+a_2x_6)x_{12},\\ b'={}&x_{12},& c'={}&x_4x_8-a_3a_9,\\ d'={}&x_{17},&e'={}&y_{13},\\ f'={}&x_4y_7-a_4y_{10}-a_3x_9,&g'={}&x_{14}y_7-a_{14}x_6,\\ r'={}&x_{14}(y_2y_7-x_3y_{10}-x_1x_8+a_3x_{11}) -x_6(a_{13}x_1+a_{14}y_2),\\ q'={}&x_{11}y_0-a_{11}x_1,\\ u'={}&x_9(x_1y_{16}-a_2x_{14}-a_1y_{13})-a_9x_1x_{17}. \end{align*} Substitution of the Laurent coordinate expressions gives $vM_vv'=F_v$ in all ten directions. These identities also follow in $R_7$ from the injective Laurent chart constructed below. \subsection{The upper coordinate ring} We first establish the initial Laurent chart. Set \[ L_0=S_7[h^{\pm1},b^{\pm1},c^{\pm1},d^{\pm1},e^{\pm1},f^{\pm1},g^{\pm1},r^{\pm1},q^{\pm1},u^{\pm1}]. \] We compute Laurent expressions in $L_0$ for the $56$ coordinates from the cyclic quadratic relations. The complete expressions are in \nolinkurl{verification/freudenthal_source_coordinates.json}, and the adjacent-chart divisibilities are recorded in \nolinkurl{verification/freudenthal_source_divisibilities.json}. We check these identities over $\mathbb Q$. The first three noninitial coordinates are \[ x_{17}=\frac{r+a_{17}c}{d},\qquad y_{13}=\frac{q+a_6f}{e},\qquad y_7=\frac{u+a_{14}eg+a_3cf+a_4df}{fg}. \] The expressions satisfy the $144$ cyclic translates of the defining quadrics in \eqref{eq:e7-defining-quadrics}, which span the $133$-dimensional space of quadratic relations. They therefore define a homomorphism $R_7\to L_0$. The eighteen frozen coordinates, seven initial linear coordinates, and three functions in \eqref{eq:e7-webs} map to the $28$ independent indeterminates of $L_0$. The image has transcendence degree $28$, so the homomorphism is injective because $R_7$ is a $28$-dimensional domain. After inverting the ten mutable functions, we obtain $R_7[(hbcdefgrqu)^{-1}]=L_0$. Every denominator is a monomial in the ten mutable coordinates; in particular, every frozen exponent is nonnegative. We also verify all $204$ negative-slice divisibilities of \eqref{eq:newton-divisibility}, including the slices with poles of order two. Thus $R_7$ is contained in all ten adjacent Laurent rings. We now check the geometric hypotheses of Lemma~\ref{lem:prime-divisor-upper}. By Lemma~\ref{lem:homogeneous-cone}, $R_7$ is factorial and each initial degree-one variable is prime. For the higher-degree variables, we use a grading by degree and the seven Cartan weights. All $56$ degree-one weights are distinct. A factor of a homogeneous torus semi-invariant is again a homogeneous semi-invariant: the connected torus cannot nontrivially permute its finitely many irreducible factors. Thus a reducible degree-two or degree-three variable has a degree-one coordinate factor. Since $r,q,u$ are units in the initial Laurent ring, such a degree-one factor must be a unit there as well. The Laurent expressions show that the only degree-one coordinates invertible in $L_0$ are $h,b,c,d,e,f,g$; the frozen coordinates remain polynomial. For each such coordinate $v$, we test $r/v,q/v,u/v$ in the relevant complete degree-one or degree-two weight space of $R_7$. Every test is negative. The degree-two spaces used here are spanned by at most six coordinate products, and are complete because $R_7$ is generated in degree one. It follows that $r,q,u$ are irreducible, hence prime. Their degree-and-weight vectors are pairwise distinct, so the ten initial variables are nonassociate. The same graded calculation proves $v\nmid v'$ for all ten first mutations. For nine of them the quotient has a weight absent from the required homogeneous degree. The remaining quotient $h'/h$ is tested against its complete six-product quadratic weight space and does not belong to it. The required weight-space calculations are over $\mathbb Q$. Lemma~\ref{lem:prime-divisor-upper} now applies over $S_7$. \begin{proposition}\label{thm:e7-upper} The rank-ten seed \eqref{eq:e7-source-cluster} gives ${\mathcal A(\Sigma_7^\circ)\subseteq \mathcal U(\Sigma_7^\circ)=R_7.}$ Every cluster variable is regular on the Freudenthal cone. \end{proposition} \subsection{Infinite type} \begin{proposition}\label{thm:e7-infinite} The rank-ten Freudenthal seed \eqref{eq:e7-source-cluster} is of infinite type. More precisely, after mutating first at $h$ and then at $g$, freezing the other eight mutable variables leaves a rank-two LP pattern of exchange degrees $(1,4)$. \end{proposition} \begin{proof} Write $H=h'$ and $G=g'$ for the two new variables. These are new independent cluster coordinates; all the other original variables keep their names. Set \begin{align*} K&=u+a_3cf+a_4df,\\ \mathsf A&=(a_{16}d+a_{17}b)P+dq(a_{11}a_{16}+a_{15}b),\\ \mathsf B&=b\bigl((a_{16}d+a_{17}b)u+dq(a_0f+a_1a_{14})\bigr),\\ \mathsf C&=deq(a_0f+a_1a_{14}),\\ \mathsf D&=a_{17}efq(a_3c+a_4d) +du\bigl(f(a_1a_6+a_2e)+a_1q\bigr). \end{align*} The initial equations have the useful decompositions \[ \left.F_h\right|_{r=0}=g\mathsf C+\mathsf D, \qquad F_r=gh\mathsf A+g\mathsf B+a_{17}bqK. \] Since $\widehat F_h=F_h/(def)$, the first mutation gives \[ F_r^{(h)}=g^2\mathsf A\mathsf C +g(\mathsf A\mathsf D+defH\mathsf B) +a_{17}bdefqHK. \] In the resulting seed $\widehat F_g=F_g=K$. Substitution $g=K/G$, clearing $G^2$, and removal of the single common factor $K$ gives the ordinary exchange polynomial \begin{equation} \label{eq:e7-ranktwo-quartic} \mathcal Q(d)=K\mathsf A\mathsf C +G(\mathsf A\mathsf D+defH\mathsf B) +a_{17}bdefqHG^2. \end{equation} There is no further factor of $K$ in \eqref{eq:e7-ranktwo-quartic}. Indeed, set $u=-2$ and all other variables to $1$; then $K=0$, whereas $\mathcal Q=-57$. Both mutations leave $F_d=r+a_{17}c$ unchanged. In the final seed the associated exchanges in these two directions are \[ \widehat F_d=r+a_{17}c, \qquad \widehat F_r=\frac{\mathcal Q(d)}{befGq}. \] All the factors $b,e,f,G,q$ become coefficient units when the eight variables $H,b,c,e,f,G,q,u$ are frozen. In particular the active exchange degrees are unchanged. From \eqref{eq:e7-ranktwo-quartic} we obtain \begin{align*} [d^4]\mathcal Q &=a_4a_{12}a_{16}e^2fq(a_0f+a_1a_{14})\ne0,\\ [d^0]\mathcal Q &=a_3a_{17}^2bcefGq (a_6a_{11}f+a_7bf+a_9a_{14}e+a_{10}ef+a_{13}ce)\ne0. \end{align*} Hence the two degrees are exactly $(1,4)$. The freezing operation, including its coefficient-unit adjustments, is the one proved compatible with LP mutation in \cite[Proposition 3.7]{LP}. By \cite[Proposition 6.2 and Theorem 6.4]{LP}, the rank-two degree pair $(1,4)$ has infinitely many denominator vectors: its real-root system has Cartan matrix \[ \begin{pmatrix}2&-4\\-1&2\end{pmatrix}. \] These variables are also cluster variables of the original rank-ten pattern. This proves infinite type. \end{proof} Here \emph{infinite type} means infinitely many cluster variables; it does not assert infinite mutation type, which concerns the exchange data up to the usual equivalence. \begin{remark} The infinite-type assertion concerns the auxiliary seed \eqref{eq:e7-source-cluster}. It does not exclude a different positive finite-type rank-ten structure as in \cite[Conjecture 6.1]{DD}. Positivity throughout this infinite mutation class and mutation equivalence of the Coxeter-translated presentations are separate questions. \end{remark} \subsection{Comparison of the Freudenthal charts} \label{supp:comparison} To relate the auxiliary seed to the primary seed, start with the ordered cluster \[ \mathbf x^\circ=(x_1,x_{11},x_9,y_{10},x_6,x_{14},x_4,r^\circ,q^\circ,u^\circ), \] where \[ \begin{aligned} r^\circ&=x_{17}y_{10}-a_{17}x_9,& q^\circ&=x_6y_{13}-a_6x_{14},\\ u^\circ&=x_{14}x_4y_7-a_{14}x_4x_6-a_3x_{14}x_9-a_4x_{14}y_{10}. \end{aligned} \] Mutation words act from left to right, with positions kept fixed. \begin{proposition} \label{prop:primary-chart-change} The word $(4,9,8,10,4)$ in the auxiliary original order gives the primary seed of Theorem~\ref{thm:primary-freudenthal-lpa}, including its stated global polynomial coordinates. It fixes each frozen coordinate and identifies both the generated and upper LP algebras. \end{proposition} \begin{proof} Apply the five LP mutations over the original coefficient ring, using the associated exchange Laurent polynomials. Direct substitution, also in the reverse direction, identifies the final cluster with the polynomial functions $r,q,u$ of the primary seed. Irreducibility of the primary original exchanges can also be checked directly by primitive linearity. In the mutable order, one may use \[ a_1,a_{11},a_4,a_0,a_1,a_1,a_4,a_2,a_0,a_2. \] The leading coefficient is a monomial coprime to the constant coefficient in each case. All normalization tests keep the frozen variables polynomial. Mutation-equivalent seeds have the same complete mutation class, since LP mutation is involutive up to the fixed unit convention. They therefore have the same generated algebra and the same intersection of Laurent rings. The assertions about functions are substitutions in this common field. \end{proof} This argument concerns the original coordinate-ring seeds only. The mirror is reconstructed and compared with the intrinsic algebra directly in Appendix~\ref{app:primary-newton}; no mutation of an auxiliary mirror is used there. \section{Spinor mirror calculations} \label{app:spinor-calculations} We prove the recurrence and triangular-coordinate identities used in Sections~\ref{sec:classical-mirror-seeds} and \ref{dnint:global-completion-section}. Throughout, $m=n-1$ and the characters and the sets $\mathcal I_a$ are those of \eqref{dnmirror:eq:chiB}--\eqref{dnmirror:eq:pairs}. \subsection{Normalized cube recurrence}\label{dnmirror:sec:sweep} We normalize each interval exchange by its first summand. Extend the character notation by \[ \rho_{s,t}=\chi_{R_{st}}\quad(s\le t\le m-2),\qquad \rho_{s,s-1}=\chi_{B_{s-1}}\quad(s\le m-2),\qquad \rho_{m-1,m-2}=\chi_Q, \] and set $\xi_s=\chi_{E_s}$ for $2\le s\le m-2$ and $\xi_{m-1}=\chi_P$. Introduce the monomials \begin{equation} \label{dnint:V-ribbons} V_{i,m-1}=\xi_i,\qquad V_{ij}=\rho_{i,j-1}\rho_{ij}V_{i+1,j+1}\quad(j\le m-2). \end{equation} Empty indexed families are omitted, in particular when $n=4$. If $A_{ij}$ is the first summand of the interval exchange $F_{ij}$, then cancellation of monomials in the interval coordinates gives \begin{equation} \label{dnint:normalized-exchanges} \begin{split} N_{ii}:=F_{ii}/A_{ii}&=1+\rho_{i,i-1}+V_{ii},\\ N_{ij}:=F_{ij}/A_{ij}&=1+\rho_{i,j-1}+V_{ij} +\rho_{i,j-1}V_{i+1,j}\quad(ib+1$, both ribbons vanish on any single axis. The old factor $P_{a-1,b-1,c-1}$ and $P_{a-1,b-1,c}$ have the same assigned target $(d,b-1)$, or are both degenerate. Their axis restrictions therefore cancel, leaving exactly the restriction of $P_{a,b,c-1}$, whose target is $(d,b)$. If $c=b+1b+1}$. If $\lambda_i$ is the value of $M_i$, the three terms on the right of \eqref{dnint:cube-normalized} have values \[ \lambda_1-\varepsilon A-B,\qquad \lambda_2-A-B,\qquad -O-\varepsilon T. \] Cancellation along the horizontal ribbon gives \[ \lambda_2-A-B=-T+\mathbf1_{s=(d,c)} +\sum_{v=b+1}^{c-1}\mathbf1_{s=(d+1,v)} +\sum_{v=b}^{c-2}\mathbf1_{s=(d-1,v)}. \] For the other ribbon, every assigned negative node after its first is cancelled by a denominator contribution of the preceding node. The two off-ribbon nodes $(d-1,b-1)$ and, when $\varepsilon=1$, $(d+1,b)$ receive positive contributions from the first node. The terminal one-factor case has $\varepsilon=0$. Consequently $\lambda_1\ge-T+B+\varepsilon A$. If $\varepsilon=1$, the third term has value $-O-T$ and the others are no smaller. If $\varepsilon=0$, the second and third terms attain the same minimum $-O-T$ between them, since $O$ and $T$ cannot both be one. Subtracting the old value $-O$ proves the asserted new value $-T$. Subtraction-free tropicalization of the cube recurrence makes this an identity of Newton minima; coefficientwise polynomial positivity is not being assumed. \end{proof} \subsection{The adjacent supports} \label{dnmirror:sec:reconstruction} The complementary coordinate $\psi_t$ has mutable leading gradient $-e_t$. Its associated slope inequality is a coordinatewise bound $0\le d_t\le\delta_{\psi_t}(\eta)$ for a correction to $\eta$; see \eqref{dnint:slope}--\eqref{dnint:finite-box} below. Lemma~\ref{dnint:indicator} gives $\delta_{\psi_t}(\mu_s)=\mathbf1_{t=s}$. \begin{lemma}\label{dnmirror:thm:two} For $\eta=\mu_a$, impose \eqref{dnint:slope} for \emph{all} regular functions $f\in\mathcal O(U_n)$. The allowable corrections are exactly \[ {d=0\quad\text{or}\quad d=e_a.} \] The resulting mirror support is ${\{\mu_a,\mu_a+c_a\},}$ and its normalized coefficient-one function is \[ z^{\mu_a}(1+z^{c_a})=\frac{\mathcal E_a}{\prod_{\rho\in\cI_a}Z_\rho}. \] \end{lemma} \begin{proof} The complementary coordinates and Lemma~\ref{dnint:indicator} restrict the bounds \eqref{dnint:finite-box} to $\{0,e_a\}$. The zero correction always satisfies the inequalities. We show that no other regular function can remove $e_a$. If $(h_f)_a\ge0$, its inequality is automatic. Suppose $(h_f)_a=-k<0$. Adjacent-chart Laurent regularity says that the coefficient of $X_a^{-k}$ in $f$ is divisible by $F_a^k$ in the Laurent ring of the other interval coordinates. Let $A_a$ be the least monomial of $F_a$. For any $\rho\in\cI_a$, the monomial $A_a\chi_\rho$ is another term of $F_a$, by \eqref{dnmirror:eq:chiB}--\eqref{dnmirror:eq:chiPQ}. Newton polytopes of products add. Since least exponents add under multiplication, the divisible slice therefore has the following Newton-polytope point (which need not be an uncancelled support monomial): $h_f+k\chi_\rho$. Here $\chi_\rho$ denotes its exponent vector in the interval character lattice. Since $\langle\chi_\rho,\mu_a\rangle=-1$ in dual coordinates, $\delta_f(\mu_a)\ge k=-(h_f)_a$. Thus $e_a$ satisfies every regular-function inequality. Finally \eqref{dnmirror:eq:mu} gives the stated Laurent polynomial. \end{proof} \subsection{Finite expansions and primitive contacts} The initial and adjacent tori cover the codimension-one points of $U_n$: its initial coordinates are nonassociate primes, and the localized chart at $a$ is $X_aw_a=F_a$. Each centre $F_a=0$ is smooth on the spectator torus, since a corner coordinate occurs linearly with a monomial coefficient. The punctured spinor cone is smooth, and boundary inversion removes its vertex. The initial logarithmic volume form extends across every rank-one chart, since the exchange replaces its wedge by its negative. It is therefore a nowhere vanishing form on $U_n$. These facts verify the smooth affine log Calabi--Yau hypotheses for the intrinsic algebra. The common birational model with a toric compactification is the blowup of $(X_a,F_a)$ along each mutable-coordinate ray, with the strict transform of its toric boundary removed. By \cite[Lemma~8.21]{KY}, skeletal curves avoid the fixed codimension-two discrepancy locus. Thus every unmarked toric contact is over one of these centres and has direction $c_a$, with positive integral multiplicity. In particular all realized cylinder bends lie in \[ \Pi=C_{\rm mut}\N^{r_0},\qquad r_0=\binom{n-2}{2}. \] The columns are independent and saturated, so $\Pi$ is a free pointed toric monoid. This verifies \cite[Assumption~20.27]{KY} for the given blowup model, including its nonbinomial centres. For a regular function $f$ with unique minimizing exponent $h_f$ in the positive $\chi$ chamber, the principal-divisor slope formula \cite[Lemma~15.3 and Theorem~15.7]{KY} gives, for a correction $d$ to an incoming label $\eta$, \begin{equation} \label{dnint:slope} h_f^{\rm mut}d\ge-\delta_f(\eta),\qquad \delta_f(\eta)=\langle h_fC^{-1},\eta\rangle -\min_{u\in\Supp f}\langle uC^{-1},\eta\rangle. \end{equation} The complementary functions therefore give the coordinatewise bound \begin{equation} \label{dnint:finite-box} 0\le d_t\le\delta_{\psi_t}(\eta)\qquad(t\in\mathcal M). \end{equation} The same slope formula excludes a nonconstant attached tropical branch rooted in this chamber. Such a branch comes from an unmarked analytic disk with one boundary point and image in $U_n$. If its vector directed from the root into the disk is $C_{\rm mut}d$, a regular function has nonnegative slope there: the slope is its number of zeros in the disk. Applying the complementary functions gives $-d_t\ge0$ for every $t$. Hence $d=0$. This argument concerns realized analytic twigs and uses no copositivity assertion about arbitrary combinatorial trees. The free contact monoid and \eqref{dnint:finite-box}, together with Lemma~\ref{lem:finite-curve-classes}, give finitely many terms and curve classes before specialization to $q^\beta=1$. The absence of rooted twigs permits the two- and three-point gluing identities in the intrinsic Laurent expansion and trace theorem of \cite{ThetaBases}, which gives the injective algebra map \[ \Phi_n:\mathscr M_n\hookrightarrow \mathscr S_n[Z_\rho^{\pm1}], \qquad M_n=\Phi_n(\mathscr M_n). \] The zero-contact term is monic. For nonnegative mutable labels the box is zero, so the initial variables and both signs of the coefficient characters are straight theta functions in $M_n$. \subsection{All primitive facet walls and the adjacent variables} \begin{proposition}\label{dnint:facet-wall} At a generic point of the positive-chamber facet $\operatorname{val}(\chi_\rho)=0$, the intrinsic wall function is $1+z^{c_{a(\rho)}}$. Moreover $\Theta_{\mu_a}=Y_a$ for every $a\in\mathcal M$. \end{proposition} \begin{proof} Take a sufficiently small transverse primitive crossing with incoming direction $-e_\rho$. The axis restriction in Lemma~\ref{dnint:axis} is the entire local face polynomial of each normalized complementary function. Its gradient changes by one precisely for $t=a(\rho)$. The slope formula for the infinitesimal cylinder consequently gives $0\le d_t\le\mathbf1_{t=a(\rho)}$. Thus only the zero-contact configuration and one simple contact at $a=a(\rho)$ can occur. Tangency is automatic because $(c_a)_\rho=0$. We count the second configuration directly. Normalize the smooth three-marked domain at $0,\infty,1$ and write the torus evaluation as $c=(c_\sigma)$. With contact parameter $\tau$, its character map is \begin{equation} \label{dnint:one-contact-map} \chi_\sigma(w)=c_\sigma w^{-\delta_{\sigma\rho}} \left(\frac{w-\tau}{1-\tau}\right)^{(c_a)_\sigma}. \end{equation} The same formula in the full basis includes the gauge characters. Every exponent of $N_a$ pairs to zero with $c_a$, since the interval exchange is independent of $X_a$. The contact factors therefore cancel in the centre equation. By \eqref{dnint:normalized-exchanges}, $N_a=A(\chi_{\ne\rho})+\chi_\rho B(\chi_{\ne\rho})$ with $A(0)=B(0)=1$, and the incidence equation is \[ A(c_{\ne\rho})\tau+c_\rho B(c_{\ne\rho})=0. \] Give $c_\rho$ valuation $\delta>0$ sufficiently small relative to the generic positive valuations of the other $\chi$ coordinates. The unique solution has valuation $\delta$, and the derivative is a unit. Applying $C^{-1}$ to \eqref{dnint:one-contact-map}, any interval coordinate has the form \[ X_s(w)=X_s(c)w^{-(C^{-1}e_\rho)_s} \left(\frac{w-\tau}{1-\tau}\right)^{\delta_{sa}}. \] Thus only $X_a$ vanishes at the unmarked contact, and it has a simple zero. Moreover $N_a(w)=A+c_\rho B/w$ has a simple zero there. Consequently $F_a/X_a$ is nonzero at the contact, and the map extends through the rank-one chart. To identify the incidence solution with a counted cylinder, consider the annulus $0<\operatorname{val}(w)<2\delta$, the principal spine has its only bend at $\operatorname{val}(w)=\delta$, as required by \cite[Definition~20.2]{KY}; the tangential displacement of this root is absorbed in the generic evaluation point. Both exterior tails are toric. There is also a uniform toric-end check: the angular valuations at the two essential ends are $\operatorname{val}(c)$ and $\operatorname{val}(c)+\delta c_a$. All their $\chi$ entries are positive after shrinking $\delta$, because $(c_a)_\rho=0$. If a terminal ray bears another blowup centre, its normalized equation evaluates to $1+$ terms of positive valuation, so the terminal position avoids that centre. Thus both marked ends lie in open essential toric strata. Start with the partial toric fan consisting of the mutable-coordinate and terminal rays, and perform the specified centre blowups. Choose a smooth compactification preserving this partial model, in particular the open strata over all these rays. The simple incidence solution makes evaluation generically etale; \cite[Lemmas~3.6 and~3.9]{KY} then allow passage to a common resolution while avoiding its fixed codimension-two loci. This establishes the converse from the unique incidence solution to a counted cylinder. The toric zero-contact curve and this primitive simple contact each have count one. Hence the primitive wall function is $1+z^{c_a}$, with no lattice or covering multiplicity. Continuation across this wall, followed by the unimodular basis change replacing $e_\rho$ by $\mu_a$, is the first LP mutation. We record why it yields Laurent regularity. The rational transform of any finite initial theta polynomial has as its only nonmonomial denominator a power of $1+z^{c_a}$. The continued local theta series has nonnegative coefficients by its enumerative definition. If $Df$ is finite, $D$ is a positive polynomial with constant term one, and $f$ is a positive series in a translate of the pointed contact monoid, then $f$ is finite: every term of $f$ already occurs in $Df$, without cancellation. Thus every $\Theta_\eta$ is Laurent in the adjacent chart. Applying this to every position and the finite LP starfish theorem \cite[Theorem~1.3]{Fei} gives $M_n\subseteq\mathcal U_n^\vee$. Now use the \emph{full} indicator Lemma~\ref{dnint:indicator} in \eqref{dnint:finite-box}, with $\eta=\mu_a$. It restricts the theta support to $\{\mu_a,\mu_a+c_a\}$. Hence $\Theta_{\mu_a}=z^{\mu_a}(1+\lambda z^{c_a})$. In any adjacent position $\rho\in\mathcal I_a$ both terms have exponent $-1$ in $Z_\rho$. Laurent divisibility by $\mathcal E_a$ forces $\lambda=1$. This recognizes $Y_a$ without a second incidence argument for a full theta label; in particular it also applies when several variables share an exchange polynomial. \end{proof} \subsection{Boundary theta functions} \label{app:spinor-marked-recognition} The functions $W_b$ are defined in \eqref{eq:classical-prefix}--\eqref{eq:classical-suffix}. \begin{proof}[Proof of the boundary identities in Proposition~\ref{dnint:local-comparison}] For a distinct frozen coordinate $b$, the complementary coordinate $\psi_t$ has order zero along $b=0$. Indeed both are distinct degree-one primes in the spinor coordinate ring; inverting the other initial coordinates cannot create a $b$ factor. Therefore its least $b$ exponent in the initial Laurent chart is zero. The leading numerator of $\psi_{(u,v)}$ is $X_{1,v-u+1}X_{u,m}$, so \eqref{dnint:finite-box} restricts the corrections for $Ce_{1k}$ to zero-one choices on \[ a_1=(2,k+1),\ a_2=(3,k+2),\ldots,\ a_\ell=(m-k,m-1). \] For $Ce_{im}$ the corresponding chain is \[ a_1=(i,m-1),\ a_2=(i,m-2),\ldots,\ a_\ell=(i,i). \] The boxes for $h$ and $X_{1m}$ are zero. Let $d_1,\ldots,d_\ell\in\{0,1\}$ be the allowed correction coefficients along either chain. For $j\ge2$ apply \eqref{dnint:slope} to the regular mutation partner $F_{a_j}/X_{a_j}$. Its least monomial is the first summand of $F_{a_j}$ divided by $X_{a_j}$. On the chain its mutable gradient is exactly $e_{a_{j-1}}-e_{a_j}$, and its frozen defect at $Ce_b$ is zero. For a prefix chain this follows from the factor $X_{u-1,v-1}/X_{uv}$; for a suffix chain it follows from $X_{u,v+1}/X_{uv}$. The singleton exchange has the same restrictions at the marked end. The other factors are off the chain. Thus $d_1\ge d_2\ge\cdots\ge d_\ell$ The allowable supports are precisely the successive prefixes in \eqref{eq:classical-prefix}--\eqref{eq:classical-suffix}. Write their unknown coefficients as $\lambda_0,\ldots,\lambda_\ell$, with $\lambda_0=1$. For the edge assigned to $a_j=(u,v)$ choose the mirror position $\rho_{u,v-1}$: it is $R_{u,v-1}$ for $u0$, each has one coordinatewise least mutable exponent, with coefficient one and a fixed coefficient-unit exponent. Dividing by that term gives a polynomial in the $\chi_i$. They also hold for $f^\sharp=F_f/(hf)$. The $56$ expansions contain $6050$ terms in total and satisfy all $144$ cyclic translates of the defining quadrics. Let $H_\psi$ be the mutable exponent of the least term of one of these $57$ functions, and let $N_\psi$ be its normalized polynomial support in the $\chi_i$. For the mutable part $m\in\mathbb Z^{20}$ of a label, set $\delta_\psi(m)=-\min_{a\in N_\psi}\langle a,m\rangle$. A permitted correction $\beta\in\mathbb Z_{\ge0}^{10}$ satisfies $H_\psi\beta\ge-\delta_\psi(m)\quad\text{for all }\psi$. The product $G=x_{15}^{\,2}x_2y_{14}x_5y_{11}x_{10}x_{13}x_0^{\,2}y_1^{\,3}$ has mutable least exponent $(-1,\ldots,-1)$, yielding the finite bound $\sum_{j=1}^{10}\beta_j\le\delta_G(m)$ We enumerate the lattice points in this simplex and retain those satisfying all the inequalities. For the $i$th primitive facet, use instead the axis degrees \[ d_{\psi i}=\max\{a_i:a\in N_\psi,\ a_j=0\ (j\ne i)\}. \] Impose $H_\psi\beta\ge-d_{\psi i}$ and the tangency equation $\sum_j C_{ij}\beta_j=0$. The weighted sum obtained from $G$ again bounds $\sum_j\beta_j$. Besides zero, the unique permitted vector is listed below. The symbols $e_h,\ldots,e_u$ denote the standard basis in the ten original mutable directions. \[ \begin{array}{c|c@{\qquad}c|c} i&\beta&i&\beta\\\hline 1,5,15,18&e_h+e_u&2&e_f\\ 6,7,12,13,20&e_u&3&e_d\\ 11,14&e_g+e_r+e_q&4&e_g\\ 16,17,19&e_r&8&e_b+e_q\\ &&9&e_c\\ &&10&e_e+e_f+e_u. \end{array} \] Mapping these two-point supports by $C$ and clearing their monomial factors gives the ordinary mirror exchanges. Their coefficient-one normalization is proved in Proposition~\ref{prop:direct-facet-counts}. For the boundary label $Ce_{a_j}$, let $m_j$ be its first twenty coordinates; its last eight coordinates give $\epsilon_j$. The full normalized supports, not their axis restrictions, give: \[ \begin{array}{c|rrrrrrrrr} j&0&1&2&3&4&5&6&7&8\\\hline \delta_G(m_j)&0&0&0&3&3&2&3&0&2\\ \text{support size}&1&1&1&3&2&3&5&1&3\\[2pt] j&9&10&11&12&13&14&15&16&17\\\hline \delta_G(m_j)&0&0&0&2&1&3&6&2&0\\ \text{support size}&1&1&1&2&2&5&5&2&1. \end{array} \] For a Laurent polynomial $P=\sum_k v_i^kP_k$ supported in one of these sets, adjacent regularity is equivalent to $E_i^{-k}\mid P_k$ for $k<0$ in the Laurent ring of the unchanged coordinates. This gives a rational linear system. In every one of the eighteen cases its kernel is one-dimensional. Normalizing at the boundary label makes every coefficient one and yields precisely $W_j$. The normalizing monomials of the LP exchanges are units in these unchanged-coordinate Laurent rings and are included in the seed checks. \subsection{Intrinsic comparison in the primary chart} \label{app:e7-direct} We use the primary coordinates and Newton characters introduced above. Put $t_i=\chi_i$ and $N=C_{\{1,\ldots,20\},\{1,\ldots,10\}}$. Its columns $n_v$, in the order $v=(h,b,c,d,e,f,g,r,q,u)$, are the contact directions of the ten mutable coordinate divisors. These columns span a saturated direct summand: the remaining eight rows of $C$ are the original characters $a_0,\ldots,a_7$, so are zero in the mutable columns. The degree-and-weight grading makes every normalized coordinate expansion independent of $a_0,\ldots,a_7$. Since the character change is unimodular, those eight invertible characters split off in the localized coordinate ring. Thus $U_{\mathrm F}=X_7\times\mathbb G_m^8$ The formulas below concern $X_7$; we restore the coefficient torus by \cite[compatibility with torus products]{ThetaBases}. \subsubsection{The initial intrinsic chart} Normalize the ordinary exchange $F_v$ of the original seed by its coordinatewise least monomial in the characters $(t_i,a_0,\ldots,a_7)$. The quotient is a polynomial $P_v(t)$ with positive coefficients and constant term one. It is independent of the $a_0,\ldots,a_7$ and satisfies $\langle\alpha,n_v\rangle=0$ for each exponent $\alpha\in\Supp P_v$. Thus $P_v=0$ is the equation of the blowup centre on the toric divisor in direction $n_v$. For instance, \[ P_h=1+t_2t_3t_4t_5t_{10}t_{14}t_{16}t_{20},\qquad P_b=1+t_2t_4t_6t_8t_{10}t_{11}t_{17},\qquad P_g=1+t_4. \] This prescription and the original exchanges define all ten centres. They need not be expanded into their $87$ monomials in the argument. The coordinate-ring theorem and the logarithmic Jacobian identity for LP mutation give a volume-preserving initial torus. They identify the rank-one charts along the ten coordinate divisors with the corresponding toric blowup charts. Their centres are smooth on the relevant tori: each original exchange is primitive linear in an invertible coordinate with monomial leading coefficient. The charts cover codimension one. Resolving their birational identifications with a smooth compactification gives a common model; its discrepancy loci have codimension at least two. Let $G$ be the regular product in Proposition~\ref{prop:primary-newton}. Its mutable leading exponent is $-\mathbf1$. The regular-function slope inequalities therefore bound every contact vector at a fixed incoming label by a finite simplex. They also exclude a nonconstant branch rooted in the positive chamber: its outgoing contact vector would satisfy $-\mathbf1\cdot\beta\ge0$, so $\beta=0$; a branch with no contact is toric and constant. Every bend is a nonnegative combination of the independent $n_v$. Lemma~\ref{lem:finite-curve-classes} now bounds the complete numerical curve classes before specialization. The intrinsic Laurent expansion and trace theorem of \cite{ThetaBases} now gives finite theta expansions with distinct coefficient-one least terms and a faithful algebra map \[ \Phi:A_{\mathrm{int}}(X_7)_1\hookrightarrow \mathbb C[v_1^{\pm1},\ldots,v_{20}^{\pm1}]. \] This theorem includes the multiplication and trace comparison. The defects vanish on nonnegative coordinate labels, so their theta functions are the corresponding monomials. Inverting the $v_i$ therefore gives an open torus of the intrinsic spectrum. The next step is local coefficient recognition, before assuming any adjacent mirror chart. \subsubsection{Initial forms and the eight mixed-contact calculations} \label{app:direct-contact-counts} At facet $i$, the inequalities with the axis-degree bounds in Appendix~\ref{app:primary-newton} allow precisely the zero vector and the vector $\beta^{(i)}$ listed there. All coordinates of $\beta^{(i)}$ are zero or one. There are therefore twelve single-contact cases, five pairs, and three triples. \begin{lemma}\label{lem:e7-single-calibration} The twelve single-contact directions have coefficient-one nonstraight terms. They are determined by the following six first-mutated source tori: \[ \begin{array}{c|c|c|c} \text{mirror directions }i&\text{source variable }v& \text{least monomial of }F_v&\text{source normalizer}\\\hline 2&f&a_{14}gu&h\\ 3&d&a_{15}bgh&bg\\ 4&g&r&1\\ 9&c&a_8fgh&g\\ 6,7,12,13,20&u&a_{13}cdeghr&dg\\ 16,17,19&r&a_5cdfghq&1 \end{array} \] The last column records the associated normalizer $M_v$ in $v'=F_v/(vM_v)$; it is retained in the original source ring. \end{lemma} \begin{proof} Let $\mu_i$ be the actual first-mutant label and let $P_v$ be $F_v$ divided by its least character monomial. The source formulas and the full unimodular $28\times28$ matrix give \[ \Supp\Theta_{\mu_i}\subseteq\{\mu_i,\mu_i+Ce_v\}, \qquad \min_{\alpha\in\Supp P_v} \langle\alpha,\mu_i\rangle=-1. \] For $g_v=v'\prod_{w\ne v}w$, its least exponent $h_v$ satisfies $\langle h_v,Ce_v\rangle=-1$. The corrected term is therefore uniquely least at the required $g_v$-value, and \cite[coefficient determination from an initial form]{ThetaBases} gives its coefficient one. The transverse normal pairing is one, so this determines the wall. The exact exponents, including all eighteen frozen characters and the associated source normalizers, are checked in \nolinkurl{verification/e7_calibration.json}. \end{proof} For directions $1,5,8,10,11,14,15,18$, the correction term has strictly larger value than the straight term for each of these ten valuations. We determine its coefficient by the following contact calculations. We compute the remaining local counts over an algebraically closed, complete non-archimedean field of residue characteristic zero. Choose $\delta>0$ and generic evaluation coordinates $c_j$ with $\nu(c_i)=\delta$ and $\nu(c_j)>100\delta$ for $j\ne i$. These valuations lie in a sufficiently small transverse neighborhood of a generic point of the facet. We choose the angular components outside the finitely many loci of cancellation in the coefficient sums. Proposition~\ref{prop:contact-interpolation} gives the character map with evaluation $c$ at $1$: \begin{equation} \label{eq:primary-contact-map} t_j(z)=c_jz^{-\delta_{ij}} \prod_{v:\,\beta_v^{(i)}=1} \left(\frac{z-r_v}{1-r_v}\right)^{(n_v)_j}. \end{equation} At contact $v$ we impose $P_v(t(r_v))=0$. Its own factor drops out because $P_v$ is invariant in direction $n_v$. The infinitesimal spine has its bend at the chosen facet. Its unmarked contacts consequently have valuation $\delta$. We will in fact check that the incidence equations have exactly one solution in the larger region $0<\nu(r_v)<2\delta$ and that this solution has all valuations $\delta$. This excludes extra roots from the local count. We check the end and compactification conditions of Proposition~\ref{prop:contact-interpolation}. The centre equations extend a solution across its distinct contacts in the rank-one charts. The two essential-end directions are $-e_i$ and $e_i-N\beta^{(i)}$. Neither is a blowup-centre ray; this follows directly from the columns of $N$. Thus both ends extend to the indicated open toric strata. The simple incidence solutions below give a generically \'{e}tale evaluation map. The smooth-evaluation and avoidance results \cite[Lemmas 3.6, 3.9 and 8.21]{KY}, on the common compactification, then give trivial logarithmic tangent pullback and avoidance of the fixed codimension-two loci. The preceding curve-class bound permits one common generic evaluation choice. On the principal annulus the tropicalization is \[ q-\lambda e_i+\min(\lambda,\delta)N\beta^{(i)},\qquad q=(\nu(c_j))_j. \] Tangency $(N\beta^{(i)})_i=0$ puts the two cuts at $0,2\delta$ on opposite sides of the facet. There are no unmarked contacts on the two outside annuli. Thus the toric-tail condition holds, and deformation invariance identifies these reduced incidence solutions with the local cylinders of \cite[Definition 20.2]{KY}. All contact types here are distinct, so the unordered-contact convention in Proposition~\ref{prop:contact-interpolation} introduces no permutation quotient. \paragraph{Pairs.} The pairs are $(h,u)$ at $i=1,5,15,18$, and $(b,q)$ at $i=8$. Write $s,t$ for the first and second contacts. Grouping the two centre polynomials gives \[ 1+Ms^{-\epsilon}\frac{1-t}{s-t}=0, \qquad U+\left(V+\frac{W}{t}\right)\frac{1-s}{t-s}=0, \] where $\epsilon=1$ for $i=5,8$ and is zero otherwise. The coefficients have the form \[ V=L V_0,\qquad W=Lc_iW_0,\qquad M=\begin{cases}L\epsilon_0,&\epsilon=0,\\Lc_i\epsilon_0,&\epsilon=1,\end{cases} \] where $U,V_0,W_0$ have residue one and $L,\epsilon_0$ are monomials in the other evaluation coordinates, of valuation greater than $100\delta$. More precisely, \[ \begin{array}{c|c|c} i&L&\epsilon_0\\\hline 1,15,18&c_{14}c_{16}c_{20}c_3c_4&c_{10}c_2c_5\\ 5&c_{14}c_{16}c_{20}c_3c_4&c_{10}c_2\\ 8&c_{11}c_{17}c_4&c_{10}c_2c_6. \end{array} \] These factorizations follow by collecting the terms with the same powers of $c_i$ and the other contact factor in $P_v$. If $\epsilon=0$, elimination gives \[ s=(1+M)t-M,\qquad t=-\frac{(1+M)W}{UM+(1+M)V}. \] Both contacts have valuation $\delta$, their difference is nonzero, and this is the unique solution. If $\epsilon=1$, the second equation gives $s=\frac{Ut^2+Vt+W}{Ut+Vt+W}$. The first reduces, after removing the inadmissible factor $t-1$, to \[ UVt^3+\bigl(M(U+V)^2+UW+V^2\bigr)t^2 +2W\bigl(M(U+V)+V\bigr)t+(1+M)W^2=0. \] Writing $\ell=\nu(L)$, the coefficient valuations, from constant to cubic, are $2\ell+2\delta,2\ell+\delta,\ell+\delta,\ell$. The Newton polygon gives one simple root of valuation $\delta$ and two roots of valuation $(\ell+\delta)/2>2\delta$. The admissible root gives $\nu(s)=\delta$, and $s-t=-M(1-t)/s\ne0$. Thus each of the five pair cases contributes one. \paragraph{Triples of types $g,r,q$.} These occur at $i=11,14$. Put $a=c_4$, and use contacts $s,r,t$ of types $g,r,q$. The $g$-equation gives $s=(1+a)r-a$. Set $V=(r-t)/(1-t)$ and $\widetilde V=(t-r)/(1-r)$. The identities obtained by this substitution in the other two centres are \[ \begin{array}{c|c|c} i&r\text{-equation}&q\text{-equation}\\\hline 11&A+K(B+Y/r)/V=0&1+c_{11}a(B+Y/t)/(t\widetilde V)=0\\[1mm] 14&A(r)+K(r)(B+L/r)/V=0&1+c_{11}a(B+L/t)/\widetilde V=0. \end{array} \] For $i=11$, the quantities satisfy \[ A\equiv1,\qquad K=\frac{a(1+c_{14}(1+a))}{1+a},\qquad \nu(B)>100\delta,\quad \nu(Y/B)>100\delta. \] For $i=14$, they satisfy \[ A(r)=A_0+T/r,\quad A_0\equiv1,\quad K(r)=\frac{a}{1+a}\left(1+\frac{(1+a)c_{14}}r\right), \] with $\nu(B),\nu(L/B),\nu(T)>100\delta$. Here congruences mean equality in the residue field. The coefficients are defined by regrouping the normalized source exchange polynomials as in the table. The coefficient groups are obtained directly from the source exchange polynomials, with all eighteen frozen variables independent; see \nolinkurl{verification/e7_calibration.json}. The valuation estimates use their monomial factors and generic nonzero leading coefficients. For both facets, let $H(r)$ be the nonconstant term of the $r$-equation before division by $V$, divided by $A$ or $A(r)$. Then $t=\frac{r+H(r)}{1+H(r)}$. Throughout $0<\nu(r)<2\delta$, we have $\nu(H(r))>2\delta$. Thus $t$ and $s$ have the same valuation as $r$. At $i=11$, the $q$-equation becomes \[ t=-c_{11}\frac{aA}{K}(1+H) \frac{B+Y/t}{B+Y/r}. \] The factor after $-c_{11}$ has residue one; hence $\nu(r)=\delta$. After writing $r=c_{11}R$, the equation reduces to $R+1=0$ modulo the maximal ideal. All its remaining coefficients, and their first derivatives on the unit annulus, have positive valuation. Hensel's lemma gives exactly one solution. At $i=14$, the equation instead becomes \[ 1+\frac{(1+a)c_{14}}r =-c_{11}(1+a)A(r)(1+H) \frac{B+L/t}{B+L/r}. \] Its right side has positive valuation, so again $\nu(r)=\delta$. Writing $r=c_{14}R$ reduces it to $R+1=0$, with the same strict Hensel condition. Thus there is precisely one solution. In both cases $s-r=-a(1-r)\ne0$. The other gap $t-r$ is nonzero and has strictly larger valuation, so the three contacts are distinct. The elimination and the simple Hensel root also prove reducedness. \paragraph{The triple $e,f,u$.} This occurs at $i=10$. Set $z=c_{10}$, $a=c_2$, and let $s,t,w$ be the three contacts. Exact grouping of the centre polynomials gives \[ \begin{aligned} &(s+zaZ)(s-w)+zaA(1-w)=0,\\ &t(t-w)+a(B+zY/t)t(1-w)=0,\\ &B+zY/w+(C+zLY/w)V+A V/U+D V^2=0, \end{aligned} \] where $U=(w-s)/(1-s)$ and $V=(w-t)/(1-t)$. Here $B$ has residue one, and $a,A,Y,Z,C,D,L$ have valuation greater than $100\delta$. Also $\nu(Z/A)>100\delta$: the polynomial $A$ contains $c_{13}+c_6$, whereas $Z=c_{11}c_{17}c_4c_8(c_{13}+c_6)$. The exact coefficients are obtained as follows: the terms of $P_e$ are $1+zaZ/s+zaA/(sV_e)$, the terms of $P_f$ are $1+a(B+zY/t)/V_f$, where $V_e=(s-w)/(1-w)$ and $V_f=(t-w)/(1-w)$. The six coefficient groups of $P_u$ are the six terms in the last equation above. This determines the notation without expanding all the centres. For $0<\nu(w)<2\delta$, the first two equations each have one solution in that annulus, with $s=w+\text{terms of valuation greater than }2\delta$ and the same assertion for $t$. Their other quadratic roots have valuation greater than $2\delta$. Moreover \[ U=\frac{zaA}{s+za(Z+A)},\qquad V=\frac{a(B+zY/t)}{1+a(B+zY/t)}. \] Substitution in the third equation gives $B(1+w/z)$ plus terms of strictly larger valuation. Thus any solution in the annulus has $\nu(w)=\delta$. For existence and simplicity, put \[ w=zW,\qquad s=w+aA S,\qquad t=w+a T. \] The first two equations, divided by $zaA$ and $at$, and the third multiplied by $S$, reduce to \[ WS+1=0,\qquad T+1=0,\qquad S+T=0. \] All omitted coefficients have positive valuation, and the denominators are units at $(W,S,T)=(-1,1,-1)$. The reduced Jacobian has determinant $-1$. Multivariate Hensel lifting therefore gives one simple solution. Conversely, the first two equations force $S,T$ to be units for every admissible solution, so this is exhaustive. Finally, $\nu(s-w)=\nu(a)+\nu(A)>\nu(a)=\nu(t-w)$, and the contacts are distinct. \begin{proof}[Proof of Proposition~\ref{prop:direct-facet-counts}] The support bounds exhaust all possibilities. Lemma~\ref{lem:e7-single-calibration} settles the twelve single-contact directions. The five pair and three triple calculations above give one reduced solution each; their interpretation by \eqref{eq:primary-contact-map} and the toric-tail condition gives the remaining local cylinder coefficients. \end{proof} \subsubsection{The first mutations and boundary functions} \label{app:direct-intrinsic-functions} By wall-crossing consistency \cite[Proposition 20.35]{KY}, the initial intrinsic expansions continue by these binomial wall transformations. The positive-series argument of Lemma~\ref{lem:positive-denominator} proves that the continuations are Laurent polynomials. The image of $\Phi$ is consequently contained in the twenty-one-chart intersection, which is the upper LP algebra. Put $b_i=N\beta^{(i)}$, and let $M_i$ be the normalizing monomial of the $i$th mirror exchange. Write $\mathbf m_i$ for its exponent vector. The first-mutation label is $m_i'=-[b_i]_- -\mathbf m_i-e_i$, where $[b]_-$ denotes the coordinatewise minimum of $b$ and zero. The full support inequalities at $m_i'$ have exactly two solutions: $0$ and $\beta^{(i)}$. The bound on the sum of their entries is $1,2$, or $3$, according to the number of contacts in $\beta^{(i)}$. Adjacent Laurent divisibility has rank one on this two-dimensional monomial space and fixes the second coefficient to one. Thus every first mutation $v_i'$ is itself a theta function in this chart. There is no additional correction term. For the eighteen boundary labels, the complete bounds and one-dimensional regularity spaces are given in Appendix~\ref{app:primary-newton}. The same reasoning identifies each intrinsic boundary function with the stated $W_j$. There is a particularly simple check of uniqueness. Every negative coordinate exponent is $-1$. In each binomial-divisibility test, the terms group in pairs differing by an odd multiple of the exchange exponent difference, so their coefficients must agree. The resulting graph on the allowed monomials is connected for every boundary function. Thus all coefficients are forced to equal the straight coefficient one. The eighteen coefficient matrices, normalized kernels, and maximal minors equal to $\pm1$ are given in \nolinkurl{verification/e7_boundary_certificates.json}. We use the full normalized supports of the $57$ regular source functions. The source inequalities bound the supports, and the Laurent divisibility equations determine their coefficients over $\mathbb Q$. Restoring the coefficient torus, let $M$ be the intrinsic image. We have \[ \mathscr S_7[v_i,v_i',W_j]\subseteq M\subseteq\mathcal U(\Sigma_7^\vee). \] Inverting the product of all $v_i$ gives the initial torus. With $d_i=\prod_{j\ne i}v_j$, localization of the starfish intersection gives \[ M[d_i^{-1}] =\frac{\mathscr S_7[v_j^{\pm1}:j\ne i][v_i,V_i]}{(v_iV_i-E_i)}, \] where $V_i$ is the monomial multiple of $v_i'$ that clears its normalizer. These are actual intrinsic open sets. The remaining problem is whether they meet every prime divisor. \subsubsection{A linear test for all prime divisors} \label{app:direct-global} Write \[ \mathscr F=\{v'_1,\ldots,v'_{20},\overline W_0,\ldots,\overline W_{17}\}, \] where the coefficient unit has been removed from $W_j$. There are $24$ distinct functions in this list. They involve only the mutable coordinates and have at most five terms each. The factors remain irreducible after adjoining the eight independent coefficient characters, on which the valuations vanish. The logarithmic form then includes their eight $d\log$ factors, while the inequalities still involve only the twenty mutable values. For a homogeneous face factor $q=A v_p+B$, with $A=cv^a$ a monomial, choose $\alpha\in\Supp q$. For each $f=v^\ell P(v)\in\mathscr F$, let $d=\deg_{v_p}P$ and collect \[ A^dP\bigl(v_p=(u-B)/A\bigr)=\sum_{j\ge0}u^jR_j(v_{\ne p}). \] By \cite[divisorial inequalities for a linear centre]{ThetaBases}, regularity forces \begin{equation} \label{eq:direct-linear-inequality} \langle\ell+e-da+j\alpha,w\rangle\ge-j \qquad(e\in\Supp R_j). \end{equation} This necessary condition follows from the divisorial valuation formula, with coefficients collected before substitution. We apply it to the following finite list of face factors. \begin{proof}[Proof of Proposition~\ref{prop:direct-rigidity}] We use the ordinary exchanges, their associated normalizers, and the eighteen boundary functions from Section~\ref{sec:e7-mirror}. First, if every monomial in every $f\in\mathscr F$ has nonnegative value, a nonnegative rational combination of these inequalities gives $\sum_iw_i\le0$. Hence $w=0$. Otherwise \cite[divisorial valuation formula]{ThetaBases} gives an irreducible factor $q$ of a negative face. To include every possible face, take \emph{all} nonempty subsets of each support, rather than finding the normal fans. After removing monomial factors, the $100$ nonmonomial subset checks give $80$ distinct irreducible factors. Each is primitive linear in some $v_p$, with monomial leading coefficient coprime to its constant coefficient. This proves absolute irreducibility and gives the hypothesis of \cite[divisorial inequalities for a linear centre]{ThetaBases}. Every irreducible factor of a weighted-homogeneous Laurent polynomial is weighted-homogeneous: the connected grading torus cannot permute its finite set of prime factors. For each $q$, we therefore impose the homogeneity equations $\langle\alpha-\alpha',w\rangle=0$ for its exponents, and impose \eqref{eq:direct-linear-inequality} for every $f\in\mathscr F$. The resulting rational inequalities give \[ \sum_iw_i\le \begin{cases} 0,&\text{for seventy factors},\\ 1,&\text{for the other ten}. \end{cases} \] The ten latter factors are exactly the ten distinct mirror exchanges. Write the inequalities as $A_qw\ge b_q$ and the homogeneity equations as $H_qw=0$. Each certificate supplies rational vectors $\lambda_q\ge0$ and $\mu_q$ with \[ \lambda_q^tA_q+\mu_q^tH_q\le-(1,\ldots,1),\qquad -\lambda_q^tb_q\in\{0,1\}. \] Since $w\ge0$, this gives exactly the asserted bound. The file \nolinkurl{verification/e7_divisor_certificates.json} contains all matrices and rational multipliers. The ancillary calculation constructs the factors and substitution matrices from the stated Laurent functions, then verifies each identity in rational arithmetic. Since $w$ is integral, the resulting bounds leave only $w=0$ or a coordinate vector. \end{proof} \subsection*{Ancillary calculations} The accompanying files give the Cayley-plane coefficient systems, the Freudenthal boundary systems, the initial-form calculations, the $56$ source-coordinate expansions, and the divisorial inequalities. They also contain exact checks of the $144$ defining quadrics, the first mutations, the $204$ adjacent-chart divisibility conditions, and the primality calculations. The conventions and verification instructions are given in \nolinkurl{verification/README.md}. \section*{Declaration of generative AI and AI-assisted technologies} During the development of the paper, the author used ChatGPT to assist in exploring and formulating most proof arguments. The author subsequently checked, revised, and independently verified all arguments and takes full responsibility for the mathematical content of the paper. \begin{thebibliography}{99} \raggedright \bibitem{Brion} M.~Brion, \emph{Lectures on the geometry of flag varieties}, in \emph{Topics in Cohomological Studies of Algebraic Varieties}, P.~Pragacz (ed.), Trends Math., Birkh\"auser, Basel, 2005, 33--85; \href{https://arxiv.org/abs/math/0410240}{arXiv:math/0410240}. \bibitem{CMP} P.-E.~Chaput, L.~Manivel, and N.~Perrin, \emph{Quantum cohomology of minuscule homogeneous spaces}, Transform. Groups \textbf{13} (2008), no.~1, 47--89; \href{https://arxiv.org/abs/math/0607492v2}{arXiv:math/0607492v2}. \bibitem{CMMM} M.-W.~Cheung, T.~Magee, T.~Mandel, and G.~Muller, \emph{Valuative independence and cluster theta reciprocity}, preprint (2025), version~2 (2026), \href{https://arxiv.org/abs/2505.09585v2}{arXiv:2505.09585v2}. \bibitem{DD} O.~Daisey and T.~Ducat, \emph{A Laurent phenomenon for the Cayley plane}, SIGMA \textbf{20} (2024), Paper No.~033, 18~pp.; \href{https://arxiv.org/abs/2310.10223v3}{arXiv:2310.10223v3}. \bibitem{Eriksson} H.~Eriksson and K.~Eriksson, \emph{Conjugacy of Coxeter elements}, Electron. J. Combin. \textbf{16} (2009), no.~2, Research Paper~4, 9~pp.; \href{https://arxiv.org/abs/1302.2842}{arXiv:1302.2842}. \bibitem{Fei} J.~Fei, \emph{Symmetric branching via Laurent phenomenon algebras}, preprint (2026), \href{https://arxiv.org/abs/2609.32425v1}{arXiv:2609.32425v1}. \bibitem{ThetaBases} J.~Fei, \emph{Theta functions and intrinsic mirrors of Laurent phenomenon algebras}, manuscript. \bibitem{HS} A.~Henriques and D.~E.~Speyer, \emph{The multidimensional cube recurrence}, Adv. Math. \textbf{223} (2010), no.~3, 1107--1136; \href{https://arxiv.org/abs/0708.2478v3}{arXiv:0708.2478v3}. \bibitem{KWY} S.~Keel, L.~White, and T.~Y.~Yu, \emph{Log Calabi--Yau mirror symmetry and non-archimedean disks}, preprint (2024), version~2 (2026), \href{https://arxiv.org/abs/2411.04067v2}{arXiv:2411.04067v2}. \bibitem{KY} S.~Keel and T.~Y.~Yu, \emph{The Frobenius structure theorem for affine log Calabi--Yau varieties containing a torus}, Ann. of Math. (2) \textbf{198} (2023), 419--536; \href{https://arxiv.org/abs/1908.09861v2}{arXiv:1908.09861v2}. Numbered references here use this arXiv version. \bibitem{KP} A.~Kuznetsov and A.~Polishchuk, \emph{Exceptional collections on isotropic Grassmannians}, J. Eur. Math. Soc. \textbf{18} (2016), no.~3, 507--574; \href{https://arxiv.org/abs/1110.5607v2}{arXiv:1110.5607v2}. \bibitem{LP} T.~Lam and P.~Pylyavskyy, \emph{Laurent phenomenon algebras}, Cambridge J. Math. \textbf{4} (2016), no.~1, 121--162; \href{https://arxiv.org/abs/1206.2611v3}{arXiv:1206.2611v3}. \bibitem{PRW} C.~Pech, K.~Rietsch, and L.~Williams, \emph{On Landau--Ginzburg models for quadrics and flat sections of Dubrovin connections}, Adv. Math. \textbf{300} (2016), 275--319; \href{https://arxiv.org/abs/1404.4844v2}{arXiv:1404.4844v2}. \bibitem{Rietsch} K.~Rietsch, \emph{A mirror symmetric construction of $qH_T^*(G/P)_{(q)}$}, Adv. Math. \textbf{217} (2008), 2401--2442; \href{https://arxiv.org/abs/math/0511124v2}{arXiv:math/0511124v2}. \bibitem{Rincon} F.~Rinc\'on, \emph{Isotropical linear spaces and valuated delta-matroids}, J. Combin. Theory Ser. A \textbf{119} (2012), no.~1, 14--32; \href{https://arxiv.org/abs/1004.4950v1}{arXiv:1004.4950v1}. \bibitem{Scott} J.~S.~Scott, \emph{Grassmannians and cluster algebras}, Proc. London Math. Soc. (3) \textbf{92} (2006), 345--380; \href{https://arxiv.org/abs/math/0311148}{arXiv:math/0311148}. \bibitem{SWExceptional} P.~Spacek and C.~Wang, \emph{Towards Landau--Ginzburg models for cominuscule spaces via the exceptional cominuscule family}, J. Algebra \textbf{630} (2023), 334--393; \href{https://arxiv.org/abs/2204.03548v2}{arXiv:2204.03548v2}. \bibitem{SWCanonical} P.~Spacek and C.~Wang, \emph{Canonical Landau--Ginzburg models for cominuscule homogeneous spaces}, preprint (2024), \href{https://arxiv.org/abs/2410.05070v1}{arXiv:2410.05070v1}. \bibitem{SWChevalley} P.~Spacek and C.~Wang, \emph{Chevalley polytopes and Newton--Okounkov bodies}, preprint (2024), \href{https://arxiv.org/abs/2411.10276v1}{arXiv:2411.10276v1}. \bibitem{SWOrthogonal} P.~Spacek and C.~Wang, \emph{Canonical mirror models for maximal orthogonal Grassmannians}, in \emph{Lie Theory and Its Applications in Physics: Varna, Bulgaria, June 2023}, V.~K.~Dobrev (ed.), Springer Proc. Math. Stat. \textbf{473}, Springer, Singapore, 2025, 353--359; \href{https://arxiv.org/abs/2312.17656v1}{arXiv:2312.17656v1}. \end{thebibliography} \end{document}