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\title{Equality of the Mirkovi\'c--Vilonen and theta bases}
\author{Qiyue Tang}
\date{}
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\begin{document}
\begin{abstract}
We prove Kamnitzer's ICM 2022 conjecture that the Mirkovi\'c--Vilonen and theta bases coincide.
\end{abstract}
\maketitle\raggedbottom
\begingroup\renewcommand{\thefootnote}{}\footnotetext{\textit{Keywords:} MV basis, theta functions, spatial fusion, logarithmic stable maps.\newline
\textit{Mathematics Subject Classification (2020):} Primary 13F60; Secondary 14M15.}\endgroup

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\section*{AI use and scholarly credit}
The initial draft was developed through seven ChatGPT-6-Pro conversations totaling approximately 23 hours and 10 minutes, interspersed with four hours of research using Pharos~\cite{Pharos}. It was then reviewed and polished with Codex and its built-in mathematical paper-writing skill before being submitted to Rethlas-OSV~\cite{RethlasOSV,Rethlas,OSVerify} for review.

This was the first time Rethlas-OSV identified a nontrivial error in my work that invalidated the proof of the main theorem. Previously, manuscripts had either passed its review or yielded editorial errors, such as incorrect reference numbers or a missing plural ending, or readily repairable mathematical errors, such as an omitted hypothesis. I submitted the review report to ChatGPT-6-Pro and asked it to spend six hours repairing the proof. After it reported completing the repair, I used a separate ChatGPT conversation for approximately three rounds of review totaling about 88 minutes. These reviews found no further errors. Codex then reviewed and polished the manuscript, expanding it from 21 to 52 pages. The revised manuscript was then reviewed using Rethlas-OSV.

The second Rethlas-OSV review identified further errors in the manuscript, including one that again blocked the proof as written. They were classified as minor because the report also supplied corrections. After these errors were corrected and the exposition was polished again, the manuscript passed Rethlas-OSV review.

By posting this manuscript on this platform, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in preparing a finished account or checking the arguments; the eventual scholarly credit should go to those who undertake that work.
\par
\endgroup

\section{Introduction}

Let $G$ be a connected complex semisimple group and $N$ a maximal unipotent subgroup. Distinguished bases of $\C[N]$ encode bases of highest-weight representations, crystal operators and tensor-product multiplicities. Three representation-theoretic constructions are the dual canonical basis, the dual semicanonical basis in simply-laced type, and the Mirkovi\'c--Vilonen (MV) basis. A fourth construction, the theta basis, comes from the cluster structure on $\C[N]$. Kamnitzer's Conjecture~6.9 in his ICM 2022 report \cite{Kam} predicts that the MV and theta bases coincide. We prove this equality for every $G$.

\subsection{Geometric bases and cluster algebras}

The canonical-basis theory of Lusztig and the crystal-basis theory of Kashiwara provide compatible bases of quantum groups and their integrable highest-weight representations \cite{Lusztig,Kashiwara}. At the classical parameter, dualizing the positive enveloping algebra gives a distinguished basis of the unipotent coordinate ring. The crystal records the leading behavior of the simple-root operators. Lusztig's semicanonical construction instead uses constructible functions on varieties of representations of preprojective algebras \cite{LusztigSemi}. Its irreducible components carry the same crystal combinatorics, while the resulting basis vectors can differ from the canonical ones.

The MV construction arises from geometric Satake. Mirkovi\'c and Vilonen identify representations of $G$ with suitable perverse sheaves on the affine Grassmannian of the Langlands dual group \cite{MV}. The irreducible components of the relevant intersections with semi-infinite orbits provide fundamental classes representing weight vectors. Baumann and Gaussent describe the crystal operations geometrically \cite{BG}; Kamnitzer describes the associated MV polytopes by tropical Pl\"ucker relations and relates their parameters to Lusztig data \cite{KamPoly}. These results give a concrete common indexing of the geometric and canonical constructions.

Passing from representations to $\C[N]$ requires compatibility as the highest weight increases. Baumann, Kamnitzer and Knutson establish this compatibility and obtain the normalized MV basis of $\C[N]$ \cite{BKK}. It is biperfect: for both left and right simple-root derivations, the highest nonzero term determines the corresponding crystal operator, with a prescribed leading coefficient. They also express multiplication of MV functions through intersection multiplicities in a spatial fusion family in the Beilinson--Drinfeld Grassmannian. This multiplication formula is the geometric input for the present comparison.

Biperfectness alone allows several bases. The examples in \cite{BKK} distinguish the MV, dual canonical and dual semicanonical constructions; their $SL_6$ calculation separates MV and dual semicanonical vectors despite the shared crystal labels. Their Duistermaat--Heckman measures detect information beyond the moment polytope. More recently, Baumann has recovered MV polytopes from the coproduct on the unipotent coordinate ring and related their extremal operations to stability decompositions \cite{BaumannPoly}. Thus the common combinatorics is well understood, while identification of the actual functions requires finer information.

Fomin and Zelevinsky introduced cluster algebras in connection with total positivity and canonical bases \cite{FZ}. Berenstein, Fomin and Zelevinsky constructed upper cluster structures on double Bruhat cells using generalized minors \cite{BFZ}. For $\C[N]$, a reduced word for the longest Weyl-group element gives an initial cluster. Its mutable variables change under exchange relations; its frozen variables remain coefficient functions. We use the original coordinate ring, in which the frozen minors are polynomial variables. Localizing at these minors produces the open cluster variety and is a separate operation in the argument.

Categorification established several links between cluster monomials and representation-theoretic bases. Geiss, Leclerc and Schr\"oer use subcategories of modules over preprojective algebras to construct cluster structures on unipotent coordinate rings in the symmetric setting, with cluster monomials in the dual semicanonical basis \cite{GLS}. Kang, Kashiwara, Kim and Oh use symmetric quiver Hecke algebras to obtain monoidal categorifications and the corresponding inclusion in dual canonical bases \cite{KKKO}. Qin's common triangular bases and injective-reachability results extend the cluster-basis comparison to the symmetrizable Lie-theoretic setting used here \cite{QinDC,QinTrop,QinAnalogs}. These results identify particular cluster families and supply the existence and triangularity inputs needed to compare complete bases.

Gross, Hacking, Keel and Kontsevich construct theta functions from consistent scattering diagrams \cite{GHKK}. A broken line begins with an integral tropical label and changes its attached monomial when it crosses a wall; summing its final monomials gives the theta function. In a fixed cluster, the function has a distinguished Laurent monomial with coefficient one, and all other exponents differ from its exponent by nonnegative combinations of exchange columns. This is pointedness. In the unipotent setting, the theta functions regular on $N$ form a basis of $\C[N]$ and include the cluster monomials. Restriction from the localized algebra is controlled by valuations at the frozen divisors. Cheung, Magee, Mandel and Muller prove general valuative independence and theta reciprocity, extending the optimized-seed arguments of \cite{GHKK} \cite{CMMM}. We give the short frozen-divisor argument needed for this ring in Section~\ref{sec:labels}.

\subsection{Kamnitzer's conjecture and subsequent work}

Kamnitzer's ICM 2022 report places the dual canonical, dual semicanonical and MV bases in a common framework of perfect bases, crystals, polytopes and measures \cite{Kam}. Its cluster-algebra section separates three assertions. Question~6.5 asks whether the theta basis is biperfect. Conjecture~6.6 asks whether the MV basis contains every cluster monomial and whether every MV element is pointed in every cluster. Conjecture~6.9 asserts equality of the MV and theta bases as functions. The last assertion determines the lower Laurent terms as well as the leading labels, and consequently identifies the entire multiplication table.

Several calculations support this comparison. Baumann, Gaussent and Littelmann prove cluster-monomial compatibility for reduced-word clusters satisfying the condition in their Proposition~7.2 \cite{BGL}. Bai, Dranowski and Kamnitzer use the Mirkovi\'c--Vybornov correspondence to compute fusion in type $A$; for the upper-unitriangular group of size four they verify the cluster exchange relations through MV products \cite{BDK}. In rank-two cluster algebras, Cheung, Gross, Muller, Musiker, Rupel, Stella and Williams identify the greedy and theta bases \cite{CGMMRSW}. That comparison is part of the evidence for the parallel between representation-theoretic and cluster bases discussed in the ICM report.

Fei develops a complementary approach through crystal structures on upper cluster algebras \cite{FeiCrystal}. Reachable frozen vertices determine crystal operators on tropical labels; under the stated hypotheses these lift to generic bases. His classification of bases compatible with both crystal actions describes the freedom remaining after the two crystal structures have been fixed. The June 2026 revision still states the corresponding theta and triangular assertions as Conjecture~9.25. In a different geometric realization, his work on universal Schubert polynomials compares generic, canonical and MV elements attached to Schubert labels \cite{FeiSchubert}. His ramified-quiver construction treats skew-symmetrizable exchange matrices admitting good realizations: under reachability, monodromy cluster characters give bases of the middle cluster algebra with frozen variables inverted, and of the upper cluster algebra when these two algebras coincide \cite{FeiRamified}. These hypotheses and localizations are relevant when transporting categorical constructions to the ring considered here.

Another recent direction uses characteristic cycles. Leroux-Lapierre relates asymptotic characters in category $\mathcal O$ for truncated shifted Yangians to equivariant multiplicities of characteristic cycles \cite{Leroux}. The proposed general change of basis from dual canonical to MV vectors appears there as Conjecture~6.3. Kamnitzer, Labelle, Leroux-Lapierre, Pinet and Weekes identify the complexified Grothendieck ring for simply-laced shifted Yangians with the Cox ring of an open bi-infinite Bott--Samelson variety and establish its cluster-algebra realization \cite{KLLPW}. This supplies a categorical setting linking affine-Grassmannian geometry, canonical bases and cluster structures. The present comparison uses the spatial-fusion multiplication of MV cycles and the curve-counting multiplication of theta functions.

The enumerative construction of theta multiplication comes from logarithmic mirror symmetry. Gross and Siebert develop the logarithmic and intrinsic mirror constructions \cite{GS,GSIntrinsic}; Abramovich, Chen, Gross and Siebert construct punctured logarithmic maps and their obstruction theories \cite{ACGSPunctured}. Keel and Yu construct the mirror algebra of an affine log Calabi--Yau variety containing a torus by counts of non-archimedean curves \cite{KY}. Johnston compares these counts with punctured logarithmic invariants \cite{JohnstonNaive}. In our setting the relevant interior is an affine Richardson variety for the Langlands dual group. Its full torus, including the frozen directions, relates these curve counts to the cluster scattering diagram.

\subsection{The coefficient comparison}

Write $\mathcal B_{\mathrm{MV}}(\C[N])$ for the normalized MV basis and $\mathcal B_\theta(\C[N])$ for the regular theta basis. Our result is
\begin{equation}\label{eq:target}
 \mathcal B_{\mathrm{MV}}(\C[N])=\mathcal B_\theta(\C[N]).
\end{equation}
Theorem~\ref{thm:main} states the equality in the original coordinate ring. We first pair the bases by their values under a reduced-word valuation. Both are homogeneous, each weight space is finite-dimensional, and the values distinguish the elements within either basis. The pairing fixes the unit and every initial minor. It remains to compare multiplication coefficients: multiplicativity then forces this pairing to fix every function, since the initial minors generate the function field.

For the geometric comparison, an actual MV cycle is represented by based polynomial loops. Keeping every coordinate of its fundamental representation vectors gives a flag map with controlled degrees. The auxiliary zeros of the frozen minors define a finite coefficient map. A double Bott--Samelson compactification of the Richardson variety has a nef divisor supported on its entire boundary. On this model, a general marked-curve and frame constraint has an entire incidence fiber consisting of smooth free maps. A pole-transfer argument recovers the original MV loop from such a map.

With two inputs, the Beilinson--Drinfeld fusion family and the logarithmic mapping space have the same generic incidence. Cartier specialization compares their classes over the collision. To extract an individual coefficient, both classes are pushed to the same finite set of actual central flag maps. The logarithmic source divisor is expanded over all separating cuts. Positive-dimensional central tropical types vanish as supported bivariant operations; the surviving central maps have weight one. The tail contribution retains the puncturing thickness, the finite logarithmic gluing degree and all admissible curve classes. Their product gives the punctured three-point coefficient at that same central map.

Finally, the full Richardson torus identifies the mirror labels with the valuation labels. We prove the weighted extension of the relevant Keel--Yu scattering comparison, using primitive center characters in every finite type. Restriction to the regular theta functions then identifies their multiplication with the coefficients obtained from fusion. The divisorial point-constraint calculation in Appendix~\ref{app:direct-point-constraint} supplies the normalization for nonprimitive outputs.

\section{An algebraic criterion}\label{sec:algebra}

\begin{lemma}\label{lem:fixed-generators}
Let $A$ be an integral algebra over a field $k$. Suppose that $(b_i)_{i\in I}$ and $(t_i)_{i\in I}$ are two $k$-bases, with corresponding unit elements, and that their finite multiplication expansions have the same coefficients:
\[
 b_i b_j=\sum_{l\in I}c_{ij}^l b_l,
 \qquad t_i t_j=\sum_{l\in I}c_{ij}^l t_l.
\]
Suppose that $x_1,\ldots,x_m$ generate $\Frac(A)$ as a field over $k$, and that each $x_a$ is a common corresponding basis element. Then $b_i=t_i$ for every $i$.
\end{lemma}
\begin{proof}
The linear bijection $P(b_i)=t_i$ preserves multiplication and the unit. It is therefore an algebra automorphism of $A$, and extends to an automorphism of $\Frac(A)$. It fixes every $x_a$, so its extension fixes $k(x_1,\ldots,x_m)=\Frac(A)$. Its restriction to $A$ is the identity.
\end{proof}

For $A=\C[N]$, one full initial cluster, including its frozen variables, generates the function field. Thus the criterion requires the identities of \emph{all} multiplication coefficients, including the vanishing coefficients. It requires only the individual initial variables to be common basis elements.

\section{Root data and prefix coordinates}\label{sec:coordinates}

We write $\N=\Z_{\geq0}$. Fix a pinning of $G$, with simple roots $\alpha_i$, simple coroots $h_i$, fundamental weights $\omega_i$, and simple reflections $s_i$, indexed by a finite set $I$. Write $e_i$ for the pinned positive Chevalley generator and $x_i(t)=\exp(te_i)$ for its root subgroup. Set $a_{ij}=\langle h_i,\alpha_j\rangle$. Let $w_0$ be the longest Weyl group element, and write $i^*$ for the index satisfying $w_0\alpha_i=-\alpha_{i^*}$. Fundamental weights and their representations may be taken on the simply connected cover; its restriction to $N$ is a pinned isomorphism.

Fix a reduced word $\mathbf i=(i_1,\ldots,i_m)$ for $w_0$, and put $w_k=s_{i_1}\cdots s_{i_k}$. For a dominant weight $\lambda$, define $\Delta_{\lambda,w\lambda}(n)=v_\lambda^*(n v_{w\lambda})$, where $v_{w\lambda}$ is the pinned extremal vector and $v_\lambda^*$ is the highest coefficient functional normalized by $v_\lambda^*(v_\lambda)=1$. The prefix minors are
\[
 X_k=\Delta_{\omega_{i_k},w_k\omega_{i_k}}\big|_N,
 \qquad 1\leq k\leq m.
\]
Let $k^-$ and $k^+$ be the preceding and following occurrences of $i_k$, with absent indices $0$ and $m+1$. The mutable set is $J=\{k:k^+\leq m\}$; the other positions are frozen.

We use the full $m\times |J|$ column matrix $B$ given by
\[
 B_{ak}=\begin{cases}
 1&a=k^-,\\
 -1&a=k^+,\\
 a_{i_a i_k}&a<k<a^+<k^+,\\
 -a_{i_a i_k}&k<a<k^+<a^+,\\
 0&\text{otherwise}.
 \end{cases}
\]
The source convention uses the opposite exchange matrix $C=-B$, required by the inverse successor coordinates on the dual variety. The exponent lattice is the full lattice $M=\Z^m$ with basis given by the $X_k$. The successor-row minor makes $C:\Z^J\to M$ injective with saturated image: order the mutable columns by increasing successor $k^+$ and select the successor rows in that order. Since $C_{ak}=0$ for $a>k^+$ and $C_{k^+,k}=1$, this auxiliary minor is upper unitriangular. Its integral inverse gives an integral left inverse to $C$. The ambient coordinate order used for valuations remains unchanged.

For $a\in\Z^m$, put $X^a=\prod_{k=1}^m X_k^{a_k}$. For $a,b\in\Z^m$, write $a<b$ in reverse lexicographic order when the last nonzero coordinate of $b-a$ is positive. For a nonzero Laurent polynomial $f=\sum_a f_aX^a$, define $v_X(f)=\min\{a:f_a\ne0\}$ in this order. Set $v_X(f/g)=v_X(f)-v_X(g)$ for nonzero Laurent polynomials $f,g$. The valuation $v_t$ is defined in the same way in the factorization coordinates below. The coefficient of $X^a$ means the scalar $f_a$, taken as zero when that monomial is absent.

For an integral exponent vector $g$, a theta function with label $g$ in this convention has an expansion with scalar coefficients $c_v$ of the form
\[
 \theta_g=X^g\left(1+\sum_{0\ne v\in\N^J}c_v X^{Cv}\right).
\]
Only labels giving regular functions in $\C[N]$ are used as basis labels below.

The right Chevalley derivation is
\[
 D_i f(n)=\left.\frac{d}{dt}\right|_{t=0}f(n x_i(t)).
\]
Its string coordinates use the \emph{reverse product} chart
\begin{equation}\label{eq:reverse-chart}
 (t_1,\ldots,t_m)\longmapsto
 x_{i_m}(t_m)\cdots x_{i_1}(t_1).
\end{equation}
This order matters: right multiplication by $x_{i_1}(u)$ sends $t_1$ to $t_1+u$ in this chart. Pinned transpose is an anti-isomorphism and reverses the factor order.

Define the lower unitriangular integral matrix
\[
 \mathsf M_{ak}=\begin{cases}
 \langle h_{i_k},s_{i_{k+1}}\cdots s_{i_a}\omega_{i_a}\rangle&k\leq a,\\
 0&k>a.
 \end{cases}
\]
The prefix-minor valuation formula of Fujita--Oya \cite[Proposition 6.3 and Theorem 6.5]{FOcluster}, after transpose and the order convention \eqref{eq:reverse-chart}, uses the rows of this matrix. The least reverse-lexicographic monomial of $X_a$ has exponent $\mathsf M_{a,*}$ and coefficient one. For row vectors, multiplication by a lower unitriangular matrix preserves reverse lexicographic order: if the last nonzero coordinate of $b-a$ is $j$, then the coordinates of $(b-a)\mathsf M$ after $j$ vanish and its $j$-th coordinate is $b_j-a_j$. Consequently different monomials $X^a$ have different leading $t$-exponents $a\mathsf M$. A finite Laurent sum has the leading exponent of its least summand, since those exponents are distinct. Taking quotients gives
\[
 v_t(f)=v_X(f)\mathsf M\qquad(f\in\C(N)^\times).
\]

\begin{remark}
For $G=SL_3$ and $\mathbf i=121$, write
\[
 n=\begin{pmatrix}1&x&z\\0&1&y\\0&0&1\end{pmatrix}.
\]
Then $(X_1,X_2,X_3)=(x,xy-z,z)$. Chart \eqref{eq:reverse-chart} gives $x=t_1+t_3$, $y=t_2$, $z=t_3t_2$, and the rows of $\mathsf M$ are $(1,0,0),(1,1,0),(0,1,1)$. 
\end{remark}

\section{Polynomial-loop representatives}\label{sec:sources}

Let $H$ be the Langlands dual group, with its dual pinning and opposite unipotent groups $U_H^+$ and $U_H^-$. The roots $\alpha_i$ of $G$ are identified with the simple coroots of $H$. Write $B_\pm=T_HU_H^\pm$ for its pinned Borel subgroups and $H_{\rm ad}$ for its adjoint form. Use the pinned representative $\dot w_0\in H$ of its longest Weyl element. Put $K=\C((z))$, $O=\C[[z]]$, and $\Gr_H=H(K)/H(O)$. For a coweight $\eta$, set $L_\eta=z^\eta H(O)$ and $S_\eta^\pm=U_H^\pm(K)L_\eta$. A stable upper MV cycle of charge $\nu=\sum_i\nu_i\alpha_i$, with $\nu_i\in\N$, is an irreducible component of the reduced closure of $S_0^+\cap S_{-\nu}^-$.

The normalized function $b_Z$ is defined by geometric Satake: for a sufficiently dominant allowed weight $\lambda$, it is the matrix coefficient $b_Z(n)=v_\lambda^*(n[z^\lambda Z])$, where the bracket denotes the fundamental cycle class in $V_G(\lambda)$ and $v_\lambda^*(v_\lambda)=1$. Stabilization makes it independent of the sufficiently large choice \cite[Section 6.1]{BKK}.

Its affine polynomial-loop part consists of points
\[
 n(z^{-1})^{-1}L_0,\qquad n(t)\in U_H^+(\C[t]),\quad n(0)=1.
\]
Ordered root coordinates give uniqueness: split off negative and nonnegative Laurent powers successively in root height; commutators change only higher roots. This construction is polynomial on each bounded pole chart and works over arbitrary coefficient algebras.

All charges used here belong to the coroot lattice of $H$. The simply connected covering group has the same unipotent subgroups and identifies its affine Grassmannian with the neutral component relevant to these charges. Fundamental representations below are taken on this cover when necessary. For each fundamental representation $V_H(\omega_i)^*$, let $v_i^-$ be its lowest vector normalized by $\langle v_i^-,v_i^+\rangle=1$, where $v_i^+$ is the pinned highest vector of $V_H(\omega_i)$. Its lowest weight is $-\omega_i$. Fix a weight basis containing $v_i^-$. The degree of a polynomial vector is the maximum of the degrees of its nonzero coordinates; this is independent of the chosen constant basis. All coordinates of $n(t)v_i^-$ are retained.

\begin{lemma}\label{lem:full-vector}
Over a geometric field, the point $n(z^{-1})^{-1}L_0$ belongs to $S_{-\nu}^-$ if and only if
\[
 \deg_t n(t)v_i^-=\nu_i\quad\text{for every }i.
\]
The inequalities with $\leq\nu_i$ are polynomial coefficient equations on every bounded chart.
\end{lemma}
\begin{proof}
Write $n^{-1}=u_-z^\mu k$ with $u_-\in U_H^-(K)$ and $k\in H(O)$. Since $u_-$ fixes the lowest vector,
\[
 nv_i^-=z^{\langle\omega_i,\mu\rangle}k^{-1}v_i^-.
\]
The last vector is integral and primitive. Its minimum coordinate order is therefore $\langle\omega_i,\mu\rangle$, which is minus the degree of the polynomial vector. Thus $\mu=-\nu$ is equivalent to the displayed degree equalities. The Iwasawa orbit label is unique, and the fundamental weights detect it. The equations for an upper degree bound are precisely the vanishing of all higher coefficients.
\end{proof}

Index fundamental degrees by these lowest-line representations, with flag target $H/B_-$. The same vectors give an actual based flag quasimap through
\[
 P_i(z)=z^{\nu_i}n(z^{-1})v_i^-.
\]
They satisfy the homogeneous Pl\"ucker relations because they do so away from $z=0$, where they come from a group element; multiplication by $z$ is injective in the polynomial coefficient ring. At a point where all vectors are nonzero they define the flag, and a common vector factor records the corresponding defect. 

Write $V=U_H^-$. To define its fundamental extremal minors, take the representation $V_H(\omega_i)$, a pinned highest vector $v_i^+$ and the coefficient functional $\phi_i^-$ of its pinned lowest vector $\dot w_0v_i^+$. Put
\[
 F_i(v)=\phi_i^-(v v_i^+)\qquad(v\in V).
\]
These are regular functions on the unipotent group, also when the representation is taken on its simply connected cover. The dual matrix-coefficient calculation pairs the row of $n^{-1}$ with $n v_i^-$; conjugation by $\dot w_0$ exchanges its highest and lowest extremal rows. This fixes the same index $i$ as in Lemma~\ref{lem:full-vector}. Let
\[
 Y=\{v\in V:F_i(v)\ne0\text{ for all }i\}.
\]
Let $\mathscr H\subset V$ be a nonempty open of constant frames
$h$ with all $F_i(h^{-1})\ne0$; it will be shrunk when finitely many
genericity conditions are imposed. Choose $h\in\mathscr H$ and define
\begin{equation}\label{eq:source-map}
 g_{n,h}(z)=\dot w_0 n(z^{-1})^{-1}\dot w_0^{-1}h^{-1}.
\end{equation}
The monic polynomials
\[
 Q_i(z)=z^{\nu_i}\frac{F_i(g_{n,h}(z))}{F_i(h^{-1})}
\]
have degree $\nu_i$. Indeed, the definitions and the contragredient action give
\[
 F_i(g_{n,h}(z))=
 \big\langle n(z^{-1})v_i^-,\dot w_0^{-1}h^{-1}v_i^+\big\rangle.
\]
Lemma~\ref{lem:full-vector} bounds its pole by $\nu_i$, and its value at infinity is $F_i(h^{-1})$. It extends to the reduced affine closure by vanishing of polynomial coefficients.

\section{Finite coefficient maps}\label{sec:finite}

For each fundamental representation the map
\[
 U_H^+\longrightarrow\prod_i V_H(\omega_i)^*,\qquad
 u\longmapsto(uv_i^-)_i
\]
is a closed immersion into the affine subspace where every lowest
coordinate is one. This follows by restricting the fundamental
Pl\"ucker embedding of $H/B_-$ to its big cell and normalizing these
coordinates. Every root coordinate is consequently a polynomial in
the complete-vector coordinates.

Let $\mathcal P_\nu$ be the affine scheme of based polynomial loops
with $\deg n(t)v_i^-\leq\nu_i$ for every $i$. It is defined in the
finite space of bounded-vector coefficients by the Pl\"ucker equations
and $n(0)=1$. These coefficient equations represent the bounded functor
over arbitrary coefficient algebras. The exact-charge criterion of
Lemma~\ref{lem:full-vector} is used over geometric fields.
Let $Z^{\mathrm{aff}}$ be the reduced closure of the corresponding
irreducible exact-charge locus in $\mathcal P_\nu$. It is integral and
has dimension $d=\height(\nu)=\sum_i\nu_i$
\cite[Proposition~7]{BG}. This explicit bounded construction also
respects the finite-stage correction in \cite{BGErratum}.

Loop rotation preserves each component, since $\Gm$ is connected.
It extends to $n(t)\mapsto n(ct)$ at $c=0$ and contracts the source
to $n=1$. Its coordinate ring is positively graded with degree-zero
part $\C$.

\begin{lemma}\label{lem:no-A1}
Suppose a smooth variety $Y$ has a smooth projective compactification $(W,E)$ with reduced simple normal crossings (SNC) boundary and $K_W+E\simeq\OO_W$. No finite-type family of nonconstant morphisms $\A^1\to Y$ covers $Y$.
\end{lemma}
\begin{proof}
Such a family would give, after a transverse parameter slice, a dominant generically finite morphism $T\times\A^1\to Y$ with $\dim T=\dim Y-1$. After shrinking smooth $T$, properness extends it to $T\times\PP^1\to W$. Indeed the indeterminacy is in $T\times\{\infty\}$ and has codimension at least two, so its image can be removed from $T$. Pullback of a nonzero logarithmic volume form is a nonzero section of $K_{T\times\PP^1}(T\times\{\infty\})$. This line bundle has degree $-1$ on every $\PP^1$ fiber and hence has no such section.
\end{proof}

For the open Richardson variety $Y$, the double Bott--Samelson compactification has these properties \cite[Section 4]{KS}.

\begin{proposition}\label{prop:finite}
For a sufficiently general frame $h$, the map
\[
 \chi_h:Z^{\mathrm{aff}}\longrightarrow\prod_i\A^{\nu_i}
\]
recording the coefficients of $z^0,\ldots,z^{\nu_i-1}$ in each monic $Q_i$ is finite and surjective. For $\nu_i=0$ this coefficient list is empty.
\end{proposition}
\begin{proof}
Use the equivalent coefficients of $F_i(g_{n,h}(t^{-1}))/F_i(h^{-1})$. Their coefficients of $t^k$ are homogeneous of positive degree $k$. If their common zero locus contained a nonvertex point for general $h$, the resulting incidence family would give the nonconstant curves
\[
 t\longmapsto\dot w_0 n(t)^{-1}\dot w_0^{-1}h^{-1}\in Y.
\]
They remain in $Y$ because all extremal minors are constant and nonzero. The incidence is of finite type because the polynomial-loop degrees have a common bound. If it dominated the frame open, one of its finitely many irreducible components would dominate; the evaluation at $t=0$ on that component would be dominant onto $Y$. Its nonvertex points give nonconstant maps because $n(t)$ is based and nonconstant. This produces exactly the family forbidden by Lemma~\ref{lem:no-A1}. Thus, on a nonempty frame open, the common zero fiber is supported at the vertex.

Let $R=\C[Z^{\mathrm{aff}}]$ and let $J$ be the ideal generated by these positive-degree coefficients. Then $R/J$ is finite-dimensional. Choose homogeneous lifts of a vector-space basis of $R/J$. For each homogeneous $f\in R$, subtract the linear combination of these lifts representing its quotient class. The remainder belongs to $J$. Express it as a sum of homogeneous coefficient generators times homogeneous elements of smaller degree and apply induction. These lifts generate $R$ over the coefficient subalgebra, so $\chi_h$ is finite. Its image has dimension $d$, equal to the dimension of its affine target; since the source is integral the image is dense, and finiteness makes it surjective.
\end{proof}

In characteristic zero the finite map $(\chi,h)$ is generically
\'etale. For finitely many sources, first choose a common dense open
of the coefficient/frame base over which all these maps are \'etale,
and then intersect it with the locus of distinct auxiliary roots
avoiding the input and unit marks. On its full root-ordering cover,
fixing the marked positions leaves $m=\dim N$ independent frame
directions.

\section{Valuations and the regular basis pairing}\label{sec:labels}

For a generic seed in the next argument, $M^\circ$ is its full
exponent lattice, $N^\circ=\operatorname{Hom}(M^\circ,\Z)$,
and $N$ is the seed lattice with basis $(e_i)$ containing $N^\circ$ with finite index.
These lattice symbols are local to the scattering argument;
the algebraic group remains the maximal unipotent subgroup of $G$.
The integers $d_i>0$ specify $N^\circ$ by its basis $d_ie_i$.
The mutable sublattice is $N_{\rm uf}=\bigoplus_{i\in I_{\rm uf}}\Z e_i$, where $I_{\rm uf}$ indexes the mutable seed vectors. For the seed skew form $\{\ ,\ \}$, define $p^*:N_{\rm uf}\to M^\circ$ by $p^*(n)(n^\circ)=\{n,n^\circ\}$ for $n^\circ\in N^\circ$. Pairings with $N$ use the rational extension of the natural pairing with $N^\circ$. Write $x^m$ for a formal Laurent monomial of exponent
$m\in M^\circ$, and $y^n$ for a positive graded Lie-algebra symbol,
where $n$ is a nonzero nonnegative combination of mutable seed
vectors. Its action is denoted $D_n$ below. The Lie bracket is
$[y^n,y^{n'}]=\{n',n\}y^{n+n'}$, as in \cite[equation~(1.1)]{GHKK}; completion is by the sum of the mutable coefficients of $n$.

The string of a normalized MV function $b$ along $\mathbf i$ is obtained by successively taking the largest nonzero powers of $D_{i_1},\ldots,D_{i_m}$ and dividing each maximal derivative by its factorial. Denote the resulting vector by $s_{\mathbf i}(b)$. More explicitly, start with $b^{(0)}=b$ and put
\[
 s_k=\max\{a\geq0:D_{i_k}^{a}b^{(k-1)}\ne0\},
 \qquad b^{(k)}=\frac{D_{i_k}^{s_k}}{s_k!}b^{(k-1)}.
\]
Right multiplication on $N$ induces this derivation; it is the left action on functions in \cite[Section~2.1]{BKK}. The maximum is finite because the derivations are locally nilpotent on $\C[N]$. The perfect-basis property and normalization in \cite[Theorems 5.4 and 6.2]{BKK} make the final endpoint equal to $1$.

\subsection{Opposite theta functions in the same function field}

\paragraph{Opposition for the full skew-symmetrizable scattering data.}
Although Appendix~A.2 of \cite{QinTrop} is introduced in the context of
skew-symmetric quivers with potential, its opposition calculation is
valid for arbitrary positive symmetrizers, as follows.  Keep the lattices $N$, $N^\circ$, $M^\circ$, all frozen
coordinates and every $d_i$ fixed.  Replacing the seed by its opposite
changes the skew form to its negative and changes $p^*$ to $-p^*$.
For the positive graded Lie algebras write
$\iota:\mathfrak g\to\mathfrak g^{\rm op}$ for the identity on the
monomial symbols $y^n$.  Then
\[
 [\iota u,\iota v]_{\rm op}=-\iota[u,v].
\]
Consequently $\iota$ induces a group \emph{anti}-isomorphism on the
completed groups defined by the Baker--Campbell--Hausdorff formula.
On the Laurent-series modules put
$\kappa(x^m)=x^{-m}$.  The two actions on monomials satisfy
\begin{align*}
 D_n(x^m)&=\langle m,n\rangle x^{m+p^*n},\\
 D_n^{\rm op}(\kappa x^m)
   &=-\langle m,n\rangle x^{-m-p^*n}
     =-\kappa D_n(x^m).
\end{align*}
For a generic path $\gamma$, let $P_\gamma$ be its ordered wall-crossing automorphism. Reflecting a path reverses its wall-crossing signs.  Combining this
sign reversal with this anti-isomorphism gives
\[
 P^{\rm op}_{\kappa\gamma}\,\kappa=\kappa P_\gamma.
\]
The reflected diagram is consistent, and its incoming walls are exactly
those of the opposite seed, with the unchanged multipliers $d_i$.
Uniqueness of consistent completion identifies it with the opposite
cluster scattering diagram. Write $\Theta(t)$ for the theta set of the seed $t$. Reflection of the defining broken lines
then identifies its initial-chamber theta set with
$\kappa P\Theta(t)$, where $P$ transports the original positive chamber
to the reachable negative chamber.  

A seed is injective-reachable if a reachable seed $t[1]$ has mutable parts of its mutable $g$-vectors equal, up to permutation, to the negatives of the initial coordinate vectors.

\begin{lemma}\label{lem:opposite-theta-functions}
Let a full-rank seed $t$ be injective-reachable. Identify the initial
variables of $t$ and its opposite $t^{\mathrm{op}}$, whose exchange
matrix is $-B(t)$, in the same rational-function field. Then the two
sets of Laurent theta functions are equal as sets of actual rational
functions. All frozen variables remain in this identification.
\end{lemma}
\begin{proof}
By \cite[Theorem~A.1.5]{QinTrop}, the full-rank injective-reachable
seed has Laurent polynomial theta functions.  The strict-cone hypothesis
in the A-variety form of the GHKK result holds because the exchange
columns are linearly independent: a rational linear functional can be
chosen to take value one on every such column.  Thus the Laurentness
hypothesis in \cite[Section~6.1 and Theorem~6.2]{KQW} is satisfied before
we apply the twist automorphism.

Set $t'=t[1]$, relabeling its mutable indices by the reachability permutation. Let $F(t)$ and $F(t')$ be their rational-function fields in all mutable and frozen variables, and let $\mu^*:F(t')\to F(t)$ be the mutation isomorphism.
Write $\psi:M^\circ(t')\to M^\circ(t)$ for the linear map of the
reachable chamber. This chamber is $\mathcal C^-$, whereas the initial
one is $\mathcal C^+$. Let $P$ denote the wall-crossing operation from
$\mathcal C^+$ to $\mathcal C^-$. For each cluster monomial of $t'$, the
monomial expression in its own chamber gives
\[
 P\circ\mu^*=\psi_*,\qquad
 P=\psi_*\circ(\mu^*)^{-1}.
\]
It suffices to check this on the cluster variables and invertible frozen
variables, which generate the field. Here a subscript $*$ means the
monomial map induced by the indicated lattice map. These identities are
the mutation/wall-crossing compatibility recalled in
\cite[Appendix~A.1]{QinTrop}.

Let $\kappa(x^m)=x^{-m}$. Reflection of broken lines, with their full
exponents, gives
\[
 \Theta(t^{\mathrm{op}})=\kappa P\Theta(t)
\]
by \cite[Proposition~A.2.5 and Lemma~A.2.6]{QinTrop}.
The variation of \cite[Proposition~5.1]{KQW} satisfies
\[
 \psi\,\operatorname{var}^M=-I,
 \qquad
 \operatorname{var}^M=
 \begin{pmatrix}I_{\rm uf}&0\\-F_{{\rm f},{\rm uf}}&-I_{\rm f}\end{pmatrix}.
\]
Here $F_{{\rm f},{\rm uf}}$ is the frozen-by-mutable block of the
reachable-chamber map
$\psi=\left(\begin{smallmatrix}-I_{\rm uf}&0\\F_{{\rm f},{\rm uf}}&I_{\rm f}\end{smallmatrix}\right)$;
the index sets ${\rm uf}$ and ${\rm f}$ denote mutable and frozen
coordinates. Thus the Donaldson--Thomas twist
$\operatorname{DT}=\mu^*\circ\operatorname{var}_*$ obeys
\[
 \operatorname{DT}^{-1}
   =(-\psi)_*\circ(\mu^*)^{-1}
   =\kappa P.
\]
Theorems~5.2 and~6.2 of \cite{KQW} say that this twist is an
automorphism preserving the theta set. Therefore
$\Theta(t^{\mathrm{op}})=\operatorname{DT}^{-1}\Theta(t)=\Theta(t)$.
 
\end{proof}

\paragraph{Frozen localization.}
In this cluster-algebra argument, $F_i$ denotes the prefix minor $X_k$ at the last occurrence of color $i$. The twist acts on these frozen variables by
$\operatorname{DT}(F_i)=F_i^{-1}$. The preceding equality is an equality
of localized theta sets in the common function field. Intersecting both
sets with $\mathbb C[N]$ gives equality of their regular subsets.
The valuation in the fixed initial variables will label these functions.

\subsection{The regular theta basis}

Put $A=\mathbb C[N]$ and $A^\circ=A[(\prod_iF_i)^{-1}]$. The all-type unipotent-cell
cluster theorem \cite[Theorem~5.3 and Appendix~B]{FOcluster} identifies
$A^\circ$ with the full upper cluster algebra. The initial seeds are
injective-reachable by \cite[Theorem~8.2.2 and Lemma~9.1.3]{QinDC},
and have full column rank. Thus \cite[Theorem~A.1.5]{QinTrop} gives a
Laurent theta basis of $A^\circ$, indexed by the full lattice $M$.
Lemma~\ref{lem:opposite-theta-functions} permits the convention $C=-B$
with the initial variables fixed.

For the restriction to $A$, each $F_i$ cuts out an irreducible
reduced divisor in $N$: it is the restriction to the big cell of the
corresponding fundamental Schubert divisor. For a fixed color $i$,
choose a reduced word for $w_0$ ending in $i$. All such prefix seeds are
mutation equivalent by \cite[Theorem~5.4]{FOcluster}. In this seed the
frozen position $f=m$ has
\[
 C_{f k}=\begin{cases}1&k^+=f,\\0&\text{otherwise},\end{cases}
 \qquad k\in J.
\]
In particular every column has nonnegative $f$-coordinate.

The geometric-type Laurent phenomenon, together with the all-type
identification of the nonlocalized cluster algebra with $A$
\cite[Theorem~9.2.3]{QinDC}, gives
\[
 A\subset
 \mathbb C[X_f,X_j^{\pm1}\ (j\ne f)].
\]
The $X_f$-adic Gauss valuation on the common function field is therefore
nonnegative on $A$. Its center contains $F_i=X_f$. The residue algebra
contains the $m-1$ algebraically independent remaining initial variables,
so its center has height one. Since $F_i$ is prime, the center is $(F_i)$. The DVR $A_{(F_i)}$ is dominated by the valuation ring; its units have valuation zero and its uniformizer $F_i$ has valuation one. Thus the valuation is exactly $\operatorname{ord}_{F_i}$ on the common function field.

For every theta function in this seed, its finite pointed expansion has
support in $g+C\mathbb N^J$ and leading coefficient one. It follows that
$\operatorname{ord}_{F_i}(\theta_g)=g_f$. Moreover, for every finite
linear combination with nonzero coefficients,
\begin{equation}\label{eq:no-frozen-cancellation}
 \operatorname{ord}_{F_i}\Bigl(\sum_g a_g\theta_g\Bigr)
      =\min_{a_g\ne0}\operatorname{ord}_{F_i}(\theta_g).
\end{equation}
To see the last assertion, keep the terms of the least $f$-degree.
On these terms, corrections have zero $f$-coordinate. Their nonzero
exponents lie in the strictly convex cone generated by the corresponding
independent columns of $C$. A total order positive on that cone makes
the least remaining leading monomial unique, so it cannot cancel.
This is the full-rank optimized-seed argument of
\cite[Proposition~9.7]{GHKK}; see \cite{CMMM} for general valuative independence.

The variety $N=\operatorname{Spec}A$ is normal, and
$N\setminus\operatorname{Spec}A^\circ$ is the union of the divisors
$F_i=0$. Equivalently,
inside $A^\circ$ a function lies in $A$ exactly when all
$\operatorname{ord}_{F_i}$ are nonnegative. Applying
\eqref{eq:no-frozen-cancellation} for every $i$ proves that
\[
 \{\theta_g:\theta_g\in\mathbb C[N]\}
\]
is a basis of $\mathbb C[N]$. 

\begin{proposition}\label{prop:pairing}
The regular theta basis and the normalized MV basis admit a unique weight-preserving pairing with equal $v_X$-values. This pairing fixes $1$ and every initial variable $X_k$.
\end{proposition}
\begin{proof}
Let $\tau_{\rm pin}:U_G^-\to N_G$ be pinned transpose, and transport
functions by $f\mapsto f\circ\tau_{\rm pin}$. Write $f_i$ for the negative Chevalley generator and $y_i(t)=\exp(tf_i)$ for its root subgroup. The action called the
left action in \cite[Section~2.1]{BKK} is
$(n\cdot f)(x)=f(xn)$, so its infinitesimal action is precisely our
$D_i$.  In the conventions of \cite[Section~3]{FOcompare},
\[
 (-f_i)\cdot(f\circ\tau_{\rm pin})
       =(D_i f)\circ\tau_{\rm pin},
\]
because $\tau_{\rm pin}(y_i(t)v)=\tau_{\rm pin}(v)x_i(t)$.
The normalized MV basis satisfies the two positivity conditions of
\cite[Section~4]{FOcompare}: the Chevalley coefficients are nonnegative
by \cite[Theorem~5.4]{BKK}, and multiplication by the simple-root basis
elements is nonnegative by \cite[Theorem~7.11]{BKK}.  Its stability under
the involution fixing the simple enveloping generators, and the exchange
of the two actions, are \cite[Proposition~6.6 and Remark~6.7]{BKK}.
The root-length factor in Theorem~5.4 cancels against the identical factor
in \cite[Proposition~5.5]{BKK}; the normalized leading Chevalley
coefficient is therefore $\varepsilon_i$ in every finite type.

Pinned transpose changes the forward negative product chart of
\cite{FOcompare} into \eqref{eq:reverse-chart}.
\cite[Proposition~3.28 and Theorem~5.1]{FOcompare} identify the highest
lexicographic exponent with the least reverse-lexicographic exponent,
after undoing the reversal in the reported vector for the latter
valuation.  In the coordinate order $(t_1,\ldots,t_m)$ both exponents
are $s_{\mathbf i}(b)$.  Their coefficient is one: successive maximal
divided derivatives $D_{i_k}^{s_k}/s_k!$ reach the normalized basis
element $1$, so the highest lexicographic monomial has coefficient one;
the lowest reverse-lexicographic monomial is the same monomial.
Consequently

\begin{equation}\label{eq:MV-valuation}
 v_X(b)=s_{\mathbf i}(b)\mathsf M^{-1}.
\end{equation}
These values distinguish MV basis elements.

The full-rank and injective-reachability hypotheses used above hold for
the prefix seeds in every finite symmetrizable type by
\cite[Theorems~8.1.2, 8.2.2, 9.2.3 and Lemma~9.1.3]{QinDC}.
Lemma~\ref{lem:opposite-theta-functions} identifies the two theta sets
as actual functions in the specified common field, with the frozen
variables retained. The preceding regularity argument selects the basis
of the original ring $\C[N]$.

Each column of $C=-B$ is positive in reverse lexicographic order: its last nonzero entry is $1$ in its successor row. The same is true for a nonzero nonnegative combination. Thus $v_X(\theta_g)=g$, with leading coefficient one. Both bases are homogeneous for the root grading, and each homogeneous part of $\C[N]$ is finite-dimensional. If a basis of a finite-dimensional space has distinct values under a valuation, the values of its nonzero linear combinations are exactly its basis values: the least term cannot cancel. Applying this observation to the two bases gives a unique weight- and unit-preserving pairing
\begin{equation}\label{eq:pairing}
 P(b_Z)=\theta_{g_Z},\qquad g_Z=v_X(b_Z).
\end{equation}

Each $X_k$ is a normalized extremal matrix coefficient and hence is an MV basis element. It is also the initial theta monomial. Its valuation is the $k$-th coordinate vector, and its extremal divided-power normalization is one. Therefore $P(X_k)=X_k$.
\end{proof}

\subsection{The actual source contact}

For a reduced word $\mathbf r=(r_1,\ldots,r_m)$ in the geometric group $H$, write
\[
 u=y_{r_1}(u_1)\cdots y_{r_m}(u_m)
\]
for the ordinary negative factorization coordinates on $V$, where $y_i(u)=\exp(uf_i^H)$ and $f_i^H$ is the pinned negative Chevalley generator of $H$. Define the full successor ratios
\[
 q_k=u_{k^+}/u_k\quad(k^+\leq m),\qquad q_k=1/u_k\quad(k^+=m+1).
\]
Here $\mathbf i=\mathbf r^*$, with color change and no word reversal.
Put $\nu^*=-w_0\nu$.  Opposition sends
$S_0^+\cap S_{-\nu}^-$ to $S_{\nu^*}^+\cap S_0^-$, so
$\dot w_0Z$ has the $\mu_-=0$ normalization of \cite{BG}.
Let $\varepsilon_i(Z)$ be the first right-string coordinate of $b_Z$ in a word starting with $i$. The operators $\widetilde e_i$ and $\widetilde f_i$ denote the corresponding crystal raising and lowering maps; superscript BG denotes the normalization of \cite{BG}. The stable endpoint formula in \cite[Section~5.2]{BKK}
and the rank-one definition of $\varphi_i^{\rm BG}$ in \cite[Section~4.1]{BG} give
\[
 \varepsilon_i(Z)
 =\max\{a\ge0:L_{-\nu+a\alpha_i}\in Z\}
 =\varphi^{\rm BG}_{i^*}(\dot w_0Z).
\]
The containment and weight characterizations of the crystal operators
then imply
\[
 \dot w_0(\widetilde e_iZ)
   =\widetilde f^{\rm BG}_{i^*}(\dot w_0Z).
\]
Thus the BG shrinking string of $\dot w_0Z$ in the word $\mathbf r$
is exactly our right string $s_{\mathbf i}(b_Z)$, with the same order
of successive operations.

The coordinate valuation statement uses the full Laurent cylinder
in \cite[Section~4.5]{BG}, which supplies a nonempty open of independent nonzero
leading coefficients for Laurent parameters of the modified string
valuations, whose Grassmannian image is dense in the cycle.  For the
fixed valuations used here, a faithful representation gives a uniform
pole bound.  Root-height Laurent splitting then shows that the
normalized polynomial-loop representative depends on only finitely
many jets of those parameters.  Hence this cylinder argument takes place in the explicit bounded
chart of Section~\ref{sec:finite}; see also \cite{BGErratum}.

The Cartan matrix of $H$ is the transpose of that of $G$. In the formula of \cite[Section 4.5]{BG}, the coefficient is the root--coroot pairing
$\langle\alpha^H_{r_k},h^H_{r_l}\rangle=a_{r_k r_l}$.
Write $s_k$ for the $k$th entry of $s_{\mathbf i}(b_Z)$. The ordinary-factorization cylinder has coordinate valuations
\[
 c_k=-s_k-\sum_{l>k}a_{r_k r_l}s_l.
\]
In the following matrix calculation, $s=(s_k)$ and $c=(c_k)$ are column vectors. Let $\mathsf D$ be the full successor-difference matrix, $(\mathsf Dc)_k=c_k-c_{k^+}$, with the absent successor term equal to zero. Taking successor ratios gives $-\mathsf Dc$. If $\mathsf K$ is upper triangular with $\mathsf K_{kk}=1$ and $\mathsf K_{kl}=a_{r_k r_l}$ for $l>k$, then $c=-\mathsf K s$ and
\[
 \mathsf D\mathsf K\mathsf M^t=I.
\]
Indeed the $k$-th row of $\mathsf D\mathsf K$ has entries $1$ at $k$ and $k^+$, the Cartan entries between them, and zero afterward. Opposition preserves the Cartan entries, so they can be written $a_{i_k i_l}$. Fix a row $a$ of $\mathsf M$. If $a<k$, the product is zero, while for $a=k$ it is one. If $k<a<k^+$, the reflection formula gives
\[
 \mathsf M_{ak}+\sum_{l=k+1}^{a}a_{i_k i_l}\mathsf M_{al}
 =\langle h_{i_k},\omega_{i_a}\rangle=0.
\]
If $k^+\leq a$, the same telescoping stopped at the next occurrence gives
\begin{align*}
 \mathsf M_{ak}+\sum_{l=k+1}^{k^+-1}a_{i_k i_l}\mathsf M_{al}
 &=\langle h_{i_k},s_{i_{k^+}}\cdots s_{i_a}\omega_{i_a}\rangle\\
 &=-\mathsf M_{a,k^+},
\end{align*}
since $i_{k^+}=i_k$. At a last occurrence only the preceding case occurs. These are all entries of $\mathsf D\mathsf K\mathsf M^t$. Hence $-\mathsf Dc=\mathsf M^{-t}s$, exactly the column form of \eqref{eq:MV-valuation}.

\begin{lemma}\label{lem:frame}
For any finite collection of generic MV sources and reduced-word tests, a single nonempty open of constant frames makes the orders of the actual successor-ratio functions in \eqref{eq:source-map} equal to $g_Z$. The assertion includes frozen positions and remains true for regular power-series frames with the same sufficiently general constant term.
\end{lemma}
\begin{proof}
Changes between ordinary reduced-word coordinates are positive birational maps in both directions \cite[Theorem 5.2 and Section 7]{BZ}. At a fixed tropical value their initial rational maps are inverse birational maps: the positive expressions do not cancel at a generic leading coefficient. A generic right multiplication by a simple root subgroup can be computed in a word ending in that root. It adds a constant to the last parameter, whose order is the negative of a string length and is nonpositive. Its leading coefficients remain generic; returning to the original word preserves the tropical value. Iterating gives the assertion for a generic constant right frame on the generic cylinder.

For a field extension $L/\C$ and a fixed representative $g\in V(L((z)))$, use polynomial root coordinates $(h_1,\ldots,h_m)$ on $V$. For a nonzero polynomial $P=\sum_\alpha a_\alpha(z)h^\alpha$, define $v_{\rm G}(P)=\min_{a_\alpha\ne0}\ord_z a_\alpha$, and extend to nonzero rational functions by $v_{\rm G}(P/Q)=v_{\rm G}(P)-v_{\rm G}(Q)$. For $f\in\C(V)^\times$, integral right multiplication of $g$ leaves $v_{\rm G}(h\mapsto f(gh))$ unchanged, because left translation by an integral element of $V$ preserves the generic special-fiber valuation of $V_{L[[z]]}$. The generic cylinder dominates the actual cycle. After a common field extension, its representative and any other generic representative differ by an element of $V(L[[z]])$. Thus the same Gauss values apply to the actual representative.

Only finitely many numerator and denominator leading polynomials are involved. Their nonvanishing can be imposed by one nonempty open of complex constant frames. For coefficients in a field extension, expand the finitely many coefficients in a complex vector-space basis; nonvanishing of one resulting complex polynomial suffices. Substituting a regular power-series frame with that constant term does not change these leading polynomials. This proves the statement.
\end{proof}

The frame open can also impose the finite tests $1+q_k$ when $\ord q_k=0$. Generic cylinder residues avoid their vanishing. Thus the generic input avoids the old binomial centers appearing in the next construction. Write $p_Z$ for the integral tropical point represented by $g_Z$ in this full chart. On a specified simple normal crossings model, $c_\rho(p_Z)$ denotes its contact order with $D_\rho$; the stratum component is the one attained by the actual generic map. These are the boundary data used below.

Fix a smooth SNC model before specializing the frame. On the generic
Laurent cylinder, the independent leading coefficients give the
monomial valuation of the full torus function field; the Gauss argument
in Lemma~\ref{lem:frame} transports that valuation to the actual source.
Near its center on the chosen model, take finitely many affine algebra
generators, local boundary equations, denominators and functions whose
nonvanishing selects the required stratum component. Impose the
nonvanishing of all their leading numerator and denominator
coefficients. This gives one dense frame open for these tests. Their
orders determine the complete boundary contact and its stratum on
this model. Repeating for the finite collection of sources and models
used below amounts to intersecting finitely many dense opens.

\section{The full torus in the Richardson variety}\label{sec:mirror}

\begin{proposition}[The full quotient torus and its toric model]
\label{prop:target-quotient-big-open}
Let $H$ be the pinned Langlands dual group, let $H_{\rm ad}$ be its adjoint
form, and let $T_{\rm ad}$ be the maximal torus of that adjoint form.  Set
\[
 Y=U_H^-\cap B_+\dot w_0B_+.
\]
For the reduced word $\mathbf r=(r_1,\ldots,r_m)$, write
$u=y_{r_1}(u_1)\cdots y_{r_m}(u_m)$ on its factorization torus and put
\[
 q_k=\begin{cases}u_{k^+}/u_k&k^+\leq m,\\1/u_k&k^+=m+1.\end{cases}
\]
With $\mathbf i=\mathbf r^*$ and the full columns $C=-B$ of
Section~\ref{sec:coordinates}, the initial $q$-torus and its singly mutated
tori embed openly in $Y$ and cover it in codimension one.  The partial toric
blowup with rays $C_k$, centers $1+q_k=0$, and strict toric boundary removed
admits an open immersion into $Y$ with complement of codimension at least
two.  This immersion preserves the specified full torus and its logarithmic
volume form.
\end{proposition}

\begin{proof}
We descend the decorated spaces of \cite{SW} while retaining the frozen variables.  Central isogenies identify pinned unipotent groups, so
we may use $H_{\rm ad}$ to describe the target.  The factorization
$B_-=T_{\rm ad}U_H^-$ restricts to an isomorphism
\[
 T_{\rm ad}\times Y\longrightarrow H_{\rm ad}^{w_0,e},
 \qquad (t,v)\longmapsto tv.
\]
In particular, the left-torus quotient is a globally trivial principal
bundle.  The quotient uses the free $T_{\rm ad}$-action; the map
$T_H\to T_{\rm ad}$ may have a finite central kernel.

For reduced words, the identification in \cite[Section~1.4]{SW} gives
\[
 \operatorname{Conf}^{w_0}_e(\mathcal A_{\rm ad})
 \simeq H_{\rm ad}^{w_0,e},
\]
and similarly for the simply connected group.  Under these
identifications the map from the simply connected decorated space to the
adjoint one is the finite etale central isogeny.  The initial A torus and
its singly mutated A tori are open by \cite[Lemma~3.37]{SW}; their union
has complement of codimension at least two by
\cite[Proposition~3.42]{SW}.  We use the all-top-word triangulation, which
is one of the triangulations used in that proof.

Propositions~3.43--3.44 of \cite{SW} identify the restriction of the central
map to the initial full seed torus as a finite monomial isogeny.  Both
propositions are needed, since the latter supplies the open-string
coordinates.  By \cite[Propositions~A.28--A.30]{SW}, this map transports
through mutation as a monomial map with the same finite lattice index.
On each singly mutated A torus its kernel translations agree on the
function field with the deck transformations of the central isogeny.
Thus that torus is invariant under the deck group.  Its finite quotient
is the corresponding X torus and is open in the adjoint decorated
space.  Finite etale descent preserves codimension, so the initial and
singly mutated X tori cover $H_{\rm ad}^{w_0,e}$ in codimension one.

We next compute the left-torus quotient on the full coordinate set.
On the SW factorization chart write
\[
 g=y_{r_1}(p_1)\cdots y_{r_m}(p_m)t,
 \qquad t_i=\alpha_i(t).
\]
Write $\mathrm{first}(i)$ for the first occurrence of color $i$. Let $L_i$ be the leftmost open string on level $i$, and let $S_k$ be the
string immediately to the right of node $k$.  The latter is closed when
$k^+\leq m$ and is a rightmost open string otherwise.  Definitions~3.12--3.13
of \cite{SW} give
\[
 L_i=p_{\mathrm{first}(i)}^{-1},\qquad
 S_k=\begin{cases}p_k/p_{k^+}&k^+\leq m,\\p_kt_{r_k}&k^+=m+1.\end{cases}
\]
Left multiplication by $s\in T_{\rm ad}$ sends
$p_k$ to $\alpha_{r_k}(s)^{-1}p_k$ and $t_i$ to $\alpha_i(s)t_i$.
It therefore multiplies $L_i$ by $\alpha_i(s)$ and fixes every $S_k$.
The simple roots form an integral character basis of $T_{\rm ad}$.
Consequently the quotient deletes precisely the leftmost open strings,
while retaining every rightmost frozen string.

Moving all Cartan factors to the left gives
\[
 u_k=p_kt_{r_k}
     =\prod_{\substack{l\geq k\\r_l=r_k}} S_l.
\]
The exponent matrix of this transformation is upper unitriangular.
Telescoping now gives $q_k=S_k^{-1}$, including the last occurrence of
each color.  Thus this is an integral identification of the full
$m$-dimensional torus.

Every mutable X coordinate is a closed-string coordinate and is invariant
under the left torus.  A mutation multiplies every other coordinate by a
rational expression in that invariant coordinate.  Hence the left open
strings keep their simple-root weights, and all retained coordinates
remain invariant, in each singly mutated chart.  These invariant open
charts descend through the principal torus bundle to open tori in $Y$.
The complement has codimension at least two because its inverse image
has codimension at least two upstairs.

For the exchange columns, the Cartan matrix of $H$ is the transpose of $(a_{ij})$. At a negative SW node
of color $i$, with left and right strings $a,b$, its contributions are
$\epsilon_{ab}=1$ and $\epsilon_{ba}=-1$.  If a string $c$ on level $j$
crosses its triangle, its contributions are
\[
 \epsilon_{ac}=\tfrac12 a_{ij},\quad
 \epsilon_{bc}=-\tfrac12 a_{ij},\quad
 \epsilon_{ca}=-\tfrac12 a_{ji},\quad
 \epsilon_{cb}=\tfrac12 a_{ji}.
\]
Sum these over the endpoints of the closed string $S_k$.  The
predecessor and successor entries are respectively $1$ and $-1$.
For $a<k<a^+<k^+$, the two half-contributions sum to
$a_{r_ar_k}=a_{i_ai_k}$; for $k<a<k^+<a^+$ they sum to its negative.
All other contributions cancel or vanish.  This calculation includes
the rightmost frozen strings, with missing successors written as $m+1$.
It gives precisely the full columns $B$.  Opposition changes colors but
preserves Cartan entries; no word reversal is used.

For an integer $b$, put $[b]_+=\max(b,0)$. SW's mutation formula is
\[
 S'_a=S_a S_k^{[B_{ak}]_+}(1+S_k)^{-B_{ak}}\quad(a\ne k),
 \qquad S'_k=S_k^{-1}.
\]
Since $q=S^{-1}$, it becomes
\[
 q'_a=q_a q_k^{[C_{ak}]_+}(1+q_k)^{-C_{ak}},
 \qquad q'_k=q_k^{-1},\qquad C=-B.
\]
In particular the dual Cartan transpose and the full frozen rows are
both retained.

Let $Y^\circ$ be the indicated partial toric blowup.  The successor minor makes each ray $C_k$ primitive.  The center character
$q_k$ is primitive and annihilates $C_k$, so its center is smooth,
connected, and reduced.  Distinct centers lie in disjoint boundary
divisors of the partial fan.  Thus $Y^\circ$ is smooth.  The inverse
mutation formula shows that an adjacent torus lifts to this blowup: along
$1+q'_k=0$ the valuation vector of $q$ is $C_k$, and the pullback of the
center ideal is invertible.  The adjacent torus meets the exceptional
divisor densely.  In the standard two-dimensional blowup factor, these tori omit the codimension-two
locus $\rho=t=0$ in
$\operatorname{Spec}\C[\rho,t,(\rho t-1)^{-1}]$.

For $f\in\OO(Y)$, its restrictions to the initial and adjacent tori are
regular.  Every prime divisor of $Y^\circ$ meets one of these tori:
the only divisors missing the initial torus are the exceptional divisors.
Normality therefore extends $f$ over $Y^\circ$.  Since $Y$ is affine,
these extensions define a morphism $Y^\circ\to Y$.

The rational form $\bigwedge_a d\log q_a$ is nonvanishing on every
quotient seed torus, since a mutation has logarithmic Jacobian
determinant $-1$.  Codimension-one coverage extends it as a nowhere-zero
top form on smooth $Y$.  It is likewise nonvanishing on $Y^\circ$:
blowing up a smooth codimension-two center in one boundary component
preserves this logarithmic form after removing the strict boundary.
The morphism pulls the target form back to the source form, since they
agree on the common torus.  Its Jacobian is therefore invertible, so it
is etale.  Being separated and birational over a normal target, it is an
open immersion by Zariski's main theorem.  Its image contains the initial
and adjacent quotient tori, and its complement has codimension at least
two.
\end{proof}

\section{A nef Richardson compactification}\label{sec:target}

The open embedding
\[
 j:Y\longrightarrow H/B_-,\qquad
 j(g)=\dot w_0^{-1}g^{-1}\dot w_0B_-
\]
identifies $Y$ with an open Richardson variety. Inversion permutes its
extremal minors, up to nonzero pinned constants. The double
Bott--Samelson construction of \cite[Sections 4.2 and 4.5, equation (11)]{KS}
gives a smooth projective compactification $X$ and a projective map
$\pi:X\to H/B_-$ extending $j$, with reduced SNC boundary
$D=(X\setminus Y)_{\mathrm{red}}=\sum_iD_i$
and $K_X+D\sim0$. Indices on $D_i$ run through all boundary components, independently of the simple-root indices on fundamental line bundles. Write $\mathcal L_i$ for the fundamental
globally generated line bundle on $H/B_-$, normalized to have
degree $\delta_{ij}$ on the $j$th simple Schubert curve. Here
$2\rho$ is twice the sum of the fundamental weights, and
$\mathcal L(2\rho)=\bigotimes_i\mathcal L_i^{\otimes2}$.
The divisor of the pulled-back boundary section of $\mathcal L(2\rho)$ is nef and has the form
\[
 A=\sum_i a_iD_i,\qquad a_i>0.
\]
Indeed, the flag boundary is an effective Cartier divisor of class
$2\rho$, and $\pi^{-1}(Y)=Y$. Its local equation therefore has positive
order at the generic point of every $D_i$. These properties persist under blowups of
logarithmic strata. We use a common logarithmic modification as $X$, chosen below.

There is a zero-dimensional boundary stratum. To see this in the
double Bott--Samelson construction, take the intersection of all standard
boundary divisors in the ordinary Bott--Samelson factor. It is the
all-identity point, whose image is the base flag. That flag belongs to
the open opposite Bruhat cell, over which the opposite factor is an
isomorphism. Near the resulting point the fiber product is consequently
the ordinary factor, and its $m$ standard boundary divisors intersect
transversely in that point. A subdivision of a maximal cone under a
logarithmic modification again has maximal cones. Thus the final model
still has a zero-dimensional boundary stratum.

Since $\mathcal O(Y)$ is a localization of a polynomial ring,
$\operatorname{Pic}(Y)=0$. Divisor localization gives
\[
 \bigoplus_i\mathbb Z[D_i]\longrightarrow
 \operatorname{Pic}(X)\longrightarrow\operatorname{Pic}(Y)\longrightarrow0.
\]
In particular, the complete boundary intersection vector determines the
numerical curve class. All lifted effective classes must still be retained
when a comparison theorem is applied.

\begin{lemma}
\label{lem:r6-volume-compatible}
Let $\omega=\bigwedge_{a=1}^m d\log q_a$ be the volume form on the
specified full factorization torus.  On the full double
Bott--Samelson compactification, and on every smooth logarithmic
modification used here, one has
\[
 \operatorname{div}_X(\omega)=-D.
\]
Thus the cluster volume form and the log Calabi--Yau volume form of
the compactification differ by a nonzero constant.  In particular,
every boundary component is essential for this specified volume form.
\end{lemma}
\begin{proof}
Proposition~\ref{prop:target-quotient-big-open} shows that $\omega$
extends to a nowhere-zero regular top form on $Y$.  At the generic
point of any prime divisor of a smooth compactification, choose a
uniformizer $t$ and write $q_a=t^{v_a}u_a$, with $u_a$ a unit in the
local discrete valuation ring.  Then
\[
 d\log q_a=v_a\frac{dt}{t}+\frac{du_a}{u_a}.
\]
In the top exterior product, at most one factor $dt/t$ occurs.
Consequently $\omega$ has at worst a simple pole along every prime
divisor.  Since it has no zero or pole in $Y$, the divisor
$\operatorname{div}_X(\omega)+D$ is effective.  It is linearly
equivalent to $K_X+D\sim0$.  An effective linearly trivial divisor
on an integral projective variety is zero, proving the displayed
identity.  The same argument applies after every smooth logarithmic
modification, which preserves $K_X+D\sim0$ and the interior $Y$.
\end{proof}

Fix an interior frame $h$ and define
\[
 \pi_h=(\dot w_0^{-1}h^{-1}\dot w_0)\cdot\pi,
 \qquad Y_h=\pi_h(Y)\subset H/B_-,
\]
where the group acts by left translation on the flag variety. Conditions on the
values of a normalized flag map below use $Y_h$ and the
corresponding translated auxiliary isomorphism loci.
For the actual MV-source map
\[
 g(z)=\dot w_0n(z^{-1})^{-1}\dot w_0^{-1}h^{-1}
\]
one has the identity
\begin{equation}\label{eq:flag-source-identity}
 \pi_h(g(z))=n(z^{-1})B_-.
\end{equation}
Thus this flag map retains every complete vector $n(z^{-1})v_i^-$.
Constant left translation preserves the fundamental line bundles and their
degrees. For $v\in V_H(\omega_i)^*$, define
\[
 \ell_{i,h}(v)=\langle v,\dot w_0^{-1}h^{-1}v_i^+\rangle.
\]
For $u\in V_H(\omega_i)^*$, invariance of the representation pairing gives
\begin{equation}\label{eq:frame-section-cancellation}
 \ell_{i,h}(\dot w_0^{-1}h^{-1}\dot w_0u)=\langle u,\dot w_0^{-1}v_i^+\rangle.
\end{equation}
The pullback divisor of this section is consequently independent of $h$.
On the source, its normalized homogeneous section is
\[
 Q_i(z)=z^{\nu_i}F_i(g(z))/F_i(h^{-1}).
\]

The boundary contact of a generic auxiliary zero of color $i$ is universal:
it is the divisorial valuation of the reduced irreducible divisor $F_i=0$
in $V$.  The fundamental extremal section has degree $\delta_{ij}$ on the
simple Schubert curves, so the corresponding Schubert divisor occurs
with multiplicity one.  Properness gives this valuation its unique center
on $X$.  The birational map $X\to H/B_-$ is an isomorphism over a dense
open of this divisor.  We retain its color and connected stratum component
as part of the contact data.

For a generic source $Z$, let $\beta_Z$ be its curve class on this target.
For contacts $(p_Z,p_W;-p_T)$, with $\nu_T=\nu_Z+\nu_W$, the universal
auxiliary contacts cancel.  Boundary localization and the contact-degree
identity consequently force the tail class
\begin{equation}\label{eq:tail-class}
 \beta_Z+\beta_W-\beta_T.
\end{equation}
This uniqueness concerns numerical classes on the specified model.
When a comparison changes models, every admissible lifted class is
retained until the proper pushforward has been taken.

\subsection{A common divisorial model}

For a projective target, $NE(X)$ denotes the monoid of effective
integral numerical curve classes, including zero. Homological classes
will be retained separately in mapping spaces. A contact is
\emph{divisorial} when it is an integral multiple of one ray of the
boundary cone complex. The coefficient $c_i(p)$ is its value on a
local equation for $D_i$, and its connected stratum is part of the data.

\begin{lemma}\label{lem:finite-model-choice}
Fix two source cycles, their total numerical curve class and their
complete external and auxiliary contacts. There is a projective
logarithmic modification on which these contacts and every nonzero
nonnegative central contact arising from an effective splitting are
divisorial.
\end{lemma}
\begin{proof}
Choose an ample divisor $H$ and integral divisor generators $L_j$.
For sufficiently large integers $M_j$, both $M_jH+L_j$ and
$M_jH-L_j$ are ample. An effective splitting
$\beta=\beta_C+\beta_T$ satisfies
\[
 |L_j\cdot\beta_C|\leq M_jH\cdot\beta.
\]
There are therefore finitely many possible numerical central classes.
For each one, its boundary contact is forced by
\[
 c_i(p_C)=D_i\cdot\beta_C-
              \sum_{a\text{ auxiliary}}c_i(p_a).
\]
Keep the nonnegative vectors and all compatible connected strata.
Insert their rays and the prescribed external rays by a common
projective subdivision, followed by a smooth subdivision.

If a nonnegative central contact appears on the new model, its
pushforward is nonnegative because the pullbacks of the old boundary
divisors have nonnegative coefficients. It satisfies the same degree
equations and belongs to the finite list. A subdivision preserves
the underlying cone-complex point, and the inserted ray persists.
Thus this one model has the asserted property.
\end{proof}

All subsequent classes and contacts refer to this fixed model.
The boundary still generates its Picard group and supports a nef
divisor with positive coefficients. Fix a very ample divisor $H$.
Genus-zero stable maps with a fixed number of markings and fixed
$H$-degree form a finite-type stack. Their integral homology class is
locally constant, so only finitely many homological classes occur.
This applies also to effective subcurves of bounded degree.
Every mapping space below includes all these homological decorations.

For a tail with fixed complete contacts, all its boundary intersections
are fixed. Expressing $H$ as an integral linear combination of the
boundary divisors fixes its $H$-degree and proves finiteness of the
possible homological tail classes. The same reasoning applies on each
refinement, using the new boundary and lifted contacts. The all-lifts
identity of \cite[Corollary~1.4 and Section~10]{JohnstonBirational} is
applied first to each downstairs homological class, and only then
summed over these finite collections.

\paragraph{Logarithmic mapping spaces.}
We write commutative monoids additively. For a monoid $Q$, its groupification is $Q^{\rm gp}$, its real dual cone is $Q^\vee_{\mathbb R}=\operatorname{Hom}(Q,\mathbb R_{\geq0})$, and $K[Q]$ is its monoid algebra over a field $K$, with basis $\chi^q$ and multiplication $\chi^q\chi^{q'}=\chi^{q+q'}$. A fine saturated monoid is finitely generated, embeds in its groupification, and contains every element of that groupification whose positive integral multiple lies in the monoid. It is sharp if its only unit is zero. A homomorphism is local if the inverse image of the units is the set of units.

For a boundary component $D_i$, put $D_i^\circ=D_i\setminus\bigcup_{j\ne i}D_j$. The boundary cone complex $\Sigma(X)$ has a cone $\mathbb R_{\geq0}^r$ for each connected stratum lying in exactly $r$ boundary components, with face maps induced by inclusions of strata; its integral coordinates record boundary orders. Write $T_X^{\log}=T_X(-\log D)$ for the vector bundle of derivations preserving the ideals of the boundary components. We use rational Chow groups. For a morphism $F\to B$,
$A^e(F\to B)$ denotes bivariant operations lowering cycle dimension
by $e$ after every allowed base change to $B$, with support in the
corresponding base change of $F$. A superscript $!$ denotes refined
Gysin pullback for a regular embedding, or the virtual pullback
specified by a relative perfect obstruction theory. Proper
pushforward includes residue-field degrees and stack stabilizers.
The class $[\mathcal L]^{\vir}$ is the virtual fundamental class
defined by that obstruction theory. A basic logarithmic map has the minimal
fine saturated (fs) base monoid determined by its contact and smoothing
data. For a tropical type $\tau$, its cone parametrizes vertex
positions and nonnegative edge lengths satisfying the contact
relations. A decoration assigns an actual curve class to each vertex.
A $\tau$-marked space includes the degenerations of this type.
The Artin fan $\mathcal X$ of $(X,D)$ retains these boundary monoids
and forgets the geometry of their strata. Evaluation enhancement at
a marking is base change from its Artin-fan evaluation to the
corresponding target stratum. Punctured maps allow a negative contact
at a specified leg; their base retains the puncturing ideal.
We use the constructions of \cite[Sections~2--5]{ACGSPunctured} with
these conventions.

The target is projective with Zariski logarithmic structure, and the
labelled boundary divisors generate its characteristic group sheaf.
Theorems~3.12 and~3.18 and Corollary~3.19 of \cite{ACGSPunctured}
therefore give properness of the stable mapping spaces with fixed
global contacts and homological class. Forgetting logarithmic
structures gives a proper map to the ordinary stable-map space.
For a divisorial negative output, its fiber over a point of the open
boundary stratum is proper. The logarithmic point constraint is finite
over that fiber by Lemma~\ref{lem:divisorial-point-constraint-cycle-direct}.

\section{Generic smooth incidence}\label{sec:generic-incidence}

For $r\geq3$, $\overline{\mathcal M}_{0,r}$ is the moduli space of stable genus-zero curves with $r$ labelled markings, and $\mathcal M_{0,r}$ is its smooth-domain locus. The map $\operatorname{st}$ forgets the target map and stabilizes the marked curve; $\operatorname{ev}_j$ evaluates it at the $j$th marking.

A component of a mapping space is horizontal when its curve/frame
map dominates the product of the smooth marked-curve space and $Y$.
A nondegenerate map has no component contained in the boundary.
A map with smooth rational domain is free here when
$f^*T_X(-\log D)$ is trivial. We first establish that all points of
a general incidence have these properties.

\subsection{The support of the incidence}

Throughout this section, $m=\dim X$, as above. For the compactifications used here, the integral affine structure needed
below is the one of \cite[Section~3, Assumption~3.1 and Lemma~3.2]{JohnstonNaive}.
Lemma~\ref{lem:r6-volume-compatible} identifies the specified torus volume
form with a generator having divisor $-D$.  Hence every cone is good,
$B=|\Sigma(X)|$, and the union $B_0$ of the relative interiors of the
cones of dimensions $m$ and $m-1$ is an open integral affine manifold.
A point of $B_0$ has an affine neighborhood in the whole support $B$. Here a cone is good when its boundary components are essential for the chosen logarithmic volume form.

For a tropical type $\alpha$, write $G_\alpha$ for its graph and $\sigma(v)$ for the target cone of a vertex $v$. At a parameter $t\in\alpha$, $h_t(v)\in\sigma(v)$ is its position; $\operatorname{ev}_v:\alpha\to\sigma(v)$ is the resulting evaluation map. An oriented edge $e:v\to w$ has integral slope $u(e)$ and length function $\ell_e$, with $h_t(w)-h_t(v)=\ell_e(t)u(e)$ in a common cone chart. We write $\operatorname{relint}$ for relative interior. A type is realizable when these positions and lengths realize its prescribed cones and graph on the relative interior of its parameter cone. It is balanced when the outgoing slopes sum to zero at each vertex in an affine chart of $B_0$.

A nontrivial broken-line type is a realizable balanced genus-zero type with two legs, a nonzero nonnegative incoming contact and a nonzero outgoing contact in a good cone. Its parameter cone has dimension $m-1$, and the outgoing leg sweeps an $m$-dimensional set. Its spine is the path joining the two legs, including those legs; see \cite[Definitions~4.1, 4.4 and~4.6]{JohnstonNaive}.

A monoid is integral if it embeds in its groupification. For integral monoids $P,Q$, a homomorphism $f:P\to Q$ is integral if every relation $f(p_1)+q_1=f(p_2)+q_2$ admits $a_1,a_2\in P$ and $q\in Q$ such that
\[
 p_1+a_1=p_2+a_2,\qquad
 q_1=f(a_1)+q,\qquad q_2=f(a_2)+q.
\]
It is exact if $(f^{\rm gp})^{-1}(Q)=P$ inside $P^{\rm gp}$, where $f^{\rm gp}$ is the induced homomorphism of groupifications.

\begin{lemma}\label{lem:r4-integral-cones}
Let $P\to Q$ be a local integral homomorphism of sharp fine saturated
monoids. It is injective and exact, and the map of real dual cones
$Q^\vee_{\mathbb R}\to P^\vee_{\mathbb R}$ is surjective.
\end{lemma}

\begin{proof}
Write $f:P\to Q$. Applying the integrality criterion with $q_1=q_2=0$, sharpness and locality give $a_1=a_2=q=0$,
proving injectivity. If $f^{\rm gp}(p_1-p_2)=q_2\in Q$, apply the
criterion with $q_1=0$. It gives $a_1=q=0$ and $p_1-p_2=a_2\in P$,
which is exactness.

Identify $P^{\rm gp}\otimes_{\mathbb Z}\mathbb R$ with its image in
$Q^{\rm gp}\otimes_{\mathbb Z}\mathbb R$. Let $\operatorname{cone}(P)$ and $\operatorname{cone}(Q)$ be the nonnegative real spans of the respective monoids in these spaces. Exactness gives
\[
 \operatorname{cone}(Q)\cap(P^{\rm gp}\otimes_{\mathbb Z}\mathbb R)
       =\operatorname{cone}(P).
\]
Indeed, clear denominators to prove this on rational points and use the
rational polyhedrality of the intersection. Restriction of nonnegative
linear functionals is consequently surjective by the polyhedral
extension theorem: the dual of its image is the displayed intersection,
and the image is closed because it is polyhedral.
\end{proof}

\begin{lemma}\label{lem:r4-unmarked-branch}
Let $\alpha$ be the tropical type of an actual genus-zero stable,
possibly punctured, logarithmic map. Suppose $v$ maps to a good zero stratum of a target of
dimension $m$, and $\dim\operatorname{ev}_v(\alpha)=m$.
Then every branch of $G_\alpha\setminus\{v\}$ contains a marked leg.
\end{lemma}

\begin{proof}
Suppose a branch contains none. If its first edge has nonzero slope,
cut it and retain the component away from $v$. Replacing the node by a
puncture gives an actual stable punctured map. Its type is realizable
and balanced by \cite[Lemma~4.3]{JohnstonNaive}, has exactly one leg of
nonzero slope, and that leg lies in the good maximal cone $\sigma(v)$.
Its leg image contains $\operatorname{ev}_v(\alpha)$, contradicting
\cite[Lemma~4.7(1)]{JohnstonNaive}.

If the first slope is zero, follow the maximal connected subtree of
zero-slope edges within this branch. All its vertices have the same
tropical position as $v$ and map to the same zero stratum. A first
nonzero edge leaving this subtree gives the preceding contradiction.
If none exists, the whole branch is a constant unmarked tree with one
attachment. For $N$ vertices it has $2(N-1)+1$ flags in total, whereas
stability would require at least $3N$. This is impossible.
\end{proof}

\begin{lemma}\label{lem:r4-balanced-face}
Suppose $K_X+D\sim0$. A face contraction of the tropical type of an actual stable
logarithmic map is realizable and balanced in the sense of
\cite[Definitions~2.2 and 4.1]{JohnstonNaive}.
\end{lemma}

\begin{proof}
Restrict the universal tropical family to the face and contract its
zero-length edges. This gives the tropical realizability required by the stated definitions.

Write $\tau\subset\alpha$ for the face, choose
$q\in\operatorname{relint}\tau$ and
$q_0\in\operatorname{relint}\alpha$, and consider
$q_\epsilon=q+\epsilon q_0$. Let $v$ be a vertex of the contracted type whose image $h_q(v)$ lies
in $B_0$.  Choose an integral affine neighborhood $W\subset B_0$ of
that point.  The finitely many vertices and internal edges of the
cluster contracted to $v$ converge to $h_q(v)$ as
$q_\epsilon=q+\epsilon q_0$ tends to $q$.  Shrinking $W$ within an
affine chart and taking $\epsilon>0$ sufficiently small puts the whole
cluster in this one chart.  Each of its vertices satisfies the
balancing equation by \cite[Lemma~4.3]{JohnstonNaive}.  Summing these
equations in the common tangent lattice cancels each internal edge
with its opposite orientation, and gives precisely the balancing
equation at $v$.  

\end{proof}

\begin{lemma}\label{lem:r4-broken-expansion}
Suppose $K_X+D\sim0$, so that all cones of $\Sigma(X)$ are good.
Let $\gamma$ be the type of an actual stable punctured map marked by a
nontrivial broken-line type $\delta$. Every spine vertex of $\gamma$
has evaluation image of dimension $m-1$. The slope of every spine
edge, and of the outgoing spine leg, is transverse to the span of
the evaluation image at its incident spine vertices. If one spine
vertex maps to the origin at a parameter value, all spine vertices
do.
\end{lemma}

\begin{proof}
This is \cite[Corollary~4.9]{JohnstonNaive}, with the following details
also accounting for zero-slope side branches and the outgoing leg.
Let $\phi:G_\gamma\to G_\delta$ be the contraction. The restriction
to $\delta$ and \cite[Lemma~4.7(2)]{JohnstonNaive} give evaluation
dimension at least $m-1$ at every spine vertex $v$. Its target cone
therefore has dimension $m-1$ or $m$. In the former case the upper
bound is automatic. In the latter case the actual component is
constant and stable, so there is an unmarked branch off the spine.
Lemma~\ref{lem:r4-unmarked-branch} excludes evaluation dimension $m$.
Thus the span $H_v$ has dimension $m-1$ and equals the span obtained
from the broken-line face.

Every off-spine slope is tangent to $H_v$. Otherwise cutting that
edge would give a balanced one-leg type whose leg sweeps a
$m$-dimensional cone, contrary to
\cite[Lemma~4.7(1)]{JohnstonNaive}. A zero slope is already tangent.
The outgoing spine leg is transverse to $H_v$, since its image on
the broken-line face has dimension $m$. Balancing at $v$ consequently
forces the next spine direction to be transverse as well. Across an
edge contracted by $\phi$, the spans at its two endpoints agree.
Across an edge not contracted by $\phi$, transversality at both ends
is the transversality proved in
\cite[proof of Lemma~2.5]{GSCanonical} for the broken-line face.
Induction along the spine gives the assertion for every edge.

For consecutive spine vertices $v,w$ and any parameter $t$, write
$h_t(w)=h_t(v)+\ell_e(t)u(e)$ in the cone containing the edge.
If $h_t(v)=0$, then $\ell_e(t)u(e)\in H_w$. Transversality implies
$\ell_e(t)=0$, hence $h_t(w)=0$. Induction proves the last assertion.
\end{proof}

\begin{lemma}\label{lem:generic-supported}
Let $(X,D)$ be a smooth projective pair over an algebraically closed
field of characteristic zero, with reduced simple normal crossings
boundary, $K_X+D\sim0$, and at least one zero-dimensional boundary
stratum. Suppose $U=X\setminus D$ is affine, contains a dense torus,
and $\sum_i a_iD_i$ is nef with every $a_i>0$.
Fix genus-zero discrete data with $r\geq3$ markings, at least one of
contact zero, denoted $x_\infty$. All other contacts are nonnegative,
and every nonzero contact is divisorial on this model $X$.
Let $\mathcal L$ be the proper stack of basic stable logarithmic maps
with these data, and set
\[
 e=(\operatorname{st},\operatorname{ev}_\infty):
 \mathcal L\longrightarrow\overline{\mathcal M}_{0,r}\times X.
\]
There is a nonempty open $B^\circ\subset\mathcal M_{0,r}\times U$
whose entire inverse image is finite \emph{\'etale} over $B^\circ$.
Its actual domains are smooth. Write the nonzero contact at $x_j$ as
$m_j$ times the primitive vector of the ray corresponding to $D_{i(j)}$.
Each map $f$ in this inverse image satisfies
\[
 f^{-1}D=\sum_{j:\,m_j>0}m_jx_j,
 \qquad f(x_j)\in D_{i(j)}^\circ.
\]
Their logarithmic tangent pullbacks are trivial.

For every geometric point $b\in B^\circ$,
\[
 b^![\mathcal L]^{\rm vir}
       =[e^{-1}(b)]\quad\text{in }A_0(e^{-1}(b))_{\mathbb Q}.
 \tag{SGI}
\]
One open suffices for finitely many fixed discrete data with the same
number of markings.
\end{lemma}

\begin{proof}
Write $\beta$ for the fixed discrete data and put
$\mathfrak A=\mathcal M^{\rm ev}(\mathcal X,\beta)$, with the
finite-type restriction of \cite[Remark~2.3]{JohnstonNaive}.
Choose a zero stratum $s$ and the degeneration $T\to X$ of
\cite[\S5, Step~3]{JohnstonNaive}. The closed log point has
characteristic $P=\mathbb N^m$, although the underlying scheme $T$
is a DVR. The dual cone map has full-dimensional image
$C_0\subset\sigma_s$. It can be chosen unimodular, by choosing a
smooth full-dimensional cone in a sufficiently fine subdivision,
and transverse to the finitely many evaluation cones.
By \cite[Lemmas~5.5--5.6]{JohnstonNaive},
$\mathfrak A_T\to T$ and $\mathfrak A_s\to s$ are integral,
and log and ordinary fiber dimensions agree.

The map $\mathcal L\to\mathfrak A$ is strict; see
\cite[Definition~5.14 and the following paragraph, p.~97]{ACGSPunctured}.
Strictness persists under the fs base changes used here.
At an actual point $z\in\mathcal L_s$, let $\alpha$ be its
unconstrained basic tropical type and $Q$ its characteristic after
the point-constraint base change. It is also the characteristic at
its image in $\mathfrak A_s$. Thus $P\to Q$ is local and integral,
and Lemma~\ref{lem:r4-integral-cones} gives a surjection
\[
 Q^\vee_{\mathbb R}
    =\alpha\mathbin{\times}_{\sigma_s}C_0\longrightarrow C_0.
\]
Consequently the output evaluation of the \emph{actual} type
$\alpha$ has dimension $m$.

Fix a smooth marked curve $c\in\mathcal M_{0,r}$ and consider an
actual incidence point over the generic point of $T$. Properness,
after a finite field extension, extends it over a DVR dominating
$T$; separatedness keeps its stabilization equal to $c$.
The output component in its special fiber maps to $s$ and is
constant. Lemma~\ref{lem:r4-unmarked-branch} shows that it has no
unmarked branches. It therefore survives stabilization, and each
branch contains exactly one of the other markings, since $c$ is
smooth. Its actual valence is exactly $r$.

Choose a generic point $\xi$ of a component of $\mathfrak A_s$
specializing to the image of $z$, and let $\tau\subset\alpha$ be
the resulting face type before the point-constraint base change.
Equality of log and ordinary fiber dimension gives
$\operatorname{rank}Q_\xi^{\rm gp}=m$. The cone
$Q_\xi^\vee=\tau\times_{\sigma_s}C_0$ has a point interior in
both factors, because ghost morphisms at points are local.
Since $C_0^{\rm gp}\to\sigma_s^{\rm gp}$ is an isomorphism,
\[
 \dim\tau=\dim\operatorname{ev}_{\rm out}(\tau)=m.
\]
Lemma~\ref{lem:r4-balanced-face} proves realizability and balancing
of $\tau$ from the actual special type $\alpha$.

Let $K$ be the connected cluster of special vertices contracted to
the generic output. For every $v\in K$, its evaluation restricted
to $\tau$ equals the output evaluation. Thus $\sigma(v)=\sigma_s$,
its actual component is constant and stable, and its full evaluation
image has dimension $m$. Lemma~\ref{lem:r4-unmarked-branch} applies
at each $v$. An unmarked generic output branch would pull back to
an unmarked branch attached to one such $v$, so the generic output
has at most $r$ flags. But contraction gives
\[
 \operatorname{val}({\rm out}_{\tau})
   =\sum_{v\in K}\operatorname{val}(v)-2(|K|-1)
   \geq r+|K|-1.
\]
Hence $K$ is the singleton consisting of the actual output, and
the generic output has exactly $r$ flags. No adjacent edge is
contracted.

Cut each nontrivial generic branch at its edge adjacent to the output,
and call its type $\tau_i$. Let $v_i$ be the vertex adjacent to the new
outgoing leg, and orient its slope $u_i$ towards the omitted output.
In compatible framed cone charts, the basic-cone description in
\cite[Proposition~2.64, equation~(2.31)]{ACGSPunctured} gives the endpoint
space
\[
 E_i=\left\{(q_i,\lambda_i)\in\tau_i\times\mathbb R_{\geq0}:
      \operatorname{ev}_{v_i}(q_i)+\lambda_i u_i\in\sigma_s\right\},
 \qquad
 \epsilon_i(q_i,\lambda_i)
      =\operatorname{ev}_{v_i}(q_i)+\lambda_i u_i.
\]
At a point induced by $\operatorname{relint}\tau$, the cut length is
positive, all retained vertices and edges have their prescribed types,
and the endpoint lies in $\operatorname{int}\sigma_s$. Thus
$q_i\in\operatorname{relint}\tau_i$, the endpoint condition imposes
only an open inequality there, and
$\dim E_i=\dim\tau_i+1$.
Restoring the root and the cut edges in the same basic-cone coordinates
identifies the whole cone, locally at this point, with the fiber product
of the $E_i$ over $\sigma_s$; a direct marked leg adds no coordinate.
Each $\epsilon_i$ has rank $m$, since the full-dimensional output
evaluation of $\tau$ factors through it. Consequently
\[
 m=\dim\tau=m+\sum_i(\dim E_i-m).
\]
Every summand is nonnegative. Hence every nontrivial branch has
$\dim\tau_i=m-1$ and outgoing-leg image of dimension $m$.
Its outgoing slope is necessarily nonzero.

If its input marking had zero contact, deleting that leg would
leave the same cone and balancing equations, producing a one-leg
type contradicting \cite[Lemma~4.7(1)]{JohnstonNaive}.
Here we delete the zero leg without stabilizing the graph.  The tropical
types in \cite[Definitions~2.1 and~4.1]{JohnstonNaive} have no stable-valence
requirement, and deletion of a zero-slope leg removes neither a basic-cone
coordinate nor a balancing term.  Thus two-valent vertices cause no
change in the cone, and \cite[Lemma~4.7(1)]{JohnstonNaive} applies to the
resulting one-leg type.
Thus all extra zero markings are directly on the generic output,
and also on the actual special output because $K$ is a singleton.
Every nontrivial remaining branch is a broken-line type: its input
is positive, its outgoing slope is nonzero, it is realizable and
balanced, and it has the two required dimensions. Both leg cones
are good because $K_X+D\sim0$.

Splitting the actual special curve at the corresponding surviving node,
followed by pre-stabilization and basicification, gives a stable punctured
map of type $\gamma_i$ marked by $\tau_i$; compatibility of splitting
with markings is \cite[Propositions~5.2 and~5.4]{ACGSPunctured}.
Replacing a node by a puncture preserves the special-point counts, so
the retained actual components remain stable.

Let $\rho\subset\alpha$ describe the original generic
fiber over the DVR. Its output lies at the origin because the
generic point constraint lies in $U$. The cut outgoing slope is
transverse to the span at the adjacent vertex by
Lemma~\ref{lem:r4-broken-expansion}, so that vertex is also at the
origin. The same lemma propagates zero along each branch spine.
Direct legs have no additional vertices. Every vertex of the
generic spine therefore lies in the zero cone.

If a generic component lay in $D$, choose the first boundary vertex
on a path from the spine to it. Its predecessor lies in the zero
cone, and the branch beyond the connecting edge has no markings.
The nef full-boundary divisor excludes this tree by
\cite[\S5, Step~1]{JohnstonNaive}. Thus every actual generic
incidence point is nondegenerate.

The required smoothness and freeness follow from the following arguments
in \cite[Proposition~3.13]{KY}. On the present minimal log Calabi--Yau
compactification, every boundary component is essential and the
prescribed nonzero contacts are divisorial.  On the nondegenerate
locus, \cite[Lemma~5.2]{JohnstonNaive} identifies logarithmic maps with
ordinary maps having exactly these boundary contacts and a unique basic
logarithmic lift.

First fix a smooth marked rational curve.  At a map with the prescribed
contacts, the tangent space with the marked domain fixed is
$H^0(\mathbb P^1,E)$, where $E=f^*T_X(-\log D)$ has degree zero.  If
evaluation at a zero-contact marking dominates $U$ on any irreducible
locally closed locus, generic smoothness supplies $m$ such sections
which span the fiber at that marking.  Their determinant is a section
of a degree-zero line bundle and is nonzero at one point, so it is
nowhere zero.  Hence $E\simeq\mathcal O_{\mathbb P^1}^{\oplus m}$.
Applying the same argument to the nonfree locus shows that its
evaluation image is not dense.  On the free locus, evaluation at
every zero-contact marking has full rank, so finitely many such
evaluations can simultaneously avoid any prescribed proper closed
subset.  This proves the evaluation assertion of \cite[Lemma~3.9(1)]{KY}.

Next, bounded-degree nonconstant $\mathbb A^1$-curves have images in a
proper closed subset $Z\subset U$.  Indeed their moduli are of finite
type, and a component whose evaluation image was dense would give a
finite-type covering family of $\mathbb A^1$-curves, excluded by the
logarithmic volume-form argument above.  For a nondegenerate stable map
whose stabilized marked domain is smooth, a component which is
unstable as a marked curve must be nonconstant.  Affineness of $U$
forces such a component to meet $D$.  The exact contact condition and
the bound of two special points then force it to have one boundary
marking and one attaching node.  Thus it is an $\mathbb A^1$-leaf.
Contracting all these leaves gives the same smooth marked
stabilization, with the former attachments now assigned contact zero.
Only finitely many subsets of the positive markings and finitely many
effective residual curve classes occur.  Apply the preceding free-map
argument to all these data and require the attaching evaluations to
avoid $Z$.  An $\mathbb A^1$-leaf would put one of them in $Z$, a
contradiction.  This is the argument of
\cite[Lemmas~3.10--3.11]{KY}.

The argument applies to every zero-contact set $I$ in \cite[equation~(3.1)]{KY}.
Varying the marked smooth curve gives the nonempty open of
$\mathcal M_{0,r}\times U$ used in
\cite[Proposition~3.13]{KY}.  Thus, when there are at least two positive
inputs, we can choose $c$ and the generic point constraint so that
every nondegenerate incidence point has stable, hence smooth, actual
domain and trivial logarithmic tangent pullback.

The cases with fewer positive inputs are immediate from the same
generic type: with one positive input, balancing at the output is
impossible; with none, all legs are at the output and there are no
other branches, leaving only the constant class. For that class
the asserted incidence is the family of constant maps.

A nonempty bad generic fiber of finite type would have a closed
point over a finite extension. The preceding argument therefore
controls the entire actual generic incidence.

Let $G\subset\mathcal L$ be the locus of nondegenerate maps with smooth
marked domains and trivial logarithmic tangent pullback. This is open:
in the smooth genus-zero family the pullback has degree zero, and its
triviality is the open condition that both cohomology groups after
twisting by $-x_\infty$ vanish. Properness of $e$ makes
$e(\mathcal L\setminus G)$ closed. The generic-constraint argument
shows that its complement in $\mathcal M_{0,r}\times U$ is nonempty;
choose $B^\circ$ in this complement.
Every geometric point above $B^\circ$ has trivial stabilizer: the smooth
rational domain has $r\geq3$ labelled markings, and the nondegenerate
map has a unique basic logarithmic lift. With its marked domain and
output evaluation fixed, the relative deformation-obstruction complex is
\[
 R\Gamma\!\left(\mathbb P^1,
 f^*T_X(-\log D)(-x_\infty)\right)
 =R\Gamma\!\left(\mathbb P^1,\mathcal O(-1)^m\right)=0.
\]
Thus $e$ is representable and \'etale above $B^\circ$. Being also proper,
it is finite \'etale. On this open the logarithmic obstruction theory is
unobstructed of the expected dimension, so the virtual class restricts
to the fundamental class.

Its refined fiber is consequently the fundamental zero-cycle of the finite
\'etale fiber. Since the entire fiber lies in this open, the
equality holds in $A_0(e^{-1}(b))_{\mathbb Q}$ itself.
Intersecting finitely many dense opens proves the final assertion.
\end{proof}

\begin{lemma}\label{lem:smooth-frame}
Let a family of smooth-domain maps with fixed marked positions and contact
data dominate the interior frame parameter.  Then the logarithmic tangent
bundle of its general map is trivial.  At every marked contact, the
leading units and the coordinates of the boundary stratum vary with full
rank as the frame varies.
\end{lemma}
\begin{proof}
At a general point choose $m$ parameter directions whose evaluations at
the frame form a basis of the target tangent space.  They give $m$ sections
of $E=f^*T_X(-\log D)$ on the already smooth $\PP^1$.  Their determinant
is nonzero at the frame.  Since $\det E\simeq\OO_{\PP^1}$, it is nowhere
zero and $E\simeq\OO_{\PP^1}^{m}$.  If locally
$x_j=z^{c_j}u_j(z)$ are the boundary coordinates and $y_l$ are stratum
coordinates, evaluation of these logarithmic sections at $z=0$ is exactly
the differential of $(u_j(0),y_l(0))$.  Its rank is $m$.  
\end{proof}

Triviality of $E$ gives
\begin{equation}\label{eq:acyclic}
H^0(\PP^1,E(-\infty))=H^1(\PP^1,E(-\infty))=0.
\end{equation}
For a rational tree $C$ and a rank-$m$ vector bundle $E$ trivial on each component, the same vanishing holds after twisting by a smooth frame point $x_\infty$. Indeed, remove a leaf $L$ not carrying $x_\infty$, attached to the remaining tree $C'$ at $p$. The normalization sequence is
\[
0\longrightarrow E(-x_\infty)\longrightarrow
 E|_{C'}(-x_\infty)\oplus E|_L
 \longrightarrow E_p\longrightarrow0.
\]
Evaluation $H^0(L,E|_L)\to E_p$ is an isomorphism and $H^1(L,E|_L)=0$. The cohomology sequence reduces both vanishings to $C'$. Induction ends with $\mathcal O_{\mathbb P^1}(-1)^m$. We use this for free maps glued at interior points.

\section{Recovering the MV source}\label{sec:inverse}

For a source of charge $\nu$, set $d=\height(\nu)$. Use one input at $0$, the old unit at $1$, the frame at $\infty$, and $d$ auxiliary markings. Initially work with all roots ordered. The full product of permutation groups on identical auxiliary types acts on both source and receiver; its quotient is used when the roots are unordered. 

Proposition~\ref{prop:finite} and Lemma~\ref{lem:smooth-frame} make
the contact leading units and stratum coordinates jointly generic in the
source family. The same lemma applies to every smooth-domain horizontal
receiver component.

\begin{lemma}\label{lem:pole-transfer}
Let $F$ be a rational function on $X$ regular on $Y$. A pole bound for $F$ on the generic complete leading family at a fixed contact stratum holds for every contact arc over a sufficiently small dense open of that stratum, including arcs over Artin coefficient rings.
\end{lemma}
\begin{proof}
Let $\Sigma$ be the specified stratum, locally cut out by boundary equations $x_1,\ldots,x_k$, and let $c_j>0$ be the prescribed order of $x_j$ along an arc centered there. Near its generic point write $F=A/(x^a v)$, where $A$ is regular, $v$ is a unit, $a\in\N^k$ and $x^a=\prod_jx_j^{a_j}$. For $b\in\Z$, let $J_b$ in $\C(\Sigma)[[x_1,\ldots,x_k]]$ be generated by the monomials $x^\alpha$, $\alpha\in\N^k$, with $\sum_jc_j\alpha_j\geq b$; thus $J_b$ is the whole ring for $b\leq0$. If $l\geq0$ is an integer, the bound $\ord_t F\geq-l$ on the generic leading family forces
\[
 A\in J_{\sum a_jc_j-l}.
\]
Otherwise the first lower-weight coefficient is a nonzero polynomial in the independent leading units and cannot vanish generically. Faithful flatness of completion gives the ideal membership in the local ring; it spreads to a dense open of the stratum. Substitution $x_j=t^{c_j}u_j$ sends this ideal into the required power of $t$ over every coefficient ring. Equality of generic orders follows by applying the argument to numerator and denominator initial terms separately.
\end{proof}

\subsection{Reconstruction from the prescribed poles}

\begin{lemma}
\label{lem:frame-independent-pole-transfer}
In the smooth-domain inverse, the complete-vector bounds may be transferred
using rational functions on the target that do not depend on the frame.

\end{lemma}
\begin{proof}
Use the coordinates of the vector
\[
 v\longmapsto \dot w_0^{-1}v^{-1}\dot w_0 v_i^-
\]
as rational functions on $X$, regular on $V=U_H^-$ and hence on $Y$.
For a source map $g=\dot w_0 n^{-1}\dot w_0^{-1}h^{-1}$, this vector is
\[
 (\dot w_0^{-1}h\dot w_0)n v_i^-.
\]
The left factor is a constant invertible matrix.  Consequently every
coordinate has pole order at most $\nu_i$ at the input.  The smooth-domain
frame determinant makes the leading units and the stratum coordinates
jointly general, so the monomial-ideal pole-transfer argument applies to
these frame-independent functions.  On a receiver map, multiply afterward
by the constant inverse matrix.  This gives the same upper bound for all
coordinates of the reconstructed $n v_i^-$.
The exact generic orders of the frame-independent extremal minors $F_i$
force the opposite inequality and hence the exact degree $\nu_i$.
The same argument at each of two separate inputs is unchanged by an
invertible integral left factor there.
\end{proof}

\begin{proposition}\label{prop:inverse}
For the complete contact data of $Z$, the smooth-domain horizontal ordinary receiver, modulo the full auxiliary relabeling group, is birational to $Z^{\rm aff}\times\mathscr H$. 
The unordered receiver has a unique smooth-domain horizontal component
for these data, and its class is $\beta_Z$.  Before taking the full
auxiliary relabeling quotient, the union of all such ordered components
is birational to the full ordered-root cover of the source.  This cover may have several components.
\end{proposition}
\begin{proof}
On an arbitrary such component, compose the map on its dense interior with $Y\subset V$ to recover a rational $V$-valued map $g(z)$. Polynomial generators of $\OO(V)$ have no poles at the auxiliary points in the actual source family. Lemma~\ref{lem:pole-transfer} transfers these bounds to the receiver. There are no other unmarked boundary preimages. Thus $g$ is regular away from $0$.

Set $h=g(\infty)^{-1}$ and
\[
 n(t)=\dot w_0^{-1}h^{-1}g(t^{-1})^{-1}\dot w_0.
\]
It is a based polynomial $U_H^+$-loop. Apply
Lemma~\ref{lem:frame-independent-pole-transfer} to the coordinates of
$\dot w_0^{-1}g^{-1}\dot w_0v_i^-$. The constant matrix of $h$ preserves
these pole bounds, giving
\[
 \deg(n(t)v_i^-)\leq\nu_i.
\]
The exact frozen orders, detected by the generic leading family, force
equality. Lemma~\ref{lem:full-vector} places the loop in an actual MV
source of the required charge.

The reconstruction retains $n,h$ and, on ordered covers, the root
ordering, and determines the original $X$-map by separatedness.
First take the quotient by the full auxiliary relabeling group.
A horizontal component still has dimension $m+d$ by
\eqref{eq:acyclic}.  Its generically injective reconstruction is
therefore dense in an actual $Z'^{\rm aff}\times\mathscr H$.
Lemma~\ref{lem:frame} and the injectivity of \eqref{eq:pairing} give
$Z'=Z$.  The forward construction and separatedness identify the
unordered component uniquely with the source birationally.
On the distinct-root open the ordered space is the full root-ordering
torsor over this unordered space.  Thus its components correspond
bijectively to the components of the full ordered-root cover of the
source; all components are retained.  The full relabeling group acts
transitively on them and does not change the underlying map, so each
has class $\beta_Z$.

\end{proof}

Specialization to the collision divisor will use
Lemma~\ref{lem:vertical-dvr-supported}.

\section{Two inputs and their Cartier fusion}\label{sec:fusion}

Let the input sources be $Z,W$, with total auxiliary count $d=\height(\nu_Z)+\height(\nu_W)$. Retain inputs at $0,a$, the old unit at $1$ and the frame at $\infty$. The cross ratio $a$ varies.

\begin{lemma}\label{lem:principal-parts}
Let $R$ be a commutative $\C$-algebra. Let $a_1,\ldots,a_r\in R$
with every $a_i-a_j$ invertible for $i\ne j$, and set
$D=\bigcup_i\{z=a_i\}\subset\PP^1_R$ and $t_i=z-a_i$.
For $q_i\in U_H^+(R((t_i)))$, there is a unique morphism
\[
 q:\PP^1_R\setminus D\longrightarrow U_H^+
\]
such that $q(\infty)=1$ and $q q_i^{-1}\in U_H^+(R[[t_i]])$
for every $i$. The construction is functorial in $R$ and is algebraic
on charts with fixed pole bounds and the finitely many required jet bounds.
\end{lemma}
\begin{proof}
Choose a central filtration of the split unipotent group whose successive
quotients are vector groups. For a vector-group quotient, sum the finite
principal parts at the sections and normalize the value at infinity to zero.
The differences of the sections are units, so each summand has a formal
expansion regular at the other sections. At each later central step, the
already constructed lower coordinates make the remaining discrepancy
additive. Sum its principal parts by the same procedure. Commutators only
involve higher filtration levels, so this finite induction constructs $q$.
All corrections are regular away from $D$, and every correction is zero at
infinity. Only finitely many coefficients of each prescribed local series
occur at a fixed bounded chart; all denominators are powers of the units
$a_i-a_j$. This also proves functoriality and the bounded algebraicity claim.

For uniqueness, if $q$ and $q'$ are two such morphisms, then at each section
\[
 q'q^{-1}=(q'q_i^{-1})(q q_i^{-1})^{-1}
\]
is regular. It is already regular away from $D$. Entrywise partial-fraction
expansion, valid over $R$ because the section differences are units, extends
this quotient to a morphism $\PP^1_R\to U_H^+$. Since $U_H^+$ is affine and
$\Gamma(\PP^1_R,\OO)=R$, this morphism is constant. Its value at infinity is
one, hence $q'=q$.
\end{proof}

Use $g=\dot w_0q^{-1}\dot w_0^{-1}h^{-1}$ and write $k_i=q q_i^{-1}\in U_H^+(R[[t_i]])$. At each input the left integral correction to $q_i$ becomes a right integral frame correction to $g$. Lemma~\ref{lem:frame} gives exactly $p_Z,p_W$. It also preserves complete vector bounds, since each $k_i$ is an invertible integral matrix in every fundamental representation. 

\begin{proposition}[Generic two-input classification]
\label{prop:generic-two-input-inverse}
On the common target constructed above, for prescribed input contacts $p_{Z_1},p_{Z_2}$ and the universal
auxiliary list of total charge $\nu=\nu_1+\nu_2$, the generic smooth
ordinary receiver, after quotienting by the full auxiliary relabeling
group, has a unique horizontal component, birational to
\[
 Z_1^{\rm aff}\times Z_2^{\rm aff}\times\mathscr H\times\{a\ne0\}.
\]
Before quotienting, the union of all generic smooth horizontal
components is birational to the full ordered-root cover of this source,
which need not be irreducible.  Every such component has ordinary
curve class $\beta_{Z_1}+\beta_{Z_2}$.

The entire generic curve/frame incidence consists of the source
fiber, and its virtual class is the fundamental zero-cycle.
\end{proposition}
\begin{proof}
For a smooth receiver, the smooth-domain frame calculation supplies
jointly general contact units and stratum evaluations.  The target map
recovers a rational $V$-map $g$ from its dense $Y$-valued locus.
Pole transfer for polynomial generators of $\mathcal O(V)$ proves
regularity at the auxiliary markings.  The supported generic-incidence
theorem excludes every other unmarked boundary preimage.  Hence $g$ is
regular away from $0,a$, and
\[
 q(z)=\dot w_0^{-1}h^{-1}g(z)^{-1}\dot w_0
\]
is a based rational $U_H^+$-map with precisely those allowed poles.
At the input $z=a_j$, apply Lemma~\ref{lem:principal-parts}
with one pole to the local Laurent series $q$, using $t_j=z-a_j$.
It gives a unique based polynomial loop $q_j(t_j^{-1})$ for which
$q_jq^{-1}$ is integral. Its inverse $k_j=(q_jq^{-1})^{-1}$ is
integral as well, and $q=k_jq_j$.
Lemma~\ref{lem:frame-independent-pole-transfer} and the exact
frozen orders give the full fundamental degrees $\nu_j$.  Thus $q_j$
belongs to an actual MV source of that charge.

Conversely, the principal-parts lemma reconstructs
one and only one global $q$ from the two local representatives, with
regularity away from $0,a$ and $q(\infty)=1$.  This construction is
algebraic on bounded charts, including over nonreduced coefficient rings.
Separatedness of $X$ determines the extension of $g$ from its dense
interior.  The inverse therefore retains the actual loop representatives,
the frame, the input positions, and the complete root ordering.

First work after the full auxiliary relabeling quotient.  A horizontal
smooth component has dimension $m+d+1$, where
$d=\operatorname{ht}(\nu_1)+\operatorname{ht}(\nu_2)$, because the
curve/frame morphism is etale; taking a finite quotient does not change
this dimension.  Its generically injective reconstruction has image
in the finite union of products of actual charge-$\nu_1$ and
charge-$\nu_2$ MV sources, times the frame and input parameter.
Every such product has the same dimension, so the image is dense in
one product.  The full source-contact calculation and injectivity of
the MV/contact pairing identify its two factors with $Z_1,Z_2$.
The unique principal-parts reconstruction and separatedness show that
there is at most one such unordered component.  On the distinct-root
open the ordered space is the full root-ordering torsor; its components
therefore correspond to all components of the full ordered-root cover
of that product.  The full relabeling group acts transitively on these
components and preserves the underlying map and its curve class.

For existence, glue general free one-input source maps at their interior
frame values, keeping their former unit markings as the new unit and
frame.  Their logarithmic tangent bundles are trivial on both components.
On the resulting rational tree the tangent bundle is trivial and, after
fixing the new frame, has vanishing $H^0$ and $H^1$.  The usual logarithmic
map deformation theory therefore gives actual smoothings dominating the
curve/frame base.  The glued node is interior. The class is the sum
of the two source classes.  The component correspondence gives that same class on every
ordered component of the generic smooth source.  Lemma~\ref{lem:generic-supported} identifies the entire general
incidence with this free source fiber. Its supported virtual class
is therefore the source fundamental zero-cycle.
\end{proof}

\subsection{A proper graph with complete vectors}\label{subsec:framed-graph}

Set $\nu_i=\nu_{1,i}+\nu_{2,i}$ and
\[
 \Delta_i(z,a)=z^{\nu_{1,i}}(z-a)^{\nu_{2,i}}.
\]
The complete vector on the separated source is
\[
 P_i(z)=\Delta_i(z,a)q(z)v_i^-
       \in H^0(\mathbb P^1,\mathcal O(\nu_i))\otimes V_H(\omega_i)^*.
 \tag{PV}
\]
Its lowest coordinate is exactly $\Delta_i$, and its value at infinity in the
standard trivialization is $v_i^-$.  

\begin{lemma}
\label{lem:bd-record-local-isomorphism}
Let $B$ be the reduced raw BD closure of the separated source, before
adjoining complete-vector records.  Adjoin the independent frame parameter.
There is an open neighbourhood $U$ of every actual central polynomial-loop
point of full fundamental degrees $\nu_i$ on which the complete vectors
\[
 P_i(z,a)=z^{\nu_{1,i}}(z-a)^{\nu_{2,i}}q(z)v_i^-
\]
are canonical regular coefficient vectors of degree at most $\nu_i$.
Their lowest coordinates are exactly
$z^{\nu_{1,i}}(z-a)^{\nu_{2,i}}$.
On this open, adjoining their projective records gives the graph of a
regular morphism; its projection to $U$ is an isomorphism.

Shrink $U$ so that the auxiliary polynomials have simple roots and the
required noncollision and interior-frame conditions hold.  The full
ordered-root cover $R\to U$ is finite \'etale.  The enriched closure with
all root orderings, restricted over $U$, is isomorphic to $R$.
These are isomorphisms before taking the collision Cartier fibre or the
retained curve/frame incidence fibre.
\end{lemma}

\begin{proof}
Let $\mathfrak h$ be the Lie algebra of $H$. At the trivial bundle
with its infinity frame, framed deformations and automorphisms are
\[
 H^1(\mathbb P^1,\mathfrak h\otimes\mathcal O(-\infty))=0,
 \qquad H^0(\mathbb P^1,\mathfrak h\otimes\mathcal O(-\infty))=0.
\]
The framed bundle functor is locally of finite presentation. Its trivial
point is therefore an etale monomorphism, hence an open immersion.
The same vanishings give a unique normalized trivialization over
square-zero extensions and thus over Artin thickenings. Work on an
affine chart $\operatorname{Spec}A$ of this open in the reduced raw closure.  The normalized
trivialization gives a regular $H$-valued map on the complement of
$z(z-a)=0$.  Its $U_H^+$ equations hold on the dense separated source,
and therefore hold on this complement: the latter is smooth over the
reduced base and is reduced.  In particular the lowest coordinate of
$q(z)v_i^-$ is one.

After shrinking the chart, choose a common integer $N$ at least as large
as the prescribed two pole orders and write each coordinate as
\[
 q(z)v_i^- =\frac{R_i(z)}{z^N(z-a)^N},\qquad R_i(z)\in A[z]\otimes V_H(\omega_i)^*.
\]
Divide $R_i$ by the monic polynomial
\[
 z^{N-\nu_{1,i}}(z-a)^{N-\nu_{2,i}}.
\]
The remainder vanishes on the dense separated source, where the
prescribed pole bounds hold.  Reducedness of $A$ makes every coefficient
of the remainder zero.  Thus $P_i$ has regular coefficients over $A$,
including at $a=0$, since the divisor is monic over $A$.  Its coefficients of
degree greater than $\nu_i$ vanish by the same argument, or by regularity
of $q$ at infinity.  Its lowest coordinate is the stated monic polynomial,
so its coefficient vector is nowhere zero and determines a regular map
to the projective record space.

The graph of this regular map is closed over $U$.  The separated source
is dense in every component of the reduced raw closure.  Consequently
the reduced closure of its graph, restricted over $U$, is exactly this
graph, scheme-theoretically.  

On the interior-frame open, applying the auxiliary linear functional to
$P_i$ and dividing by its nonzero frame value gives a monic polynomial
with regular coefficients.  On its discriminant open the scheme of all
orderings of its roots is finite \'etale.  Include the open conditions
that the roots avoid the input and unit markings.  Finite \'etale maps
are open, so the inverse image of the dense separated source is dense
in every component of this full ordering cover.  Since the construction
retains every ordering, its reduced closure is the whole cover $R$.
The preceding canonical-record argument then identifies the enriched
closure over $U$ with $R$.
\end{proof}

Let $S$ be a smooth chart for the ordered marked positions and the
cross-ratio $a$. Write $c$ for the $d$ local coordinates of the retained
$(d+3)$-pointed curve obtained by forgetting the second input. We choose
the chart with local parameters $(c,a)$ so that $a=0$ has partition
\[
 \{\mathrm{input}_1,\mathrm{input}_2\}
 \mid\{\mathrm{unit},\mathrm{frame},\mathrm{auxiliary\ markings}\}.
\]
The auxiliary markings remain distinct. Retain the frame as a base
parameter and put $S_{\rm fr}=S\times\mathscr H$.

Let $\mathcal M_S$ be the ordinary stable-map space for the fixed
markings and the finite collection of curve classes, pulled back to $S$
by stabilization; it is proper over $S$. Define the framed ordinary space by
\begin{equation}\label{eq:framed-ordinary-space}
 \mathcal M_{\rm fr}
 =\{(f,h)\in\mathcal M_S\times\mathscr H:
                  \ev_\infty(f)=h^{-1}\text{ in }X\}.
\end{equation}
Inversion identifies $\mathscr H$ with the open subset
$\mathscr H^{-1}=\{h^{-1}:h\in\mathscr H\}$ of $Y\subset X$.
The displayed condition is the pullback of the closed diagonal of $X$,
so $\mathcal M_{\rm fr}$ is proper over $S_{\rm fr}$. Its projection
to $\mathcal M_S$ identifies it with the open locus whose frame
evaluation lies in $\mathscr H^{-1}$. Pull every positive logarithmic
mapping space over this chart back to $\mathcal M_{\rm fr}$.
These are open restrictions of the actual mapping spaces, with their
restricted virtual classes.

Choose a bounded projective BD ambient space containing the separated
sources. Uniform polynomial-loop bounds, including inverse-matrix bounds
in a faithful representation, supply such a space. Base change it to
$S_{\rm fr}$ and add the projective complete-vector records
\[
 \prod_i\mathbb P\bigl(H^0(\mathbb P^1,\mathcal O(\nu_i))\otimes V_H(\omega_i)^*\bigr).
\]
Let $\widetilde{\mathcal B}$ be the reduced closure of the full
ordered-root source in this product, retaining all components of the
root-ordering cover. It is projective over $S_{\rm fr}$.
The source identity $q(\infty)=1$ gives
$\ev_\infty(g)=h^{-1}$. Thus its ordinary map takes values in
$\mathcal M_{\rm fr}$. Let $\Gamma$ be the reduced closure of its graph in
\[
 \widetilde{\mathcal B}\times_{S_{\rm fr}}\mathcal M_{\rm fr}.
\]
A subscript $0$ denotes the fiber at $a=0$. Write $p_{\rm BD}$ and $p_{\rm ord}$ for the two projections.
Each ambient projection is a base change of a proper morphism;
its restriction to the closed graph is therefore proper. The
smooth-source inverse gives generic degree one on every component,
so the fundamental-class pushforwards are the fundamental classes of
the corresponding source closures. These proper morphisms remain proper after
fixing $(c,h)$ and after collision base change.
Stabilizing the first input, unit, and frame gives the actual domain
a degree-one projection to $\mathbb P^1$. Composition with
$\pi_h:X\to H/B_-$ gives the common stable flag graph.

\begin{lemma}
\label{lem:bd-graph-log-support}
The ordinary projection of $\Gamma$ factors through the scheme-theoretic
proper image in $\mathcal M_{\rm fr}$ of the positive logarithmic mapping space with the fixed
external data and frame condition \eqref{eq:framed-ordinary-space}.
\end{lemma}
\begin{proof}
On the separated smooth source, the prescribed boundary intersections
provide its canonical basic logarithmic enhancement.  All inputs are
positive and divisorial on the common model.  Thus its ordinary map
factors through the framed positive logarithmic space. Its forgetful map
to $\mathcal M_{\rm fr}$ is proper by base change from the ordinary
forgetful map, so its scheme-theoretic image is closed.
Pull the ideal of this image to the reduced graph $\Gamma$.  It vanishes
on the dense separated part of every irreducible component, hence vanishes
on all of $\Gamma$.  This proves the factorization, which persists under incidence base
change, including over nonreduced bases.
\end{proof}

\begin{lemma}
\label{lem:bd-no-hidden-boundary}
Let $q_0$ be one of the finite genuine central flag maps of total
fundamental degrees $\nu_i$.  Every point of $\Gamma_0$ above $q_0$
lies in the actual framed polynomial-loop open of
$\widetilde{\mathcal B}_0$, and its BD point is the loop reconstructed
from $q_0$.  Conversely, every point above that actual central BD point
has retained flag map $q_0$.  These assertions hold on opens containing
the entire corresponding fibres.
\end{lemma}
\begin{proof}
The generic source identities between its flag lines and (PV) are closed
incidence equations on the universal degree-one graph.  They therefore
hold on $\Gamma$.  On a special fibre over $q_0$, the projective record
$[P_i]$ is a nonzero polynomial vector.  On the degree-one core, away from
its finitely many zeros, its line is the line of $q_0$.
The lowest coordinate of $P_i$ is a scalar multiple
$\lambda_i z^{\nu_i}$.  If $\lambda_i=0$, the lowest coordinate of the
rational flag line would vanish identically.  This contradicts the fact
that the core meets the big cell and that its normalized frame flag is
the lowest flag.  Hence $\lambda_i\ne0$; normalize it to one.

Let $B_i$ be the common factor of the coordinates of $P_i$ as a section
of $\mathcal O(\nu_i)$.  The associated rational flag line has degree
$\nu_i-\deg B_i$.  The retained map has degree $\nu_i$, so $B_i$ is
constant.  Thus no fundamental vector has a base point.  Nonvanishing
of $\lambda_i$ and absence of a common vector zero are open conditions
in the projective coefficient records.  On this open the full Plucker
relations give a genuine flag morphism on $\mathbb P^1$, and its
normalized lowest coordinates are $\Delta_i(z,a)$.

Away from the input divisor all these coordinates are invertible.
The flag therefore lies in $U_H^+B_-/B_-$ and has its unique $U_H^+$
lift $q(z)$, with $q(\infty)=1$.  This defines an actual BD modification
of the trivial bundle, functorially on that open.  It agrees with the
original BD projection on the dense separated part of the reduced graph.
Separatedness of the bounded BD space makes the two morphisms equal
there.  In particular every point over $q_0$ has the stated actual BD
point, including graph boundary points.

Conversely, over that actual BD point the normalized complete polynomial
vectors are those of $q_0$ and have no base points.  They determine the
flag map on the degree-one component.  Its fundamental degrees already
exhaust the total degrees $\nu_i$.  Nefness of every fundamental flag
bundle makes every other flag component constant.  The generic auxiliary
isomorphism conditions and the prescribed two-input boundary partition
then leave only the input subtree, as in the finite-receiver construction.
After forgetting the second input it contracts, and the retained flag
map is $q_0$.  This proves the assertions on the full fibers.
\end{proof}

\subsection{The BD coefficient in the supported finite fibre}

Define the MV multiplication coefficients by
\[
 b_{Z_1}b_{Z_2}=\sum_T m_{Z_1Z_2}^{T}b_T.
\]
The sum ranges over MV cycles $T$ of charge $\nu_1+\nu_2$; it is finite by homogeneity. For fixed ordered auxiliary positions and frame, write $Q_T$ for the coefficient-map fiber of $T$, with that root ordering.

\begin{proposition}[Cartier coefficient at an actual central point]
\label{prop:bd-supported-q-coefficient}
Choose general ordered auxiliary positions $c$ and frame $h$ for which
all the charge-$\nu$ coefficient fibres are finite etale, disjoint from
intersections of distinct MV source components, and satisfy the genuine
flag conditions.  Let $Q$ be their finite union.  Form all supported
incidence classes before reducing any special fibre.  Then the pushforward
to $A_0(Q)$ of the BD graph's collision class is
\[
 \sum_T\;\sum_{q_0\in Q_T}m_{Z_1Z_2}^{T}[q_0].
 \tag{BQ}
\]
A point with empty graph fibre has coefficient zero.
\end{proposition}
\begin{proof}
By \cite[Theorem~7.11]{BKK}, the collision Cartier cycle on the raw actual
central MV open is
\[
 0^![B]=\sum_T m_{Z_1Z_2}^{T}[T].
\]
Pull back along the smooth frame factor $\mathscr H$ and restrict to the open of
Lemma~\ref{lem:bd-record-local-isomorphism} and then take its full
ordered-root \'etale cover.  On this neighbourhood the enriched space is
that cover itself, so flat pullback of the raw BKK formula is the
Cartier formula for the enriched space there.  

Lemma~\ref{lem:bd-no-hidden-boundary} puts the entire graph inverse image
of the actual finite set $Q$ inside this neighbourhood.  Near $Q$ the
coefficient map and the root-ordering map are \'etale.  The supported
pullback fixing $(c,h)$ therefore takes each $[T]$ to
$\sum_{q_0\in Q_T}[q_0]$, with coefficient one at every point.  Proper
pushforward of the graph class and its two refined pullbacks is taken
on this full inverse image.  Its ordinary and BD maps to $Q$ agree, so this gives (BQ) in $A_0(Q)$.

On $S_{\rm fr}$, fixing $(c,h)$ is a regular embedding of codimension
$d+m$, leaving the cross-ratio parameter free. Form the one-cycle
\[
 \widetilde\gamma=(c,h)^![\Gamma].
\]
For either proper projection $p=p_{\rm BD}$ or $p=p_{\rm ord}$,
let $p_{c,h}$ be its incidence base change. Proper pushforward gives
\[
 (p_{c,h})_*\widetilde\gamma=(c,h)^!p_*[\Gamma]
\]
in the Chow group of the corresponding incidence fiber. The regular
pullbacks fixing $(c,h)$ and setting $a=0$ commute on this common
smooth base, and commute with these proper pushforwards. The BD
projection consequently gives (BQ) for the same supported collision class.

Lemma~\ref{lem:bd-no-hidden-boundary} says that the preimage of the good
ordinary receiver and the preimage of the actual finite BD fibre $Q$
are the same open in $\Gamma_0$.  On this open their maps to $Q$ agree.
Their proper pushforwards therefore identify the left hand side of
(BQ) with the class used in the ordinary receiver comparison.  All
Cartier lengths, normalization valuations and residue-field degrees
are already retained by the refined pullback and proper pushforward.
One may use reduction-invariance of Chow groups only after this class
has been formed.

The source intersection locus for two distinct
charge-$\nu$ MV components has dimension less than
$\operatorname{ht}(\nu)$.  Finiteness of $\chi_h$ gives it proper image
in the coefficient base.  A common general choice avoids all those
images.  Thus every $q_0$ is attached to one actual $T$.  Over a nonclosed
base field one makes a common finite separable extension splitting $Q$;
the same base change is made on both sides.  The equality therefore compares coefficients at individual geometric points.
\end{proof}

\section{Central flag maps and a finite receiver}\label{sec:finite-receiver}

Over $S_{\rm fr}$, let $O^{\rm pre}\subset\mathcal M_{\rm fr}$ be
the scheme-theoretic union of the proper images of the framed positive
logarithmic mapping spaces and the BD graph. Include all homological
decorations of the total numerical datum. The prescribed auxiliary
contacts factor evaluations through their closed divisors. This is a
closed receiver proper over $S_{\rm fr}$, and $p_{\rm ord}$ factors as
a proper map $p_{O^{\rm pre}}:\Gamma\to O^{\rm pre}$.
Define $O$ by fixing the retained curve and frame $(c,h)$ and pulling
back to the cross-ratio trait at $a=0$. Then $O$ is proper over that
trait. All refined classes use this same base change. Under the
identification of $\mathcal M_{\rm fr}$ with its open image
in $\mathcal M_S$, fixing $h$ is exactly the original refined frame insertion
$\ev_\infty=h^{-1}$, of Gysin degree $m$.

\begin{lemma}\label{lem:divisorwise-common-support}
Every geometric map in this receiver has a positive logarithmic
lift after a field extension. If a nondegenerate component $C_0$
has an unmarked attached tree $T$, and $c_i$ is the contact of
$C_0$ with $D_i$ at the attachment, then
\[
 D_i\cdot\beta_T=-c_i\qquad\text{for every }i.
\]
The same identity holds in a punctured central space for a tree
carrying none of its markings.
\end{lemma}
\begin{proof}
The logarithmic image is closed by properness, and every geometric
point of it lifts after a field extension. Lemma~\ref{lem:bd-graph-log-support}
puts the graph image in that same closed image, scheme-theoretically.
These facts persist after incidence base change. Sum the
contact-degree identity for $D_i$ over the vertices of $T$.
The internal edge terms cancel. There are no marked legs on $T$,
and its only external edge contributes $-c_i$. This proves the
identity, also when a puncture lies on the complementary component.
\end{proof}

\begin{lemma}
\label{lem:flag-degree-exhaustion}
Let $f:C\to X$ be an ordinary coarse map with the fixed external data of
total charge $\nu$. Suppose the unit, frame, and all auxiliary markings lie on one smooth
component $C_0$, and the complement is a connected subtree containing the
two input markings. Suppose the frame is interior and the color-$i$
auxiliary markings are the $\nu_i$ distinct prescribed zeros of the section
in \eqref{eq:frame-section-cancellation}. Then
\[
 \deg_{C_0}(\pi_hf)^*\mathcal L_i=\nu_i,
 \qquad
 \deg_{C'}(\pi_hf)^*\mathcal L_i=0
 \quad(C'\ne C_0).
\]
Every component off $C_0$ is constant as a map to $H/B_-$.
\end{lemma}
\begin{proof}
The section on $C_0$ is nonzero at the frame and vanishes at the $\nu_i$
distinct auxiliary points. Its degree is therefore at least $\nu_i$.
The total degree is $\nu_i$ and $\mathcal L_i$ is nef. Equality follows,
as does degree zero on every other component. A positive sum of the
fundamental bundles is ample; hence all these other components have
constant flag image. They may still be nonconstant in an exceptional fiber
of $X\to H/B_-$.
\end{proof}

Compose a stable $X$-map with $\pi_h$, forget the second input marking, and
stabilize as a stable map to $H/B_-$. This defines a proper morphism
with $d+3$ retained markings (the first input, unit,
frame, and $d$ auxiliary markings),
\begin{equation}\label{eq:canonical-flag-receiver}
 \Phi:O_0\longrightarrow
 \overline{\mathcal M}_{0,d+3}(H/B_-,\nu).
\end{equation}
On the locus of Lemma~\ref{lem:flag-degree-exhaustion}, its value is the
central flag map. Here $\nu$ specifies the flag curve class by its fundamental degrees $\nu_i$. Stabilization defines this morphism functorially in families.

\begin{lemma}
\label{lem:flag-to-actual-MV}
Let $q$ be a smooth central flag map with the fixed frame and auxiliary
data. Suppose its rational lift $g$ is regular in $V$ away from the input,
has its stated generic auxiliary contacts, and has complete flag degrees
$\nu_i$. Then
\[
 n(t)=\dot w_0^{-1}h^{-1}g(t^{-1})^{-1}\dot w_0
\]
is a based polynomial loop in $U_H^+$ of semi-infinite charge $\nu$.
It belongs to an actual MV source, and its coefficient map has exactly the
prescribed ordered auxiliary roots.
\end{lemma}
\begin{proof}
The displayed expression is regular for finite $t$ and equals one at
$t=0$. Each complete vector $n(t)v_i^-$ has its lowest coordinate identically
one. Consequently its coordinates have no common polynomial factor, and
their maximum degree is the degree of the associated flag line bundle,
namely $\nu_i$. The complete-vector characterization of the semi-infinite
orbit now gives the charge $\nu$. The irreducible-component description of
the resulting finite-dimensional orbit intersection supplies an actual MV
source. The section identity above identifies its auxiliary roots with
the prescribed ones.
\end{proof}

\begin{lemma}
\label{lem:nef-cut-side-trees}
On the fixed separating-curve fiber, let $G$ be the relative open locus
where the unit, frame, and all auxiliary markings lie on one smooth component $C_0$,
every auxiliary evaluation is in the prescribed generic $V$-divisor locus
where $X\to H/B_-$ is an isomorphism, and unit/frame evaluations lie in
$Y$. These opens are taken after imposing the closed incidence. On the common ordinary receiver, $C\setminus C_0$ is one connected subtree containing
both inputs.
\end{lemma}
\begin{proof}
The property defining $G$ survives smoothing edges of the domain graph,
so it is open in the finite collection of domain strata. The fixed stable
curve has the partition $\{\text{input 1,input 2}\}\mid
\{\text{all other marks}\}$. Hence the inputs lie in the same connected
component of $C\setminus C_0$; every other such component is an unmarked
tree $T$. The frame makes $C_0$ nondegenerate. Write $c_i\geq0$ for its
ordinary contact with $D_i$ at the attaching point of $T$.

Lemma~\ref{lem:divisorwise-common-support} gives
$D_i\cdot\beta_T=-c_i$ for every boundary divisor.

If some $c_i>0$, nefness contradicts
$A\cdot\beta_T=-\sum_i a_i c_i<0$. If every $c_i=0$, the attachment lies
in $Y$. The first component of $T$ has degree zero against the effective
full-support divisor $A$, so it stays in $Y$ and is constant by affineness.
Repeat along the connected tree. An unmarked constant tree with one
attachment cannot be stable. Thus $T$ does not exist.
\end{proof}

\subsection{A finite proper receiver for individual coefficients}

Let $G\subset O_0$ be the open from
Lemma~\ref{lem:nef-cut-side-trees}. Its complement to the central
component is exactly one connected input subtree.
Let $Q$ be the finite reduced union of the general
coefficient-map fibers over all actual MV sources of charge $\nu$, regarded
as actual normalized smooth central flag maps. The finite-map theorem and
generic etaleness give the asserted finiteness and reducedness. For every point of $G$, the divisorwise restrictions exclude unmarked
boundary intersections, and the nef cut leaves only the input subtree.
The auxiliary evaluation conditions put its other boundary values in $V$.
Its rational $V$-map is therefore regular away from the input. Fundamental
degree exhaustion and Lemma~\ref{lem:flag-to-actual-MV} reconstruct an
actual source of charge $\nu$ with the prescribed coefficient fibre.
Consequently $\Phi(G)\subset Q$.

Choose the general incidence so that every $q\in Q$ takes values
in $Y_h$ off the input and auxiliary markings, and its auxiliary
values lie in the $\pi_h$-images of the prescribed generic
$V$-divisor loci where $\pi_h$ is an isomorphism. These are simultaneous generic conditions on the
finitely many source families; the smooth-domain frame argument supplies
the auxiliary genericity.

\begin{lemma}
\label{lem:finite-flag-receiver}
Under the preceding incidence choices, every geometric fiber
$\Phi^{-1}(q)$, $q\in Q$, lies in $G$. There is an open subspace
$O_{\rm good}\subset O_0$ containing these fibers and a
proper map
\[
 (O_{\rm good})_{\rm red}\longrightarrow Q.
\]
It induces a proper pushforward
$A_0(O_{\rm good})\to A_0(Q)$, where $A_0(Q)$ is freely generated
by the actual points $q$.
\end{lemma}
\begin{proof}
A map over $q$ has exactly one component surviving as the smooth marked
domain of $q$; stable-map stabilization never contracts a nonconstant flag
component. This formulation includes charge zero. Every other component is mapped into a fiber of $\pi_h$. Away from
the input, that fiber is a point: $q$ takes values in $Y_h$ or in
the translated auxiliary isomorphism loci. The fixed separating marked curve puts the two
inputs over its input point. Thus an off-input subtree has no input marking
and at most one non-input marking. A tree of constant components with at
most one marking and one attaching point cannot be stable. Hence every
non-input marking lies on the main component and every remaining component
is attached at the input. The prescribed separating marked curve puts
both inputs in the same connected subtree there. A nodal curve cannot
attach a second subtree at the same smooth point of its main component.
This proves $\Phi^{-1}(q)\subset G$.

Properness of $\Phi$ makes $\Phi(O_0\setminus G)$ closed and
disjoint from $Q$. Put
\[
 O_{\rm good}=\Phi^{-1}\!\left(
 \overline{\mathcal M}_{0,d+3}(H/B_-,\nu)
 \setminus\Phi(O_0\setminus G)\right).
\]
Then $O_{\rm good}\subset G$ and its image is contained in $Q$.
Its reduction factors through the finite reduced $Q$ and is proper over
$Q$, by properness of the corresponding base change of $\Phi$. Chow
groups are unchanged by nilpotent thickening, which gives the displayed
pushforward. The cycle to be pushed is formed before invoking this
nilpotent-invariance statement: Cartier-fiber multiplicities are retained.
\end{proof}

The same open can be chosen for the finitely many central punctured
spaces arising from cuts with zero tail flag degree. Their ordinary
images are proper. In each of them, impose the same auxiliary
isomorphism loci and interior conditions to define $G_C$.
The preceding tree argument applies to branches without the input
marking, and the degree argument gives $\Phi_C(G_C)\subset Q$.
Every fiber over $Q$ lies in $G_C$, because off the input the fibers
of $\pi_h$ are points and a constant tree with at most one marking
and one attachment is unstable. Delete the finite union of the
proper closed bad images from the flag space. Its complement
contains $Q$ and works for the total and every central receiver.
For zero tail flag degree, flag stabilization after gluing factors
through the central map; hence this common-open restriction is
pulled back from the central factor.

\section{Supported Cartier specialization}\label{sec:common-receiver}
\begin{lemma}\label{lem:vertical-dvr-supported}
Let $O$ be proper over a discrete valuation ring $R$, with closed point
$0$, and let $\alpha,\beta$ be relative one-cycle classes whose generic
restrictions agree.  Then
\[
 0^!\alpha=0^!\beta\quad\hbox{in }A_0(O_0)_{\mathbb Q}.
\]
Suppose $\Phi:O_0\to Z$ is proper, $V\subset Z$ is open, and
$\Phi(O_0)\cap V$ is contained in a finite reduced closed scheme
$Q\subset Z$ which is contained in $V$.  The equality restricts to
$O_V=\Phi^{-1}(V)$ and then
pushes to an equality in $A_0(Q)_{\mathbb Q}$.
\end{lemma}

\begin{proof}
Localization writes $\alpha-\beta=i_*\gamma$ for a class supported on
$O_0$.  The refined pullback is the base change of the regular embedding
$0\to\operatorname{Spec}R$.  The self-intersection rule gives
$0^!i_*\gamma=c_1(N_{0/R})\cap\gamma=0$, because the normal line is
trivial.

By properness the image inside $V$ is closed.  Its support is finite,
so $O_V$ is also closed in $O_0$.  After the Cartier pullback has been
formed, reduction preserves Chow groups and its already-computed
coefficients.  The reduced map factors through $Q$, and is proper:
it is the restriction of a proper map to a closed finite receiver.
Restrict and push the displayed equality.  It is enough that the image
be \emph{contained} in $Q$.  A candidate point with empty inverse image
has coefficient zero.  Over a nonclosed residue field one may make one
finite separable extension splitting the finite \emph{\'etale} candidate
set; all stack stabilizers and base-change degrees are retained on both
sides.  The equality then compares the coefficients of each geometric point.
\end{proof}

The auxiliary-divisor conditions are part of the prescribed
contact data. The auxiliary isomorphism loci define opens within
that closed incidence. The actual insertions fix the retained curve
and frame, of codimensions $d$ and $m$. They reduce the two-input
virtual dimension $m+d+1$ to one; collision then gives a zero-cycle.

\section{Logarithmic collision}
\label{sec:direct-log}

We compare the coefficients directly using logarithmic virtual classes.
The evaluation-enhanced gluing theories are those of \cite[Sections~4.2 and~5.3]{ACGSPunctured} and
\cite[Theorems~5.1, 5.3, 5.5]{GrossGluing}, always on the original
scheme target. The Artin-stack Chow operations and their rational
proper descent are those of \cite[Appendices~B--C]{BSS}.

\begin{lemma}
\label{lem:small-artin-base-bivariant}
Work with rational Chow groups over a characteristic-zero field.
Let $B$ be a finite-type algebraic stack stratified by global quotient
stacks, with $\dim B\leq b$.  Let $F\to B$ be a morphism with $F$
a finite-type Deligne--Mumford stack, and let
\[
 c\in A^e(F\longrightarrow B)
\]
be a bivariant class on finite-type quotient-stratified test stacks:
its operations commute with representable projective
pushforward, flat pullback, and refined pullback for representable lci
morphisms.  If $e>b$, then $c=0$.
\end{lemma}

\begin{proof}
Take the reduction of $B$ and apply functorial resolution of singularities
to it.  More explicitly, \cite[Theorem~5.1.1]{TemkinResolution} extends the smooth-functorial
sequence of ordinary blowups from a smooth atlas and its groupoid to a
sequence of blowups of the reduced Artin stack.  Thus there is a projective
representable surjection
\[
 r:\widetilde B\longrightarrow B
\]
with $\widetilde B$ smooth, whose connected components have dimension
at most $b$.  Compose the resolution of the reduction with its closed immersion
in $B$.  Properness, representability,
and surjectivity hold after every base change.  Since $r$ is
representable, $\widetilde F=F\times_B\widetilde B$ is still
Deligne--Mumford.  The pullback of $c$ to any smooth pure-dimensional
component $B'$ of $\widetilde B$, of dimension $b'\leq b$, satisfies
\[
 c_{B'}[B']\in A_{b'-e}(F\times_B B')_{\mathbb Q}=0.
\]
Here negative-dimensional rational Chow groups vanish because the
support is Deligne--Mumford.

The supported smooth-base duality formula is as follows.
If $Y$ is smooth of pure dimension $b'$ and $a\in A^e(H\to Y)$,
then, for every $g:U\to Y$ and $\eta\in A_*(U)$,
\[
 a_g(\eta)=\Gamma_g^!\bigl(\eta\times a_{\mathrm{id}}[Y]\bigr)
       \quad\hbox{in }A_*(H\times_Y U).
\]
Here $\Gamma_g:U\to U\times Y$ is representable lci.  It need not
be a regular closed immersion for an Artin target; its virtual
codimension is $b'$, which may be negative.  The formula follows by
flat exterior product and compatibility with refined lci pullback:
first prove it for a fundamental cycle, then use representable projective
pushforward and homotopy invariance to pass to Kresch Chow classes.
These are the operations in the proof of \cite[Theorem~C.6]{BSS},
with support in $H$ retained at every step.  Consequently
$c_{B'}[B']=0$ implies $c_{B'}=0$, and hence $r^*c=0$.

For an arbitrary test stack $U\to B$, set
$\widetilde U=U\times_B\widetilde B$.
The map $\widetilde U\to U$ is projective, surjective, and representable.
\cite[Proposition~B.19]{BSS} gives a surjection on rational
Chow groups in every dimension.  Lift $\eta\in A_*(U)$ to
$\widetilde\eta\in A_*(\widetilde U)$.  The projective-pushforward axiom
then gives
\[
 c_U(\eta)=r_{F,U*}\bigl((r^*c)_{\widetilde U}
                      (\widetilde\eta)\bigr)=0.
\]

\end{proof}

\begin{lemma}
\label{lem:positive-tropical-central-vanishing}
Let $(X,D)$ be a smooth projective simple-normal-crossings log
Calabi--Yau pair of dimension $m$, and let $\tau_C$ be a realizable
genus-zero decorated type with $d+3$ labelled legs.  Enhance its
Artin-fan mapping stack at precisely one distinguished cut leg:
\[
 B_C=\mathfrak M^{\mathrm{ev}}(\mathcal X,\tau_C),\qquad
 \epsilon_C:L_C=\mathcal M(X,\tau_C)\longrightarrow B_C.
\]
Here the Artin base uses the underlying undecorated type; the actual
mapping stack retains the specified homological decorations, by
\cite[Proposition~5.21 and diagram~(5.22)]{ACGSPunctured}.
Use a finite-type restriction containing the actual maps under
consideration, and the compatible enhanced perfect obstruction theory
of ACGS Sections~4.2 and~5.3.  Let $P$ be the smooth curve/frame
parameter space of dimension $m+d$, let $b\in P$, and let $L_{C,b}$
be the corresponding actual incidence.  Then
\[
 c_C:=b^!\epsilon_C^!
       \in A^{m+d}(L_{C,b}\longrightarrow B_C)
\]
vanishes if $\dim\tau_C>0$.
In particular, after every base change in the stated test category it annihilates
every gluing-base cycle, including cycles not expressible as products of vertex classes.
\end{lemma}

\begin{proof}
ACGS Proposition~3.30 and Gross's dimension calculation in the proof
of Theorem~5.3 give
\[
 \dim B_C=(d+3)-3-\dim\tau_C+m=m+d-\dim\tau_C.
\]
The enhancement is smooth of relative dimension $m$, by ACGS
Definition~5.14. Write $\pi:\mathcal C\to L_C$ and $f:\mathcal C\to X$ for the universal curve and map, and $p_{\rm cut}$ for its cut section. Its compatible relative obstruction complex is
\[
 (R\pi_*f^*T_X^{\log}(-p_{\rm cut}))^\vee.
\]
Its virtual rank is
$\chi(f^*T_X^{\log})-m=m-m=0$: genus zero and
$c_1(T_X^{\log})=0$ are used, and the sign of a puncture does not
alter this Euler characteristic.  The incidence pullback has degree
$\dim P=m+d$.  Thus $c_C$ has the displayed degree.  The actual
stable punctured mapping stack and its incidence are Deligne--Mumford.
Its virtual pullback, composed with the regular incidence pullback,
has all the bivariant compatibilities in the preceding lemma
\cite[Theorems~4.3--4.4]{Manolache}.  The genus-zero prestable-curve
automorphism groups are affine.  The kernel contributed by logarithmic
maps to the Artin fan is a subgroup of the product of the finitely many
torus automorphism groups of the ghost charts, with finite graph
automorphisms; see \cite[Remark~3.27 and diagrams~(3.8)--(3.9)]{ACGSPunctured}.  Thus $B_C$ has affine geometric stabilizers; evaluation
enhancement is representable and preserves this property.  The
finite-type restrictions are therefore stratified by global quotient
stacks, by \cite[Proposition~3.5.9]{Kresch}.
Lemma~\ref{lem:small-artin-base-bivariant} proves the claim.

For gluing, first choose the enhanced obstruction theories on the
central and tail factors, and on their product choose the external
direct sum, including its arrow to the relative cotangent complex.
The proof of \cite[Theorem~5.19(1)]{ACGSPunctured}, in its final
paragraph, then permits the dashed arrow for the glued enhanced
obstruction theory to be chosen so that this theory is the pullback
of that product theory.  Indeed, the proof fixes the split arrow and
adjusts the glued arrow through the ambiguity described in
\cite[Remark~4.4]{ACGSPunctured}.  The resulting glued theory still
belongs to the compatible triangle of Section~4.2, so its comparison
with the unenhanced theory is preserved.

Write $B_T$ for the tail Artin base enhanced at its cut marking, and $\epsilon_T:L_T\to B_T$ for its actual mapping space with the compatible obstruction theory. Write $G=B_{\rm gl}^{\rm ev}$ for the glued enhanced Artin base and
$j:G\to B_C\times B_T$ for its splitting map.  By
\cite[Proposition~5.17]{ACGSPunctured}, the square defining the actual
glued stack is Cartesian in the category of ordinary stacks.  Since
the retained curve/frame incidence is pulled back from the central
factor, it identifies the actual glued incidence with
\[
 L_{{\rm gl},b}
 \simeq G\times_{B_C\times B_T}(L_{C,b}\times L_T).
\]
Let $\epsilon_{T,G}:U_T\to G$ be the base change of $\epsilon_T$ along $G\to B_T$. For any $\alpha\in A_*(G)$, set
\[
 U_T=G\times_{B_T}L_T,
 \qquad
 \eta=\epsilon_{T,G}^!(\alpha)\in A_*(U_T).
\]
The stack $U_T$ is an admissible test stack for $c_C$: it is finite
type on the chosen restrictions, and $U_T\to G$ is of DM-type, so
it inherits quotient stratification.  Hence
\[
 (c_C)_{U_T}(\eta)=0
 \quad\hbox{in }A_*(L_{{\rm gl},b}).
\]
The split compatible triangle and
\cite[Theorem~4.8]{Manolache} identify the virtual pullback for the
product obstruction theory with the composite of the tail and
central virtual pullbacks; \cite[Theorem~4.3]{Manolache} permits
commutation with the central incidence pullback.  With the compatible
glued arrow chosen above, the last displayed zero is therefore exactly
$b^!\epsilon_{\rm gl}^!(\alpha)$.

Thus the glued virtual-incidence operation vanishes on its actual
carrier for every class on the glued enhanced base.  
\end{proof}

\subsection{The complete source Cartier cycle and its tails}

\paragraph{The source boundary.}
There are $d+4$ marked points before the two distinguished
inputs collide.  Work over the curve chart on which stabilization
forgets the second input to give the retained $(d+3)$-pointed curve,
and the remaining coordinate is the cross-ratio $a$.
At $a=0$ the two input marks lie on the three-point tail of the stabilized
curve; its other component carries the unit, frame, and all auxiliary
markings. Forgetting the second input contracts that tail and transfers
the first input label to its attachment on the retained curve.
Let $\mathfrak A$ be the Artin-fan mapping stack for the full positive
\emph{undecorated} global datum: genus, labelled markings, and complete
global contact vectors.  Homology classes are imposed on the actual mapping spaces.
At this stage the full global-contact stack precedes curve/frame incidence.
Use a finite-type restriction containing every actual map occurring
over the proper ordinary receiver under discussion.  Let
\[
 \epsilon:\mathcal L\longrightarrow\mathfrak A
\]
be the positive stable-log-map morphism with its usual relative
obstruction theory, taking the disjoint union over the entire bounded
collection of actual homological classes when needed.

The full positive Artin base is logarithmically smooth over the
ground field.  We compute its canonical idealized structure from the global datum.  Write $\bar\beta$ for the undecorated class
datum: it has one vertex, no edges, and all reference strata equal
to the origin, as in ACGS Definition~2.44(4).  For a geometric point
$w$ of the full mapping base, let $\tau_w$ be its actual type and
$\tau_w\to\bar\beta$ its marking contraction.  Every vertex is
sent to the origin and every internal edge is contracted.  Thus the
face in ACGS equation~(2.21) is
\[
 Q^\vee_{\bar\beta\tau_w}
 =\{(V_v,\ell_e)\in Q^\vee_{\tau_w}:
                 V_v=0\text{ for all }v,\ 
                 \ell_e=0\text{ for all }e\}
 =\{0\}.
\]
Consequently the target sharp monoid $Q_{\bar\beta\tau_w}$ of
the localization in equation~(2.22) is zero.  All three sets of
generators in ACGS Definition~3.20 are therefore empty: in (i),
each reference cone is zero, so
$P_x\setminus\chi_x^{-1}(0)=\varnothing$; in (ii), no node is
preserved by the marking; and in (iii),
$Q_{\bar\beta\tau_w}\setminus\{0\}=\varnothing$.
Furthermore, the prescribed external contacts are all nonnegative.
The prestable puncturing hence has no negative contact generator,
and its puncturing log ideal is empty by ACGS Definitions~2.49
and~2.55 and the generator description in the proof of
Proposition~2.57. The canonical idealized structure of Definition~3.23
is empty.

The curve base for this one-vertex marking has no marked nodal
ideal either.  ACGS Theorem~3.25 consequently gives an ordinary
logarithmically etale morphism from the full positive Artin base to
the prestable-curve stack with its nodal log structure.  The latter
is logarithmically smooth over the ground field, so the same is
true of the former.  In particular, toroidal smooth charts are
normal and reduced and have dense locus of trivial characteristic.
The pulled-back stabilization parameter $a$ is, on each such chart,
a unit times a monomial and is a nonzerodivisor.

\begin{lemma}
\label{lem:direct-log-source-all-cuts}
The Cartier cycle $[a=0]$ on $\mathfrak A$ is the sum of its
one-dimensional tropical-type cycles, with coefficient
\[
 \lambda_\tau=\sum_{e\in E_{\rm sep}(\tau)}\ell_e(v_\tau).
\]
Here $v_\tau$ is the primitive generator of the lattice of the
one-dimensional cone $\tau$. The set $E_{\rm sep}(\tau)$ consists of
the edges whose removal separates both input markings from the unit,
frame, and all auxiliary markings. Consequently its virtual pullback, followed by
the retained curve/frame insertion, is a sum of terms indexed by a
type $\tau$ and one chosen separating edge $e$, with weight
$\ell_e(v_\tau)$.  Each term cuts exactly one edge.

The equality holds on the actual special mapping stack and commutes
with proper pushforward to the ordinary receiver.  
\end{lemma}

\begin{proof}
The edges inducing this partition form the chain contracted to the
separating node of the stabilized $(d+4)$-pointed curve. On a chart of
the prestable-curve stack their smoothing parameters satisfy
\[
 a=u\prod_{e\in E_{\rm sep}}s_e,
\]
where $u$ is invertible.  This follows by successively contracting the
two-point components in the unique separating chain; for two adjacent
nodes their product is the parameter of the resulting node.  Nodes
off this chain do not occur in the stabilization parameter.  Pullback
to the basic positive Artin-fan map stack writes each $s_e$ as a unit
times the monomial $\chi^{\ell_e}$ in its basic monoid.  Thus
\[
 a=u'\chi^{\sum_{e\in E_{\rm sep}}\ell_e}.
\]
This is an equality of Cartier equations.  It retains contacts of
arbitrary integral multiplicity.

Every codimension-one component of this monomial divisor has generic
sharp characteristic monoid $\mathbb N$; its generic tropical type
has dimension one.  The multiplicity on that component is the value
of its exponent on the primitive valuation, which gives the asserted
sum.  Conversely every component of the divisor is detected this
way, since the positive base is normal and the equation is a
nonzero monomial on each toroidal chart.

First use undecorated tropical
types $\tau$ and the full $\tau$-marked
mapping stacks, which contain the exact-type locus as a dense open;
ACGS
Proposition~3.30 (v3, pp.~71--72) gives reducedness and pure
dimension for the full marked stack of a realizable type, and
Remark~3.31 gives its finite unramified forgetful morphism to the
full mapping base.  Its image is the appropriate type-stratum
closure.  Generic multiplicities and degrees are computed on the
dense exact-type open, and the cycles are pushed from these full
marked stacks.
One can either use their quotients by type automorphisms or divide
their pushed cycles by the generic type-marking degree.  The displayed
cycle identity is on these quotients.  At this Artin-base stage the
denominator is the finite marking degree
$|\operatorname{Aut}\tau|$ of the undecorated type, and the sum
includes all separating edges.
After virtual pullback to an actual mapping space of a specified
total class, ACGS Proposition~5.21 and diagram~(5.22) identify the
base change with the disjoint union of all compatible curve-class
decorations of the vertices, with compatible obstruction theories.
Applying orbit--stabilizer
to the action of $\operatorname{Aut}\tau$ on these decorations
converts the coefficient into
$1/|\operatorname{Aut}\boldsymbol\tau|$ for each decorated type
$\boldsymbol\tau$.  This is the orbit--stabilizer calculation in ACGS Theorem~5.25, equations~(5.24)--(5.25), applied after Proposition~5.21.  Equivalently, the
one-cut sum is over isomorphism classes of pairs
$(\boldsymbol\tau,e)$ with denominator
$|\operatorname{Aut}(\boldsymbol\tau,e)|$.
Gluing and the edge-length functions are equivariant for these
automorphisms, so the same quotient factors enter the gluing calculation.

The definition of the logarithmic virtual class is
$\epsilon^![\mathfrak A]$.  Refined Cartier pullback commutes with
the virtual pullback and with the regular curve/frame insertion.
The equality of Cartier cycles therefore gives the equality stated
on the actual special mapping stack.  When switching to an
evaluation-enhanced base for gluing, ACGS Definition~5.14 and the
compatible obstruction-theory triangle of its Section~4.2 identify
the resulting virtual pullback with the same class.  Finally proper
pushforward of these operations gives the equality on any common
ordinary receiver.  
\end{proof}

\paragraph{Restriction to the finite receiver.}
Use the good flag open and finite set $Q$ of
Lemma~\ref{lem:finite-flag-receiver}.  A tail of positive flag degree has empty inverse image of this
open.  For a zero flag degree tail, the entire tail tree is flag
constant, so flag stabilization of the glued map factors through the
central factor.  The retained curve/frame insertion is also pulled
back from the central factor: after forgetting the second input,
stabilization contracts the two-input tail and transfers the first
input label to the cut mark. The retained curve is therefore the
stabilization of the central curve with that label at its cut leg.

For each one-cut term, let $\tau_C$ denote its induced central
punctured type. Its legs are the $d+2$ original non-input markings
and the cut leg, giving $d+3$ labelled legs.  The vanishing assertion of
Lemma~\ref{lem:positive-tropical-central-vanishing} applies with
evaluation enhanced at that one cut leg.  In conjunction with the
ordinary Cartesian square of ACGS Proposition~5.17 and the compatible
pullback obstruction theory of Theorem~5.19(1), it kills the gluing
operation itself whenever $\dim\tau_C>0$.

A realizable type with a positive edge length, or with a nonzero
vertex degeneracy, has positive-dimensional cone: simultaneous
positive scaling supplies a nonzero ray.  Hence a surviving central
type has one nondegenerate vertex and no edges.  Its generic source
is smooth, and all its contacts are nonnegative.  The type-marked
mapping space still includes more degenerate types. Apply Lemma~\ref{lem:generic-supported} to this whole positive
central mapping space; it identifies its entire actual incidence
with the finite reduced free maps $q\in Q$.  Its virtual
class there is $[q]$, of weight one.  Proposition~\ref{prop:inverse} recovers the full source $T$, the complete contact $p_T$, and its
class $\beta_T$. For the nonunit products in question $p_T\ne0$. Let $\rho$ be the primitive generator of its ray on the chosen model, let $k$ be its positive integral multiplicity, and let $p_{\rm cut}$ be the cut marking. Then
\[
 p_T=k\rho,\qquad k\geq1,
 \qquad z=q(p_{\rm cut})\in D_\rho^\circ.
\]
The vanishing also
removes every other cut farther down a separating chain whose
central side is still nodal.  

\begin{lemma}
\label{lem:direct-log-one-cut-tail-coefficient}
Fix a surviving free central point $q$, with complete output $k\rho$,
and a tail curve class.  Let $\widehat{\mathfrak B}_T$ be the full
one-negative tail Artin base enhanced at its output, and
$\widehat{\mathcal M}_T$ the corresponding actual punctured mapping
space. Write $[\widehat{\mathcal M}_T]^{\rm ref}$ for the virtual pullback of the puncturing base cycle with its monomial multiplicities, as in Appendix~\ref{app:direct-point-constraint}. Restrict evaluation to $D_\rho^\circ$ for the local calculation. The sum of all surviving one-cut terms, with their
source Cartier weights, pushes on the same ordinary glued-map
receiver to
\[
 [q]\;\times\;z^![\widehat{\mathcal M}_T]^{\rm ref}.
\]
The equality includes every top-dimensional component of the full tail base with its puncturing multiplicity.
\end{lemma}

\begin{proof}
The full tail has three markings and exactly one negative
divisor-marking pair.  Its universal puncturing condition is one
monomial Cartier equation.  On the output open it is log smooth
over the logarithmic evaluation stack, as in \cite[Theorem~3.16]{GSIntrinsic}.  Smooth toroidal charts of its enhanced base therefore
have the form
\[
 \operatorname{Spec}K[Q]/(\chi^h),\qquad h\in Q\setminus\{0\},
\]
with $Q$ fs and $K[Q]$ normal, times smooth factors.  Its refined
base cycle is its fundamental cycle with these monomial
multiplicities, as established in the scheme-target point-constraint
argument.

This enhanced base is pure-dimensional of dimension $m-1$.
For the full marked stack of a realizable tail type $\tau_T$, its
dimension is $m-\dim\tau_T$ by ACGS Proposition~3.30 and evaluation
enhancement.  The negative output forces a positive offset $h$, so
$\dim\tau_T\geq1$.  Every top-dimensional component is thus an
image of a component of a full type-marked stack with
$\dim\tau_T=1$, and every such component is retained.
The finite maps of ACGS Remark~3.31 supply the proper cycle carriers;
their dense exact-type opens are used only for the generic
calculation.  At such a generic point let $t$ be the primitive lattice
parameter and write
\[
 h=b\,t,\qquad b>0.
\]
Its coefficient in the full refined base cycle is $b$.  This coefficient retains the nonreduced thickness of the full base.

For schematic gluing, the virtual diagonal formula applies as follows.  Both enhanced evaluations, central and tail,
factor through the same smooth stratum $D_\rho^\circ$.
On the Artin fan the corresponding open ray stratum is
$B\mathbb G_m$, and the enhanced base is the pullback of
$D_\rho^\circ\to B\mathbb G_m$.  Every morphism to
$B\mathbb G_m$ over the field is flat: after its smooth point atlas
it is a $\mathbb G_m$-torsor over a stack over a field.
Thus these enhanced evaluations to $D_\rho^\circ$ are flat.
Equivalently, \cite[Theorem~5.3]{GrossGluing} applies to the permitted open:
every allowed contraction keeps the cut-leg cone equal to $\rho$,
so the stratum-dimension increment in that theorem is zero, and
$\dim\tau'\geq\dim\tau$.  The calculation is on the evaluation open $D_\rho^\circ$.  The diagonal of
$D_\rho^\circ$ is lci.  \cite[Theorem~5.1 and Remark~5.2]{GrossGluing} now
give the schematic gluing class by the ordinary refined diagonal.
After central incidence this is precisely the tail point pullback
at $z$ paired with the point $q$.

The remaining logarithmic gluing is finite and representable.
The central tropical lattice is zero.  At the one cut the gluing
lattice homomorphism of \cite[Definition~4.2]{GrossGluing} is
\[
 \Psi:\mathbb Zt\oplus\mathbb Z\ell\longrightarrow\mathbb Z\rho,
 \qquad (t,\ell)\longmapsto bt-k\ell.
\]
It has finite cokernel of order
$g=\gcd(b,k)$.  Therefore the gluing is tropically transverse;
the flatness just checked and \cite[Theorem~5.5]{GrossGluing} show that the finite
map is dominant, hence surjective, and that its degree is $g$.
The primitive generator $s$ of the glued one-dimensional cone
satisfies
\[
 t=(k/g)s,\qquad \ell=(b/g)s.
\]
Consequently the contribution of this cut to the source Cartier
parameter has weight $b/g$.  Its product with the logarithmic
gluing degree is
\[
 \underbrace{b/g}_{\text{source Cartier weight}}
 \underbrace{g}_{\text{finite gluing degree}}=b.
\]
After a separable residue extension there are $g$ unit branches;
without that extension their residue-field degrees sum to $g$.
Existing stack stabilizers and graph automorphisms are already in
the degree and the quotient type cycles.    This calculation agrees with the one-negative point
constraint's saturated chart
$bt=k\ell$, whose branches have transverse length $b/g$.
For the surviving cut, this statement also follows directly from
\[
 \operatorname{Aut}(\tau,e)\simeq\operatorname{Aut}(\tau_T).
\]
Indeed the central vertex is the unique vertex carrying the unit,
frame, and all auxiliary legs; its class and complete contact data are fixed, and
the chosen edge is fixed.  Deleting that vertex and edge identifies
its automorphisms with those of the tail with its three labelled
legs.  Thus the source marked-cut denominator is exactly the tail
type denominator.  The fixed actual free central curve has no
additional automorphisms because its markings are retained.

ACGS Theorem~5.19, with the compatible choices of enhanced
obstruction theories, transports this finite-base-cycle equality
to the actual mapping spaces.  The finite logarithmic gluing and
the schematic diagonal are over the same ordinary nodal map.
The equality consequently holds after proper pushforward to the
ordinary receiver, before taking its degree.  Summing over the
reduced tail-type components gives exactly the multiplicities $b$
of $[\widehat{\mathfrak B}_T]$ and hence the full refined tail
class, with its output point insertion.

It remains to verify coverage of the full tail class.  From
any one-dimensional tail type with output in $D_\rho^\circ$, attach
the nondegenerate central vertex of $q$ and replace the two matching
legs by the edge of length $h/k>0$.  All other legs and decorations
are unchanged.  This is a realizable positive total type of
dimension one.  Its total class is $\beta_T+\beta_{\rm tail}$.
The finite gluing is surjective by the preceding transversality
and flatness, so every matching geometric point of each full marked tail carrier
is included. The coefficient $b$ retains its scheme thickness.  Conversely,
deleting the surviving central vertex and the single cut of any
surviving total type gives exactly such a tail type.  The relation
$h=k\ell$ identifies their real cones; hence a surviving total
Cartier ray has a one-dimensional tail cone, and no other
top-dimensional tail component is missed.

This correspondence is applied separately to every curve class in
the entire bounded comparison collection.  The already proved
degree and boundary-localization identities impose the required
total class and zero tail flag degree.  More explicitly, the complete
boundary degree equations determine the numerical tail class
$\beta_Z+\beta_W-\beta_T$, and each fundamental flag line bundle
has degree $\nu_{Z,i}+\nu_{W,i}-\nu_{T,i}=0$ on it.  Therefore
every admissible tail in the reverse construction is flag constant,
component by component, and the whole proper tail fiber has flag
stabilization equal to the fixed ordinary point $q$ after gluing.
Thus every tail component maps to the same finite receiver.  
\end{proof}

\section{Multiplication coefficients}\label{sec:compare}

Products containing the paired unit already agree. For this section fix
nonunit $Z,W$ and put $\nu=\nu_Z+\nu_W$, $d=\sum_i\nu_i$.
Use the direct logarithmic formula of Section~\ref{sec:direct-log}.

Choose the common model for the total positive data and all effective
nonnegative central contacts as above. Choose the retained
$(d+3)$-pointed curve $c$ and frame $h$ generally.  The choices avoid the
closed bad images for all finitely many central data and lie in the
projection of the good two-input open.  Thus the generic value of the
remaining cross-ratio $a$ belongs to that two-input open.  Work over the
trait at $a=0$ with coefficient field generated by these general parameters.
One finite separable extension may be made to split the finite ordered
coefficient fibers.

Let $O$ be the base change of $O^{\rm pre}\subset\mathcal M_{\rm fr}$
along this fixed $(c,h)$ trait in $S_{\rm fr}$. It is proper over the
trait by Section~\ref{sec:finite-receiver}. Write $\alpha$ for the pushed whole logarithmic virtual class after
the supported $(c,h)$ insertion, summed over all actual homological
decorations of the fixed numerical total datum. This collection is
finite by the bounded-degree argument above. Proposition~\ref{prop:generic-two-input-inverse} identifies every generic map with the actual two-input
BD source; decorations with no such generic source have zero
generic class. The vertical-DVR argument therefore also controls
their specialization, and none is discarded from the gluing sum.
Use the graph $\Gamma$ over $S_{\rm fr}$ from
Section~\ref{subsec:framed-graph}, with its complete-vector records.
Write $\Gamma_{c,h}$ for the fixed $(c,h)$ graph over the same cross-ratio
trait and $p_O:\Gamma_{c,h}\to O$ for the base change of the proper map
$p_{O^{\rm pre}}$. Then
\[
 \gamma=(c,h)^!(p_{O^{\rm pre}})_*[\Gamma]
        =(p_O)_*\bigl((c,h)^![\Gamma]\bigr).
\]
Both expressions are the same refined one-cycle on $O$.
By the smooth inverse and Lemma~\ref{lem:generic-supported},
$\alpha_\eta=\gamma_\eta$ on the generic cross-ratio fiber.
Here (SGI) is applied at a geometric generic point. The resulting
rational equivalence descends to a finite extension of the generic
coefficient field. Restriction followed by proper transfer multiplies
rational Chow classes by that extension degree, so the equality
holds over the coefficient field of the trait itself. Hence
Lemma~\ref{lem:vertical-dvr-supported} gives
\begin{equation}\label{eq:whole-supported-specialization}
 0^!\alpha=0^!\gamma\quad\hbox{in }A_0(O_0)_{\mathbb Q}.
\end{equation}

The flag morphism and the degree-exhaustion argument isolate the finite
candidate set $Q$ of actual central maps.  Restrict
\eqref{eq:whole-supported-specialization} to the good open and push to
$A_0(Q)_{\mathbb Q}$.  Candidate points with empty fibers are retained
with coefficient zero.  The group $A_0(Q)_{\mathbb Q}$ has the actual points as an independent basis.

For an actual homological class $A$, denote by
$N_{p_Z,p_W}^{p_T,A}(X,D)$ the degree of the Gross--Siebert
three-point punctured virtual class with inputs $p_Z,p_W$ and
negative output $-p_T$, with its logarithmic output point constraint.
Appendix~\ref{app:direct-point-constraint} compares this constraint
with ordinary point insertion when $p_T$ is divisorial.

\begin{proposition}[Individual coefficient comparison]
\label{prop:individual-coefficient}
For every actual output source $T$ of charge $\nu_Z+\nu_W$, one has
\begin{equation}\label{eq:coefficients}
 m_{ZW}^T
 =\sum_{\beta_T^{\rm tail}}N_{p_Z,p_W}^{p_T,\beta_T^{\rm tail}}(X,D),
 \qquad \nu_T=\nu_Z+\nu_W,
\end{equation}
where every admissible tail class on the chosen model is retained.
Its numerical class is $\beta_Z+\beta_W-\beta_T$ on this model. If
$\sigma:X\to X^{(0)}$ is a preceding comparison modification, its
pushforward is $\sigma_*(\beta_Z+\beta_W-\beta_T)$; all admissible
lifts are retained when identifying the coefficient on $X^{(0)}$.
The same argument includes zero coefficients.
\end{proposition}
\begin{proof}
Apply Lemma~\ref{lem:direct-log-source-all-cuts} to the logarithmic side.
For each distinguished cut, positive tail flag degree is excluded by
the good ordinary receiver. In the remaining splittings, the receiver
and the retained incidence factor through the whole central mapping
space. Lemma~\ref{lem:positive-tropical-central-vanishing} annihilates
all positive-dimensional central tropical types on their actual gluing
carriers. The zero-dimensional central type has positive contacts;
Lemma~\ref{lem:generic-supported} identifies its whole incidence with
the finite smooth points $q$. The inverse construction recovers at
each $q$ its actual MV source $T$, complete contact $p_T$ and class
$\beta_T$.

If the numerical class $\beta_Z+\beta_W-\beta_T$ has no effective
homological representative, the complete
boundary-degree equations make every punctured tail space with
output $p_T$ empty. For such a candidate, apply
Lemma~\ref{lem:direct-log-source-all-cuts} and
Lemma~\ref{lem:positive-tropical-central-vanishing} before invoking
any divisorial tail formula: a surviving one-cut term over
$q\in Q_T$ would have a smooth central map reconstructed as $T$
and an effective tail of numerical class $\beta_Z+\beta_W-\beta_T$,
a contradiction.
Here Lemma~\ref{lem:generic-supported} is applied only to central
data of actual effective splittings, which
Lemma~\ref{lem:finite-model-choice} makes divisorial; no divisoriality
assertion is needed for the empty candidate itself. Thus the
logarithmic specialization has coefficient zero at $q$, and
\eqref{eq:whole-supported-specialization} together with
Proposition~\ref{prop:bd-supported-q-coefficient} gives $m_{ZW}^T=0$.
For every remaining candidate, an effective representative of
$\beta_Z+\beta_W-\beta_T$
gives an effective splitting in Lemma~\ref{lem:finite-model-choice};
hence the chosen target makes $p_T=k\rho$ divisorial. This does not
require the corresponding tail mapping space to be nonempty.
Lemma~\ref{lem:direct-log-one-cut-tail-coefficient} identifies the full
contribution over that same $q$ with the one-negative refined tail
point class, including every tail-type component, homological
decoration, monomial thickness and torsion-unit degree.

For each full tail point fiber let $\xi$ be its refined zero-cycle
and $\rho_Q$ its proper map to $Q$. Its flag image is the fixed point
$q$, so $(\rho_Q)_*\xi=\deg(\xi)[q]$. By
Proposition~\ref{prop:divisorial-point-constraint-direct}, this degree
is the Gross--Siebert point-constrained scalar. Thus the scalar
appears after the proper pushforward of the entire tail point class.

Proposition~\ref{prop:bd-supported-q-coefficient} gives coefficient
$m_{ZW}^T$ for the BD side at the same actual point. Equality in the
free zero-cycle group $A_0(Q)_{\mathbb Q}$ proves the asserted identity,
including zero coefficients. The all-lifts identity of
\cite[Corollary~1.4 and \S10]{JohnstonBirational} gives the stated sum
on a preceding model. It is applied to each downstairs homological
class and then summed over its finite collection of decorations.
The auxiliary contacts cancel by color and charge, so boundary
localization gives precisely the stated numerical tail class.
\end{proof}

The MV basis is homogeneous.  The theta functions are homogeneous in the
root grading: every exchange column has weight zero, so every broken-line
correction to an initial exponent has the same weight.  The pairing \eqref{eq:pairing} preserves this grading.  Consequently the two multiplication
coefficients both vanish when $\nu_T\ne\nu_Z+\nu_W$.  Charge zero gives
the unit.  This is compatible with the geometric observation that a
reconstructed unipotent map regular on all of $\PP^1$ is constant.

\section{Weighted scattering and the affine mirror}\label{sec:affine-mirror-dictionary}

We use the full lattice, including the frozen coordinates. In nonsimply-laced types, the primitive normal of a binomial center and the covector supplied by the alternating form differ by a root-length factor. We incorporate this factor in the scattering comparison of \cite[Section~21]{KY}.

Let $M$ be a free integral lattice of finite rank, put
$M^*=\operatorname{Hom}(M,\mathbb Z)$ and
$M_{\mathbb Q}=M\otimes_{\mathbb Z}\mathbb Q$ and $M_{\mathbb R}=M\otimes_{\mathbb Z}\mathbb R$, and choose a basis
$(n_k)$ of $M^*$. Let $J$ be the
mutable set.  Let $C_k\in M$, $k\in J$, be independent vectors spanning a
saturated sublattice, with $n_k(C_k)=0$.  Write $C_{jk}=n_j(C_k)$.  If
$\bar d_k>0$ is a conventional row symmetrizer, then
\[
 \bar d_j C_{jk}=-\bar d_k C_{kj}\qquad(j,k\in J).
\]
Choose positive integral row symmetrizers and put
$d_k=\operatorname{lcm}_{j\in J}(\bar d_j)/\bar d_k$ for $k\in J$. Consequently
\begin{equation}\label{eq:weighted-KY-symmetrizer}
 \frac{n_j(C_k)}{d_j}=-\frac{n_k(C_j)}{d_k}.
\end{equation}
These $d_k$ are the GHKK multipliers in the convention used below; they are
reciprocal, up to common scale, to the row symmetrizers.  

More precisely, in the notation of \cite[Appendix~A]{GHKK}, our $M$ is
$\mathsf M^\circ$, the seed vector is $e_k\in\mathsf N$, and
\begin{equation}\label{eq:weighted-KY-integral-data}
 n_k=d_ke_k\in\mathsf N^\circ=\operatorname{Hom}(M,\mathbb Z),
 \qquad C_k=p_1^*(e_k).
\end{equation}
To complete the fixed data, choose $d_f=1$ for each frozen coordinate index $f$,
put $e_f=n_f$, and take
\[
 \mathsf N=\bigoplus_{i=1}^{\operatorname{rank}M}\mathbb Z e_i,\qquad
 \mathsf N^\circ=\bigoplus_{i=1}^{\operatorname{rank}M}\mathbb Z n_i,\qquad
 \mathsf N_{\rm uf}=\bigoplus_{k\in J}\mathbb Z e_k.
\]
Define
\[
 \{e_k,e_j\}=\frac{C_{jk}}{d_j}\quad(k\in J),
\]
extend by skew-symmetry, and set the frozen--frozen pairings equal to
zero. Equation~\eqref{eq:weighted-KY-symmetrizer} makes this definition
consistent when both indices are mutable. The relations
\[
 \{\mathsf N_{\rm uf},\mathsf N^\circ\}\subset\mathbb Z,\qquad
 \{\mathsf N,\mathsf N_{\rm uf}\cap\mathsf N^\circ\}\subset\mathbb Z
\]
follow from the integral columns $C_k$ (for a frozen first index,
$\{e_f,d_ke_k\}=-d_k C_{fk}$). These are the integrality conditions in
\cite[Appendix~A]{GHKK}. Moreover
$p_1^*(e_k)=C_k$, so the injectivity assumption follows from the
independence of the full columns. This is an explicit realization of
the scattering data; the lattices retain their specified finite index.

The initial wall normal primitive in $\mathsf N^\circ$ is therefore $n_k$.
These equations, or equivalently the primitive initial automorphisms below,
fix the integral convention.

Let $T_M=\operatorname{Spec}\mathbb C[M^*]$. Its characters are denoted $z^n$ for $n\in M^*$; the mirror monomials $z^a$, $a\in M$, belong instead to the group algebra of $M$. Form the toric variety whose fan
has the zero cone and the rays $\mathbb R_{\geq0}C_k$.  In the open divisor
with ray $C_k$, blow up
\[
 Z_k=\{1+z^{n_k}=0\},
\]
and remove the strict transform of the toric boundary.  Denote this open
toric blowup by $Y^\circ$.

We use the following tropical terminology. A spine is the marked
metric tree obtained from a skeletal analytic curve by retaining
the paths between its prescribed ends, with its piecewise integral
affine map to the skeleton. A twig is an attached part omitted from
this spine; its monomial is its outgoing integral direction at the
attachment. A toric tail extends a prescribed end in the toric
model. A pointed monoid $P$ here is an additive submonoid of the
direction lattice whose real cone contains no line.
For a spine $S$ and effective curve class $\gamma$, $N(S,\gamma)$
denotes the degree of the corresponding finite analytic incidence
fiber, with the toric-tail classes subtracted from its complete
curve class, as in \cite[Sections~9--11]{KY}.

\begin{lemma}
\label{lem:weighted-straight-count}
Use a compactification and toric model satisfying
\cite[Assumption~2.4]{KY}. Suppose every nonempty twig under consideration
has monomial in $P\setminus\{0\}$, where $P$ is strictly convex.
Let $S$ be a transverse spine without bending vertices. Assume also that
its extension by the prescribed affine toric tails is fan-generic:
its image avoids the codimension-two skeleton of the toric fan, and the
inverse image of the codimension-one skeleton is a finite set of
two-valent points. Auxiliary contracted evaluation legs are attached
away from the codimension-one skeleton. These are the conditions in
\cite[Lemma~7.2(1--2)]{KY}.
Then exactly one curve class $\gamma_S$ has nonzero count, and
\[
 N(S,\gamma_S)=1.
\]
The class $\gamma_S$ need not be zero. The zero-class conclusion requires
the additional fan-avoidance condition in \cite[Lemma~12.3]{KY}.
\end{lemma}

\begin{proof}
Append the prescribed toric tails using \cite[Construction~9.3]{KY}.
At a vertex of the original spine the sum of its outgoing vectors is
zero. Balancing of the complete tropical curve therefore makes the sum
of its attached twig monomials zero. Since every nonempty twig has a
nonzero monomial in the pointed monoid $P$, there are no such twigs.
The toric-tail condition, equivalently \cite[Lemma~11.3(4)]{KY}, excludes
twigs on the appended portions. Thus the complete tropicalization is the
extended spine.

As in \cite[Lemma~12.3 and Proposition~20.33]{KY}, every contributing
complete curve has image in the common toric isomorphism locus $W$ of
\cite[Lemma~2.6]{KY}. Conversely, the toric skeletal curve avoids the
complement of $W$ in the toric model: that complement is a closed subset
of codimension at least two, and \cite[Lemma~8.21(1)]{KY} applies to the
toric target. Hence the finite moduli fibers on the two sides are
identified through $W$, including their scheme structures.

First suppose the metric and evaluation data are rational. Their
Berkovich point has a completed residue field with discrete value group;
finite extensions used to split the finite etale fiber still have
discrete value group. The fan conditions allow the application of
\cite[Lemma~9.4 and Proposition~7.11]{KY}. They give one map of multiplicity
one, in the determined class after subtracting the prescribed tail
classes. Specifically, $\gamma_S$ is the class assigned to the original spine map $h$ by \cite[Definition~7.9]{KY} on the fixed toric model. Write $\delta_{\widehat h}$ for the class of its extension by the prescribed tails, indexed by a finite set $F$, and $\delta_j$ for the class of the $j$th tail. Additivity gives
\[
 \delta_{\widehat h}-\sum_{j\in F}\delta_j=\gamma_S.
\]
This is the unique class in the rational calculation; in particular it
is effective. Fix the directions and the open fan cones containing
the endpoints of each affine domain. By \cite[Remark~7.10]{KY}, the
same class $\gamma_S$ is constant throughout a sufficiently small
neighborhood of these data.

For real metric or evaluation data, perform this computation after an
arbitrarily small rational perturbation within the same finite fan and
wall combinatorial chamber. Such a perturbation exists: the affine
constraints have rational coefficients, and the required transversality
and fixed open-cell conditions are open. For each fixed class, choose a
finite wall bound large enough for that class before perturbing.
The count in that class is unchanged by \cite[Theorem~11.9]{KY}, while
the class of each appended affine tail is unchanged by
\cite[Remark~7.10]{KY}. Every such perturbation therefore computes the
same class $\gamma_S$, independently of which class was fixed
for the deformation argument. Consequently the count for $\gamma_S$
is one and the count for each other class is zero at the original real
data. Equivalently, one can use the constant degree of the finite
etale fibers in \cite[Lemma~9.11]{KY}. This use of deformation is necessary
because the curve-class computation in \cite[Section~7]{KY} is first
made over discretely valued fields.

\end{proof}

\begin{lemma}
\label{lem:weighted-fan-generic-pieces}
Let $\Sigma$ be a finite rational fan in $M_{\mathbb R}$, and consider
a finite collection of transverse piecewise affine paths with nonzero
rational directions, rational wall hyperplanes, and a common endpoint $x$.
At each wall $H=\ker n$, assume the directions $v,w$ on the two sides
satisfy
\[
 n(v)=n(w)\ne0.
\]
For each fixed wall itinerary there is a finite union of proper rational
linear subspaces of $M_{\mathbb R}$ outside which every affine line supporting a
segment avoids the codimension-two skeleton of $\Sigma$ and is not
contained in the span of a codimension-one cone. The same assertion
holds simultaneously for finitely many itineraries.

In particular, an endpoint outside every proper rational hyperplane
satisfies these assertions for every finite collection. Such endpoints
exist in the interior of the positive cluster chamber. We use them as real skeletal points.
\end{lemma}

\begin{proof}
For a nonzero rational direction $v$, write $L_v=M_{\mathbb R}/\mathbb Rv$ for the
space of affine lines parallel to $v$. If $H=\ker n$ is transverse to
$v$, the restriction $H\to L_v$ is a rational linear isomorphism.
At a crossing where both $v$ and $w$ are transverse to $H$, the common
crossing point therefore identifies $L_v$ and $L_w$ by a rational linear
isomorphism. Composing these identifications backwards along the path
expresses the offset of every supporting line as a rational linear
isomorphism applied to the offset of the final line through $x$.

For a cone $\sigma$ of codimension at least two, the offsets of lines
parallel to $v$ that meet $\sigma$ lie in the proper rational subspace
\[
 \operatorname{im}(\operatorname{span}_{\mathbb R}\sigma\longrightarrow L_v),
\]
whose dimension is at most $\dim M_{\mathbb R}-2$. If a codimension-one span $H_\sigma$
contains $v$, the offsets of lines contained in $H_\sigma$ similarly form
the proper subspace $H_\sigma/\mathbb Rv$ of $L_v$. Their inverse images
in the endpoint space, under the preceding rational isomorphisms and
$M_{\mathbb R}\to L_{v_{\rm final}}$, are proper rational linear subspaces.
There are finitely many cones, segments and chosen itineraries.
This proves the first assertion. The complement of all rational
hyperplanes meets every nonempty real open set, which proves the last
assertion.

A supporting line satisfying these conditions meets each fan facet
transversely in at most one point. Cut points and contracted evaluation
legs can therefore be chosen away from the finite collection of such
intersections. Extending a resulting straight piece by affine tails
keeps it on the same supporting line, so it satisfies precisely the
fan conditions of Lemma~\ref{lem:weighted-straight-count}.
\end{proof}

\begin{proposition}[Weighted affine mirror comparison]
\label{prop:weighted-affine-mirror}
Suppose $Y$ is smooth affine and $Y^\circ\hookrightarrow Y$ is a big open
immersion preserving the specified torus and its logarithmic volume form.
Then the Keel--Yu scattering diagram of $Y$,
after sending every curve-class monomial to $1$, is equivalent to the GHKK
diagram with initial walls
\[
 \bigl(n_k^\perp,1+z^{C_k}\bigr),\qquad k\in J,
\]
and positive wall crossing
\begin{equation}\label{eq:weighted-KY-initial-action}
 z^a\longmapsto z^a(1+z^{C_k})^{\langle n_k,a\rangle}.
\end{equation}
The basis identification indexed by the full lattice
$M=\operatorname{Sk}(Y,\mathbb Z)$ identifies the specialized Keel--Yu
mirror algebra with the formal broken-line theta algebra for the GHKK
diagram.  Regular functions will be selected in Lemma~\ref{lem:formal-theta-regular-restriction}.
In particular, every GHKK multiplication coefficient is the finite sum of
the corresponding analytic coefficients over curve classes.
\end{proposition}

\begin{proof}
We extend the skew-symmetric comparison of
\cite[Section~21]{KY} using the specified primitive normals.

\paragraph{The empty mutable set.}
If $J=\varnothing$, the partial model is the torus $T_M$ itself.
The big-open hypothesis and normality give
$\mathcal O(Y)=\mathcal O(T_M)=\C[M^*]$; since $Y$ is affine, its specified
open torus is all of $Y$.  The comparison is then the toric calculation of
\cite[Lemma~9.4]{KY}, with empty scattering diagram and Laurent monomial
basis.  We may therefore assume $J\ne\varnothing$ below.

\smallskip\noindent
\emph{A nondegenerate alternating form.}
Put $E=\operatorname{span}_{\mathbb Q}\{C_k:k\in J\}$.
For $v\in M_{\mathbb Q}$, prescribe $\omega_0(C_k,v)=n_k(v)/d_k$.
Equation~\eqref{eq:weighted-KY-symmetrizer} makes this prescription skew-symmetric on $E\times E$. Choose $M_{\mathbb Q}=E\oplus F$ and extend it to an alternating form by putting $\omega_0|_{F\times F}=0$.
If $R=\operatorname{rad}(\omega_0)$, then $R\cap E=0$: an element
$\sum a_k C_k$ in this intersection would give
$\sum(a_k/d_k)n_k=0$, and the coordinate normals are independent.

Choose a rational vector space $K$ of dimension $\dim R$ and a map
$j:K\to M_{\mathbb Q}^*$ whose image annihilates $E$ and pairs perfectly with $R$.
Such a map exists because $R\cap E=0$. On $M_{\mathbb Q}\oplus K$ put
\[
 \omega((v,k),(w,l))
 =\omega_0(v,w)+j(l)(v)-j(k)(w).
\]
This form is nondegenerate.  Indeed, a radical vector $(v,k)$ pairs with
$R\oplus0$ to force $k=0$, then with $M_{\mathbb Q}\oplus0$ to force $v\in R$, and
finally with $0\oplus K$ to force $v=0$.  It satisfies
\begin{equation}\label{eq:weighted-KY-form}
 \omega(C_k,(v,l))=\frac{n_k(v)}{d_k}.
\end{equation}
Choose a lattice $K_{\mathbb Z}$ in $K$ and set $\widetilde M=M\oplus K_{\mathbb Z}$.

For a vector $a$, write $a^{\perp_\omega}=\{v:\omega(a,v)=0\}$; for several vectors this denotes the intersection of their orthogonal subspaces. A rational hyperplane can be written
as $a^{\perp_\omega}$ for an integral vector $a\in\widetilde M$, after multiplying the
rational inverse image by a positive denominator.  Thus the genericity
argument in \cite[Lemma~21.8]{KY} applies.  The primitive center characters and wall normals remain those of the
fixed integral lattice.

Temporarily replace $Y$ by $\widetilde Y=Y\times T_{K_{\mathbb Z}}$.
The primitive ray and center data remain $(C_k,0)$ and $(n_k,0)$.
The center remains the reduced binomial $1+z^{n_k}=0$.  The following twig and wall arguments take place in $\widetilde M$ and $\widetilde Y$; the original $C_k,n_k$ denote their inclusions with zero auxiliary coordinates.

\smallskip\noindent
\emph{The twig argument.}
The monoid $P=\sum_{k\in J}\mathbb N C_k$ is free, saturated and strictly
convex, since the $C_k$ extend to an integral basis.  Its degree is
$\deg(\sum a_kC_k)=\sum a_k$.

The common-model argument of \cite[Lemma~21.16]{KY} applies unchanged to
the present centers.  After toric refinements, the birational comparison
with the partial toric blowup is an isomorphism away from a codimension-two
subset of the interior and essential boundary.  Skeletal curves avoid
that subset by \cite[Lemma~8.21(1)]{KY}.  An unmarked toric contact therefore
lies over some $Z_k$.  Since $z^{n_k}=-1$ there, its tropical end lies in
$n_k^\perp$, and its outgoing vector is a positive integral multiple of
the primitive ray $C_k$.  By~\eqref{eq:weighted-KY-form},
\[
 n_k^\perp=C_k^{\perp_\omega}.
\]
These are exactly the end conditions in \cite[Definition~21.5]{KY}.

Balancing gives every edge vector as a nonzero element of $P$.
Induction from the ends places that edge in its $\omega$-orthogonal
hyperplane: at a vertex the incident vectors add, and
$\omega(v,v)=0$.  This is \cite[Lemma~21.6]{KY}.  The genericity propagation
in \cite[Lemmas~21.7--21.8]{KY} uses nondegeneracy to identify ranks with
codimensions of rational orthogonals; the form constructed above has exactly
these properties.

For two walls with monomials $v_1,v_2$ and
$\omega(v_1,v_2)\ne0$, a sum wall is supported on
\[
 \mathfrak d_1\cap\mathfrak d_2
       -\mathbb R_{\geq0}(v_1+v_2).
\]
If it contained $v_1+v_2$, a positive multiple of this vector would lie in
$(v_1,v_2)^{\perp_\omega}$, contradicting the nonzero pairing.
For parallel monomials, an incoming sum requires both summands incoming.
Thus the induction of \cite[Construction~21.13 and Lemmas~21.14--21.15]{KY}
produces no new incoming wall.  Every bend belongs to $P$, so
\cite[Assumption~20.27]{KY} holds.  The finiteness statements
\cite[Lemmas~20.29--20.31]{KY} permit forgetting curve classes at every
finite $P$-adic order.  The generic-wall argument of
\cite[Lemmas~21.17--21.19]{KY} then shows that a wall function contains
only positive multiples of its single monomial direction.  The resulting
diagram is Kontsevich--Soibelman consistent by
\cite[Proposition~20.36]{KY}, which follows there from the theta-function
consistency of \cite[Proposition~20.35]{KY}.

\smallskip\noindent
\emph{The primitive incoming wall count.}
At a generic point of an incoming wall of direction $C_k$, every contributing
twig has total monomial a positive multiple of $C_k$.  Since the $C_j$ are
part of a basis, every end of such a twig has type $k$.
The reduction in \cite[Claim~21.26 and Lemma~21.25]{KY} therefore takes place
in the single blowup at $Z_k$.

Choose $v\in M$ with $n_k(v)=1$.  Because $C_k$ is primitive in
$\ker n_k$, the pair $C_k,v$ extends to an integral basis with all remaining
vectors in $\ker n_k$, and adjoining a basis of $K_{\mathbb Z}$ gives a basis of $\widetilde M$. The local target is consequently
\[
 \operatorname{Bl}_{(0,-1)}(\mathbb A^1\times\mathbb G_m)
       \times\mathbb G_m^{\operatorname{rank}\widetilde M-2}.
\]
For a cylinder used to compute the wall function, the input exponent can
be taken to be $v$, by \cite[Lemma~20.12]{KY}; its output differs by a
multiple of $C_k$.  The other torus projections are constant.  The local calculation is as follows.

\paragraph{The primitive local count and higher covers.}
The rank-two calculation is uniform in the root system. In the
basis beginning with $C_k,v$ chosen above, let $x$ be the torus
character dual to $C_k$, and put $y=z^{n_k}$, the character dual
to $v$. Thus $x$ vanishes on the toric divisor and $y$ is a unit
there. The center is
$(x,y)=(0,-1)$. For the cylinder with input
$v$ and output $v+rC_k$, $r\geq0$, its complete three-pointed skeletal
curve has one simple zero and one simple pole of $y$. Thus $y$ has degree
one and is a coordinate on the source. The other torus factors are
constant. The boundary condition forces every remaining zero of $x$
to lie at $y=-1$, and forces its pole to be at $y=\infty$. Consequently
\[
 x=c(1+y)^r\qquad(c\ne0).
\]
For $r>0$, on the blowup chart with coordinates
$t=1+y$ and $\rho=x/(1+y)$, this curve is
$\rho=ct^{r-1}$. The strict transform of the old divisor $x=0$ is
$\rho=0$. It has been removed, so $r\geq2$ is impossible, whereas
$r=1$ is allowed and meets the exceptional divisor off the removed
boundary. The case $r=0$ is the straight toric curve. Evaluation at a
general framed point determines $c$ uniquely in each allowed case.
KY's finite etale fiber property gives length one; the degree-one
$y$-map also rules out nontrivial multiple covers. Thus the wall
function is exactly $1+z^{C_k}$. 

\smallskip\noindent
\emph{The scattering Lie algebra.}
For $a=(a_k)\in\mathbb N^J\setminus\{0\}$ write
\[
 m_a=\sum a_kC_k,
 \qquad u_a=\sum\frac{a_k}{d_k}n_k=\sum a_ke_k.
\]
For $u\in M^*\otimes_{\mathbb Z}\mathbb Q$, define the logarithmic derivation on the group algebra
by $\partial_u(z^m)=\langle u,m\rangle z^m$ for $m\in M$; it acts trivially on the auxiliary characters.
Then $\omega(m_a,-)=u_a$. The homogeneous log derivation space is
$\mathfrak g_a=\mathbb Q z^{m_a}\partial_{u_a}$.  Its bracket is
\[
 [z^{m_a}\partial_{u_a},z^{m_b}\partial_{u_b}]
 =\omega(m_a,m_b)z^{m_{a+b}}\partial_{u_{a+b}}.
\]
These are exactly the homogeneous spaces and bracket of
\cite[Section~1]{GHKK}, since
\[
 m_a=p_1^*\left(\sum a_ke_k\right).
\]
For any noninitial wall, its primitive geometric normal is the primitive
positive vector on $\mathbb R_{\geq0}u_a$ in
$M^*=\mathsf N^\circ$.  This is the vector $n_0'$ of
\cite[Definition~1.2]{GHKK}; all nonprimitive root factors are therefore
retained on both sides.  For an explicit integral normalization at every wall, put
\[
 D=\operatorname{lcm}_{k\in J}(d_k),\qquad
 h_a=\gcd_{k\in J}\left(\frac{Da_k}{d_k}\right)
 \quad(a\in\mathbb N^J\setminus\{0\}),
\]
where zero entries do not affect the gcd.  The primitive positive normal is
\[
 n'_a=\sum_{k\in J}\frac{Da_k}{h_a d_k}\,n_k
      =\frac{D}{h_a}u_a\in M^*=\mathsf N^\circ.
\]
Its integral coordinates have gcd one, so it is precisely the generator
of $\mathbb R_{\geq0}u_a\cap\mathsf N^\circ$ used in
\cite[Definition~1.2]{GHKK}.  This remains the same primitive vector
when $a$ is replaced by a positive integral multiple.  Taking $a$ to be the $k$-th coordinate vector gives $n'_a=n_k$, so the initial automorphism contains its
root-length multiplier exactly once. For a general wall with attached function $f$,
$\log(f)\partial_{n'_a}$ has every homogeneous part in the previously
identified GHKK graded Lie algebra.  

Every geometric wall belongs to this same completed
group.  The incoming walls and their automorphisms agree, all other walls
are outgoing, and the diagram is consistent.  Hence
\cite[Theorem~1.12]{GHKK} identifies the diagrams up to equivalence at every
finite degree.

\paragraph{Wall and curve-class bounds.}
The geometric walls and the toric fan serve different purposes.
Choose a real point $x\in\widetilde M_{\mathbb R}=\widetilde M\otimes_{\mathbb Z}\mathbb R$ in the inverse image of the positive cluster chamber under $\widetilde M_{\mathbb R}\to M_{\mathbb R}$, outside every proper rational hyperplane of this enlarged space. This is allowed in
\cite[Definition~20.20]{KY}: by \cite[Lemma~2.2]{KY}, the whole space
$\widetilde M_{\mathbb R}$ is the Berkovich skeleton of $\widetilde Y$, and
\cite[Constructions~9.2--9.3 and Lemma~9.11]{KY} take finite etale
fibers over the completed residue fields of its points. 

Fix an input $m\in M\subset\widetilde M$ and an order $\ell\in\mathbb N$. Every retained intermediate exponent
belongs to the finite set
\[
 \{m+Ca:a\in\mathbb N^J,\ \sum_{k\in J}a_k\leq\ell\}.
\]
The case $m=0$ is the unit and is treated separately. Starting with
$m\ne0$, a transverse bend cannot produce a zero exponent: its incoming
and outgoing exponents have the same nonzero pairing with the crossing
normal. Thus all directions used in the supporting-line lemma are
nonzero.
Every nonzero bend consumes positive $P$-degree, so a retained path has
at most $\ell$ bends. The finite-degree twig-wall construction
\cite[Construction~21.13 and Lemma~21.15]{KY} gives finitely many possible
wall cones and directions through this degree. Hence there are only
finitely many candidate itineraries. Each itinerary, together with the
endpoint, determines its supporting lines by the rational crossing maps
in Lemma~\ref{lem:weighted-fan-generic-pieces}.

For the geometric counts, \cite[Lemmas~20.29--20.31]{KY} give finitely
many cylinder and local-theta classes at these orders, and the proof of
\cite[Proposition~20.21]{KY} gives finitely many contributing spines.
Include the classes in the products of the retained local factors on
the finitely many candidate itineraries as well. This is still a finite
set: cylinder classes are finite by Lemma~20.29, and a straight-piece
class depends only on its direction and the open fan cones of its
endpoints, by \cite[Definition~7.9 and Remark~7.10]{KY}. These choices
are finite. Include every effective numerical decomposition of these
total classes. There are only finitely many: intersection with an ample
divisor bounds the degree of every summand, and effective numerical
classes of bounded ample degree form a finite set in the numerical
lattice. On a common compactification, write $\pi_{\rm tor}$ for its map to
the toric model, whose fan lies in $\widetilde M_{\mathbb R}$. Write $D_{\rm tor}$ for the reduced toric boundary, and
$D_{\rm cmp}$ for its own reduced boundary. For a contributing curve of class $\gamma$, its total twig degree is $\gamma\cdot(\pi_{\rm tor}^*D_{\rm tor}-D_{\rm cmp})$ by \cite[Proposition~4.24]{KY}. Choose an integer $A\geq0$ bounding this degree for every retained class, as in \cite[Definition~4.28]{KY}.
Here $A$ is a degree bound, local to this scattering calculation.
Write $\mathrm{Wall}_A$ for the finite polyhedral wall set constructed
from twigs of degree at most $A$ in \cite[Construction~4.16]{KY}. Appending
the toric tails does not require an unbounded further enlargement,
by \cite[Lemma~9.12]{KY}.

Use $\mathrm{Wall}_A$ to cut paths into short crossing pieces and
pieces avoiding these walls. Separately, use the finite intersections
with the codimension-one skeleton of this toric fan to choose all cut points
and evaluation legs away from that skeleton. The supporting-line lemma
already excludes codimension-two fan intersections and lines contained
in fan facets. Thus every straight piece to which the count-one lemma
is applied satisfies the full hypotheses of KY Lemma~7.2, including
its affine tails. Any additional subdivision caused by enlarging $A$
only splits the already retained total classes into effective summands,
which were included when choosing $A$; the chosen bound remains valid.

At a crossing, the bend is tangent to its wall, so the equality
$n(v)=n(w)\ne0$ needed above follows from
\cite[Lemma~20.3]{KY} (and from the same wall-crossing formula on the
GHKK side). For a retained output with a nonzero $P$-increment, a
nonempty twig is necessary, and its image meets $\mathrm{Wall}_A$.
Hence a wall-free piece has only its unchanged exponent through order
$\ell$. That unchanged-exponent spine is straight, and its
coefficient is one after summing classes by
Lemma~\ref{lem:weighted-straight-count}. This statement is taken modulo terms of $P$-degree greater than $\ell$.

The gluing formula \cite[Theorem~12.4]{KY} applies with this single $A$
to every retained class decomposition. Crossing pieces give the
previously identified wall automorphisms, and straight pieces give
identity transmission. Conversely, each retained broken-line choice
uses one of the included itineraries and these same local factors, so
the gluing formula reconstructs exactly its geometric coefficient.
This proves equality of the local expansions through order $\ell$.
If rational data are used to calculate any straight factor, the small
perturbations remain in this same finite wall and fan chamber;
\cite[Theorem~11.9 and Remark~7.10]{KY} preserve its classwise count and
tail classes. The coefficient is therefore the coefficient at the original real point $x$.

The endpoint $x$ can be fixed for all $\ell$, and truncations merely
discard positive $P$-degree. The identities are therefore compatible
as $\ell$ grows. The structure-coefficient comparison is then the
gluing argument in the proof of \cite[Theorem~21.28]{KY}, with the
weighted diagram already identified. For a fixed product the sum over
output labels and curve classes is finite by \cite[Theorem~1.6]{KY}.

\smallskip\noindent
\emph{Removing the auxiliary torus.}
All counts used above have a definition by complete rational curves.
For a disk count this is \cite[Definition~1.5]{KY}, which appends the
opposite output contact and a toric tail; for finite-ended spines it is
\cite[Construction~9.3]{KY}.  Use a product compactification with
$(\mathbb P^1)^{\operatorname{rank}K}$ in the new factors.
For labels $(p,0)$, every original and every appended contact has zero
new coordinate.  The complete curve has no other boundary preimages.
Each new torus character therefore pulls back to a regular invertible
function on a complete connected rational curve, so is constant, including
on nodal domains.  For a family $\pi_C:C\to S$ of connected genus-zero curves, $(\pi_C)_*\mathcal O_C=\mathcal O_S$, so the auxiliary torus morphism factors through $S$, also over nonreduced bases. The frame evaluation fixes that constant. This proves
an identification of the finite moduli fibers with those for $Y$, with
degree-one torus factor and zero new curve class.  

On the scattering side, the initial walls are pulled back from $M_\mathbb R$
and all functions and normals have zero new coordinate.  Uniqueness of the
consistent completion gives the pulled-back diagram.  Broken lines of
label $(p,0)$ have constant extra coordinates and zero extra monomial
exponents, hence give the original $\theta_p$.
The comparison therefore descends to the original full lattice $M$.
\end{proof}

\begin{corollary}\label{cor:nef-target-theta}
In the situation of Proposition~\ref{prop:weighted-affine-mirror}, let
$(W,E)$ be a smooth projective SNC log Calabi--Yau compactification of the
same affine $Y$.  Suppose a nef divisor with strictly positive coefficient
on each component of $E$ is supported on $E$. Then, for full lattice labels $p,q,r$, the GHKK structure coefficient with inputs $p,q$ and output $r$ equals
\[
 \sum_\beta N_{p,q}^{r,\beta}(W,E),
\]
where $N_{p,q}^{r,\beta}$ is the point-constrained punctured invariant defined in Section~\ref{sec:compare}.
The sum is finite. If the complete contact-degree constraints admit
only one numerical curve class $\beta_0$, all other numerical terms
are empty. The surviving coefficient is still the finite aggregation
of every actual homological class with numerical class $\beta_0$.
For our Richardson target, the boundary-generated Picard group and
the displayed tail-class calculation establish precisely this condition.
\end{corollary}

\begin{proof}
\cite[Corollary~5.10]{JohnstonNaive} compares the punctured logarithmic
coefficients on $(W,E)$ with the analytic disk coefficients of $Y$, including
the toric tail condition.  Apply
Proposition~\ref{prop:weighted-affine-mirror} and the finiteness in
\cite[Theorem~1.6]{KY}.  The labels are the same valuations of the fixed torus;
Lemma~\ref{lem:r6-volume-compatible} identifies its volume form and
essential skeleton with those of the chosen Richardson target.

\end{proof}

\begin{lemma}
\label{lem:formal-theta-regular-restriction}
Let $P=\sum_{k\in J}\N C_k\subset M$. Write $\C[[P]]$ for the completion of $\C[P]$ at the ideal generated by $z^{C_k}$, $k\in J$, and put
\[
 \widehat L=\C[[P]]\otimes_{\C[P]}\C[M].
\]
Suppose the weighted mirror comparison identifies the specialization of
the Keel--Yu algebra with the formal broken-line theta algebra in
$\widehat L$, with basis indexed by $M$.  Suppose also that the
regular theta functions
\[
 \{\theta_g:g\in R\},\qquad
 R=\{g\in M:\theta_g\in\C[N]\},
\]
form a basis of $\C[N]$.  Then, for $p,q\in R$, the specialized
Keel--Yu coefficient with output $r$ equals the multiplication
coefficient in this regular basis when $r\in R$, and is zero when
$r\notin R$.
\end{lemma}

\begin{proof}
The successor minor makes the columns $C_k$ a basis of a
saturated direct summand of $M$.  Hence $\C[M]$ embeds in $\widehat L$;
after choosing a complementary lattice, the latter is a power-series
ring in $|J|$ variables with those variables inverted, tensored with the
group algebra of the complement.

The local theta expansion, equivalently \cite[Lemma~20.34]{KY}, is
\[
 \theta_g=z^g\left(1+
 \sum_{0\ne a\in\N^J}c_{g,a}z^{Ca}\right).
\]
Every nonzero $Ca$ is positive in the fixed reverse lexicographic order.
For a nonzero finite linear combination of these series, the least label
with a nonzero coefficient contributes its monomial $z^g$ with coefficient
one; no series with a greater label can contribute that monomial.  Thus
the formal theta functions indexed by all of $M$ are linearly independent
for finite linear combinations.

For $p,q\in R$, write their finite expansion in the regular basis as
\[
 \theta_p\theta_q=\sum_{r\in R}a_r\theta_r.
\]
The initial-torus restriction embeds $\C[N]$ into $\C[M]$, so this is an
equality in the same ring $\widehat L$.  By the weighted mirror comparison
the Keel--Yu multiplication gives another equality there,
\[
 \theta_p\theta_q=\sum_{r\in M}\kappa_r\theta_r.
\]
Here $\kappa_r$ is the sum of the analytic three-point coefficients
with output $r$ over their effective curve classes $\beta$. The sum
over pairs $(r,\beta)$ is finite by \cite[Theorem~1.6]{KY}.
Subtracting the two finite expansions and applying the linear independence
just proved gives $\kappa_r=a_r$ for $r\in R$ and $\kappa_r=0$ otherwise.
\end{proof}

\section{Equality of the bases}\label{sec:conclusion}

\begin{theorem}\label{thm:main}
The normalized MV and regular theta functions agree in the original coordinate ring $\C[N]$ for every connected complex semisimple group $G$.
\end{theorem}
\begin{proof}
The pairing \eqref{eq:pairing} is a weight- and unit-preserving bijection. Proposition~\ref{prop:individual-coefficient}, Corollary~\ref{cor:nef-target-theta}
and Lemma~\ref{lem:formal-theta-regular-restriction} identify its multiplication
coefficients, including the unit and unequal-weight cases. Its linear
extension is therefore multiplicative. Proposition~\ref{prop:pairing} shows that it fixes the initial variables, which generate $\C(N)$, so Lemma~\ref{lem:fixed-generators} makes it the identity on $\C[N]$. Central covers do not change the pinned unipotent subgroup, and products give tensor products of all the constructions. 
\end{proof}

\appendix

\section{Divisorial point constraints}
\label{app:direct-point-constraint}
\label{subsec:divisorial-point-constraint-direct}

Let $(X,D)$ be a smooth simple-normal-crossings pair over an algebraically
closed field of characteristic zero.  Write $D=\sum_i D_i$ and fix
genus-zero discrete data $\beta$ with precisely one negative
divisor-marking pair: the output has contact $-k\rho$, where $k\geq1$
and $\rho$ is the primitive ray of $D_1$, and all other contacts are
nonnegative.  Set
\[
 D_1^\circ=D_1\setminus\bigcup_{i\ne1}D_i
 \quad\hbox{and choose }z\in D_1^\circ .
\]
All finite-type restrictions below are open neighborhoods of the
relevant actual-map images.

We distinguish full mapping spaces from the open spaces used in the
local calculation.  Let $\widehat{\mathcal M}$ be the full actual
punctured mapping stack, $\widehat{\mathfrak A}$ the corresponding
Artin-fan mapping stack, and $\widehat{\mathfrak B}$ its evaluation
enhancement at the output.  The underlying output evaluation on
$\widehat{\mathcal M}$ and $\widehat{\mathfrak B}$ factors through $D_1$.
Let $\mathcal M$ and $\mathfrak B$ be their restrictions to evaluation
in $D_1^\circ$, and let $\mathfrak A$ be the corresponding restriction
of the Artin-fan mapping stack to the open ray stratum.
The morphisms
\[
 \widehat\epsilon:\widehat{\mathcal M}\longrightarrow
       \widehat{\mathfrak B},
 \qquad
 \epsilon:\mathcal M\longrightarrow\mathfrak B
\]
are strict.

Choose an evaluation-enhanced relative perfect obstruction theory
compatible with the unenhanced theory and the smooth morphism
$\widehat{\mathfrak B}\to\widehat{\mathfrak A}$, as in \cite[Section~4.2]{ACGSPunctured}.  The isomorphism class of the enhanced
complex and the resulting virtual classes do not depend on
the compatible arrow chosen as in its Remark~4.4.
All obstruction theories below are restrictions or base changes of
this one chosen compatible construction.  If $\pi:C\to\mathcal M$ and $f:C\to X$ are the universal curve and universal map, and $p=x_{\rm out}$ is the output section, its relative complex is
\[
 (R\pi_*f^*T_X^{\log}(-p))^\vee .
\]

On the output open stratum, the logarithmic evaluation stack has
underlying stack
\[
 \underline{\mathcal P(X,k\rho)}
       =D_1^\circ\times B\mathbb G_m.
\]
Here and below this notation denotes its restriction to $D_1^\circ$.
Let
\[
 Z_k:=\mathcal P(X,k\rho)|_{\{z\}\times B\mathbb G_m}
\]
with the induced log structure.  Thus $Z_k\hookrightarrow
\mathcal P(X,k\rho)$ is strict.  In the universal-torsor convention
of \cite[Proposition~3.8]{GSIntrinsic}, let $\mathcal U$ be the universal line bundle on $B\mathbb G_m$. The log
generator on $Z_k$ has torsor and zero section
$(\mathcal U^{\otimes k},0)$.
Let $W=B\mathbb G_m^\dagger$ have generator $(\mathcal U,0)$.
The point-constraint morphism
\[
 W\longrightarrow Z_k\longrightarrow\mathcal P(X,k\rho)
\]
has underlying first map the identity of $B\mathbb G_m$ and
characteristic map $\mathbb N\to\mathbb N$, $1\mapsto k$.
This specifies the actual logarithmic map, after the allowed choice
of units and of the boundary-line trivialization at $z$.

Define
\[
 \begin{split}
 \mathfrak B_0&=\mathfrak B\times_{D_1^\circ}\{z\}
             =\widehat{\mathfrak B}\times_{D_1}\{z\},\\
 \mathcal M_0&=\mathcal M\times_{D_1^\circ}\{z\}
             =\widehat{\mathcal M}\times_{D_1}\{z\},\\
 \mathfrak B_{\log}
       &=\mathfrak B\times^{\rm fs}_{\mathcal P(X,k\rho)}W,
 \qquad
 \mathcal M_{\log}
       =\mathcal M\times^{\rm fs}_{\mathcal P(X,k\rho)}W .
 \end{split}
\]
The strict inclusion defining $Z_k$ identifies $\mathfrak B_0$,
with its pulled-back basic log structure, with
$\mathfrak B\times^{\rm fs}_{\mathcal P(X,k\rho)}Z_k$.
The ordinary fiber product with $W$ has the same underlying stack
$\mathfrak B_0$: the $B\mathbb G_m$ factor is retained on both
sides, and its identity map cancels in the fiber product.

\begin{lemma}
\label{lem:divisorial-point-constraint-cycle-direct}
The natural morphism
$\nu:\mathfrak B_{\log}\to\mathfrak B_0$ is representable and finite.
Its pushforward on fundamental cycles is
\begin{equation}\label{eq:direct-point-base-cycle}
 \nu_*[\mathfrak B_{\log}]=[\mathfrak B_0].
\end{equation}
Moreover, in the Chow group of $\mathfrak B_0$,
\[
 [\mathfrak B_0]=z^![\mathfrak B]
               =z^![\widehat{\mathfrak B}],
\]
where the last equality uses the identification of the two point
fibers and locality of refined Gysin pullback around $z$.
\end{lemma}

\begin{proof}
Because the pair is simple normal crossings and $D_1$ is a labelled
irreducible component, the open ray stratum of its Artin fan is
$\mathfrak D_1^\circ\simeq B\mathbb G_m$.
There is no characteristic-monoid monodromy on this stratum:
the sharp monoid is $\mathbb N$ and its automorphism group is trivial.
The morphism $D_1^\circ\to B\mathbb G_m$ classifies the normal line bundle, including its twisting.  Evaluation enhancement gives
\begin{equation}\label{eq:direct-enhanced-base}
 \mathfrak B=\mathfrak A\times_{B\mathbb G_m}D_1^\circ
\end{equation}
as underlying stacks, with the basic log structure pulled back
strictly from $\mathfrak A$.

Every morphism of algebraic stacks over the ground field to
$B\mathbb G_m$ is flat: after the smooth surjective atlas
$\operatorname{Spec}k\to B\mathbb G_m$ its source is a
$\mathbb G_m$-torsor and its target is $\operatorname{Spec}k$.
Consequently $\mathfrak B\to D_1^\circ$ is flat.
Regular point Gysin pullback therefore gives
$z^![\mathfrak B]=[\mathfrak B_0]$.
The full-space statement follows since the point fiber is contained
in this open restriction.

The map $W\to Z_k$ is integral, as its characteristic map is
$\mathbb N\xrightarrow{\times k}\mathbb N$.
Thus the fine logarithmic fiber product has underlying stack
$\mathfrak B_0$.  Saturation is finite and representable, including
for logarithmic algebraic stacks; see \cite[Section~2.1.4 and Remark~2.1]{GSIntrinsic}.
This proves the corresponding assertions for $\nu$.
Saturation is applied to the base log structure.

Since every other contact is nonnegative, \cite[Theorem~3.16]{GSIntrinsic} says that
$\mathfrak B\to\mathcal P(X,k\rho)$ is log smooth.  Its strict base change
$\mathfrak B_0\to Z_k$ is therefore log smooth.
Choose strict smooth atlases and frames for the log lines.
The logarithmic chart criterion then gives, locally, a smooth map
to a toric hypersurface
\[
 S=\operatorname{Spec}k[Q]/(\chi^h),
 \qquad h\in Q\setminus\{0\},
\]
where $Q$ is a sharp fs monoid and the map from the target
characteristic monoid sends $1$ to $h$.
The case that $h$ is a unit gives an empty chart.
For the saturated fiber the corresponding chart is
\[
 S'=\operatorname{Spec}k[P^{\rm sat}]/(\chi^w),
 \qquad
 P=Q\mathbin{\oplus_{\mathbb N}}\mathbb Nw,\qquad h=kw.
\]
Indeed, before saturation the equation $\chi^w=0$ also implies
$\chi^h=(\chi^w)^k=0$; after saturation it is still the equation $\chi^w=0$
that is imposed.  These chart constructions commute with the
strict smooth base changes just used.

After splitting the finite torsion units, $k[P^{\rm sat}]$ is a
product of normal toric domains.  Their real cones have the same
dimension and face structure as the cone of $Q$: the additional
real coordinate is uniquely $h/k$.  On each such domain $\chi^w$
is a nonzero, nonunit monomial.  Its zero locus is a Cartier
divisor of pure codimension one.  Its irreducible components
correspond to exactly the same rays on which $h$ is positive
as do the components of $(\chi^h=0)$.
Thus every irreducible component of $S'$ dominates a component
of $S$, and the same assertion holds after the indicated smooth
base changes.  In particular, $\nu$ has no additional
irreducible components supported over proper closed subsets
of components of $\mathfrak B_0$.

It now suffices to compare the multiplicities at each generic
component of $\mathfrak B_0$.  Invert its toric face.
The resulting sharp characteristic monoid is $\mathbb Nq$,
and $h=aq$ for some $a>0$.
After passing to the generic residue field and choosing frames,
the transverse ring is $K[x]/(x^a)$.
The saturated pushout is obtained from
\[
 \mathbb Nq\mathbin{\oplus_{\mathbb Np}}\mathbb Nw,
 \qquad p\longmapsto aq,\quad p\longmapsto kw.
\]
Put $d=\gcd(a,k)$.
The groupification of this pushout is
$\mathbb Z\oplus\mathbb Z/d\mathbb Z$.
Its saturation has sharp quotient $\mathbb Nu$, with
\[
 q=(k/d)u,\qquad w=(a/d)u
\]
up to torsion units.
After an etale extension splitting the units, its underlying
finite algebra has $d$ branches.  Since $\chi^w=0$ on the
point constraint, each branch has transverse ring
\[
 K[u]/(u^{a/d}).
\]
The old equation $\chi^{aq}=0$ pulls back to
$u^{ka/d}=0$ and is redundant.
The total generic cycle coefficient is consequently
\[
 d\cdot(a/d)=a,
\]
the coefficient of the original component.
Without splitting the units, the sum of residue-field degrees
gives the same factor $d$.

This computation takes place on one strict smooth atlas and
its base change.  Representability of $\nu$ means that these torsion units describe branches of a finite algebra. Existing
stabilizers and type automorphisms descend with the atlas calculation.
Together with the absence of vertical components proved above,
this proves the equality of fundamental cycles
\eqref{eq:direct-point-base-cycle}.
\end{proof}

\begin{proposition}[Ordinary divisor-point insertion and the log point constraint]
\label{prop:divisorial-point-constraint-direct}
Let
$\nu_{\mathcal M}:\mathcal M_{\log}\to\mathcal M_0$ and
$\widehat j:\mathcal M_0\to\widehat{\mathcal M}$ be the natural maps, and write $\widehat{\operatorname{ev}}_{D_1}:\widehat{\mathcal M}\to D_1$ for output evaluation.
For the refined class on the full punctured mapping stack and
the Gross--Siebert point-constrained punctured class, one has
\begin{equation}\label{eq:direct-point-virtual-cycle}
 (\nu_{\mathcal M})_*[\mathcal M_{\log}]^{\rm vir}
   =z^![\widehat{\mathcal M}]^{\rm ref}
       \quad\hbox{in }A_*(\mathcal M_0)_{\mathbb Q}.
\end{equation}
Consequently
\begin{equation}\label{eq:direct-point-insertion}
 (\widehat j\circ\nu_{\mathcal M})_*
           [\mathcal M_{\log}]^{\rm vir}
   =\widehat{\operatorname{ev}}_{D_1}^{\,*}[z]
           \cap[\widehat{\mathcal M}]^{\rm ref}.
\end{equation}
These identities commute with capping by classes pulled back
from the full mapping stack, including ordinary descendants: if $x_j$ is the $j$th marked section and $\omega_\pi$ is the relative dualizing line bundle of the universal curve, these are powers of $\psi_j=c_1(x_j^*\omega_\pi)$.
Equation~\eqref{eq:direct-point-virtual-cycle} may be pushed to
any common receiver to which the morphism from $\mathcal M_0$
is proper and preserves the underlying ordinary stable map.
\end{proposition}

\begin{proof}
The total puncturing rank is one.  The universal puncturing
substack is therefore a Cartier divisor in its normal toroidal
Artin fan, and its refined class equals its fundamental class,
with its nonreduced multiplicities.  Concretely its local equation
is a nonzero monomial in a normal toric domain, hence a
nonzerodivisor; where the offset is zero in the characteristic
monoid the zero locus is empty.  This is the rank-one case of
\cite[Section~1.5.1, equation~(8), and Definitions~1.11--1.13]{BNRNegativeAudited}.
Smooth pullback to the mapping base and the chosen compatible
obstruction theories consequently give
\[
 [\widehat{\mathcal M}]^{\rm ref}
      =\widehat\epsilon^![\widehat{\mathfrak B}],
 \qquad
 [\mathcal M]^{\rm ref}=\epsilon^![\mathfrak B].
\]

The comparison of the two obstruction theories includes the
morphisms to their relative cotangent complexes.
The smooth morphism $\mathfrak B\to\mathfrak A$ is obtained
from $X\to\mathcal X$, where $\mathcal X$ is the Artin fan.
Its relative tangent bundle pulls back to $p^*f^*T_X^{\log}$.
Over $D_1^\circ$ it is the Atiyah extension
\[
 0\longrightarrow\mathcal O_{D_1^\circ}
 \longrightarrow T_X^{\log}|_{D_1^\circ}
 \longrightarrow T_{D_1^\circ}\longrightarrow0,
\]
with the line convention determined by $X\to\mathcal X$.
Its rank is $\dim X$.
The evaluation exact sequence gives the compatible triangle
\[
 (p^*f^*T_X^{\log})^\vee\longrightarrow(R\pi_*f^*T_X^{\log})^\vee
 \longrightarrow(R\pi_*f^*T_X^{\log}(-p))^\vee\longrightarrow .
\]
\cite[Section~4.2, diagram~(4.15), Remark~4.4, and Proposition~4.5]{ACGSPunctured} construct a morphism from this triangle to
the cotangent triangle for
$\mathcal M\to\mathfrak B\to\mathfrak A$.
One chooses a compatible final arrow in that construction;
its resulting virtual class is independent of that choice.

Strictness of $\epsilon$ makes
\[
 \begin{matrix}
 \mathcal M_{\log}&\longrightarrow&\mathcal M_0\\
 \downarrow&&\downarrow\\
 \mathfrak B_{\log}&\xrightarrow{\nu}&\mathfrak B_0
 \end{matrix}
\]
cartesian on underlying algebraic stacks.
Write $\epsilon_0$ and $\epsilon_{\log}$ for the base changes of $\epsilon$ to $\mathfrak B_0$ and $\mathfrak B_{\log}$. Both vertical arrows carry the pullback of the chosen enhanced perfect obstruction theory.
By \cite[Definition~3.10 and Proposition~3.12]{GSIntrinsic}, the
class on the upper left is
$\epsilon_{\log}^![\mathfrak B_{\log}]$.
The log smoothness used in the lemma also gives the required
pure-dimensional base; compare their Proposition~3.19.
The proper-pushforward property of the virtual pullback class,
the lemma, and its compatibility with the regular point Gysin
map now give
\[
 \begin{split}
 (\nu_{\mathcal M})_*[\mathcal M_{\log}]^{\rm vir}
   &=\epsilon_0^!\nu_*[\mathfrak B_{\log}]\\
   &=\epsilon_0^![\mathfrak B_0]\\
   &=z^!\epsilon^![\mathfrak B]\\
   &=z^![\widehat{\mathcal M}]^{\rm ref}.
 \end{split}
\]
The last equality uses locality near the point fiber.
Pushing by $\widehat j$ proves
\eqref{eq:direct-point-insertion}.

Fine pullback of a puncturing preserves the underlying curve
and prestability; see \cite[Proposition~2.8 and Corollary~2.10]{ACGSPunctured}.  The source puncturing is preserved. Hence the ordinary stable map and its cotangent
lines are the same after the indicated base changes.
The descendant statement and any proper push to an ordinary
receiver follow from the projection formula.
\end{proof}

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\noindent\textsc{Zhili College, Tsinghua University, Beijing 100084, China}\par
\noindent\textit{Email address:}\enspace\mbox{\href{mailto:tangqy24@mails.tsinghua.edu.cn}{\texttt{tangqy24@mails.tsinghua.edu.cn}}}\par
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