$\begin{split}\int_{[\mathscr{M}(X_{r},\beta)]^{vir}}\psi_{x_{out}}^{a}ev_{x_{out}}^{*}([pt])\prod_{i=1}^{l}\psi_{x_{i}}^{a_{i}}\alpha_{i}&=b_{1}^{m_{1,-}}\prod_{i>1}b_{i}^{m_{i,-}}\int_{\small{[\mathscr{M}^{orb}((X_{r})_{\textbf{b}},\beta)]^{vir}}}\psi_{x_{out}}^{a}ev_{x_{out}}^{*}([pt])\prod_{i=1}^{p}\psi_{x_{i}}^{a_{i}}\alpha_{i}\\\
&=\frac{1}{k^{m_{1,-}}}\langle\prod_{i>1}\psi_{x_{i}}^{a_{i}}[\alpha_{i}],\psi_{x_{out}}^{a}[pt]\rangle_{\textbf{A}}.\end{split}$

By the second part of Lemma 7.1, the vanishing hypothesis gives the left hand
side of the above expression is zero for all $\alpha_{i}$, hence the righthand
side vanishes, as required for the first claim of the corollary. The claimed
equality of structure constants follows by multiplying both sides of Equation
7 by $k$, and combining the resulting equality with the equality deduced in
Corollary 7.3.

∎

Using the corollary above, Theorem 1.1 follows in a straightforward manner
from the birational invariance of the invariants $N_{p,q,r}^{\textbf{B}}$
proven in [Joh22a] Corollary $1.6$:

###### Proof of Theorem 1.1.

Letting $p,q,r\in B(\mathbb{Z})$ and $\textbf{B}\in NE(X)$, after packing this
discrete data into the type $\beta$, consider the moduli space
$\mathscr{M}(X,\beta)$ of punctured log maps. Pick a slope sensitive
modification $\pi:\tilde{X}\rightarrow X$ with respect to $\beta$, i.e. a log
étale modification such that any lift of the discrete data $\beta$ to
$\tilde{X}$ is slope sensitive. After potentially further blowups, we may
assume $r$ is contained in a $1$-dimensional cone of $\Sigma(\tilde{X})$.
Thus, the requirements of Corollary 7.4 are satisfied, and for any
$\textbf{A}\in NE(\tilde{X})$, and we have the equality:

(7.4) $N_{p_{1},p_{2},r}^{\textbf{A}}=N_{p_{1},p_{2},r}^{orb\textbf{A}}$

Moreover, by Corollary $1.6$ of [Joh22a], we have:

(7.5)
$N_{p_{1},p_{2},r}^{\textbf{B}}=\sum_{\pi_{*}(\textbf{A})=\textbf{B}}N_{p_{1},p_{2},r}^{\textbf{A}}$

Substituting the right hand side of Equation 7.4 for terms on the right hand
side of 7.5 gives the desired result. ∎

We now assume the pair $(X,D)$ satisfies condition $1$ of Assumption 4.1.
Under this assumption, Tseng and You proved that the invariants above give the
structure constants of an algebra over $S_{I}$. Using Theorem 1.1, together
with birational invariance of the structure constants used in the definition
of the logarithmic mirror algebra, we give a proof of associativity of the log
mirror algebra with the same assumptions.

###### Proof of Corollary 1.2 with Assumtion 4.1(1).

Let $p_{1},p_{2},p_{3}\in B(\mathbb{Z})$, and consider the
$z^{\textbf{A}}\vartheta_{r}$ term of the two following products of theta
functions:

(7.6)
$\displaystyle\vartheta_{p_{1}}(\vartheta_{p_{2}}\vartheta_{p_{3}})[z^{\textbf{A}}\vartheta_{r}]$
$\displaystyle=\sum_{\textbf{A}=\textbf{A}_{1}+\textbf{A}_{2}\text{, }s\in
B(\mathbb{Z})}N_{p_{1},s,r}^{\textbf{A}_{1}}N_{p_{2},p_{3},s}^{\textbf{A}_{2}}$
(7.7)
$\displaystyle(\vartheta_{p_{1}}\vartheta_{p_{2}})\vartheta_{p_{3}}[z^{\textbf{A}}\vartheta_{r}]$
$\displaystyle=\sum_{\textbf{A}=\textbf{A}_{1}+\textbf{A}_{2}\text{, }s\in
B(\mathbb{Z})}N_{p_{1},p_{2},s}^{\textbf{A}_{1}}N_{p_{3},s,r}^{\textbf{A}_{2}}.$

By Proposition 4.5, only finitely many choices of $s\in B(\mathbb{Z})$ for a
given choice of splitting $\textbf{A}_{1}+\textbf{A}_{2}=\textbf{A}$ could
possibly yield a non-zero contribution to right hand side of the equalities
above. Hence, only finitely many moduli space of punctured log maps appear in
defining this collection of structure constants. We may therefore pass to a
sufficiently fine subdivision of $(X,D)$, $(\tilde{X},\tilde{D})$, such that
each of the resulting types $\beta$ are slope sensitive. By birational
invariance of the structure constants, we find the two products of $\vartheta$
functions above are respectively equal to:

(7.8)
$\vartheta_{p_{1}}(\vartheta_{p_{2}}\vartheta_{p_{3}})[z^{\textbf{A}}\vartheta_{r}]=\sum_{\textbf{A}=\textbf{A}_{1}+\textbf{A}_{2}\text{,
}s\in
B(\mathbb{Z})}\sum_{\pi_{*}(\textbf{B}_{i})=\textbf{A}_{i}}N_{p_{1},s,r}^{\textbf{B}_{1}}N_{p_{2},p_{3},s}^{\textbf{B}_{2}}$
(7.9)
$(\vartheta_{p_{1}}\vartheta_{p_{2}})\vartheta_{p_{3}}[z^{\textbf{A}}\vartheta_{r}]=\sum_{\textbf{A}=\textbf{A}_{1}+\textbf{A}_{2}\text{,
}s\in
B(\mathbb{Z})}\sum_{\pi_{*}(\textbf{B}_{i})=\textbf{A}_{i}}N_{p_{1},p_{2},s}^{\textbf{B}_{1}}N_{p_{3},s,r}^{\textbf{B}_{2}}.$

By associativity of the orbifold mirror algebra i.e. Theorem $37$ of [TY20],
after fixing $\textbf{B}\in NE(\tilde{X})$ such that
$\pi_{*}(\textbf{B})=\textbf{A}$, we have:

(7.10) $\sum_{\textbf{B}=\textbf{B}_{1}+\textbf{B}_{2}\text{, }s\in
B(\mathbb{Z})}N_{p_{1},s,r}^{\textbf{B}_{1},orb}N_{p_{2},p_{3},s}^{\textbf{B}_{2},orb}=\sum_{\textbf{B}=\textbf{B}_{1}+\textbf{B}_{2}\text{,
}s\in
B(\mathbb{Z})}N_{p_{1},p_{2},s}^{\textbf{B}_{1},orb}N_{p_{3},s,r}^{\textbf{B}_{2},orb}.$

As before, for a given splitting $\textbf{B}=\textbf{B}_{1}+\textbf{B}_{2}$,
there are finitely many choices of $s\in B(\mathbb{Z})$ which could possibly
yield a non-zero contribution to either side of the equality above. In fact,
the moduli stack used to define the invariant
$N_{p_{1},p_{2},s}^{\textbf{B}_{1}}$ satisfies the condition of Proposition
4.5 required for non-emptiness only if the moduli stack used to define
$N_{p_{1},p_{2},s}^{\pi_{*}(\textbf{B})}$ satisfies the condition of
Proposition 4.5. Hence, both side of Equation 7.10 only include invariants
associated with discrete data potentially contributing to either equation 7.8
or 7.9. In particular, by our choice of blowup and Corollary 7.4, we have
$N_{p,q,r}^{\textbf{B}}=N_{p,q,r}^{\textbf{B},orb}$ for all invariants
appearing in the relation of orbifold invariants above.

By summing over all choices of curve classes B with
$\pi_{*}(\textbf{B})=\textbf{A}$ and appropriately rearranging terms using the
birational invariance of the structure constants, we derive the desired
equality. More explicitly, by Corollary $1.6$ of [Joh22a], we have

$N_{p_{1}p_{2}r}^{\textbf{A}}=\sum_{\pi_{*}(\textbf{B})=\textbf{A}}N_{p_{1}p_{2}r}^{\textbf{B}}$

Using this equation, we have the following equality:

$\begin{split}\sum_{\pi_{*}(\textbf{B})=\textbf{A}}\sum_{\textbf{B}=\textbf{B}_{1}+\textbf{B}_{2}\text{,
}s\in
B(\mathbb{Z})}N_{p_{1},s,r}^{\textbf{B}_{1}}N_{p_{2},p_{3},s}^{\textbf{B}_{2}}&=\sum_{\textbf{A}=\textbf{A}_{1}+\textbf{A}_{2}}(\sum_{\pi_{*}(\textbf{B}_{1})=\textbf{A}_{1}}N_{p_{1},s,r}^{\textbf{B}_{1}})(\sum_{\pi_{*}(\textbf{B}_{2})=\textbf{A}_{2}}N_{p_{2},p_{3},s}^{\textbf{B}_{2}})\\\
&=\sum_{\textbf{A}=\textbf{A}_{1}+\textbf{A}_{2}}N_{p_{1},s,r}^{\textbf{A}_{1}}N_{p_{2},p_{3},s}^{\textbf{A}_{2}}\\\
&=\vartheta_{p_{1}}(\vartheta_{p_{2}}\vartheta_{p_{3}})[z^{\textbf{A}}\vartheta_{r}].\end{split}$

Similarly, we have:

$\sum_{\pi_{*}(\textbf{B})=\textbf{A}}\sum_{\textbf{B}=\textbf{B}_{1}+\textbf{B}_{2}\text{,
}s\in
B(\mathbb{Z})}N_{p_{1},p_{2},s}^{\textbf{B}_{1}}N_{p_{3},s,r}^{\textbf{B}_{2}}=\sum_{\textbf{A}=\textbf{A}_{1}+\textbf{A}_{2}}N_{p_{1},p_{2},s}^{\textbf{A}_{1}}N_{p_{3},s,r}^{\textbf{A}_{2}}=(\vartheta_{p_{1}}\vartheta_{p_{2}})\vartheta_{p_{3}}[z^{\textbf{A}}\vartheta_{r}].$

Since the left hand sides of the two above equations are equal by Equation
7.10, the right hand sides are equal, as required.

∎

##  8\. WDVV, TRR, and intrinsic mirror symmetry for log Calabi-Yau varieties

To prove our main theorems for all log Calabi-Yau pairs considered in [GS19],
we recall the following two relations which hold in the large $r$ genus $0$
orbifold Gromov-Witten theory for any simple normal crossings pair $(X,D)$,
see [TY20] Propositions $27$ and $26$ respectively:

######  Proposition 8.1 (WDVV relation).

Let $\textbf{A}\in NE(X)$, and a collection of $m$ pairs
$(s_{i},[\gamma_{i}]_{s_{i}})$ with $s_{i}\in\Sigma(X)(\mathbb{Z})$ and
$[\gamma_{i}]_{s_{i}}\in H^{*}(D_{s})$. After picking a graded basis
$\\{T_{s,k}\\}\subset H^{*}(D_{s})$, with a dual basis $\\{T_{s}^{k}\\}\subset
H^{*}(X_{s})$ under the intersection pairing on $H^{*}(X_{s})$, then by
summing over all tuples
$(\textbf{A}_{1},\textbf{A}_{2},s,S_{1},S_{2},T_{s,k})$ with
$\textbf{A}_{1}+\textbf{A}_{2}=\textbf{A}$, $S_{1}\sqcup
S_{2}=[m]\setminus[4]$, $s\in\mathbb{Z}^{m}$ and $T_{s,k}$ an element of the
chosen basis $H^{*}(X_{s})$.

(8.1)
$\begin{split}\sum_{(\textbf{A}_{1},\textbf{A}_{2},S_{1},S_{2},s,T_{s,k})}\langle[\gamma_{1}]_{s_{1}},[\gamma_{2}]_{s_{2}},\prod_{j\in
S_{1}}[\gamma_{j}]_{s_{j}},T_{-s,k}\rangle_{\textbf{A}_{1}}\langle
T^{k}_{s},[\gamma_{3}]_{s_{3}},[\gamma_{4}]_{s_{4}},\prod_{j\in
S_{2}}[\gamma_{j}]_{s_{j}}\rangle_{\textbf{A}_{2}}\\\
=\sum_{(\textbf{A}_{1},\textbf{A}_{2},s,S_{1},S_{2},T_{s,k})}\langle[\gamma_{2}]_{p_{2}},[\gamma_{3}]_{p_{3}},\prod_{j\in
S_{1}}[\gamma_{j}]_{s_{j}},T_{-s,k}\rangle_{\textbf{A}_{1}}\langle
T^{k}_{s},[\gamma_{1}]_{p_{1}},[\gamma_{4}]_{s_{4}},\prod_{j\in
S_{2}}[\gamma_{j}]_{s_{j}}\rangle_{\textbf{A}_{2}}.\end{split}$

######  Proposition 8.2 (Topological recursion relation).

With notation as above but with $S_{1}\sqcup S_{2}=[m]\setminus[3]$, and
$a_{1},\ldots,a_{l}\in\mathbb{N}$, we have the following equality of large $r$
orbifold Gromov-Witten invariants:

(8.2)
$\begin{split}\langle\prod_{i}[\gamma_{i}]_{s_{i}}\psi^{a_{i}}\rangle=\sum_{(\textbf{A}_{i},s,S_{i},T_{s,k})}\langle\psi^{a_{1}-1}[\gamma_{1}]_{s_{1}},\prod_{j\in
S_{1}}\psi^{a_{j}}[\gamma_{j}]_{s_{j}},T_{s,k}\rangle_{\textbf{A}_{1}}\langle
T_{-s}^{k},\psi^{a_{2}}[\gamma_{2}]_{s_{2}},\psi^{a_{3}}[\gamma_{3}]_{s_{3}},\prod_{j\in
S_{2}}\psi^{a_{j}}[\gamma_{j}]_{s_{j}}\rangle_{\textbf{A}_{2}}.\end{split}$

As before, in the logarithmic setting, we will have access to analogous
relations only if we assume all the moduli spaces determined by the discrete
data appearing in equations above are slope sensitive. More explicitly, after
fixing the initial data of the above two propositions, let us say a decorated
tropical type $\beta=(\textbf{A}_{1},((s_{i}),s))$ is a _potentially
contributing type_ if the type is balanced in the sense of Proposition 4.5 and
there exists a curve class $\textbf{A}_{2}$ such that
$\textbf{A}_{1}+\textbf{A}_{2}=\textbf{A}$. By Proposition 4.5, the space of
orbifold maps with discrete data $(\textbf{A}_{1},((s_{i}),s))$ is non-empty
only if the discrete data is potentially contributing. Moreover, there are
only finitely many potentially contributing types. Indeed, by fixing an ample
line bundle $L$ on $X$ and defining $deg(\textbf{B})=\textbf{B}\cdot
c_{1}(L)$, there are only finitely many curves $\textbf{A}^{\prime}$ such that
$deg(\textbf{A}^{\prime})\leq deg(\textbf{A})$. Moreover, by Proposition 4.5,
there are only finitely many choices of contact order $s$ after having fixed
$\textbf{A}_{1}$ and $s_{i}$.

As a result, there are only finitely many non-zero possible terms in either of
the above relations. Therefore, by Section $3.5$ of [BNR22], there exists a
smooth projective resolution $\tilde{X}\rightarrow X$ such that any lift of a
potentially contributing tropical type to $\Sigma(\tilde{X})$ is slope-
sensitive. In addition, it is straightforward to check that any potentially
contributing type for $\tilde{X}$ pushes forward to a potentially contributing
type for $X$. Hence, all potentially contributing types to the WDVV and TRR
equations for $\tilde{X}$ are slope sensitive. In particular, we produce
relations in the log Gromov-Witten theory for $\tilde{X}$ by the main result
of [BNR24]. We record this fact in the following theorem:

######  Theorem 8.3.

Given a curve class $\textbf{A}\in NE(X$), and integral points
$s_{i}\in\Sigma(X)(\mathbb{Z})$ for $1\leq i\leq m$, there exists a smooth
projective log étale modification $\pi:\tilde{X}\rightarrow X$ such that after
a choice of lift $s_{i}^{\prime}\in\Sigma(\tilde{X})(\mathbb{Z})$, for any
class B with $\pi_{*}(\textbf{B})=\textbf{A}$, equations 8.1 and 8.2 hold
after replacing all orbifold Gromov-Witten invariants with corresponding
refined punctured log Gromov-Witten invariants defined in [BNR24].

######  Remark 8.4.

When the target pair is toric, the genus $0$ log GW theory is entirely
captured by tropical geometry. As a result, when $X$ is a surface, one should
recover results of Gathman, Markwig and Rau in [GM08], [MR09], and for general
toric $X$ the WDVV and TRR relations investigated in [Rau16]. The latter paper
including a hypothesis of $\Theta$ directionality, which appears to be implied
by the assumption that all moduli spaces involved are slope sensitive. The
source of complexity requiring this extra condition comes from issues with
splitting the diagonal of the target, analogous to the problems found in
gluing formulas for log GW invariants.

In order to use these formula, one needs to know a log étale modification of
the target space in which the tropical types are slope sensitive, which
typically requires a good understanding of the tropical types associated to
strata of the relevant moduli space. Assuming this is known however, the extra
cohomology classes appearing in the formula come correspond to piecewise
polynomial functions on the tropicalization, see for instance [HMP+22] Section
$6.6.2$. In this setting however, the relevant classes will be fundamental
classes of strata as well as their Poincaré dual point classes, leading to a
universal formula that assumes no further geometric input aside from
assumption of log Calabi-Yau.

For our application of interest, we will be considering on the one hand $m=4$,
with $p_{1},p_{2},p_{3},-r\in\Sigma(X)(\mathbb{Z})$ and insertions
$[1]_{p_{1}},[1]_{p_{2}},[1]_{p_{3}},[pt]_{-r}$, and on the other hand $m=k+1$
with $p_{0}=0$, $p_{i}\in\Sigma(X)(\mathbb{Z})$ for $i\not=0$, and insertions
$[pt]_{p_{0}}\psi^{k-2},[1]_{p_{1}},\ldots,[1]_{p_{k}}$. In both cases of
interest, equations 8.1 and 8.2 do not yield the desired relations, but only
after showing various terms contributing to summed expressions vanish. To
deduce the necessary vanishing of orbifold invariants, we will show an
associated log invariant vanishes, and use the log-orbifold correspondence of
[BNR22] to prove the necessary vanishing of orbifold invariants.

###### Proof of Corollary 1.2.

Consider the WDVV relation described following Equation 8.1:

(8.3)
$\begin{split}\sum_{(\textbf{A}_{1},\textbf{A}_{2},s,k)}\langle[1]_{p_{1}},[1]_{p_{2}},T_{-s,k}\rangle_{\textbf{A}_{1}}\langle
T^{k}_{s},[1]_{p_{3}},[pt]_{-r}\rangle_{\textbf{A}_{2}}\\\
=\sum_{(\textbf{A}_{1},\textbf{A}_{2},s,k)}\langle[1]_{p_{2}},[1]_{p_{3}},T_{-s,k}\rangle_{\textbf{A}_{1}}\langle
T^{k}_{s},[1]_{p_{1}},[pt]_{-r}\rangle_{\textbf{A}_{2}}.\end{split}$

As only finitely many orbifold invariants are involved in the above relation,
we pass to a smooth projective log modification $\tilde{X}\rightarrow X$ in
which all tropical lifts of all tropical types corresponding to strata of the
associated log moduli problems are slope sensitive, and $r\in\Sigma(X)^{[1]}$.
By Theorem 8.3, we know all contributing combinatorial types to the orbifold
WDVV relation for $\tilde{X}$ are slope sensitive.

We must show that the only possible terms which give a non-zero contribution
to either side of equation 8.3 for the target $\tilde{X}$ have insertions
either $[1]_{s}$ or $[pt]_{s}$, with one of $\pm s$ a positive contact order.
Since each is a product of two orbifold Gromov-Witten invariants, we will show
that for terms with insertion not of this form, then at least one of them is
zero. We prove this statement using both the log/orbifold correspondence
established in [BNR24], and a modification of the proof Lemma $7.14$ of
[GS19], referred to as the no-tail lemma in loc. cit.

######  Lemma 8.5.

If $\langle T_{s}^{k},[1]_{p},[pt]_{-r}\rangle_{\textbf{A}_{2}}\not=0$, then
$s\in B(\mathbb{Z})$ and $T_{s}^{k}=[1]_{s}$.

###### Proof.

We bundle the degree $\textbf{A}_{2}$ and contact orders $-r,p,s$ into the
type $\beta$, and consider the associated moduli space of punctured log maps
$\mathscr{M}(X,\beta)$. By assumption, we have $r\in\Sigma^{[1]}(X)$. Thus, by
Corollary 7.4, to show $\langle
T_{s},[1]_{p},[pt]_{-r}\rangle_{\textbf{A}_{2}}=0$, it suffices to consider
the point constrained moduli spaces $\mathscr{M}(X,\tau,z)$ for all realizable
tropical types $\tau$ marked by $\beta$, and show that the virtual classes
$[\mathscr{M}(X,\tau,z)]^{vir}$ are all zero. We will show this is the case
unless we have $s\in B(\mathbb{Z})$.

To begin, note that by the point constraint, $G_{\tau}$ must contain a vertex
$v_{0}$ containing the leg with contact order $-r$ mapping into the cone
$\sigma_{r}$ with $r\in\sigma_{r,\mathbb{N}}$. In particular, this vertex must
map to a cone in $\mathscr{P}$. If $v_{0}$ were only $2$-valent, then we may
cut along the unique edge containing $v_{0}$ to produce two tropical types
which glue to give $\tau$, one of which consists of a single vertex with $2$
legs. By the vanishing tail lemma of [GS19] i.e. Theorem $7.14$ of loc cit, we
must have $[\mathscr{M}(X,\boldsymbol{\tau},z)]^{vir}=0$.

Thus, we suppose $v_{0}$ is at least $3$-valent. We define the spine of $\tau$
be the subgraph of $G_{\tau}$ given by the convex hull of the legs with
contact orders $-r$ and $s$. We will now show that each edge in this spine is
contained in $B$:

######  Lemma 8.6.

Let $\tau$ be a tropical type with legs $L,l_{out},l_{2}$, with
$u_{l_{out}}=-r$ and $u_{l_{2}}=p$, and a vertex $v\in V(G_{\tau})$ with
$\boldsymbol{\sigma}(v)\in\mathscr{P}(X)$ and contained in the legs
$l_{out},l_{2}$. If $[\mathscr{M}(X,\boldsymbol{\tau},z)]^{vir}\not=0$, then
for all edges and legs $e\in E(G_{\tau})\cup L(G_{\tau})$ contained in the
convex hull of $v$ and $L$, we have $\boldsymbol{\sigma}(e)\subset B$.

###### Proof.

We prove the lemma by induction on length of the chain of edges and legs in
the convex hull of $L$ and $l_{out}$. To that end, enumerate the edges in this
convex hull by $l_{out}=e_{0},e_{1},\ldots,e_{J}=L$, with $e_{1}$ the unique
edge in the convex hull which contains $v_{0}$ which is not equal to $L_{out}$
nor $l_{2}$. As a base case, note that $r\in B(\mathbb{Z})$, and since
$v_{0}\in B$ by the point constraint, the leg with contact order $-r$ must be
contained in $B$, i.e.
$\boldsymbol{\sigma}(l_{out}),\boldsymbol{\sigma}(l_{2})\subset B$. Now for
the induction hypothesis, we suppose we have an edge $e_{k}$ in the spine such
that $\boldsymbol{\sigma}(e_{k-1})\subset B$. In particular,
$D_{i}^{*}(u_{e_{k-1}})=0$ for all bad divisors $D_{i}$. Letting $v\in
V(G_{\tau})$ be the shared vertex of $e_{k}$ and $e_{k-1}$, consider the two
tropical types produced by cutting $\tau$ along $e_{k}$, and let $\tau_{1}$
the resulting tropical type containing $v$. By the induction hypothesis, we
may assume $\boldsymbol{\sigma}(e_{k-1})\in B$, so in particular $v\in B$.
Hence, $D_{i}^{*}(u_{e_{k}})\geq 0$ for any bad divisor $D_{i}$. Note now that
since $c_{1}(T_{X}^{log})=\sum_{i}-a_{i}D_{i}$, with $a_{i}\geq 0$ and
$a_{i}=0$ if and only if $\sigma_{D_{i}}\in B\subset\Sigma(X)$, and [GS19]
Corollary $1.14$, we have:

$A(\boldsymbol{\tau}_{1})\cdot c_{1}(T_{X}^{log})=\sum_{D_{i}\text{ }bad\text{
}divisor}-a_{i}D_{i}^{*}(u_{e_{k}}+p-r)=\sum_{D_{i}\text{ }bad\text{
}divisor}-a_{i}D_{i}^{*}(u_{e_{k}})\leq 0.$

Hence we must have $A(\tau_{1})\cdot c_{1}(T_{X}^{log})\leq 0$, with
$A(\tau_{1})\cdot c_{1}(T_{X}^{log})=0$ if and only if
$D_{i}^{*}(u_{e_{k}})=0$ for all bad divisors $D_{i}$. Since $v\in B$, this
would also imply $e_{k}\in B$. Thus, suppose $A(\tau_{1})\cdot
c_{1}(T_{X}^{log})<0$. Letting $x_{out}$ and $x$ correspond to the special
sections associated with the legs $l_{out}$ and $L$ respectively, note that by
construction and Corollary $7.3$ of [ACGS20], we have the following cartesian
diagram, with the right vertical arrow equipped with the product of
obstruction theories:

${\mathscr{M}(X,\tau,z)}$${\mathscr{M}(X,\tau_{1},z)\times\mathscr{M}(X,\tau_{2})}$${\mathscr{M}(X,\tau_{1})\times\mathscr{M}(X,\tau_{2})}$${\mathfrak{M}^{ev(x_{out})}(\mathcal{X},\tau,z)}$${\mathfrak{M}^{ev(x_{out},x)}(\mathcal{X},\tau_{1},z)\times\mathfrak{M}^{ev(x)}(\mathcal{X},\tau_{2})}$${\mathfrak{M}^{ev(x_{out},x)}(\mathcal{X},\tau_{1})\times\mathfrak{M}^{ev(x)}(\mathcal{X},\tau_{2}).}$

Moreover, the obstruction theories for the middle and left vertical arrow is
the pullback of the obstruction theory of the right vertical arrow. By [GS19]
Theorem $A.13$, if we show $\mathscr{M}(X,\boldsymbol{\tau}_{1},z)$ has
negative virtual dimension, we will have shown that $\boldsymbol{\tau}$ cannot
contribute.

With this end in mind, we note that the perfect obstruction theory for the
morphism
$\mathscr{M}(X,\boldsymbol{\tau}_{1},z)\rightarrow\mathfrak{M}^{ev(x_{out},x)}(\mathcal{X},\tau_{1},z)$
gives $\mathscr{M}(X,\boldsymbol{\tau}_{1},z)$ the following relative virtual
dimension over $\mathfrak{M}^{ev(x_{out},x)}(\mathcal{X},\tau_{1},z)$:

$\chi(f^{*}T_{X}^{log}(-x_{out}-x))=A(\boldsymbol{\tau}_{1})\cdot
c_{1}(T_{X}^{log})-dim\text{ }X<-dim\text{ }X.$

The equality above follows from Riemann-Roch and the inequality follows from
assumption. Thus, it suffices to show that $dim\text{
}\mathfrak{M}^{ev(x_{out},x)}(\mathcal{X},\tau_{1},z)\leq dim\text{ }X$. Note
that this dimension equals:

$dim\text{
}\mathfrak{M}^{ev(x_{out},x)}(\mathcal{X},\tau_{1},z)=dim(\mathfrak{M}^{ev(x_{out})}(\mathcal{X},\tau_{1})\times_{\mathscr{P}(X,r)}^{fs}W)\times_{\underline{\mathcal{X}}}\underline{X}=dim(\mathfrak{M}^{ev}(\mathcal{X},\tau_{1})\times_{\mathscr{P}(X,r)}^{fs}W)+dim\text{
}X.$

On the other hand, since $r$ is contained in the $1$-skeleton of $B$,
$dim\text{ }\boldsymbol{\sigma}(v_{out})\leq 1$. Since the tropical evaluation
map $ev_{v_{out}}^{gp}:\tau_{1}\rightarrow\boldsymbol{\sigma}(v_{out})$ must
intersect the interior of $\boldsymbol{\sigma}(v_{out})$, the evaluation must
be surjective. Thus, we have $dim\text{ }\tau_{1}\geq codim\text{ }X_{r}$. By
[ACGS20] Proposition $3.28$, we therefore have $dim\text{
}\mathfrak{M}^{ev}(\mathcal{X},\tau_{1})\leq dim\text{ }X-codim\text{
}X_{r}=dim\text{ }X_{r}$.

To bound the dimension of $\mathfrak{M}^{ev}(\mathcal{X},\tau_{1},z)$, we note
that after removing all stratum
$\mathfrak{M}_{\tau_{1}^{\prime}}^{ev}\subset\mathfrak{M}^{ev}(\mathcal{X},\tau_{1})$
with
$\mathfrak{M}^{ev}_{\tau_{1}^{\prime}}\times_{\mathscr{P}(X,r)}W=\emptyset$,
we have $\mathfrak{M}^{ev}(\mathcal{X},\tau_{1})$ is isomorphic to either
$\mathfrak{M}(\mathcal{X},\tau_{1})\times X^{\circ}$ or a
$\mathbb{G}_{m}$-torsor over the product
$\mathfrak{M}(\mathcal{X},\tau_{1})\times X_{r}^{\circ}$, depending whether or
not $r=0$. Indeed, the first isomorphism is by definition. For the second
isomorphism, we let $\mathcal{U}$ be the universal $\mathbb{G}_{m}$-torsor
over $B\mathbb{G}_{m}$, we consider the morphism
$(ev_{x_{out}}\circ\pi_{1})^{*}(\mathcal{U})\otimes(p\circ\pi_{2})^{*}(\mathcal{U}^{\vee}):\mathfrak{M}(\mathcal{X},\tau_{1})\times
X_{r}^{\circ}\rightarrow B\mathbb{G}_{m}$ with $p:X_{r}^{\circ}\rightarrow
B\mathbb{G}_{m}$ the morphism classifying the log structure on $X_{r}$. The
second isomorphisms follows by observing the following is a cartesian diagram,
with the bottom horizontal morphism given above:

${\mathfrak{M}^{ev}(\mathcal{X},\tau_{1})}$${Spec\text{
}\mathds{k}}$${\mathfrak{M}(\mathcal{X},\tau_{1})\times
X_{r}^{\circ}}$${B\mathbb{G}_{m}}$

Using this description, it follows that after removing the relevant strata
from the source that $\mathfrak{M}^{ev}(\mathcal{X},\tau_{1})\rightarrow
X_{r}$ is flat, and hence
$\mathfrak{M}^{ev}(\mathcal{X},\tau_{1})\times_{X_{r}}z$ has dimension

$dim\text{ }\mathfrak{M}^{ev}(\mathcal{X},\tau_{1})\times_{X_{r}}z=dim\text{
}\mathfrak{M}^{ev}(\mathcal{X},\tau_{1})-dim\text{ }X_{r}.$

Additionally, it is straightforward to check that the previous fiber product
of stacks is also the stack theoretic fiber product
$\mathfrak{M}^{ev}(\mathcal{X},\tau_{1})\times_{\underline{\mathscr{P}(X,r)}}\underline{W}$.
Thus, we have the following dimension bound:

$dim\text{
}\mathfrak{M}^{ev}(\mathcal{X},\tau_{1})\times_{\underline{\mathscr{P}(X,r)}}\underline{W}\leq
dim\text{ }X_{r}-dim\text{ }X_{r}=0.$

Since the dimension of fs fiber products is bounded above by the dimension of
the fiber product of underlying stacks, the fs fiber product has dimension
bounded above by $0$. From this inequality, we deduce $dim\text{
}\mathfrak{M}^{ev(x_{out},x)}(\mathcal{X},\tau_{1},z)\leq dim\text{ }X$,
implying the desired negative virtual dimension.

∎

Thus, the spine of $\tau$ must map into $B\subset\Sigma(X)$. Note that if
$[\mathscr{M}(X,\beta,z)]^{vir}\not=0$, we must have
$[\mathscr{M}(X,\boldsymbol{\tau},z)]^{vir}\not=0$ for some realizable
tropical type $\tau$ marked by $\beta$. As was just shown, this can only be
the case if $D_{i}^{*}(s)=0$ for all bad components $D_{i}$ of $D$. As a
result, we have $c_{1}(T_{X}^{log})\cap\textbf{A}_{2}=0$. This restriction
allows us to remove the nef/antinef assumption appearing in Theorem $37$ of
[TY20], and the argument for this theorem in loc. cit. now shows
$D_{i}^{*}(s)\geq 0$ for all components $D_{i}$ of $D$. Indeed, recall from
Equation 3.1 that the virtual dimension of the moduli space
$\mathscr{M}^{orb}(X,\beta)$ is

$vdim\text{
}\mathscr{M}^{orb}(X,\beta)=c_{1}(T_{X}^{log})\cdot\textbf{A}_{2}-3+n+dim\text{
}X-\sum_{i}e_{i}=dim\text{ }X-\sum_{i}e_{i}.$

Since the contact order $r$ satisfies $D_{i}^{*}(r)<0$ for only one divisor by
the slope sensitivity condition, and the contact order $p_{3}$ satisfies
$D_{i}^{*}(p_{3})\geq 0$ for all components $D_{i}\subset D$, letting
$e_{s}=|\\{D_{i}\text{ }\mid{}D_{i}^{*}(s)<0\\}|$, we have:

$vdim\text{ }\mathscr{M}^{orb}(X,\beta)=dim\text{ }X-1-e_{s}.$

If $\langle T_{s},[1]_{p_{1}},[pt]_{-r}\rangle_{\textbf{A}_{2}}\not=0$, the
virtual dimension of $\mathscr{M}^{orb}(X,\beta)$ must be at least $dim\text{
}X-1$, implying $e_{s}=0$ and $T_{s}=[1]_{s}$. ∎

The above lemma allows us to conclude all the additional terms in the WDVV
equation appearing on both sides of equation 8.3 vanish in the non-minimal
setting as well, and the resulting equality gives the desired associativity of
the log mirror algebra. ∎

######  Remark 8.7.

While the examples given in Section $5$ show that the structure constants used
to define $R_{(X,D)}$ and $R_{(X,D)}^{orb}$ are distinct, their difference is
precisely because of the fact the latter algebra do not behave nicely under
base-change, unlike the birational invariance property that $R_{(X,D)}$
established in [Joh22a].

Additionally, it is noted in [GS19] Remark $4.17$ that the no-tail lemma is
crucial for associativity of the product, as it was in the above proof of
associativity, and the failure of associativity in general should be encoded
in a non-zero differential used in the definition of the analogue of
symplectic cohomology in this setting.

######  Example 8.8.

To better understand how the proof of associativity in this work relates with
the proof found in [GS19], we consider the extended example $4.5$ of [GS19].
This example features the blowup of $\mathbb{P}^{2}$ at an interior point on a
coordinate axis, and $D$ is given by the strict transform of the toric
boundary, we refer to the resulting log Calabi-Yau as $(X,D)$. The cone
complex $\Sigma(X)$ can be identified with the cone complex associated with
the fan of $\mathbb{P}^{2}$. Letting $D_{1}$ be the boundary component of the
boundary which intersects the exceptional divisor, and $D_{2}$ and $D_{3}$ the
remaining components, we consider the product of the theta functions
$\vartheta_{p_{1}},\vartheta_{p_{2}},\vartheta_{p_{3}}$, with:

$p_{1}=D_{1}+2D_{2},\text{ }p_{2}=D_{2},\text{ }p_{3}=2D_{3},$

We are particularly interested in the $\vartheta_{D_{2}}$ term of the
resulting product of theta functions. This term is understood by intersection
theory on the moduli space $\mathscr{M}(X,\beta,z)$, where $\beta$ fixes
contact orders $p_{1},p_{2},p_{3}$ and $D_{2}^{*}$ and curve class $A=2L-E$, L
being the total transform of a line and $E$ the class of the exceptional
curve. This moduli space admits a stabilization map
$\mathscr{M}(X,\beta,z)\rightarrow\overline{\mathscr{M}}_{0,4}$. The fiber
over the boundary point of $\overline{\mathscr{M}}_{0,4}$ associated with the
partition $(x_{1},x_{2}|x_{3},x_{out})$ is a non-reduced point of length $2$,
while the fiber over the boundary point associated with the partition
$(x_{2},x_{3}|x_{1},x_{out})$ is $\mathbb{P}^{1}$. The tropical types
associated with the generic points of the fiber in the latter case is depicted
in Figure 8.1:

Figure 8.1. The tropical curve associated with the generic point of
$\mathscr{M}_{2}$ Figure 8.2. Tropical curves associated to lower dimensional
strata of $\mathscr{M}_{2}$ which contribute to
$\vartheta_{p_{1}}\vartheta_{p_{2}}\vartheta_{p_{3}}[\vartheta_{D_{2}}z^{A}]$.

This discrepancy is fixed in [GS19] by imposing a logarithmic condition on the
image under the stabilization map which requires the length of the unique edge
in the tropicalization to be large. This results in an integral on
$\mathscr{M}_{2}$ which is localized to the two strata associated with the
tropical data depicted in Figure 8.2.

These two strata are also identified from the perspective taken in this paper,
as the natural cutting edge in Figure 8.1 does not have a non-negative contact
order in the blowup required to ensure slope sensitivity, while the tropical
types depicted in Figure 8.2 do have cutting edges with non-negative contact
orders.

In a similar fashion, we also deduce the weak Frobenius structure theorem for
all targets considered in this section. The basic structure of the argument is
similar to that for the associativity of the mirror algebra, except instead of
considering the WDVV relation, we instead consider the following topological
recursion relation.

(8.4)
$N^{\beta}_{p_{1},\ldots,p_{m},0}=\sum_{\textbf{A}_{1}+\textbf{A}_{2}=\textbf{A}\in
H_{2}(X),\text{ }s\in\mathbb{Z}^{n},\text{
}k,S_{1},S_{2}}\langle[pt]_{0}\psi^{m-3},\prod_{j\in
S_{1}}[1]_{p_{j}},T_{-s,k}\rangle_{\textbf{A}_{1}}\langle
T^{k}_{s},[1]_{p_{1}},[1]_{p_{2}},\prod_{j\in
S_{2}}[1]_{p_{j}}\rangle_{\textbf{A}_{2}}.$

In the above equation, the pair $(S_{1},S_{2})$ vary over all pairs of
disjoint subsets of $\\{1,\ldots,m\\}$ such that $S_{1}\cup
S_{2}=\\{3,\ldots,m\\}$. As before, we may assume that all of the
corresponding moduli spaces of punctured log maps are all slope sensitive. We
wish to show the following:

######  Lemma 8.9.

$\langle[pt]_{0}\psi^{m-3},\prod_{j\in
S_{1}}[1]_{p_{j}},T_{-s,k}\rangle_{\textbf{A}_{1}}=0$ unless
$T_{-s,k}=[1]_{-s}$, $-s$ is a positive contact order, and $|S_{1}|=m-2$.

###### Proof.

By Corollary 7.4, we may show the given orbifold invariant involving a point
constraint vanishes by considering an integral on the moduli space of point
constrained log stable maps $\mathscr{M}(X,\beta,z)$, with $z\in X\setminus D$
generic, and $\beta$ a decorated tropical type specifying contact order
$0,-s$, and $p_{j}$ for $j\in S_{1}$. As before, we consider the virtual
pullback along the map
$\mathscr{M}(X,\beta,z)\rightarrow\mathfrak{M}^{ev}(\mathcal{X},\beta,z)$ of
strata of $\mathfrak{M}^{ev}(\mathcal{X},\beta,z)$, and
$\eta\in\mathfrak{M}^{ev}(\mathcal{X},\beta,z)$ the generic point of the
statum. Letting $\tau$ be the tropical type of the log map
$C_{\eta}\rightarrow\mathcal{X}$ associated with $\eta$, note that for the
vertex $v_{out}\in V(G_{\tau})$ containing the leg with contact order $0$, the
point constraint ensures that $\tau$ is contributing only if
$\boldsymbol{\sigma}(v_{out})$ is the zero cone in $\Sigma(X)$. We
additionally define an modification of the obstruction theory associated with
the morphism
$\mathscr{M}(X,\beta,z)\rightarrow\mathfrak{M}^{ev}(\mathcal{X},\beta,z)$ to
be the dual of

$R\pi_{*}f^{*}T_{X}^{log}(-x_{out})\oplus L_{x_{out}}^{m-3}[1],$

with $L_{x_{out}}$ the cotangent line to $x_{out}$. The new virtual
fundamental class $[\mathscr{M}(X,\beta,z)]^{vir^{\prime}}$ is related to the
original virtual fundamental class by

$[\mathscr{M}(X,\beta,z)]^{vir^{\prime}}=[\mathscr{M}(X,\beta,z)]^{vir}\cap\psi_{x_{out}}^{m-3}.$

In particular, the invariant of interest is the degree of the Chow class
$[\mathscr{M}(X,\beta,z)]^{vir^{\prime}}$.

We now prove the following lemma, which is a slight modification of Lemma 8.6
above:

######  Lemma 8.10.

Let $\tau$ be a tropical type with legs $L,l_{out},\ldots,l_{m}$, $m>1$,
having contact orders $D_{i}^{*}(u_{l_{i}})=0$ for all bad divisors $D_{i}$,
and a vertex $v\in V(G_{\tau})$ with
$\boldsymbol{\sigma}(v)\in\mathscr{P}(X)$, contained in the legs
$l_{out},\ldots,l_{m}$. If
$[\mathscr{M}(X,\boldsymbol{\tau},z,C)]^{vir}\not=0$, then for all edges and
legs $e\in E(G_{\tau})\cup L(G_{\tau})$ contained in the convex hull of $v$
and $L$, we have $D_{i}^{*}(u_{e})=0$ for any bad divisor $D_{i}$.

###### Proof.

As in Lemma 8.6, we prove the proposition by induction on the length of a
chain of edges in the spine, with the labeling of edges and the base case as
in Lemma 8.6. Assuming the inductive hypothesis for the chain of length $k-1$,
we cut $\tau$ along an edge $e_{k}\in E(G_{\tau})$ in the spine, producing two
tropical type $\tau_{1}$ and $\tau_{2}$ with $\tau_{1}$ containing the vertex
$v$. By the same argument at the start of Lemma 8.6, we have $A(\tau_{1})\cdot
c_{1}(T_{X}^{log})\leq 0$. Moreover, we observe by Corollary $7.3$ of [ACGS20]
that we have the following cartesian diagram:

${\mathscr{M}(X,\tau,z)}$${\mathscr{M}(X,\tau_{1},z)\times\mathscr{M}(X,\tau_{2})}$${\mathfrak{M}^{ev(x_{out})}(\mathcal{X},\tau,z)}$${\mathfrak{M}^{ev(x_{out},x)}(\mathcal{X},\tau_{1},z)\times\mathfrak{M}^{ev(x)}(\mathcal{X},\tau_{2}).}$

Unlike in Lemma 8.6, the obstruction theories on the left vertical arrow is
the one defined preceeding this lemma, and the obstruction theory for the
right vertical arrow is the product obstruction theory, with the usual one for
the moduli stacks associated with $\tau_{2}$, and the dual of
$R\pi_{*}f^{*}T_{X}^{log}(-x_{out}-x)\oplus L_{x_{out}}^{m-3}[1]$ for
$\tau_{1}$. Since the cotangent line for $x_{out}$ on
$\mathscr{M}(X,\tau_{1},z)$ clearly pulls back to the cotangent line for
$x_{out}$ on $\mathscr{M}(X,\tau,z)$, the obstruction theories are compatible.
Furthermore, we have the relative virtual dimension of
$\mathscr{M}(X,\boldsymbol{\tau}_{1},z)\rightarrow\mathfrak{M}^{ev(x_{out},x)}(\mathcal{X},\tau_{1},z)$
is less than or equal to $-dim\text{ }X-m+3$, with equality only when the the
conclusion of the lemma holds. Hence, we must show as before that
$\mathfrak{M}^{ev(x_{out},x)}(\mathcal{X},\tau_{1},z)\leq dim\text{ }X+m-3$.
This follows from the analogous argument given in Lemma 8.6, with the only
difference being Proposition $3.28$ of [ACGS20] gives $dim\text{
}\mathfrak{M}^{ev}(\mathcal{X},\tau_{1})\leq dim\text{ }X+m-3$. We conclude
that $vdim\text{ }\mathscr{M}(X,\tau_{1},z)<0$ when there is a bad divisor
$D_{i}$ such that $D_{i}^{*}(u_{e_{k}})\not=0$, and hence by [GS19] Theorem
$A.13$, $[\mathscr{M}(X,\boldsymbol{\tau},z)]^{vir}=0$. ∎

By the above lemma, we must have $D_{i}^{*}(-s)=0$ for all bad divisors
$D_{i}$. In particular, since $c_{1}(T_{X}^{log})=\sum_{i}-a_{i}D_{i}$ with
$a_{i}\geq 0$ and $a_{i}>0$ for only $i$ such that $D_{i}$ is a bad divisor,
by Corollary $1.14$ of [GS19], we have
$c_{1}(T_{X}^{log})\cdot\textbf{A}_{1}=0$. Now recall the virtual dimension of
$\mathscr{M}^{orb}(X,\beta)$ is:

$vdim\text{
}\mathscr{M}^{orb}(X,\beta)=c_{1}(T_{X}^{log})\cdot\textbf{A}_{1}-3+n+dim\text{
}X-\sum_{i}e_{i}=-3+|S_{1}|+2+dim\text{ }X-\sum_{i=1}^{k}e_{i}.$

For the invariant $\langle[pt]_{0}\psi^{m-3},\prod_{j\in
S_{1}}[1]_{p_{j}},T_{-s,k}\rangle_{\textbf{A}_{1}}$ to be non-zero, we must
have

$vdim\text{ }\mathscr{M}(X,\beta)\geq m-3+dim\text{ }X.$

Since $|S_{1}|\leq m-2$, we must have $e_{i}=0$ for all $i$ and $|S_{1}|=m-2$.
With these constraints, the only possible insertion we can have is
$T_{-s,k}=[1]_{-s}$ as required. ∎

Using the above lemma, Equation 8.4 reduces to:

$N^{\textbf{A}}_{p_{1},\ldots,p_{m},0}=\sum_{(\textbf{A}_{1},\textbf{A}_{2},s)}N^{\textbf{A}_{1}}_{p_{3},\ldots,p_{m},s,0}N_{p_{1},p_{2},s}^{\textbf{A}_{2}}$

A straightforward induction on size of $m$ now shows:

$N^{\textbf{A}}_{p_{1},\ldots,p_{m},0}=\sum_{(\textbf{A}_{1},\ldots,\textbf{A}_{m-1},s_{1},\ldots,s_{m-2})}N_{p_{1},p_{2},s_{1}}^{\textbf{A}_{1}}(\prod_{i=2}^{m-2}N^{\textbf{A}_{i}}_{s_{i-1},p_{i+1},s_{i}})N_{s_{m-2},p_{k},0}^{\textbf{A}_{m-1}}.$

The right-hand side of the above relation is the expression for the
$z^{\textbf{A}}\vartheta_{0}$ term of
$\vartheta_{p_{1}}\cdots\vartheta_{p_{m}}$ found by repeatedly applying the
product rule for theta functions, giving the desired equality of Theorem 1.3.

######  Remark 8.11.

The utilities of both relative theories can be observed in this paper. On the
orbifold side, there are straight out of the box relations which sidestep many
otherwise delicate gluing arguments on the log GW side of the correspondence.
On the other hand, the orbifold theory is not log étale invariant unlike the
log theory, leading to the counterexample to log étale invariance presented in
this paper. On the logarithmic side, the relationship between the virtual
geometry of the moduli spaces of log stable maps and the corresponding
tropical moduli problems allow us to refine the invariants of interest by
specifying tropical behavior, which is not possible using only orbifold
Gromov-Witten invariants. As a result, we believe the correct takeaway is that
both theories yield interesting and distinct insights into the Gromov-Witten
theory of pairs $(X,D)$, and there is much potential in combining techniques
from these two areas.

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