By the estimates in <ref> and <ref>, any unramified special fiber in all situations (with or without residue constraints) contributes at least $1/3$ to the age. Therefore, if $|S_{\text{un}}|
\geq 3$, we thus obtain that $\age (\bftau) \geq 1$. Moreover, the case $|S_{\text{ram}}|=0$ is impossible since the sum of entries of $\mu$ (which is $-2$) is not divisible by $3$. Hence we only need to consider the cases $|S_{\text{un}}| = 1$ or $2$ and $|S_{\text{ram}}| = 1$ or $2$.
If $|S_{\text{un}}| = 2$ and
$|S_{\text{ram}}| = 1$ or $2$, then by the dimension reason there is
an extra subspace of periods (besides the two unramified fiber contributions) on which $\tau$ acts with eigenvalues $1$ or $(1,1)$ or
$(\zeta, \zeta^2)$ since the total determinant of the $\tau$-action is one.
One checks that $\age(\tau) \geq 1$ in all these cases.
Consider now the case $|S_{\text{un}}| =1$ and $|S_{\text{ram}}| =2$. Assume
there is a totally ramified zero. If the second totally ramified fiber is
a (GRC free) pole or a zero, then besides the eigenvalues $\zeta, \zeta^2$
from the unramified fiber we get an eigenvalue $1$ from the residue or
a relative period, since the determinant of the unprojectivized action
of $\tau$ is one. This gives $\age(\tau) \geq 1$. If the second ramified fiber
is a (GRC constrained) pole with $d=1$, we get $\omoduli[0]^\frakR(3m_1+m_2-2, -m_2,
\{-m_1,-m_1,-m_1\})$, with $m_i>0$ and $\frakR=\{r_3+r_4+r_5=0,\ r_2=0\}$,
which is Case (R1).
Suppose both ramified fibers are poles. Then the unramified special fiber
must consist of zeros, hence it gives signatures of type $(\{m, m, m\}, -m_1, -m_2)$
with $3m - m_1 - m_2 = -2$ and $m_1, m_2 > 0$, where the unramified zero fiber
contributes eigenvalues $\zeta$ and $\zeta^2$. If any of the two poles can have
a nonzero residue, then the residue gives an extra eigenvalue $1$, which together
with $\zeta$ and $\zeta^2$ makes $\age(\tau) \geq 1$. The remaining case leads to
the GRC-constrained  stratum with $\frakR=\{r_4 = 0\}$ or $\frakR=\{r_4 = r_5 = 0\}$
listed as Case (R2).
Finally if $|S_{\text{un}}| =1$ and $|S_{\text{ram}}| =1$, it gives
$\omoduli[0](\{m,m,m\},-3m-2)$ with $m \in \bZ$, which is Case (2).

Subcase $\mathbf{k = 4}$
Suppose first an unramified fiber
consists of poles that are subject to residue conditions. As noticed above, these
are vertical edges joined to $d$ higher connected components with $d > 1$.
Then combining inequalities <ref> and <ref>
gives $\age(\bftau) \geq 1$ in this case.
Now we can assume that any unramified special fiber consists of either zeros or poles without extra residue constraints (i.e. $d=1$). Using <ref>, <ref>, or <ref>, the relative periods or residue cycles from this fiber contribute at least $3/4$ to $\age(\tau)$. Hence we can further assume that $|S_{\text{un}}|= 1$. If $|S_{\text{ram}}|=2$, then $X$ has genus $g = 0$ with six zeros and poles, and hence the (unprojectivized) stratum of $X$ has dimension equal to four. Besides the eigenvalues $\zeta, \zeta^2, \zeta^3$
from the special unramified fiber, the remaining eigenvalue must be $1$
or $\zeta^2 = -1$, since the determinant of the $\tau$-action
is $\pm 1$ (from an invertible integral matrix). In both cases one checks that
$\age(\tau) \geq 1$. Therefore, the remaining possibility is
$|S_{\text{un}}| = |S_{\text{ram}}|=1$, i.e. $\bP\omoduli[0](\{m, m, m, m\}, -4m-2)$
where the four zeros (or poles) are permuted by $\tau$ and the pole (or zero)
is fixed, with (unprojectivized) eigenvalues $\zeta, \zeta^2, \zeta^3$
and (projectivized) $\age(\tau) = 3/4 < 1$ if $a = 1$. This is Case (8).

§.§.§ Case $\mathbf{g' = 0,\ k\geq 3,\ b \geq 3}$

It is well-known (e.g. <cit.>) that any
primitive $k$-th root of unity appears as an eigenvalue with multiplicity
$b-2 \geq 1$ for the action of $\tau$ on the $2g$-dimensional absolute
part $H^1(X; \bbC)$, and hence $g\geq 1$.

Subcase $\mathbf{k \not \in \{2,3,4,6\}}$

For  $k \not \in \{2,3,4,6\}$, using <ref> and its notation
we find that $\age (\tau) \geq (b-2)\phi_k(a) / k \geq 1$.
Hence the remaining cases are cyclic covers of $\bP^1$ with
degree $k = 2, 3, 4, 6$, with $b \geq 3$ branch points, and $g \geq 1$.

Subcase $\bf{k=3}$
In this case the cover is totally
ramified at $b = g+2$ points. If moreover $g \geq 3$, since $\phi_3(a) \geq 1$
for any $a$, we find $\age (\tau) \geq (b-2) / 3 \geq 1$. Hence we only need to
consider $g=2$ and $g=1$.
Consider first $g = 2$. Then the absolute periods already give a contribution
of at least $2/3$ to $\age(\tau)$. If $|S_{\text{un}}|\geq 1$, then an unramified special fiber
contributes to the age at least $1/3$ by the estimates in <ref> and <ref>, and hence $\age (\bftau) \geq 1$ in this case. We can thus assume that $|S_{\text{un}}|=0$. Moreover, if
there are two ramified zero fibers, then the relative paths joining them give an eigenvalue $1$. If there is any
ramified pole fiber whose residue is not constrained to be zero, then its residue gives an eigenvalue $1$. In
both cases $\age (\tau) \geq 1$ since $1, \zeta, \zeta^2$ all appear as eigenvalues. The remaining cases
are $|S_{\text{ram}}|=2, 3, 4$ with exactly one ramified zero fiber and all residues constrained to be zero, giving
Cases (3), (R3) and (R4).
Now we deal with $g = 1$. The absolute periods give eigenvalues $\zeta, \zeta^2$,
and hence contribute at least $1/3$ to $\age (\tau)$. Note that any two special zero fibers contribute an eigenvalue $1$
using their relative periods, which makes the total age at least one together with
the absolute periods. Similarly if there is a totally ramified pole fiber whose
residue is not constrained to be zero, then its residue gives an eigenvalue $1$,
which again makes the total age at least one. We can thus assume that there is
exactly one special zero fiber and all ramified poles are constrained to have zero
residue. Moreover if $|S_{\text{un}}|\geq 2$, since each unramified fiber contributes
at least $1/3$ to the age  by the estimates in <ref> and
<ref>, then altogether we obtain $\age (\bftau)\geq 1$. Hence we
only need to consider $|S_{\text{un}}| = 1$ or $0$.
Suppose $|S_{\text{un}}| = 1$, which contributes at least $1/3$ to the age as explained above. If this is an unramified pole fiber giving three edges that permute three connected components in upper level, then <ref> contributes $1$ to the age, hence we can assume that the unramified fiber is either a zero fiber or a pole fiber not constrained by the GRC. The remaining cases give signatures  $(\{0,0,0\})$ or $(\{m, m, m \}, -3m)$
without residue conditions and those with residue conditions as follows:

* $(\{m, m, m \}, -m_1, -m_2)^{\frakR}$ with $\frakR = \{r_4 =0\}$ or
$\frakR = \{r_4 = r_5 = 0\}$,

* $(\{m, m, m \}, -m_1, -m_2, -m_3)^{\frakR}$ with residue condition
$\frakR = \{r_4 = r_5 = 0\}$ or  $\frakR  = \{r_4 = r_5 = r_6 = 0\}$,

* $(3m_1+m_2, \{-m_1, -m_1, -m_1 \}, -m_2)^{\frakR}$ with $\frakR = \{ r_5 = 0\}$,

* $(3m_1+m_2+m_3, \{-m_1, -m_1, -m_1 \}, -m_2, -m_3)^{\frakR}$ with
$\frakR = \{ r_5 = r_6 = 0\}$.

The first two cases without residue conditions correspond to Cases (6) and (7).
However in the cases with non-trivial $\frakR$, the vertex $v$ already contributes
at least $2/3$ to $\age (\bftau)$. If the three edges in the unramified special fiber
join three vertices in $\Gamma$ that are permuted by $\bftau$, then
$\age (\bftau) \geq 1$ by the argument in the proof of <ref> below.
If they join the same vertex $v'$, then their relative periods or residues
(without GRC) for $v'$ contribute at least $1/3$ to $\age (\bftau)$, thus making
$\age (\bftau) \geq 1$. Hence these cases do not appear in the tables.
Suppose $|S_{\text{un}}| = 0$, i.e. there is no unramified special fiber. If there is
no pole, then the special fibers consist of a unique zero fiber, thus giving the
signature $\mu = (0)$ listed as Case (4).
If there are (ramified) poles, then each of them is constrained to have residue zero
as explained above, hence we obtain signatures  $(m, -m)$  and
$(m_1+m_2, -m_1, -m_2)^{\frakR}$ with $\frakR=\{r_2=0\}$ or $\{r_2=r_3=0\}$, which
correspond to Cases (5) and (R5).

Subcase $\bf{k=4}$
We have $\phi_4(a) \geq 2$
for any $a$, and hence $\age (\tau) \geq 2(b-2) / 4 \geq 1$ for $b\geq 4$.
We thus need to consider the case $b = 3$. Since $2g+ 6 = \sum_{i=1}^3(4-s_i) \leq 9$ with $s_i$ a proper divisor of $4$, we conclude that $g\leq 1$. Since there exist
eigenvalues $\zeta$ and $\zeta^3$  from the absolute homology, we only need to
consider the case $g = 1$.
In this case $\pi$ has two totally ramified fibers and the third ramified fiber consists of two simply ramified points. Since the absolute periods
already contribute eigenvalues $\zeta$ and $\zeta^3$ (thus at least $1/2$ to the age), if there exists an unramified special
fiber, it contributes by <ref> enough to make $\age(\bftau)
\geq 1$. If there are two special zero fibers, then their relative periods give an eigenvalue $1$, which contributes enough to make $\age(\bftau)
\geq 1$. Similarly if there is a totally ramified pole without residue constraint, then its residue gives an eigenvalue $1$, which also makes
$\age(\bftau) \geq 1$. Hence we can assume in the sequel that $|S_{\text{un}}|= 0$, there is a unique special zero fiber, and any totally ramified pole is constrained to have residue zero.
If the fiber with two simply ramified points is not special, we get the strata $\bP\omoduli[1](0)$ and $\bP\omoduli[1](m,-m)$
which are listed as Cases (9) and (10).
If the fiber consisting of two simply ramified points is a special zero fiber, then it contributes at least an eigenvalue $\zeta^2$. As said, any remaining special fiber must be a ramified pole fiber with residue constraint. We thus get $(\{0,0\})$, $(\{m, m \}, -2m)$ and $(\{m, m \}, -m_1, -m_2)^{\frakR}$ with $\frakR = \{ r_3 = 0\}$ or $\{ r_3 = r_4 = 0\}$, listed as Cases (11), (12) (with two permuted zeros) and (R6).
Now suppose the fiber with two simply ramified points is a special pole fiber.
A GRC with the case $d=2$ for this fiber
(as defined in <ref> (i)) gives enough to make $\age(\bftau) \geq 1$
by <ref>. The case $d=1$ leads to the strata
$\bP\omoduli[1] (\{-m, -m\}, 2m)$ and $\bP\omoduli[1]^\frakR(2m_1+m_2, \{-m_1,-m_1\},-m_2)$ with $\frakR=\{r_4=0\}$ or $\frakR = \{r_2+r_3=0\}$,
listed as Cases (12) (with two permuted poles) and (R7).
If the fiber with two simply ramified points consists of simple poles adjacent
to other vertices, then the residues at the two nodes give
eigenvalues $\pm 1$. Together with the eigenvalues $\zeta, \zeta^3$ from the absolute periods
we obtain $\age(\tau) \geq 1$ for any $\zeta^a$ used for projectivization. If the fiber with two simply ramified points has two simple poles that form a self-node, then $\omega$ is $\tau$-anti-invariant, i.e. $a=2$.
Projectivization of the eigenvalues of the absolute periods already
gives $\age(\tau) \geq 1$.

Subcase $\bf{k=6}$
In this case $\phi_6(a) \geq 4$
for any $a\neq 5$ and $\phi_6(5) = 2$, hence $\age (\tau) \geq 2(b-2) / 6
\geq 1$ for $b\geq 5$ and any $a$. Moreover,
$\age(\tau) \geq 1$ for $b = 4$ and any $a \neq 5$.
Consider first $b = 4$ and $a = 5$. Then the eigenvalues of $\tau$ from
the absolute homology of $X$ contain $\zeta, \zeta, \zeta^5, \zeta^5$,
contributing $2/3$ to $\age(\tau)$, and $g\geq 2$ in this case. If $g > 2$,
an additional conjugate pair of eigenvalues from the absolute part of
type $(\zeta^{a_i}, \zeta^{6-a_i})$ for $1 \leq a_i \leq 3$ or $(1,1)$,
after dividing by $\zeta^5$, can contribute at least $1/3$ to $\age(\tau)$,
which is enough. For $g = 2$, if there is a special zero or pole fiber with cardinality $2$ or $3$ or $6$,
using the respective estimates in <ref> and <ref>, we can still obtain
that $\age(\bftau) \geq 1$.
If all zeros and poles are totally ramified, then by Riemann–Hurwitz the only case is $\bP\omoduli[2](2)$ for $s_1 = 1$ and $s_2 = s_3 = s_4 = 3$ where the unique zero $z$ is totally ramified. But in that case $\tau^2$ induces a triple cover of $\bP^1$ totally ramified at the Weierstrass point $z$, which is impossible because the linear system $|3z|$ has a base point at $z$.
Next consider $b = 3$. By Riemann–Hurwitz $g=1$ or $2$ in this case.
For $ b = 3$ and $g = 1$, we get $s_1 + s_2 + s_3 = 6$ with $1\leq s_i \leq 3$
dividing $6$. The only possibilities are $(2,2,2)$ or $(1,2,3)$. The former
is impossible for a connected cyclic cover of $\bP^1$ as the gcd of $s_1, s_2, s_3$
is not relatively prime to $6$. For the latter, note that the
eigenvalues of $\tau$ from the absolute homology of $X$ contain
$\zeta$ and $\zeta^5$. If the concerned stratum for $X$ has any extra dimension (from non-absolute periods or residues), since the  determinant of the $\tau$-action is $\pm 1$, the extra eigenvalue(s) together with $(\zeta, \zeta^5)$ will make $\age(\tau) \geq 1$. If there is no extra dimension, then there is a unique zero and all poles are constrained to have zero residue. Moreover if $d$ polar edges in a special fiber joining $d$ higher connected components get permuted for $d > 2$, then we obtain $\age{\bftau} \geq 1$ by <ref>, and for $d =2$ we obtain an extra eigenvalue $-1 = \zeta^3$ which combined with the absolute eigenvalues still makes $\age{\bftau} \geq 1$. Therefore, the only remaining case is $\bP\omoduli[1](0)$ where the marked point is totally ramified, which gives Case (13).
For $ b = 3$ and $g=2$, we get $s_1 + s_2 + s_3 = 4$ with $1\leq s_i \leq 3$
dividing $6$. The only possibility is $(1,1,2)$. The eigenvalues of $\tau$
from the absolute homology of $X$ are $(\zeta, \zeta^5)$ together with
another conjugate pair.  Since the target is $\bP^1$, there is no invariant
absolute period, hence the other pair is either $(\zeta^2, \zeta^4)$
or $(\zeta^3, \zeta^3)$. Both cases give $\age (\tau) \geq 1$
after taking projectivization by $\zeta^a$ for any $a$.

In the sequel we need to bound the number of vertices in a level graph that can be permuted by an automorphism of small age.

Suppose $\bftau$ is an automorphism of a multi-scale differential
of type $\mu$ (possibly meromorphic) with $\age(\bftau)<1$. Then any $\bftau$-orbit of cyclicly permuted vertices in the associated level graph has cardinality at most two. Moreover if two vertices are swapped, then $\bftau^2$ acts trivially on their underlying stable curves.

Suppose there is an orbit of $d>1$ cyclicly permuted
vertices (which have to be on the same level). We need to show that $d = 2$. By <ref> $\age(\bftau)$ is at least one
if $d \geq 3, M>0$ or $d = 2, M > 1$, where recall that $M$ is the dimension of the multi-vertex RC-independent subspace. Therefore, any permuted vertex must have genus zero with at most two zero edges (otherwise $M \geq 2$ from the relative periods of the zeros). Moreover if $d\geq 3$, then each of the permuted vertices can admit only one zero edge (otherwise $M\geq 1$).
Suppose there exists an orbit of $d \geq 3$ permuted vertices. Among all of such orbits we choose the largest $d$, and further choose the highest level containing the orbit if there are multiple orbits of $d$ permuted vertices. We denote the chosen $d$ permuted vertices by $v_1, \ldots, v_d$. If they admit horizontal edges (in all situations of self-loops, or joining between them, or joining some other vertices on the same level), then there exist at least $d$ independent horizontal plumbing parameters that are cyclicly permuted, making $\age (\bftau) \geq 1$
by <ref> (using $d\geq 3$ and $M\geq 1$), which contradicts the assumption. We can thus assume that all polar edges of the $v_i$ are vertical only. By the preceding paragraph, each $v_i$ admits only one zero edge and hence at least two vertical polar edges (by stability and since it does not have simple poles).  Take a polar edge $e_1$ of $v_1$ joining to a higher level and consider part of its $\bftau$-orbit $e_1, e_2, \ldots, e_d$ (where the next one $e_{d+1}$ does not have to be $e_1$ as it might be another polar edge of $v_1$).  Suppose the upper ends of these $d$ edges are adjacent to $d'$ vertices $v'_1, \ldots, v'_{d'}$ that are permuted by $\bftau$, where $d = d' k'$.  By assumption $d' < d$ (as we chose the $d$ permuted vertices to be both largest and highest) and hence each $v'_j$ admits at least $k' = d/d'$ zero edges joining to some of the $v_i$.  Using $M' \geq k'-1$ from the relative periods of these zero edges of $v'_j$, we obtain at least $(d'-1)(k'-1)/2$ for the age by <ref>.  Since $k' = d/d' > 1$, if $d' \geq 3$, then $\age (\bftau)\geq 1$ which contradicts the assumption.  Hence the only possibilities are $d' = 1$, or $d' = 2$ with $k' = 2$.
Consider first $d' = 1$, i.e. $k' = d$. Then $e_1, \ldots, e_d$ are the zero edges of a single higher-level vertex and they are permuted by $\bftau$. Then we can apply <ref> to conclude that $d \leq 4$.  The case $d=4$ is only possible when $e_1, \ldots, e_4$ are the zero edges of a single higher-level vertex in Case (8) of <ref>, which contributes at least $3/4$ to the age.  Since each of the $v_i$ admits at least another polar edge, consider its orbit with $d' = 1$ or $d'=k'=2$ in the same notation.  The former contributes at least another $3/4$ and the latter contributes at least an extra $1/2$, both of which make the total age bigger than one, contradicting the assumption.  For the case $d = 3$, the estimate in <ref> gives at least $1/3$.  Since each of $v_1, v_2, v_3$ admits at least another polar edge, consider its orbit with $d' = 1$ in the same notation and we gain another $1/3$ (here $d=3$ is not divisible by $2$, so $d' = 2$ cannot occur).
Moreover, $v_i$ cannot admit more than two polar edges (as we have already got $2/3$ for the age).  Then the residue cycles $r_i$ of the two polar edges (up to sign) on the $v_i$ can either freely vary or are constrained by $r_1 + r_2 + r_3 = 0$ only, hence the action of $\bftau$ on the subspace generated by $r_1, r_2, r_3$ contains eigenvalues $\zeta_3, \zeta_3^2$, contributing an extra $1/3$ and making the total age $\geq 1$, leading to a contradiction.  Alternatively, the three zero edges of $v_1, v_2, v_3$ go down to another three permuted vertices (otherwise the adjacent single vertex in lower level contributes at least $1/3$ by checking the cases of $k=3$ in <ref> and <ref> which makes the total age $\geq 1$). These three new permuted vertices can only admit one zero edge each. Hence we can go down along them again, until we find a single vertex with three permuted edges, which leads to the same contradiction as before.
Consider the remaining possibility that $d' = k' = 2$, and hence $d = 4$.  In this case the age contribution from $v'_1, v'_2$ is already at least $(d'-1)(k'-1)/2 = 1/2$.  Since each $v_i$ admits at least another polar edge, consider its orbit and the associated upper adjacent vertices (again with $d' = k' = 2$ as the only possibility).  Then we obtain another contribution $1/2$, which altogether makes the total age $\geq 1$, leading to a contradiction.
We have thus proved that $d=2$, i.e., any $\bftau$-permuted vertices must appear in pairs. Take two mutually permuted vertices $v_1$ and $v_2$. Next we will show that $\bftau^2$ acts trivially on this pair, i.e., $\bftau^2$ fixes every edge of $v_1$ and $v_2$. Prove by contradiction.
First suppose $\bftau^2$ does not fix all zero edges in the pair. Then we can find an orbit of zero edges
$z_1 \mapsto z_2 \mapsto z'_1 \mapsto z'_2$ ($\mapsto z_1$, since we have seen that each permuted vertex can admit at most two zero edges), where $z_i$ and $z'_i$ are the zero edges of $v_i$ for $i = 1,2$. If these four zero edges belong to a single higher vertex, then it can only be Case (8) in <ref> which contributes at least $3/4$ to the age. On the other hand, the permuted pair contributes at least $1/2$ from the relative periods of the zero edges, which altogether makes the total age $\geq 1$ and contradicts the assumption.  The remaining possibility is that $v_i$ joins a lower vertex $v'_i$ via $z_i$ and $z'_i$, and hence the two banana pairs $(v_1, v'_1)$ and $(v_2, v'_2)$ are swapped by $\bftau$.  But the upper relative periods and the lower residues each contribute at least $1/2$, in total making the age $\geq 1$ and thus leading to a contradiction. The same argument can be used to show that $\bftau^2$ fixes all vertical polar edges of $v_1$ and $v_2$.
For any horizontal edge of $v_i$, it cannot be a self-loop at $v_i$ (otherwise its residue cycle and plumbing parameter can make the multi-vertex RC-independent dimension $M \geq 2$). Similarly we can rule out the case that two or more horizontal edges of $v_i$ are permuted by $\bftau$. Therefore, every horizontal edge of $v_1$ joins $v_2$, which is fixed by $\bftau$ (with the two ends swapped). Consequently $\bftau^2$ fixes the ends of every horizontal edge at both $v_1$ and $v_2$. In summary, we have shown that $\bftau^2$ fixes all zero and polar edges on each $v_i$ (whose number is at least three by stability since $v_i$ has genus zero). Hence $\bftau^2$ acts trivially on $v_1$ and $v_2$.

Suppose $v_1$ and $v_2$ are two vertices of genus zero such that ${\bftau}$ swaps them and ${\bftau}^2$ is the identity restricted to each of them.
If their relative periods and residues contribute zero to $\age (\bftau)$, then the above proof implies that each $v_i$ admits a unique zero edge, and moreover, any two permuted (vertical) polar edges must be constrained by the GRC to have the sum of the residues equal to zero. We call such permuted $(v_1, v_2)$ of age zero a trivial pair. In particular if we view a trivial pair as a 'single' vertex, then it behaves the same as hyperelliptic vertices of age zero as described in <ref>.

Finally we can show the following result about automorphisms of multi-scale differentials with small age.  Recall the coordinates
$\bA_{\rel}(X)$, $\bC_{\hor}$ and $\bC_{\lev}$ introduced in <ref>, and $\bA = \bA_{\rel}(X) \times \bC_{\hor}\times \bC_{\lev}$.

Let $\bftau$ be a lift to $\bA$ of an automorphism of a multi-scale differential
of type $\mu$ (possibly meromorphic).
If $\age(\bftau)<1$ and $\bftau$
does not induce a trivial action or a quasi-reflection on $\bA$, then the action of $\bftau$ on the subspace $\bC_{\lev}$ of level parameters
is non-trivial for all $\mu$ in $g \geq 2$ except for Case (3) in $g=2$ in <ref>.

Suppose that $\bftau$ acts trivially
on $\bC_{\lev}$. We remark that in this case any two $\bftau$-fixed
vertices connected by vertical edges must have the same $\bftau$-restricted order.
To see it, let $X$ and $Y$ be two $\bftau$-fixed vertices joined by vertical
edges $e_1, \ldots, e_d$ that are permuted in one orbit (there could be more edges
between them in other orbits).  Suppose the $\bftau$-orders restricted to $X$ and $Y$
are $k_1$ and $k_2$ respectively, and $k_i = d \ell_i$ where $\ell_i$ is the
ramification multiplicity of the quotient map of $\bftau$ restricted to each end
of the edges. Each $e_j$ gives a relation $x_j  y_j$ equal to some product of the
level parameters (as in <ref>), where $x_j$ and $y_j$ are local
standard coordinates at $e_j$. Note that $\bftau^d$ maps $x_1$ to $\zeta_{\ell_1} x_{1}$
and maps $y_1$ to $\zeta_{\ell_2}  y_{1}$ for some primitive $\ell_i$-th roots of unity.
Since by assumption $\bftau$ acts trivially on the level parameters, it implies that
$\zeta_{\ell_1}  \zeta_{\ell_2} = 1$, and hence $\ell_1 = \ell_2$. Since $k_i = d \ell_i$,
we thus conclude that $k_1 = k_2$.
If $\bftau$ permutes any vertices, by <ref> $\bftau^2$ fixes every vertex,
acts trivially on all permuted pairs, and acts trivially on the level parameters (since $\bftau$ does). Applying the remark in the preceding paragraph to $\bftau^2$, it implies that $\bftau^2$ is a trivial action. Therefore, $\bftau$ has order two and it induces either a trivial action or a quasi-reflection (i.e., with age $0$ or $1/2$) as the only possibilities of having age smaller than one, which is ruled out by the assumption.
From now on suppose that $\bftau$ has order at least three and that $\bftau$ fixes every vertex. If there are at least three permuted horizontal
edges, then by <ref> we get $\age (\bftau) \geq 1$. Moreover if a
horizontal edge $e$ is fixed at a vertex $v$, then the local standard form $dx/x$
is invariant at $e$, hence $\omega$ restricted to $v$ is an invariant form, which
cannot occur for any cases in <ref> and <ref>,
except for possibly hyperelliptic vertices or vertices on which $\bftau$ acts
trivially. If two horizontal edges (or the two ends of a horizontal self-loop) are
permuted, looking at the cases, only hyperelliptic vertices are possible. In summary,
horizontal edges are only adjacent to vertices on which the restricted
$\bftau$-orders are one or two. Combining with the remark that vertical edges join
vertices that have the same restricted $\bftau$-order, we thus conclude
that $\bftau$ restricted to every vertex has order exactly $k$ for some $k\geq 3$.
From the Cases of $k\geq 3$ in <ref>
and <ref> we see that the age contribution is at least $1/3$ from
any $\bftau$-fixed vertex of order $k\geq 3$. Since $\age(\bftau) < 1$, there can be at most two vertices in the graph.  If there is a single vertex, the only case with $g\geq 2$ in <ref> is Case (3) in genus two. Suppose there are exactly two vertices $v_1$ and $v_2$. Then the only possibilities are
Cases (2), (4), (5), (R1), (R2), (R5) for $k=3$, and Case (13) for $k = 6$. The latter cannot appear twice as its signature does not have a pole.  Hence we can assume that $k=3$. Since all zeros and poles of the ambient stratum are labeled, any permuted zeros and poles in these Cases must be edges. Then Cases (4), (5), (R5) can appear in pairs and Cases (2), (R1), (R2) can appear in pairs (including possibly self-pairing). Due to the congruence conditions and residue conditions in these Cases, the only possibilities are (4) paired with (R5) and (2) paired with (2). In both cases the signatures of the ambient strata belong to Case (3).

§ SINGULARITIES OF THE COARSE MODULI SPACE

The purpose of this section is to control the singularities of the coarse
moduli space $\bP\MScoarse$ in order to show that the usual strategy for
proving general type—the canonical bundle is ample plus effective—can
be used after an appropriate modification due to non-canonical singularities
at the boundary as stated in <ref>, which
we will prove at the end of this section.
We start with the interior of the moduli space and the digression on the
logarithmic viewpoint:

For any signature $\mu$, except
of type $\mu = (m, 2-m)$
in genus $g=2$ for $1\neq m \equiv 1 \mod 3$ (Case (3) in <ref>),
the interior of the coarse moduli space $\bP\MScoarse$ has canonical singularities.
The pair $(\bP\MScoarse, \mathbf{D})$ consisting of the coarse moduli space
and the total boundary $\mathbf{D} = \partial \bP\MScoarse$ is a log canonical pair.
Both statements hold as well for the strata with unlabeled zeros and poles,
after further ruling out Cases (6), (7), (8), (11), (12) in <ref>.

This situation is quite parallel to the moduli space of curves $\overline{M}_{g}$.
The singularities in the interior $M_{g}$ are also canonical, as shown by
<cit.>. The pair $(\overline{M}_{g}, \partial M_{g})$ being
log canonical is a general fact for the coarse moduli space of a Deligne–Mumford stack
with a normal crossings boundary divisor (see e.g., <cit.>).
The proof of <ref> is given in <ref>.
In contrast, there can be non-canonical singularities at the boundary, see
<ref> for an easy example of such a singularity induced by
ghost automorphisms. These are discussed in general in <ref>.
The effect of curve automorphisms and ghost automorphisms are combined in
<ref>. There we define the compensation divisor $D_{\NC}$ and
prove <ref>.

§.§ Canonical sheaf and singularities on quotient stacks

We recall several well-known facts on singularities and the canonical
sheaf of an irreducible normal variety $W$, with a focus on the case of coarse
moduli spaces of smooth Deligne–Mumford stacks. In particular, the spaces
we consider are $\bQ$-factorial, i.e., every Weil divisor is $\bQ$-Cartier.
On a singular variety there are three competing definitions of sheaves
of differential forms. First, $\Omega^1_W$ denotes the sheaf of
Kähler differentials and $\Omega^p_W = \wedge^p \Omega^1_W$
its tensor power. It is badly behaved near the singular points and plays
hardly any role in the sequel. Second, let
$i\colon U:= W_{\mathrm{reg}} \to W$ be the inclusion of the regular part
and let
Ω^[p]_W    :=  i_*(Ω^p_U) = (Ω^p_W)^**
be the sheaf of reflexive differential forms. Its top power
$\omega_W = \Omega^{[p]}_W$ for $p = \dim(W)$ is a line bundle, called
Grothendieck's dualizing sheaf (even though $W$ is not
Cohen–Macaulay in general, so Serre duality does not hold with $\omega_W$).
Third, let $ \pi\colon \widetilde{W} \to W$ be a resolution of singularities
and we define
Ω^p_W π̄_* Ω^p_W
which is useful for computing global sections on $\widetilde{W}$.
Finally we define the canonical divisor $K_W = \pi_* K_{\widetilde{W}}$ using the
pushforward of cycles. Note that $\omega_W = \cO(K_W)$.
We now discuss the logarithmic situation, or more generally the case of pairs.
Let $W$ still be a normal variety and $\Delta = \sum a_i D_i$ a sum of prime
divisors with $a_i \in \bQ$. Choose the smooth resolution $\pi$
so that moreover the preimage $\widetilde{{D}} = \pi^{-1} {D}$
is a normal crossings divisor. Now let $i\colon U \to W$ be the inclusion of
the open subset where $W$ is smooth and ${D}$ is normal crossings.
As above we define the sheaf of reflexive logarithmic differentials and
the pushforward of logarithmic differentials on the resolution to be
Ω^[p]_W(logD)   :=  i_*(Ω^p_U(logD|_U))
and   Ω^p_W(logD)
π̄_* Ω^p_W(logD) .
The logarithmic canonical divisor is defined to be $K_W + {D}$.
This is consistent with the above notation since $\cO(K_W + {D})
= \Omega^{[p]}_W(\log {D})$ for $p = \dim(W)$.
Next we review some types of singularities of pairs. Recall that the
discrepancy $\discrep(W,\Delta)$ is the infimum over all exceptional
divisors $E$ in all birational morphisms $\widetilde{W} \to W$ of the
coefficient $a(E,\Delta,\widetilde{W})$ of $E$ in the pullback of $K_W
+ \Delta$. The pair $(W,\Delta)$ has canonical singularities if
$\discrep(W,\Delta) \geq 0$ and logarithmic canonical singularities
if $\discrep(W,\Delta) \geq -1$. More details can be found in
In particular if $W = (W,\emptyset)$ has canonical singularities, then
sections of the canonical bundle restricted to the regular locus of $W$ can
be extended across the singularities (<cit.>).

§.§ Singularities from curve automorphisms

We apply this discussion to $W = \bP\MScoarse$, keep the notation $D_\Gamma$ for
the boundary divisors of the stack and write $\mathbf{D}_\Gamma$ etc for the
boundary divisors of the coarse moduli space. Since  $\bP\MScoarse$ has only
finite quotient singularities, every subvariety of codimension one is $\bQ$-Cartier.
Thus the rational Picard groups of $\bP\MScoarse$ and $\bP\LMS$ are identical and
computations are performed mostly in terms of the classes of $D_\Gamma$ in the sequel.

For the first statement we examine the cases in <ref>.
Except Case (3), all the strata in that table either involve
permuted marked points (not allowed for labeled strata), or are one-dimensional
with smooth quotients.
To prove the second statement we use that $(\bP\MScoarse, \mathbf{D})$ is the coarse moduli space
associated with a smooth Deligne–Mumford stack with normal crossings boundary divisor
(<cit.>) and that $\mathbf{D}$ has each boundary term with coefficient one.
In fact, <cit.> shows that in this situation
there is some boundary divisor $\Delta$ such that the pullback of
$m(K_{\bP\MScoarse} + \Delta)$ equals $m(K_{\bP\LMS} + D)$ and furthermore
that $(\bP\MScoarse, \Delta)$ is log canonical. Since all boundary divisors in $D$
appear with coefficient one, the same holds for $\Delta$, i.e., it implies
that $\Delta = \mathbf{D}$. This reflects the fact that log canonical divisors are
insensitive to branching (see <cit.>  for a general
comparison formula).
Finally for unlabeled strata we only need to rule out those of (projectivized) dimension at least two in <ref>.

Consider the left slanted cherry $\Delta = \Delta_\ell$ from
<ref>, i.e., with the twist $p = 2$ on the short edge. The stack
structure is given by the group $K_\Delta = \bZ/3\bZ$. As we verified there, the
generator $(2,1)$ of this group acts by $(\zeta_6^2, \zeta_3^1)
= (\zeta_3, \zeta_3)$ on the coordinates corresponding to opening up the
levels. (See also Equations (6.7) and (12.6) in <cit.> for the construction.
The local parameter called $t_i$ in loc. cit.  raised to the lcm of $p_e$
for all edges $e$ crossing the level passage gives the coordinate called $s_i$
that rescales the differential.) This group action does not satisfy the
Reid–Tai criterion and <cit.> shows explicitly why
the corresponding singularity is not canonical. We will elaborate on
this in the next subsection.

Besides <ref> we need to show that non-canonical
singularities occur in the boundary of all strata with possible exceptions in
low genus only. A more elaborate version
of <ref> can be used to show that as long as
$g \geq 5$. Consider a triangle graph with one vertex at each level,
the top vertex in the stratum $\omoduli[2,2](0,2)$, the middle
vertex in the stratum $\omoduli[2,2](-2,4)$ and the bottom level vertex
with the remaining genus and all the marked points. The prongs are $3$
on the long edge from the top to bottom level, and $1$ and $5$ on the short
edges. The element $(\zeta_{3}^1, \zeta_{15}^5)$ fixes all prongs, and it
defines a ghost automorphism with $\age < 1$.

§.§ Singularities induced by ghost automorphisms

The singularities induced at the boundary of $\bP\LMS$, say at
an enhanced level graph $\Gamma \in \LG_L$, stem from the action
of the ghost automorphisms $K_\Gamma = \Tw/\sTw$. These are toric
singularities. We explain
here how to fit the data of the graph and the enhancements into
the standard framework of toric geometry. The goal is to give a
formula for the discrepancies and, more generally, a formula for
the pullback of torus-invariant divisors in terms of these graph
We start by recalling some well-known toric terminologies.
An affine toric variety is given by a $\bZ$-module $N$ that we
view as a lattice inside $N_{\bR} = N \otimes_{\bZ} \bR$ and a
convex rational polyhedral cone $\sigma \subset N_{\bR}$. We let $M = N^\vee
= \Hom(N,\bZ)$ and view the dual cone $\sigma^\vee$ as a subset of $M_{\bR}$.
Then the group algebra $\bC[M \cap \sigma^\vee]$ is a finitely generated
algebra and the associated (affine) toric variety is defined as
X_N,σ Spec([M ∩σ^∨]) . The spanning rays of $\sigma$ generated by the primitive elements
$r_1,\ldots,r_L$  are in bijection to
the torus-invariant divisors $D_1,\ldots,D_L$ of $X_{N,\sigma}$.
We omit $\sigma$ from the notation, if it is the positive cone
for some implicitly chosen basis of $N_{\bR}$.
The affine toric variety $X_{N}$ is non-singular if $\sigma\cap N$ is generated
by a subset of a basis of $N$. If this is not the case, we can resolve
the singularities by subdividing $\sigma$ as a union of subcones such that
each of the subcones satisfies the above condition. Let $F$ be the fan obtained
from the cone subdivision.
The additional rays of the subcones are given by the primitive
interior points in $\sigma^{\circ} \cap N$, which we list as $v_1,\ldots,v_s$.
Each of the rays $v_i$ corresponds to a torus-invariant exceptional
divisor $E_i$ in the resolution $\pi\colon \widetilde{X}_{F} \to X_N$.
We state the next proposition in the case of interest to us, namely
that $X_{N}$ has only abelian finite quotient singularities, which is equivalent
to $\sigma$ being a simplicial cone by <cit.>, i.e.,
$L = \dim N_{\bR}$. Consequently there are elements $m_{\sigma,i} \in M_\bR$
such that $\langle m_{\sigma,i}, r_j \rangle = \delta_{ij}$.
Let $m_\sigma = \sum_{i=1}^L m_{\sigma,i}$. We denote the non-exceptional
torus-invariant divisors, the strict transforms
of the $D_i$ by $\widetilde{D}_i$.

The canonical divisor $K_{X_N} = - \sum_{i=1}^L D_i$
is the negative sum of the torus-invariant divisors. Moreover if $X_{N}$ has
only abelian quotient singularities, then  the discrepancy
of $E_i$ is given by
K_X_F - π^* K_X_N ∑̄_j=1^s (⟨m_σ, v_j
⟩-1) E_j
and more generally
π^* D_i - D_i ∑̄_j=1^s ⟨m_σ,i, v_j
for all  i=1,…,L .

The claims follow from combining <cit.> Theorem 8.2.3 on the description
of the canonical bundle, Lemma 11.4.10 for its pullback to resolutions,
Theorem 4.2.8 for the conversion of divisors into support functions
and Proposition 6.2.7 for the pullback of divisors written in these

In general, not all pluricanonical forms extend from $X_N$ to its
resolution $\widetilde{X}_{F}$, since they can acquire poles along the exceptional
divisors $E_i$. However, we can consider only a subset of pluricanonical forms
having high enough order of vanishing along the divisors $D_i$. We now show
a criterion about how high the order of vanishing of pluricanonical forms
along $D_i$ has to be in order to ensure that they extend to the resolution.

Let $(b_1,\ldots,b_L) \in \bN^L$ be a tuple such that

\begin{equation}\label{eq:conditionextension}
\sum_{i=1}^L (b_i + 1)
\langle m_{\sigma,i}, v_j \rangle\geq 1,\quad \text{for } j=1,\dots, s.
\end{equation}

Then for all $a \in \bN$ we have the inequality

\[	h^0(\widetilde{X}_F, aK_{\widetilde{X}_F})
\,\geq \, h^0\Bigl(X_N, a\Big(K_{X_N} - \sum_{i=1}^L b_i D_i\Big)\Bigr)\,.
\]

It suffices to show that the exceptional
divisors $E_j$ occur with non-negative coefficients in the difference

K_X_F  -  π^*(K_X_N - ∑_i=1^L b_i Di)

∑̄_i=1^L b_i D_i
∑_j=1^s (⟨m_σ, v_j ⟩-1) E_j
∑_j=1^s E_j ·∑_i=1^L b_i ⟨m_σ,i, v_j ⟩

∑̄_i=1^L b_i D_i
∑_j=1^s E_j ·(-1  + ∑_i=1^L (b_i + 1)
⟨m_σ,i, v_j ⟩) ,
which is ensured by the standing assumption.

Recall that an inclusion of lattices $N' \hookrightarrow N$ with quotient
group $G$, and the same cone $\sigma$ in both lattices, gives rise to
a quotient map $X_{N'} \mapsto X_{N'}/G = X_{N}$, see e.g.<cit.>.
We focus now on toric varieties obtained by the the quotient of affine space
via a cyclic group of order $n$. We say that a singularity is of
type $\tfrac1n(a_1,\ldots,a_L)$ if it is the quotient of $\bC^L$ by a
cyclic group $G=\langle\tau\rangle$  of order $n$ acting by $\zeta_n^{a_i}$
on the $L$ coordinates.  Consider then the case $X_{N'}=\bC^L$, so $N'$ is a
lattice in $\bR^L$  generated by vectors $e_i$ and $\sigma$ is the standard
cone generated by the basis $(e_i)$ of $N'$. If we define $N$ to be the
lattice generated by the basis of $N'$ and by $v_\tau=\sum_{i=1}^L a_i/n\cdot e_i$,
then $N/N'=G$ and $X_{N}=\bC^L/G$.
We specialize further to the toric varieties $X_{N}$. Since the cone $\sigma$
is generated by the coordinate vectors $e_i=v_i$, using the notation previously
introduced, the $m_{\sigma,i}$ are simply the dual vector $e_i^{\vee}$. The only
primitive interior points in $\sigma^{\circ} \cap N$ are the vectors $v_{\tau^j}$,
for $j=1,\dots,n$. Hence, in this setting, <ref>
specializes to the following statement.

Let $X=\bC^d/\langle\tau\rangle$, where $\tau$ acts by multiplication
of $\zeta_n^{a_i}$ on the $i$-th coordinate. If

\begin{equation}\label{eq:conditionextensioncyclic}
\sum_{i=1}^L \frac{a_i}{n}(b_i+1)\geq 1
\end{equation}

then the inequality

\[	h^0(\widetilde{X}, aK_{\widetilde{X}})
\,\geq \, h^0\Bigl(X, a\Big(K_{X} - \sum_{i=1}^d b_i D_i\Big)\Bigr)\,
\]

holds for all $a \in \bN$, where $D_i$ are the image in $X$ of the
divisors $\{x_i=0\}\subset \bC^d$ and where $\widetilde{X}$ is a smooth
resolution of $X$.

Note that if the age of $\tau$ is indeed greater or equal to one, then we
take $b_i=0$, and so the singularities of $X_{N}$ are canonical.
The resolution $\pi\colon \widetilde{X} \to X$ of a singularity of
type $\tfrac13(1,1)$ has a single exceptional divisor $E$. In this case we have
K_X - π^* K_X -̄1/3 E
and   π^*D_i - D_i %̄s̄/̄%̄s̄13 E
where $D_i$ for $i=1,2$ are the two coordinate axes. If we consider for example $b_1=1$ and $b_2=0$, then by <ref> we have that all sections of $K_{X}-D_1$ extend to the resolution $\widetilde{X}$.
The resolution $\pi\colon \widetilde{X} \to X$ of a singularity of
type $\tfrac14(1,2)$ also has a single exceptional divisor $E$. In this case we have
K_X - π^* K_X -̄1/4 E
and   π^*D_1 - D_1 %̄s̄/̄%̄s̄14 E,
π^*D_2 - D_2 %̄s̄/̄%̄s̄12 E .
In this case, if we set for example $b_1=1$ and $b_2=0$, by <ref> we have that all sections of $K_{X}-D_1$ extend to the resolution $\widetilde{X}$.
This is a special case of the resolutions in <ref>.

Even though not needed in the sequel, it is instructive to describe in standard
toric geometry language the structure near a boundary component of the orderly
blow-up  $\bP\MSgrp$, where only ghost automorphisms are quotiented out. Recall
indeed that the map $\varphi_1\colon \bP\LMS \to \bP\MSgrp$ to the orderly blowup
is locally given by the map $[U/K_\Gamma] \to U/K_\Gamma$, where $U$ is a
neighborhood of a generic point of a boundary component $D_\Gamma$
and $K_\Gamma= \text{Tw}_\Gamma/\text{Tw}^s_\Gamma$ is the group of ghost automorphisms.
Recall from <ref> the coordinate system
near the boundary. We analyze the toric geometry of the part
$\bC^{\lev}$ of this coordinate system. This is the affine toric
variety with the (dual) lattice
M'     ⟨1/p_e ·w_i,   e ∈E(Γ),  $i$-th level passage crossed by $e$

⟨1/ℓ_i ·w_i,   i = 1, …, L ⟩_
and $\sigma^{\vee}$ the positive (dual) cone generated by the $w_i$ in
$M'_{\bR} \cong \bR^L$, where $w_i$ is the $i$-th unit vector and $\ell_i$ is the $\lcm$ of
all enhancements $p_e$ crossing the
$i$-th level passage.

N' (̄M')^∨[Γ] ⟨̄ℓ_i ·e_i,  i=1,…, L ⟩_
is the simple twist group by definition, where $e_i$ is the dual vector of $w_i$.
In order to define the twist group similarly, recall from
<cit.> that it depends only on the level passages
crossed by the edges and the enhancements, not on the vertices the
edges are attached to. For $0 \leq i<j \leq L$ we let

\[w_{-j}^{-i} \= (0,\ldots,0,1,\ldots,1,0,\ldots,0) \in \bR^L\]

be the vector where the string of ones goes from $i+1$ to $j$. Then

\[
N \,:= \Tw[\Gamma] \= (M')^\vee \quad \text{and} \quad
M \= \Bigl \langle
\frac{1}{p_e} \cdot w^{e^+}_{e^-}, \quad e \in E(\Gamma)
\Bigr \rangle_{\bZ}\,
\]

where $e^\pm$ are the upper and lower ends of $e$. Note that the
explicit computation of a basis of $M$ is in general not possible
in closed form, which requires working with gcds, i.e., computing
a Smith normal form.

We continue with the running example of the cherry graph, now generalizing
to enhancements $a$ on the short edge and $b$ on the long edge. We
let $\ell_1 = \lcm(a,b)$ and $\ell_2=b$. Then
N'    [Γ]      ⟨(1/a, 0),
(1/b, 0), (0, 1/b )   ⟩^∨    ⟨(ℓ_1, 0),
(0,ℓ_2 )  ⟩ .

N    [Γ]      ⟨(1/a, 0),
(1/b, 1/b )  ⟩^∨
⟨(a, -a ), (0,ℓ_2 )  ⟩
so that $n:= [N:N'] = \ell_1/a = b/\gcd(a, b)$.
We restrict moreover to $b \geq a$. We see
that this is a cyclic quotient singularity of order $n =\ell_1/a$ and
of type $\frac1n (1, q)$ where $q = \tfrac{b-a}{\gcd(a, b)}$. This generalizes
<ref>. To resolve this singularity minimally we have
to insert the rays generated by boundary points of the lower convex hull
of $N$ in the positive quadrant $\sigma = \sigma'$. These are the
rays generated by

\[v_j=j\cdot (\tfrac1n, \tfrac{q}n)\in \sigma^{\circ}\cap N\]
for $j=1,\ldots,q'$ with $0< q' < n$ and $qq' \equiv 1 \mod n$, see
<cit.> or <cit.> for another version
of this resolution ('Hirzebruch-Jung continued fraction').

§.§ Singularities at the boundary

Here we combine the previous two subsections to analyze the singularities
at the boundary with the goal of proving <ref>.
We start with the definition of the non-canonical compensation divisor $D_{\NC}$.
First we distinguish several special edge types in a two-level graph (for vertical edges only).
If an edge corresponds to a separating node, we say that it is of compact type (CPT). Otherwise we say that it is of non-compact type (NCT).
If the lower part of the graph separated by a CPT edge consists of a single rational vertex, the edge type is called a rational bottom tail (RBT).
Recall that a (vertical) dumbbell (VDB) graph
is defined to be a graph of compact type with a unique (separating)
edge which is vertical. If the graph contains a unique (vertical) edge (i.e., a VDB graph) and if one end of the edge is of genus one, we say that it is an elliptic dumbbell  (EDB).
An edge of compact type which is neither RBT nor EDB is called other compact type (OCT).
Let $E_\Gamma$ be the number of (vertical) edges of a two-level graph $\Gamma$. We then define (for $g\geq 2$) that

D_NC      :=  ∑_Γ∈_1 b_^Γ[D_Γ]  :=  ∑_Γ∈_1
(ℓ_ΓR_^Γ-1) [D_Γ]    where

R_^Γ   =  ∑_NCT 1/21/p_e +
∑_RBT 1/p_e  + ∑_OCT
2/ p_e +∑_EDB 4/p_e  .
In the above each sum runs over the edges of the corresponding type. Note that the edge type EDB is exclusive, i.e., if it appears in a two-level graph, then the graph has a unique edge and all other edge types do not appear. In particular, an edge cannot be both RBT and EDB due to the assumption that $g\geq 2$.

baseline=([yshift=0.5ex]current bounding box.center),
scale=2,very thick,
bend angle=30]
[circled number] (T) [] $4$;
[] (T-1) [right=of T] ;
[circled number] (T-2) [right=of T-1] $3$;
[] (T-3) [right=of T-2] ;
[] (T-4) [left=of T] ;
[circled number, below=of T-1] (B) $6$;
[comp, below=of T-3] (C) ;
[circled number, below=of T-4] (D) $1$;
(T) – (T-1) node[ midway, below=of T,xshift=-10pt,yshift=-32pt] (BL) ;
(T-1) – (T-2) node[ midway, below=of T,xshift=10pt,yshift=-32pt] (BR) ;
(T-1) – (B) node[] (B-1) ;
(T-2) – (B) node[] (B-2) ;
(T-2) edge
node [order top left,xshift=3pt,yshift=14pt]
node [order bottom right,xshift=15pt,yshift=-13pt] $5$ (C);
(T) edge
node [order top left,xshift=3pt,yshift=14pt]
node [order bottom right,xshift=15pt,yshift=-13pt] $7$ (B);
(B) edge [bend left]
node [order bottom right, xshift=-3,yshift=0]
node [order top left, xshift=-5,yshift=-14] $1$ (T-2)
edge [bend right]
node [order bottom right, xshift=-5,yshift=-4]
node [order top right, xshift=-8,yshift=-15] $1$ (T-2);
(T) edge [bend left]
node [order bottom left, xshift=0,yshift=-20] $1$
node [order top left, xshift=-1,yshift=-13]  (D)
edge [bend right]
node [order bottom right, xshift=-20,yshift=-12] $1$
node [order top right, xshift=-6,yshift=4]  (D);
(T) edge
node [order top left,xshift=3pt,yshift=14pt]
node [order bottom right,xshift=-15pt,yshift=-15pt] $1$ (D);

(B) – +(270: .25) node [xshift=0,yshift=-5] $22$;
(C) – +(225: .25) node [xshift=-2,yshift=-5] $2$;
(C) – +(-45: .25) node [xshift=-2,yshift=-5] $2$;
(D) – +(270: .25) node [xshift=0,yshift=-5] $6$;

In this graph there are five NCT edges with prong $1$, one OCT edge with prong $7$, and one RBT edge with prong $5$.

For certain range of genera and signatures, one can alter the definition of $R_{\NC}^\Gamma$ in <ref> slightly. Indeed for certifying general type for the minimal strata in low genera, we need another version of $R_{\NC}^\Gamma$ (see <ref> and <ref>).

We are now ready to present the main proof of this section.

The content of the proposition is that global sections of
$a(K_{\bP\MScoarse} - D_{\mathrm{NC}})$ extend to $a$-canonical sections on
a smooth resolution $\tMScoarse$ of $\bP\MScoarse$, for any $a\in \bN$. We revisit the
argument <cit.> for this purpose.
Suppose such a section $\eta$ does not extend to $\tMScoarse$ and
suppose this happens near some multi-scale differential
$(X,\bfomega,\bfz, \bfsigma)$ compatible with some level graph $\Pi$, in fact
necessarily in the boundary by <ref>, i.e. $\Pi$ is
non-trivial The
section $\eta$ thus acquires poles near a divisor $E$ of $\tMScoarse$. Using the
same notation as in <ref>, we consider the local covering $\bA$
of $\bP\MScoarse$, where  $\bA =  \bC_{\lev} \times \bA_\rel(X) \times  \bC_{\hor} $,
the tangent space to the orbifold chart near $(X,\bfomega,\bfz,\bfsigma)$.
Consider finally the normalization of $\tMScoarse$ in the function field
of $\bA$, i.e., the normalization of the corresponding
fiber product. For each component $E'$ of the preimage
of $E$ in this normalization, the stabilizer (in the full Deck group of the
cover, the extension $ \mathrm{Iso}(X,\bfomega)$ of $\Aut(X,\bfomega)$
by $K_\Pi$ as in <ref>) is cyclic,
say generated by an element $\bftau$, a lift of an automorphism in
$\Aut(X,\bfomega)$ to an automorphism acting on $\bA$ as considered in
<ref>. Consequently the pullback of $\eta$ to
$\bA/\langle \bftau \rangle$ does not extend to its smooth resolution.
We will show that this does not happen for sections under consideration, i.e.,
with enough vanishing along some boundary divisors, using <ref>.
If $\bftau$ is a quasi-reflection on $\bA$, the quotient is
smooth and the extension of $a$-canonical sections is automatic. In the case
of $\age(\bftau) \geq 1$, the singularities
of $\bA/ \langle \bftau \rangle$  are canonical and all $a$-canonical sections
of $K_{\bP\MScoarse}$ extend to $\bA/ \langle \bftau \rangle$ by the original argument
of Tai's criterion.
In the remaining cases, by <ref> and as our genus and signature
hypothesis, $\bftau$ acts non-trivially on at least one of the level
coordinates $t_i$. We may thus assume that $\bftau$ acts by $\exp(2\pi i a_j/k)$
in an appropriate basis of $\bA_\rel(X) \times  \bC_{\hor}$ and by
$\exp(2\pi i c_j/k_j)$ on the coordinates $t_j$, where at least one of the
entries $c_i$ of $n_\bftau$ is non-zero by our assumption.  Using
<ref> with $b_i=b_{\NC}^{\delta_i(\Gamma)}$ for $i=1,\dots,L$
(where $\delta_j(\Gamma) \in \LG_1$ is the undegeneration of a level graph $\Gamma$
obtained by compressing all level passages but the $j$-th one) and with $b_i=0$
for $i=L+1,\dots, d$, we need to show that

\begin{equation}\label{eq:condition}
c(\bftau)\,:=\,\sum_{i=1}^L \frac{c_j}{k_j}\left(b_{\NC}^{\delta_i(\Gamma)}+1\right)
+\sum_{i=L+1}^d \frac{a_j}{k}\geq 1\,.
\end{equation}

This is the statement of <ref> below, which is long
and technical so we separate it.

It remains to introduce and justify <ref> used in the above
proof. This requires some additional preparation. Denote by

\[0\leq s_j \= c_j / k_j < 1\]

the (rational) argument of the action of $\bftau$ on $t_j$ (mod $2\pi i$) as used
in <ref>. For a (vertical) edge $e$, denote by $[e]$ the interval
of level passages crossed by $e$. We say that a level passage is non-trivial
if the corresponding $s_j > 0$ and that an edge is non-trivial if it crosses
a non-trivial level passage. We also say that a vertex has order $k$ if the
order of $\bftau$ restricted to that vertex has order $k$. Finally for an edge $e$
we define its contribution

$$ c_{e} \= \sum_{j\in [e]} (\ell_j/p_e) s_j $$

which depends on $\bftau$ but we skip it in the notation when the context is clear.

Let $e$ be an edge fixed by $\bftau$ and joining two vertices $v_1$ and $v_2$ where each $v_i$ has order $k_i$. Suppose either $k_1 \neq k_2$, or $k_1 = k_2$ and $e$ is non-trivial. Then $ \lcm (k_1, k_2) c_e$ is a positive integer.  In particular,

\begin{eqnarray}
\label{eq:nc-edge-fix}
c_e \,\geq\, \frac{1}{\lcm (k_1, k_2)}\,.
\end{eqnarray}

Recall the local equation $x_1 x_2 = \prod_{j \in [e]} t_j^{\ell_j / p_e}$ at the node represented by $e$ in the universal family over the moduli space of multi-scale differentials, where $x_i$ is a standard coordinate at $e$ in $v_i$. Consider first the case $k_1 \neq k_2$. Since $\bftau(x_i) = \zeta_{k_i}x_i$ where $\zeta_{k_i}$ is a primitive $k_i$-th root of unity, $\bftau(x_1x_2)$ differs from $x_1x_2$ by a non-trivial root of unity of order at most $\lcm (k_1, k_2)$. Hence in this case $c_e \geq 1/\lcm (k_1, k_2)$. If $k_1 = k_2 = k$, then $\bftau(x_1x_2)$ differs from $x_1x_2$ by a root of unity of order at most $k$, which implies that $c_e$ is either zero or at least $1/k$. However the former is impossible since by assumption $e$ crosses some non-trivial level passage, i.e., at least some $s_j > 0$ in the sum. This thus verifies the inequality <ref>.

Let $e_1, \ldots, e_h$ be edges which are cyclically permuted by $\bftau$ and which  join two vertices $v_1$ and $v_2$ fixed by $\bftau^h$, where each $v_i$ has order $k_i$. Suppose either $k_1 \neq k_2$, or $k_1 = k_2$ and the $e_i$ are non-trivial. Then $\lcm (k_1, k_2) c_{e_i}$ is a positive integer. In particular

\begin{eqnarray}
\label{eq:nc-edge-permute}
c_{e_i} \geq \frac{1}{\lcm (k_1, k_2)}\,
\end{eqnarray}

for all $i$.

Note that $\bftau^h$ fixes each $e_i$ and the two vertices, and that it has order $k_i / h$ on each $v_i$. Then
the proof of the previous lemma implies that $h c_e$
is a multiple of $1/\lcm (k_1/h, k_2/h)$. It is in fact a non-zero multiple
by the previous proof in the case of $k_1 \neq k_2$, and directly by the existence of
a non-trivial level passage in the case of $k_1 = k_2$.

Finally we let

$$r_{j} = b_{\rm NC}^{\delta_j(\Gamma)} + 1$$

and rewrite the contribution $c(\bftau)$ in <ref>
in terms of the above notations as

$$c(\bftau) \= \sum_{j=1}^L r_j s_j + \age (\bftau)|_{\bA_{\rel} \times \bC_{\hor}}\,$$

where by definition $\age (\bftau)|_{\bA_{\rel} \times \bC_{\hor}}=\sum_{i=L+1}^d \frac{a_j}{k}$.

PROOF OF R_^Γ   =  ∑_NCT 2/(E+2)1/p_e +
∑_RBT 1/p_e  + ∑_OCT
2/ p_e +∑_EDB 4/p_e  .

For $g\geq 2$, suppose that $\bftau$ does not induce a quasi-reflection on $\bA$
and that not all level passages are trivial under $\bftau$. Then $c(\bftau)\geq 1$.

If $\bftau$ permutes some vertices, then $\age (\bftau)|_{\bA_{\rel} \times \bC_{\hor}}\geq 1$ unless the permuted vertices consist of two vertices of genus zero swapped by $\bftau$  as described in <ref>. For such a permuted pair we can combine the two vertices as one 'hyperelliptic' vertex, which does not affect the analysis of edge contributions when we apply <ref> (for $h = 2$). Moreover, a trivial pair of age zero described in <ref> behaves the same as a single hyperelliptic vertex of age zero. In this sense from now on we can assume that $\bftau$ fixes every vertex in the graph.
Let $\mathrm{H}$ be a non-trivial level passage such that it is crossed by the maximum number of edges among all non-trivial level passages, where we denote by $E_H$ the number of edges crossing $\mathrm{H}$. By <ref> or <ref>, any non-trivial edge joining two vertices of order $k_1$ and $k_2$ contributes at least $c_e \geq 1/\lcm (k_1, k_2)$ times the corresponding $(\ell/p_e)$-coefficient in <ref>, which is then at least $1/(E_{\mathrm{H}}\lcm (k_1, k_2))$ for all edge types (as $ 2/ (E_{\rm H} + 2) \geq 1/E_{\rm H} $ for NCT edges with $E_{\rm H} \geq 2$ and the CPT coefficients are at least one for $E_{\rm H} = 1$). Hence we can sum up the contributions of the $E_{\rm H}$ edges and obtain that $c(\bftau) \geq 1$ if $k_1 = k_2 = 1$ for these edges. From now on we can assume that there is at least one vertex of order $k_i > 1$. In this case the edge contribution can become smaller, but the vertex age can make an extra contribution to $c(\bftau)$ by using Figures <ref> and <ref>. We thus need to analyze a number of cases depending on the orders of vertices.

§.§ Case that all vertices are of order one or two

Assume that all vertices are of order one or two, and there is at least one vertex of order two. For any such vertex, we can assume that it is hyperelliptic under $\bftau$ (otherwise the vertex age is at least one). In this case an edge is either fixed or permuted with another edge under the hyperelliptic involution. Since the total edge contribution is at least $1/2$ from the preceding analysis (for $k_1, k_2 \in \{1,2\}$), we can assume that the age of every hyperelliptic vertex is zero. By <ref> and <ref>, a hyperelliptic vertex of age zero (including the age from $\bC_{\rm hor}$) does not admit horizontal edges and, either its poles are constrained by GRC, or it can have at most one vertical fixed edge or one vertical pair of permuted edges going in the same direction in the level graph. Moreover, we can assume that all edges are non-trivial after collapsing trivial level passages and merging vertices of the same order joined by trivial edges. As before, let $\mathrm{H}$ be the level passage crossed by the maximum number of edges $E_{\mathrm{H}}$. Since in this case each edge gives a contribution $c_e \geq 1/2$, if the total number $E_\Gamma$ of edges  in $\Gamma$ is at least $E_{\mathrm{H}} + 2$, then $c(\bftau)\geq1$.

Suppose that $E_\Gamma$ is either $E_{\mathrm{H}} +1$ or $E_{\mathrm{H}}$, which implies that
$\Gamma$ has at most two level passages.  If there is a CPT edge
which is not RBT, or if there are at least two RBT edges, then we obtain at least one from the edge contribution. If there is
one RBT edge, we obtain for $ E_{\mathrm{H}} \geq 4$ at least
$1/2 + (E_{\mathrm{H}}-1)/(E_{\mathrm{H}}+2)
\geq 1$. If $E_{\mathrm{H}}\leq 3$ and the RBT edge does not cross the level passage ${\rm H}$, then we still obtain enough. There are only two possibilities left for the remaining cases of a two-level graph with $E_\Gamma\leq 3$ and one RBT edge. If $E_\Gamma=3$, then there is only one top vertex with three edges attached, which cannot be hyperelliptic, and we obtain enough from the edge contribution. If $\Gamma$ has only one RDB edge, then either the top vertex is trivial or it is hyperelliptic. In the first case we obtain enough from the edge contribution, while in the second case $\bftau$ is a quasi-reflection. Therefore, from now on we can assume that all edges are NCT.
Let $v_0$ be a lowest vertex.  If $v_0$ is hyperelliptic and joins at least two upper vertices, then by GRC (for $\age(v_0) = 0$) at least one of the edges is CPT, and so we obtain enough age. If $v_0$ is hyperelliptic and its edges join a unique upper vertex $v_1$, either the vertices are joined by a CPT edge which contributes enough, or by a pair of permuted edges which contributes enough unless $v_1$ is a hyperelliptic vertex of age zero. In the last case either $\Gamma$ is a banana graph and $\bftau$ acts as a quasi-reflection, or $v_1$ admits at least two non-CPT edges going up which makes  $E_\Gamma>E_{\mathrm{H}}+1$ and contradicts the assumption.
Finally suppose all bottom level vertices are trivial.  If one of the trivial bottom
vertices $v_0$ joins at least two upper hyperelliptic vertices, then both of them need to further go up to avoid having CPT edges, hence making the total number of edges more than assumed. If there are at least two edges crossing the lowest level passage and joining two trivial vertices, since $c_e=1$ for these edges, the total edge contribution is at least $2(2+(E_{\mathrm{H}}-2)/2)/(E_{\mathrm{H}}+2)=1$. The remaining case is that the bottom level contains a unique trivial vertex joining at most one hyperelliptic vertex or one trivial vertex on a higher level (with a unique edge joining them in the latter case). Since we have excluded CPT edges, the only possibility is a triangle graph where the top and bottom vertices are trivial and the middle vertex is hyperelliptic. In this case the long edge contributes at least $2/(E_{\mathrm{H}}+2)=1/2$ and each of the two short edges contributes at least $1/4$, hence the total contribution is at least one.
From now on we assume that besides vertices of order one and two, all other vertices are of order three, four, or six from Figures <ref> and <ref>.

§.§ Case that a vertex is of order four

We start by assuming that there is at least one vertex  of order four. Since all vertices of order four have age at least $1/2$, we can assume that there is a unique vertex $v$ of order four, which rules out Case (8) as it has four cyclicly permuted edges and thus requires another vertex of the same type to pair with it, giving enough vertex age.  If $v$ has a pair of permuted edges, then they can only join a hyperelliptic vertex of age zero. Since $\age(v) = 3/4$ in the relevant Cases (11), (12), (R6) and (R7), we can assume that all other vertices are of order one or two. As before, since the $E_{\mathrm{H}}$ edges contribute at least $1/4$, we conclude that $c(\bftau) \geq \age(v) + 1/4 = 1$.
Now suppose all edges of $v$ are fixed. The relevant Cases are (9) and (10), for which $\age(v)=1/2$. Case (9) has genus one with a unique edge $e$ going down. Suppose the lower end $v'$ of $e$ has order $k'$ which can be one, two, or three (as Case (13) of order $6$ cannot be a lower vertex). Since $e$ is CPT (and not RBT as $g\geq 2$), we obtain that $c(\bftau) \geq \age(v) + \age(v') + 2/\lcm(4, k') \geq 1$ for any $k' = 1,2,3$.
For Case (10), if it admits one edge and one leg, then the same argument works as in Case (9). Suppose it admits two edges $e_1$ and $e_2$, one going up to $v_1$ and the other going down to $v_2$ where $v_i$ is of order $k_i\neq 4$. As before we can reduce to the case that all non-trivial edges are not CPT, any hyperelliptic vertex has age zero, and there is at most one other vertex with non-zero age. Note that $e_1$ and $e_2$ cross different level passages. If there is no vertex of order three or six, then $c(\bftau) \geq \age(v)  + 2(1/ 4 + 1/4 + (E_{\mathrm{H}}-1)/2) / (E_{\mathrm{H}}+2) \geq 1$. If there is exactly one vertex of order three or six, then $c(\bftau) \geq \age(v) + 1/3 + 2(1/12 + 1/4 + (E_{\mathrm{H}}-1)/6) / (E_{\mathrm{H}}+2) \geq 1$.

§.§ Case that a vertex is of order six

From now on we assume that all vertices are of order one, two, three, or six, and there is at least one of order six. Consider Case (13) for a vertex $v$ of order six where $\age(v) = 1/3$ and $v$ admits a unique edge $e$ going down. Let $v'$ be the lower end of $e$ with order $k'$. Since $e$ is CPT (and not RBT as $g\geq 2$), its contribution to $c(\bftau)$ is $2c_e\geq 2/\lcm (6, k') \geq 1/3$, hence combining with $\age(v)$ we can assume that all vertices (except $v$) are of order one or two and that any hyperelliptic vertex has age zero. If there are other non-trivial edges besides $e$, then similarly as before we obtain $c(\bftau) \geq 2/3 + 2(E_{\mathrm{H}}-1)/2
(E_{\mathrm{H}}+2) \geq 1$. Otherwise it reduces to EDB, which implies that $c(\bftau)\geq \age (v) + 4/\lcm(1,6) = 1$.

§.§ Case that a vertex is of order three

Finally we assume that all vertices are of order one, two, or three, and there exists at least one vertex $v$ of order three.  If $v$ admits three cyclicly permuted edges, then it must be paired with another vertex $v'$ of order three, which includes Cases (2), (6), (7), (R1) and (R2).  The only combination for $\age(v) + \age (v') < 1$ is where $v$ and $v'$ are of type (2), (R1) or (R2), and then $\age(v) + \age (v') = 2/3$. We can assume that any hyperelliptic vertex has age zero and that there are no CPT edges.  Either the three edges do not cross the level passage ${\rm H}$, and then
$c(\bftau)\geq 2/3+2(1/6+1/6+(E_{\mathrm{H}}-1)/2)/(E_{\mathrm{H}}+2)\geq 1$, or we obtain that
$c(\bftau) \geq 2/3 + 2(1/3+1/3+1/3 + (E_{\mathrm{H}}-3)/2)
/(E_{\mathrm{H}}+2) \geq 1$ since $E_{\mathrm{H}}\geq 3$ in this case.
Now suppose all vertices of order three have fixed edges only, which includes
Cases (3), (4), (5), (R3), (R4) and (R5).  For Cases (3), (R3) and (R4), $\age (v) = 2/3$, and they can be treated similarly as in the preceding case. For Case (4), the vertex $v$ admits a unique edge and we can use the same argument as in Case (13).
For Case (5), if $v$ admits one leg and one edge, then the argument is still the same as Case (13). Next suppose $v$ admits two edges $e_1$ and $e_2$, going up and down respectively to $v_1$ and $v_2$.  Again we only need to consider the case that all other vertices (except $v$) are of order one or two and any hyperelliptic vertex has age zero. It follows that $c(\bftau) \geq 1/3 + 2(1/6 + 1/6 + (E_{\mathrm{H}}-1)/2 ) /
(E_{\mathrm{H}}+2) \geq 1$ for $E_{\rm H} \geq 5$. For the remaining cases, consider first $2 \leq E_{\mathrm{H}}\leq 4$. If $E_\Gamma>
E_{\mathrm{H}}+1$, then we obtain at least an extra $1/(E_{\mathrm{H}}+2)$ contribution to
the previous expression which gives enough. If $E_\Gamma = E_{\mathrm{H}}+1$, then $\Gamma$ is a pseudo-triangle three-level graph where the number of long edges ranges from one to three. In this case the top and bottom level vertices cannot be hyperelliptic since they need to have zero age, and then we conclude that $c(\bftau) \geq 1/3 + 2(1/3 + 1/3 + (E_{\mathrm{H}}-1) ) / (E_{\mathrm{H}}+2) \geq 1$. Finally consider the case that  $E_{\mathrm{H}} = 1$. Then the graph reduces to a chain where both $e_1$ and $e_2$ are CPT. Moreover if $v_2$ is hyperelliptic, then $e_2$ cannot be RBT by stability of $v_2$. Therefore, the contribution of $e_i$ to $c(\bftau)$ is at least $1/3$, which is from either $2c_{e_i} \geq 2/\lcm (2,3) = 1/3$ or $c_{e_i} \geq 1/\lcm (1,3) = 1/3$. It follows that  $c(\bftau) \geq \age(v) + 1/3 + 1/3 = 1$.
For Case (R5) a similar argument as the previous one works.

We will need the following refinement to certify general type for minimal strata with
for $27 \leq g \leq 44$. We call a graph with two vertices (on two levels) and
any number of edges $E$ a multibanana graph, the case $E=2$ being the usual
banana graph.

In the case of minimal strata $\mu=(2g-2)$ with $g\geq 8$, if $\Gamma$ is a
banana graph with rational bottom level and prongs $(1,2g-3)$ or $(2,2g-4)$,
we can improve and set $R_{\NC}^\Gamma= P_{-1}/(E_\Gamma+1)$, where
$P_{-1}=\sum_{e\in E(\Gamma)} 1/p_e$. We can do the same if $\Gamma$ is a multibanana
with three edges, rational bottom and at least one prong of order one.

We use the same notation as the proof of <ref>. Since in the original proof we have only used the contribution of edges crossing $\mathrm{H}$, which is at least $c_e/E_\mathrm{H}$ for each edge, if $E_\mathrm{H}\geq 4$ the original proof suffices. Moreover, also if $E_\mathrm{H}=1$ we don't need to argue, since there cannot be multibanana type level passages. Hence from now we assume $2\leq E_\mathrm{H}\leq 3$. We start by considering the case with only trivial vertices.  Since any edge in $\Gamma$ yields a contribution of at least $1/(E_\mathrm{H}+1)$, the $E_\mathrm{H}$ edges crossing $\mathrm{H}$ give a contribution of at least $E_\mathrm{H}/(E_\mathrm{H}+1)$. If $E_\Gamma>E_\mathrm{H}$, then we gain the additional contribution to reach age one, and otherwise $\Gamma$ is a two-level graph with trivial order vertices, hence contained in the smooth locus. Consider the case where there is at least one vertex of order two. If $E_\Gamma=E_H\in\{2,3\}$, we reach enough contribution since $\Gamma$ would be a special multibanana and cannot have hyperelliptic vertices. Consider first the case $E_\mathrm{H}=2$, so each edge gives a contribution of at least $1/6$. Hence we can reduce to investigate the cases of $3\leq E_\Gamma\leq 5$. Since we get at least a contribution of $1/2$ from the edges, we can assume that all hyperelliptic vertices have age zero. Consider the special edge $e$ with prong one or two and let $v^\pm$ be its upper and lower vertices. Note that $v^+$ cannot be hyperelliptic of age zero. If $v^-$ is hyperelliptic, then the other special banana edge cannot join $v^-$ and this vertex  has to join a lower level via at least two permuted edges, making $E_\mathrm{H}>2$. Then we can assume that $v^\pm$ are trivial, so it gives a contribution of at least $1/3$. Hence $ E_\Gamma\in \{3,4\}$ and by inspecting the possibilities without CPT edges, we can reduce to the case of a triangle graph where $e$ is the long edge. In this case, if we denote by $p\geq 12$ the prong of the other special banana egde $e_p$,  we can then compute $c_e\geq p/2 c_{e_p}\geq 6\cdot 1/2$.
The case with $E_\mathrm{H}=3$ can be treated similarly, and the cases where there is at least one vertex of order greater than two are even easier to treat.

For $g\geq 2$, suppose that $\bftau$ does not induce a quasi-reflection on $\bA$
and that not all level passages are trivial under $\bftau$. Then $c(\bftau)\geq 1$.

Before showing the proof of the above proposition, we present an alternative version of $R_{\NC}^\Gamma$ which we will need to use to prove  <ref> in low genera. (More precisely, we will need the following version of $R_{\NC}^\Gamma$, together with the improvement in <ref>, to show that the minimal strata with odd spin parity are of general type for $13\leq g\leq 43$.)

For $g\geq 2$,  suppose that $\bftau$ does not induce a quasi-reflection on $\bA$
and that not all level passages are trivial under $\bftau$. Let $v^\top$ be the number of top level vertices in $\Gamma$.
Then substituting $R_{\NC}^\Gamma$ in the definition <ref> with

\[R_{\NC}^\Gamma =\, \sum_{\rm NCT} \frac{1}{E_\Gamma}\frac{1}{p_e} +
\sum_{\rm RBT} \frac{1}{p_{e}}  + \sum_{\rm OCT}
\frac{2}{ p_{e}} +\sum_{\rm EDB} \frac{4}{p_{e}}+(v^\top-1) \,\]

still satisfies that $c(\bftau)\geq 1$.

Our strategy is to prove first <ref>. Since $1/E_\Gamma\leq 1/2$ in the presence of NCT edges, the same proof will work for <ref> if we can show that the additional $v^\top$-term, which is not present in <ref>, is not needed if the coefficient for NCT edges is $1/2$ instead of  $1/E_\Gamma$.

We denote by $\Pi$ the level graph on which we perform the analysis. If $\bftau$ permutes some vertices of $\Pi$, then $\age (\bftau)|_{\bA_{\rel} \times \bC_{\hor}}\geq 1$ unless the permuted vertices consist of two vertices of genus zero swapped by $\bftau$  as described in <ref>. For such a permuted pair we can combine the two vertices as one 'hyperelliptic' vertex, which does not affect the analysis of edge contributions when we apply <ref> (for $h = 2$). Moreover, a trivial pair of age zero described in <ref> behaves the same way as a single hyperelliptic vertex of age zero. In this sense from now on we can assume that $\bftau$ fixes every vertex of $\Pi$. Moreover, we can also assume that $\Pi$ has no horizontal edges (since higher order vertices with $\age < 1$ do not
admit such edges according to the tables and as a consequence of the
trivial pair discussion in the proof of <ref>).
Let $\mathrm{H}$ be a non-trivial level passage of $\Pi$ such that it is crossed by the maximum number of edges among all non-trivial level passages, where we denote by $E_H$ the number of edges crossing $\mathrm{H}$. By <ref> or <ref>, any non-trivial edge joining two vertices of order $k_1$ and $k_2$ contributes at least $c_e \geq 1/\lcm (k_1, k_2)$ times the corresponding $(\ell/p_e)$-coefficient in <ref>, which is then at least $1/(E_{\mathrm{H}}\lcm (k_1, k_2))$ for all edge types. Hence we can sum up the contributions of the $E_{\rm H}$ edges and obtain that

\begin{equation}
\label{eq:levelHcontrib}
c(\bftau) \geq \frac{E_{\mathrm{H}}}{E_{\mathrm{H}}\lcm (k_1, k_2)}=\frac{1}{\lcm (k_1, k_2)}\,.
\end{equation}

If $k_1 = k_2 = 1$ for these edges, then we obtain enough contribution to $\age(\bftau)$.
Another preliminary remark is that, if there is a level passage $j_0$ whose
corresponding two-level graph has $v^\top=v_{j_0}^\top>1$ vertices on top level, then
there is a special edge $e$ crossing this level passage such that all the level
passages $j\in [e]$ satisfy $v_j^\top>1$, where $v_j^\top$ denotes the
number of top level vertices of $\delta_j(\Pi)$. To see the existence of such a special edge $e$, consider all
non-backtracking paths that start and end with an edge crossing $j_0$,
with exactly these two edges crossing $j_0$, and that connect two disjoint
connected components above $j_0$.
By hypothesis this set is nonempty, and we can orient the paths such that the
starting level of the paths is not above the ending level.
This means that the first edge $e_1$ of the path has the property that
all level passages in $[e_1]$ above $j_0$ satisfy $v_j^\top>1$.
Consider now a path where the lowest level touched by the path is as high as
possible among all paths. Then the first edge $e_1$ of this path also has the property
that all level passages in $j\in [e_1] $ satisfy $v_j^\top>1$. Indeed if there
is a level passage $j'\in [e_1]$ below $j_0$ with  $v_{j'}=1$, then we would find
another path with $j'$ as the lowest touched level. In summary, one can use $e = e_1$ as a special edge. Note that if the starting level passage $j_0$ is non-trivial, e.g., $j_0=H$, then the special edge $e$ is non-trivial, since it crosses $j_0$. Hence by <ref> or <ref>,
we obtain a contribution of at least

\begin{equation}
\label{eq:vtopcontrib}
\end{equation}
where $k_i$ are the order of the
vertices joined by $e$. We call this  the $v^\top$-contribution of the
special edge.
From now on we can assume that there is at least one vertex of order $k_i > 1$. In this case the edge contribution can become smaller, but the vertex age can make an extra contribution to $c(\bftau)$ by using <ref> and <ref>. We thus need to analyze a number of cases depending on the orders of vertices.

§.§ Case that all vertices are of order one or two

Assume that all vertices are of order one or two, and there is at least one vertex of order two. For any such vertex, we can assume that it is hyperelliptic under $\bftau$ (otherwise the vertex age is at least one). In this case an edge is either fixed or permuted with another edge under the hyperelliptic involution. Since the total edge contribution is at least $1/2$ from the contribution <ref> (for $k_1, k_2 \in \{1,2\}$), we can assume that the age of every hyperelliptic vertex is zero, since $\age(\bftau) \geq 1$ and we are done otherwise. By <ref> and <ref>, a hyperelliptic vertex of age zero (including the age from $\bC_{\rm hor}$) does not admit horizontal edges, has at most one fixed zero or one pair of permuted zeros, has every fixed pole with residue zero constrained by GRC, and has every permuted pair of poles with the zero sum of the residues constrained by GRC. Therefore, we can assume that every hyperelliptic vertex belongs to a hyperelliptic (banana) tree of age zero, which is a connected subgraph consisting of hyperelliptic vertices of age zero whose edge configurations are described above. If a non-hyperelliptic vertex is adjacent to one end of a hyperelliptic tree, we say that the connecting edge is a handle.
Among the $E_{\mathrm{H}}$ edges crossing ${\mathrm{H}}$, suppose $E_1$ edges are fixed
and have both endpoints trivial (type one), $E_2$ edges are fixed and have at
least one endpoint hyperelliptic (type two), and $2E_3$ edges are permuted in
pairs with both endpoints hyperelliptic (type three), where
$E_1 + E_2 + 2E_3 = E_{\mathrm{H}}$.
By <ref> and <ref>,
edges $e_1$, $e_2$, $(e_3, e'_3)$ of the three types satisfy that $c_{e_1} \geq 1$,
$c_{e_2} \geq 1/2$, and $c_{e_3} + c_{e'_3} \geq 1/2 + 1/2 = 1$, respectively. If there are edges of type two or three constrained by GRC, then the undegeneration of the level passage $\mathrm{H}$ gives rise to a two-level graph with $v_{\mathrm{H}}^\top>1$, and the $v^\top$-contribution <ref> is at least $1/2$ which is enough. Hence we can assume that for every edge of type two which is not a handle, or for every pair of edges of type three, there is at least one associated CPT handle or at least two NCT handles.
Note that a handle $e$ of a hyperelliptic tree has endpoints of order one and
two respectively, hence $c_e \geq 1/2$. In addition, any permuted edge of type three
is not a handle.
Consider first the case that every handle of a hyperelliptic tree is not CPT. Then every hyperelliptic tree has at least two handles, one at each end.
It follows that $c(\bftau) \geq E_1 / E_{\mathrm{H}}  + (1/2 + 1/2) E_2 / E_{\mathrm{H}}  + (1+1/2+1/2) E_3 / E_{\mathrm{H}}  = 1$. Next consider the case that there are at least two CPT handles. Then $c(\bftau) \geq 1/2 + 1/2 = 1$. Finally suppose there is exactly one CPT handle $e$. Then $c(\bftau) \geq 1/2 + E_1 / E_{\mathrm{H}} + E_2  / E_{\mathrm{H}} + 2E_3 / E_{\mathrm{H}} - 1/E_{\mathrm{H}} \geq 1$ for $E_{\mathrm{H}}\geq 2$, where we subtract $1/E_{\mathrm{H}}$ for not counting redundantly the contribution from the hyperelliptic tree with the handle $e$. If $E_{\mathrm{H}} = 1$, then the graph consists of a single hyperelliptic tree (modulo trivial level passages), hence either $c(\bftau) \geq 1$ or it induces a quasi-reflection.

From now on we assume that besides vertices of order one and two, all other vertices are of order three, four, or six from <ref> and <ref>.

§.§ Case that a vertex is of order four

We start by assuming that there is at least one vertex  of order four. Since all vertices of order four have age at least $1/2$, we can assume that there is a unique vertex $v$ of order four, which rules out Case (8) as it has four cyclicly permuted edges and thus requires another vertex of the same type to pair with it, giving enough vertex age.  If $v$ has a pair of permuted edges, then they can only join a hyperelliptic vertex of age zero. Since $\age(v) = 3/4$ in the relevant Cases (11), (12), (R6) and (R7), we can assume that all other vertices are of order one or two. As before, since the $E_{\mathrm{H}}$ edges contribute at least $1/4$, we conclude that $c(\bftau) \geq \age(v) + 1/4 = 1$.
Now suppose all edges of $v$ are fixed. The relevant cases are (9) and (10), for which $\age(v)=1/2$. Case (9) has genus one with a unique edge $e$ going down. Suppose the lower end $v'$ of $e$ has order $k'$ which can be one, two, or three (as in Case (13) of order $6$ cannot be a lower vertex). Since $e$ is CPT (and not RBT as $g\geq 2$), we obtain that $c(\bftau) \geq \age(v) + \age(v') + 2/\lcm(4, k') \geq 1$ for any $k' = 1,2,3$.
For Case (10), if it admits one edge and one leg, then the same argument works as in Case (9). Suppose it admits two edges $e_1$ and $e_2$, one going up to $v_1$ and the other going down to $v_2$ where $v_i$ is of order $k_i\neq 4$. As before we can reduce to the case that all non-trivial edges are not CPT, any hyperelliptic vertex has age zero, and there is at most one other vertex with non-zero age. Note that $e_1$ and $e_2$ cross different level passages. If there is no vertex of order three or six, then $c(\bftau) \geq \age(v)  + (1/ 4 + 1/4 + (E_{\mathrm{H}}-1)/2) / E_{\mathrm{H}} = 1$. If there is exactly one vertex of order three or six, then $c(\bftau) \geq \age(v) + 1/3 + (1/12 + 1/4 + (E_{\mathrm{H}}-1)/6) / E_{\mathrm{H}} \geq 1$.

§.§ Case that a vertex is of order six

Now we assume that all vertices are of order one, two, three, or six, and there is at least one of order six. Consider Case (13) for a vertex $v$ of order six where $\age(v) = 1/3$ and $v$ admits a unique edge $e$ going down. Let $v'$ be the lower end of $e$ with order $k'$. Since $e$ is CPT (and not RBT as $g\geq 2$), its contribution to $c(\bftau)$ is $2c_e\geq 2/\lcm (6, k') \geq 1/3$, hence combining with $\age(v)$ we can assume that all vertices (except $v$) are of order one or two and that any hyperelliptic vertex has age zero. If there are other non-trivial edges besides $e$, then either they are CPT and we obtain enough age, or similarly as before we obtain $c(\bftau) \geq 2/3 + (E_{\mathrm{H}}-1)/(2
E_{\mathrm{H}}) \geq 1$ for $E_{\mathrm{H}}\geq 3$. Otherwise it reduces to EDB, which implies that $c(\bftau)\geq \age (v) + 4/\lcm(1,6) = 1$.

§.§ Case that a vertex is of order three

Finally we assume that all vertices are of order one, two, or three, and there exists at least one vertex $v$ of order three.  If $v$ admits three cyclicly permuted edges, then it must be paired with another vertex $v'$ of order three, which includes Cases (2), (6), (7), (R1) and (R2).  The only combination for $\age(v) + \age (v') < 1$ is where $v$ and $v'$ are of type (2), (R1) or (R2), and then $\age(v) + \age (v') = 2/3$. We can assume that any hyperelliptic vertex has age zero and that there are no CPT edges.  Either the three edges do not cross the level passage ${\rm H}$, and then
$c(\bftau)\geq 2/3+(1/6+1/6+(E_{\mathrm{H}}-1)/2)/E_{\mathrm{H}}\geq 1$, or we obtain that
$c(\bftau) \geq 2/3 + (1/3+1/3+1/3 + (E_{\mathrm{H}}-3)/2)
/E_{\mathrm{H}} \geq 1$ since $E_{\mathrm{H}}\geq 3$ in this case.
Now suppose all vertices of order three have fixed edges only, which includes
Cases (3), (4), (5), (R3), (R4) and (R5).  For Cases (3), (R3) and (R4), $\age (v) = 2/3$, and they can be treated similarly as in the preceding case. For Case (4), the vertex $v$ admits a unique edge and we can use the same argument as in Case (13).
For Case (5), if $v$ admits one leg and one edge, then the argument is still the same as Case (13). Next suppose $v$ admits two edges $e_1$ and $e_2$, going up and down respectively to $v_1$ and $v_2$.  Again we only need to consider the case that all other vertices (except $v$) are of order one or two and any hyperelliptic vertex has age zero. It follows that $c(\bftau) \geq 1/3 + (1/6 + 1/6 + (E_{\mathrm{H}}-1)/2 ) /
E_{\mathrm{H}}$. If there is an edge crossing ${\mathrm{H}}$ constrained by GRC, then the corresponding two-level graph has $v_{\mathrm{H}}^\top>1$, and so the $v^\top$-contribution <ref> gives an additional $1/2$, which  is enough. Hence, as in the case of vertices of order two, the edges crossing the level ${\mathrm{H}}$ and the handles of the hyperelliptic trees crossing ${\mathrm{H}}$, give a contribution of at least $(E_{\mathrm{H}}-1)/E_{\mathrm{H}}+1/6$, where we subtract one since the contributing edge might be $e_1$ or $e_2$. If $E_{\mathrm{H}} >1$, then the preceding estimate together with the vertex age is enough.
Consider finally the case that  $E_{\mathrm{H}} = 1$. Then the graph (modulo trivial level passages) reduces to a tree where both $e_1$ and $e_2$ are CPT. Moreover if $v_2$ is hyperelliptic, then $e_2$ cannot be RBT by stability of $v_2$. Therefore, the contribution of $e_i$ to $c(\bftau)$ is at least $1/3$, which is from either $2c_{e_i} \geq 2/\lcm (2,3) = 1/3$ or $c_{e_i} \geq 1/\lcm (1,3) = 1/3$. It follows that  $c(\bftau) \geq \age(v) + 1/3 + 1/3 = 1$.
For Case (R5) a similar argument as the previous one works.

A slight modification of the previous proof can be used to show <ref>.

Since $1/E_\Gamma\leq 1/2$ in the presence of NCT edges, the same proof will work for <ref> if we can show that the $v^\top$-contribution <ref> is not needed if we change the NCT coefficient to $1/2$ as in <ref>. Indeed in the proof of <ref>, the $v^\top$-contribution is only used in two instances, one in the case of vertices of order one or two and the other in the case of vertices of order three, to ensure that if there are edges crossing $\mathrm{H}$ constrained by GRC, then we obtain a contribution of at least $1/2$. In both cases since each edge crossing $\mathrm{H}$ constrained by GRC gives a contribution of at least $(1/2)\cdot (1/2)= 1/4$ and there are at least two such edges, these edges contribute at least $1/2$, hence giving enough without using the $v^\top$-contribution.

In order to prove <ref> for low genera (the general type result for the minimal odd strata with $g\leq 43$), we need to further reduce the $(\ell/p_e)$-coefficients in $R_{\rm NC}^{\Gamma}$ for certain graphs as follows.
We say that a two-level graph for the minimal stratum in genus $g$ is
a rational multi-banana (RMB) if it has two vertices only, one
on top and one on the bottom,
joined by $E$ edges for $E\geq 2$, where the bottom vertex is of genus zero.

For the minimal strata $\mu=(2g-2)$, we can refine the $R_{\NC}^\Gamma$
coefficients of <ref> by setting
$R_{\NC}^\Gamma= 1/\ell_\Gamma$ for  RMB graphs with prongs of type $(1^{E_\Gamma-1}, p)$
where $p\geq 2E_\Gamma-3$ or with prongs of type $(1^{E_\Gamma-2}, 2, p)$ where $p\geq 2E_\Gamma-2$, and by
setting  $R_{\NC}^\Gamma= P_{-1}/(E_\Gamma+1)$ for RMB graphs with at least one
prong of order one and at least one prong of order greater than seven, where
$P_{-1}=\sum_{e\in E(\Gamma)} 1/p_e$.

We will use the same notation as in the proof of <ref>.

§.§ Case that there exists an RMB with prongs
$(1^{E_\mathrm{R}-1}, p)$ or $(1^{E_\mathrm{R}-2}, 2, p)$

Consider first the case in a minimal stratum that $\Pi$ has a level passage $\mathrm{R}$ whose undegeneration gives an RMB graph of prong type $(1^{E_\mathrm{R}-1}, p)$ with $p \geq 2$. Let $e_p$ be the edge of prong $p$. Let $v_i^{\pm}$ be the upper and lower ends of the edges of prong order one and $u^{\pm}$ the upper and lower ends of $e_p$, where some of the vertices can coincide. We make some observations first. We can assume that
$\mathrm{R}$ is non-trivial, since otherwise we can apply the initial argument. Since in the minimal strata $\Pi$ has no local minima other than the unique bottom level vertex, any vertex $v_i^+$ lower than $u^+$ admits a unique polar edge of prong one. It follows that $p$ and $1$ are the only prong values for the level passages between $\mathrm{R}$ and $u^+$. On the other hand, the subgraph below $\mathrm{R}$ is a tree with rational vertices where each vertex goes down to the bottom vertex via a unique path. In particular, other than $p$, the maximum prong value between $\mathrm{R}$ and $u^-$ can be at most $2 E_{\mathrm{R}} - 3$, which is less than or equal to $p$ by assumption.  Moreover the prong values between $\mathrm{R}$ and $u^-$ are
all greater than two. It follows that $P_{-1,j}/E_j \geq 1/p$ for $j \in [e_p]$ and any
level passage in $[e_p]$ cannot be RMB with at least one prong of order one (except for
the prong type $(1^{E-1}, p)$ where $1/\ell_{\mathrm{R}}=1/p$ is the desired no-compensation
coefficient). Since $e_p$ crosses the non-trivial level passage $\mathrm{R}$, we thus conclude from <ref> and <ref> that

\[c(\bftau) \geq \sum_{j\in [e_p]} \frac{ \ell_jP_{-1,j}}{E_j} s_j \geq c_{e_p} \geq 1/\lcm (k^+, k^-)\]

where $k^{\pm}$ are the vertex orders of $u^{\pm}$ under $\bftau$.
If $u^{\pm}$ are both of order one, then $c_{e_p}\geq 1$ and we are done. If one of them is hyperelliptic, then $c_{e_p} \geq 1/2$, and we can assume that any hyperelliptic vertex is of age zero and that $e_p$ is not constrained by GRC, since otherwise the $v^\top$-contribution <ref> would give another $1/2$. First suppose $u^-$ is hyperelliptic. Since it has age zero and $e_p$ is not constrained by GRC, the vertex $u^-$ admits a unique polar edge (which is $e_p$) and a unique zero edge (or leg), which contradicts stability as $u^-$ is of genus zero. Next suppose that $u^+$ is hyperelliptic and let $e_q$ be a polar edge of maximal prong $q$ (among all the polar edges of $u^+$). Using as before that there is no local minima other than the bottom level vertex, we conclude that all prong values in $[e_q]$ are $q$ and $1$. Therefore, if $e_q$ is not trivial, we obtain from these level passages that $c_{e_q} \geq 1/2$ and hence $c(\bftau) \geq c_{e_p} + c_{e_q} \geq 1$. If $e_q$ is trivial, then its upper vertex $u_1^+$ is hyperelliptic (of age zero), and all other vertices of order one connected to $u^+$ are on higher level than $u_1^+$. Hence we can iterate the same argument we used for $u_+$ to $u_1^+$, and continue this procedure until we reach a non-trivial edge. The procedure has to stop since the top level of $\Pi$ cannot have hyperelliptic vertices of age zero (since those have to be of genus zero).
The argument for an RMB graph of prong type $(1^{E_\mathrm{R}-2}, 2, p)$ with $p \geq 2$ is similar. Indeed similarly as before,  $p$, $1$ and $2$ are the only prong values for the level passages between $\mathrm{R}$ and $u^+$. Moreover, arguing as before, the maximum prong value between $\mathrm{R}$ and $u^-$ can be at most $2 E_{\mathrm{R}} -2$, which is less than or equal to $p$ by assumption.
One additional observation is that in this case $-2 (E_\mathrm{R} - 2) - 3 - (p+1) + 2g-2 = -2$, hence $p$ is even. In particular $1/\ell_{\mathrm{R}}=1/p$ as before. So the previous case of having only order one vertices and the previous case where $u^-$ is hyperelliptic are clear.  If  $u^+$ is hyperelliptic of age zero, then under the above notation assume that $e_q$ is non-trivial. Then $e_q$ is the only polar edge and $q$ is even, since  $-q - 1 + p-1 = - 2$ and $p$ is even. Indeed, if there are polar edges constrained by GRC, there is a non-trivial level passage with $v^\top>1$, and hence the $v^\top$-contribution <ref> gives another $1/2$. If there are a pair of permuted edges with prong $q$, we have $2(-q-1)+p-1= -2$, which is impossible since $p$ is even. It follows that $q\geq 2$ and $q$ is still the largest prong for the level passages in $[e_q]$ (since there might be edges with prongs of order $2$ crossing a level passage in $[e_q]$, we had to rule out the case of $q=1$ in order for the previous sentence to be true).  If $e_q$ is trivial, as before we can iterate the procedure until we find a non-trivial edge giving a contribution of $1/2$.
Consider now RMB with prongs of type $(1^{E_\mathrm{R}-1}, p)$ or $(1^{E_\mathrm{R}-2}, 2, p)$ in the situation of having higher order vertices from <ref> and <ref>.
First we treat the cases in <ref> with GRC of order four. If there is such a vertex, then $\Pi$ has a non-trivial level passage whose corresponding divisor has $v^\top>1$. Then in this case the $v^\top$-contribution <ref> is at least $1/\lcm(k_1,k_2)$, where $k_i$ are the orders of the vertices joined by the special edge with the property that all the level passages crossed by it have $v^\top>1$. It is easy to check that this contribution, together with the age contribution from the vertices, is enough.
Next consider the cases in  <ref>. Since the Cases (4), (9) and (13) give CPT edges, which cannot cross $R$, we do not need to consider these cases. Indeed we have seen that, using the above notation, it is enough to consider the contribution from the edges $e_p$ and $e_q$. Since $e_p$ cannot be a CPT edge, the only possibility would be that $e_q$ is the CPT edge, but in this case $c_{e_q}\geq 2/6$, which, together with $c_{e_p}$ and the age contribution from the vertices, is enough. Moreover, Case (8) needs to join another vertex of order four, thus giving enough vertex age. The remaining cases of order four are (10) and (11). Since these cases correspond to vertices of positive genus, the vertex $u^{-}$ cannot be of this type. If $u^{+}$ is not of type (10) or (11), then we obtain enough age from $c_{e_p}\geq 1/2$ and the vertex age. If $u^{+}$ is of type (10) or (11), we obtain that $c_{e_p}+c_{e_q}+\age(u^{+})\geq 1/4+1/4+1/2=1$.
We are left with considering the cases of order three. The only cases of genus zero are (R1) and (R2), both having three permuted edges which need to join another vertex of order three. Then the age contribution of at least $2/3$ from these two vertices, together with the additional $v^\top$-contribution <ref> of at least $1/6$ and the edge contribution of at least $1/6$, is enough. The remaining cases are all of positive genus, hence they cannot correspond to the vertex $u^{-}$. If $u^{-}$ is hyperelliptic, by the argument above we have $\age(u^{-})\geq 1/2$. Then the vertex age is at least $1/2+1/3$ and $c_{e_p}\geq 1/6$, hence we obtain  enough age. Consider finally the case where $u^{-}$ is of order one, and then $c_{e_p}\geq 1/3$. If  the vertex age is at least $2/3$,  we obtain enough age. Hence we are left to consider the case where only one  vertex of order three is present and it has age $1/3$, which means Cases (5) and (R5) (Case (4) admits a non-trivial CPT edge which gives enough contribution). If the upper end of $e_q$ is of order one, then we obtain $c_{e_p}+c_{e_q}\geq 2/3$, which together with the vertex age, is enough. If the upper end of $e_q$ is hyperelliptic (necessarily of age zero), then by the same analysis for $u^+$ we obtain another edge giving a contribution of at least $1/2$, hence we obtain enough together with the contribution of $c_{e_p}+c_{e_q}\geq 1/3+1/6$ and the vertex age.

§.§ Case that there exists an RMB with a prong of order one and a prong of large order

Consider a graph $\Pi$ such that one of its undegenerations is an RMB graph with at least one prong of order one and one prong of order greater than seven. We can also assume that there is no special RMB level of prong type $(1^{E_\mathrm{R}-1}, p)$
or $(1^{E_\mathrm{R}-2}, 2, p)$. As before, denote by $\mathrm{H}$ the level passage with the largest number of edges crossing and by $\mathrm{R}$ the RMB level passage, which we can assume to be non-trivial.
Consider first the case of having only vertices of order one. Since any non-trivial edge in $\Pi$ yields a contribution of at least $1/(E_\mathrm{H}+1)$, the $E_\mathrm{H}$ edges crossing $\mathrm{H}$ give a contribution of at least $E_\mathrm{H}/(E_\mathrm{H}+1)$. If there is a non-trivial level passage different from $\mathrm{H}$,  then we gain the additional contribution to reach age one. If $\mathrm{H}$ is the only non-trivial level passage, then by <ref> and <ref> we obtain $s_\mathrm{H}\ell_{\mathrm{H}}/p_e\in \bN$ for all edges $e$ crossing $\mathrm{H}$, which is impossible for $0<s_{\mathrm{H}}<1$.
Consider now the case where the vertices have order one or two. Since the edge contribution is at least $E_\mathrm{H}/2(E_\mathrm{H}+1)$, if there is a contribution of $1/2$ from a vertex of order two or from a non-trivial level passage with $v^\top>1$, we can argue similarly as before. Indeed any edge yields a contribution of at least $1/2(E_\mathrm{H}+1)$. Hence if there is a non-trivial level passage different from $\mathrm{H}$, we obtain enough. As before we can also rule out the case that  $\mathrm{H}$ is the only non-trivial level passage. Therefore, it suffices to consider the case where all vertices of order two are hyperelliptic of age zero and all non-trivial level passages have $v^\top=1$.
Since we can assume that $v_\mathrm{H}^\top=1$, using the same analysis as in the general case, i.e., the case with vertices of order two in the proof of <ref>, we obtain a contribution of at least $c(\bftau)\geq E_\mathrm{H}/(E_\mathrm{H}+1)$ by considering the contributions of the $E_\mathrm{H}$ edges together with the contributions of the handles of hyperelliptic trees crossing $\mathrm{H}$. We thus need to find an additional contribution of at least $1/(E_\mathrm{H}+1)$.
Let $e$ be the special edge with  prong one and let $v^\pm$ be the vertices at its upper and lower
ends.  If $e$ is a handle of type two, i.e., it is a fixed edge and at least one of its endpoints is hyperelliptic, then $v^+$ cannot be hyperelliptic of age zero and hence $v^-$, being hyperelliptic of age zero,  has a pair of permuted zero edges of prong one. This is impossible since the graph below $v^-$ is a tree with rational vertices by the RMB assumption for the level $\mathrm{R}$.  Hence if $e$ does not cross $\mathrm{H}$, we obtain the additional contribution we need. If $e$ crosses $\mathrm{H}$ and is of type three, i.e., it belongs to a pair of permuted edges with prong one, then $v^+$  admits a single polar edge $e_1$, which is a handle of prong one. Moreover, $v^-$ also admits a single zero edge  $e_2$ of prong three, which (modulo trivial level passages) is a handle since the graph below $v^-$ is a tree.  It follows that $v^+$ (resp. $v^-$) is (modulo trivial level passages) on the top (resp. bottom) level of $\Pi$ (since otherwise we would gain an additional contribution from edges joining trivial vertices or hyperelliptic trees not crossing $\mathrm{H}$). Moreover, they are the only vertices on the top and bottom levels, since $v_{\mathrm{H}}^\top=1$ and $\Pi$ has a unique bottom level. Let $e_p$ be the special edge crossing $\mathrm{R}$ with prong $p\geq 8$. If $e_p$ crosses only the level passages crossed by $e$ or the upper handle $e_1$ (which have both prong order one), then $c_e+c_{e_1}\geq pc_{e_p}\geq p/2\geq 2$. Since the original calculation only used a contribution of $1$ for $c_e+c_{e_1}$, we have found the desired missing contribution. Therefore, we can assume that $e_p$ joins the top and bottom vertices (since otherwise we would get enough contribution from additional edges joining trivial vertices or hyperelliptic trees). In this case $e_p$ joins two trivial vertices and since $p\geq 8$, then $c_e+c_{e_1}+c_{e_2}\geq pc_{e_p}/3\geq p/3\geq 3/2+1$. This is enough since in the original computation the contribution given by $c_e+c_{e_1}+c_{e_2}$ was $3/2$.
Now we have reduced to the situation where $e$ crosses $\mathrm{H}$ and it is of type one, i.e., the vertices $v^\pm$  are trivial. In the following arguments we can assume that edges joining trivial vertices are non-trivial (e.g., by collapsing those trivial edges and merging the corresponding trivial vertices). If $v^+$ admits a polar edge going up  to a hyperelliptic vertex of age zero, then the hyperelliptic tree associated to this vertex does not cross $\mathrm{H}$ and hence yields an additional contribution of at least $1/(E_\mathrm{H}+1)$ given by its (at least) two handles.
If $v^+$ admits a polar edge going up to another trivial vertex, then we gain the desired extra $1/(E_\mathrm{H}+1)$ from this extra edge, which is neither a handle nor crossing $\mathrm{H}$.
From now on we can assume that $v^+$ admits zero edges only.  If there is an edge crossing $\mathrm{H}$ not joining $v^+$,  since $v^\top_\mathrm{H}=1$ and since $v^+$ is above $\mathrm{H}$ (as $e$ crosses $\mathrm{H}$), then there is a path completely above $\mathrm{H}$ joining the upper vertex of this edge and $v^+$.  Then this path gives enough contribution either because there is one edge joining two trivial vertices or because it is a hyperelliptic tree. Hence we can  assume that $v^+$ is the common upper vertex for all edges crossing $\mathrm{H}$. Suppose $v^-$ is the lowest vertex reached by edges crossing $R$. In this case, since by assumption there is an edge $e_p$ with prong $p \geq 8$ crossing $\mathrm{R}$, using that $e$ has prong one we obtain $c_e \geq p  c_{e_p} \geq 8  (1/2) \geq 2$. Since in the original estimate we only used $c_e\geq 1$, we gain the extra contribution we needed. We are left to show that if there is an edge crossing $\mathrm{R}$, then its ending vertex is above $v^-$ or we get enough compensation. If $v^-$ is on the bottom level of $\Pi$, then we are done.  If not, $v^-$ joins a lower level via an edge $e'$.   If the lower end of $e’$ is trivial, we obtain the desired extra contribution of at least $1/(E_\mathrm{H}+1)$.  If the lower end of $e’$ is hyperelliptic (which cannot be the lowest level of $\Pi$), then the hyperelliptic tree starting with handle $e’$ gives enough contribution.
Consider finally the situation where higher order vertices appear. If $v_\mathrm{H}^\top>1$, we obtain a contribution of at least $(E_{\mathrm{H}}/(E_{\mathrm{H}}+1)+1)/\max(\lcm(k_1,k_2))$, where $k_i$ run among the order of the vertices. First, the case of $\mathrm{H}$ being the only non-trivial level passage is impossible since $\mathrm{R}$ is non-trivial with $v_\mathrm{R}^\top=1$. If now there is at least another non-trivial level passage, we have an additional edge contribution of $1/(E_\mathrm{H}+1)/\max(\lcm(k_1,k_2))$. By inspecting <ref> and considering also the vertex age contribution,  the only possibility is to have at most one higher order vertex with age $1/3$, which means Cases (5) and (R5) (Cases (4) and (13) can be excluded since a non-trivial CPT edge yields enough contribution). In this case the $v^\top$-contribution is at least $1/6$. Let $v_1$ and $v_2$ be the two vertices joined to the special vertex of order three. If $v_1, v_2$ are both trivial and there are in total at least $E_\mathrm{H} + 2$ non-trivial edges, then we have the age estimate $c(\bftau)\geq 1/3 + 1/6 + (1/3 + 1/3 + E_\mathrm{H}/2) / (E_\mathrm{H}+1) > 1$.  If there are exactly $E_\mathrm{H} + 1$ non-trivial non-CPT edges, then $v_\mathrm{H}^\top=1$.
Suppose one of the $v_i$  is hyperelliptic and the other is trivial.  Then the hyperelliptic vertex needs to go up or down further (to avoid CPT and since the bottom minimal rational vertex cannot be hyperelliptic).  Hence there are in total at least $E_\mathrm{H} + 2$ non-trivial edges.  Then we have $c(\bftau)\geq 1/3 + 1/6 + (1/3 + 1/6 + E_\mathrm{H}/2) / (E_\mathrm{H}+1) = 1$.
Suppose finally that $v_1$ and $v_2$ are both hyperelliptic.  Then they need to go up and down respectively for the same reason. Hence there are in total at least $E_\mathrm{H} + 3$ non-trivial edges, and we have $c(\bftau)\geq 1/3 + 1/6 + (1/6 + 1/6 + (E_\mathrm{H}+1)/2) / (E_\mathrm{H}+1) > 1$.
At this point we have reduced to the situation of $v_\mathrm{H}^\top=1$ for any level passages, and we can argue similarly as in the non-special RMB situation, and use the same notation as in the case of vertices of order two in the proof of <ref>, where we considered three types  of edges. Besides these  three types, there can be now $E_4$ edges of a fourth type joining at least one vertex of order greater than two. If $E_4=0$ and the handles of the hyperelliptic trees crossing $\mathrm{H}$ do not join higher order vertices, then as before the level passage $\mathrm{H}$ gives a contribution of at least $ E_\mathrm{H}/(E_\mathrm{H}+1)\geq 2/3$ for $ E_\mathrm{H}\geq 2$, which is enough together with the additional age from the higher order vertices. One can argue similarly if there are some handles that join higher order vertices. If $E_\mathrm{H}=1$, then we have only CPT edges, and we have already discussed this case. If $E_4>0$, then we can only have $E_4=1,2,3$. If $E_4=3$, then there are at least two vertices of order three and the vertex age is at least $2/3$, which together with $ (E_\mathrm{H}-E_4+3/3)/(E_\mathrm{H}+1)\geq 1/3$ for $E_\mathrm{H}\geq 4$ is enough. If $E_\mathrm{H}=E_4=3$, then ${\rm H}$ cannot be ${\rm R}$, since ${\rm R}$ is crossed by at least two edges with different prongs. If $E_4=2$, one can argue similarly. If $E_4=1$, then we get a contribution from ${\rm H}$ of at least $ (E_\mathrm{H}-E_4)/(E_\mathrm{H}+1)$, which is at least $1/3$ for $E_\mathrm{H}\geq 2$ and at least  $2/3$ for $E_\mathrm{H}\geq 5$. Hence if  $E_\mathrm{H}\geq 5$ or $E_\mathrm{H}=1$ we get enough contribution. If $2\leq E_\mathrm{H}\leq 4$, the only two possibilities left are Cases (5) and (10) (as before Cases yielding CPT edges do not need to be considered). For Case (10), which gives a vertex age of $1/2$, we obtain a refined estimate $ (E_\mathrm{H}-1+1/4)/(E_\mathrm{H}+1)+1/E_\mathrm{H}\geq 1/2$ for $E_\mathrm{H}\geq 2$, where the last $1/E_\mathrm{H}$ term comes from the additional edge attached to the special vertex. Similarly for Case (5), which gives a vertex age of $1/3$, we obtain $ (E_\mathrm{H}-1+1/6)/(E_\mathrm{H}+1)+1/E_\mathrm{H}\geq 2/3$ for $E_\mathrm{H}\geq 2$.

We give some examples to illustrate that the choices of the coefficients in $R_{\rm NC}^{\Gamma}$ are delicate.
Take three vertices, each on a different level, where the middle vertex joins the top and the bottom each by a single edge of prong $p_1$ and $p_2$, respectively, the top joins the bottom by $E-1$ edges of prong all equal to $p_3 = p_1 + p_2$, and $p_1, p_2$ are relatively prime.
Suppose $\bftau$ acts trivially on the vertices and acts on the level parameters $t_1$ and $t_2$ by $t_i\mapsto e^{2\pi i / p}t_i$, i.e., the arguments of the action (mod $2\pi i$) are $s_1 = s_2 = 1/p_3$ as in the proof of <ref>.
In this case $\ell_1 = p_1p_3$, $\ell_2 = p_2p_3$, and they satisfy the requirements that $(\ell_1 / p_1)  s_1 = 1 $, $(\ell_1 / p_2) s_2 = 1$, and  $(\ell_1 / p_3)  s_1 + (\ell_2 / p_3)  s_2 = 1$. We have $\ell_1 (1/p_1 + (E-1)/p_3) s_1 + \ell_2 (1/p_2 + (E-1)p_3) s_2 = E+1$, hence we cannot use a coefficient smaller than $1/(E+1)$ (and indeed we use $1/E$ for general NCT edge types in <ref>).
For the RBT coefficient, since $\ell = p$ for the unique RBT edge and $\ell / p - 1 = 0$, i.e., we do not make any compensation, it is clearly sharp.
For the general CPT coefficient (i.e., OCT), take Case (5) of order 3 and age $1/3$ as the middle vertex sitting in between two hyperelliptic trees of age zero. Let  $s_1 = s_2 = 1/6 = 1/\lcm (2,3)$ for the action on the level parameters of the two handles.  Then $c(\bftau)  = 1/3 + 2\cdot (1/6+1/6) = 1$, which implies that we cannot reduce the OCT coefficient to be smaller than $2$. Note that in this case the genus of the top or the bottom can be almost arbitrary after a divisorial undegeneration (except that it cannot be zero since Case (5) of genus one is contained in a middle level, which singles out RBT).
For the EDB coefficient, take Case (5) as a top or a bottom vertex joining a hyperelliptic tree of age zero, and let $s = 1/6$ for the action on the level parameter of the handle. Then $c(\bftau)  = 1/3 + 4\cdot (1/6) = 1$, which shows that the EDB coefficient $4$ is necessary.
For the minimal strata RMB coefficients in <ref>, consider a triangle graph with one top vertex $v_1$, one middle vertex $v_2$, and one bottom vertex $v_3$, where $v_2$ is hyperelliptic of age zero, $v_1$ and $v_3$ are trivial, and moreover $v_3$ is rational with the unique marked zero.  Let $p_1$ be the higher short edge, $p_2$ the lower short edge, and $p_3$ the long edge satisfying that $2p_3 = p_1 + p_2$.  Suppose $p_1, p_2, p_3$ are odd and pairwise relatively prime.  Then $\ell_1 = p_1p_3$ and $\ell_2 = p_2p_3$. The top level undegeneration is a non-RMB banana graph (as $v_2$ cannot be rational with only two edges and no legs) and the bottom level undegeneration is an RMB.  Take $s_1 = s_2 = 1/(2p_3)$ which satisfies the half-integer requirement along the short edges and the integer requirement along the long edge.  Suppose we use the coefficient $1/2$ for the top non-RMB banana graph and the coefficient $1/(E_{\rm R}+1) = 1/3$ for the bottom RMB.  Then we obtain  $c(\bftau) = (1/2) \cdot \ell_1 s_1 (1/p_1 + 1/p_3) + (1/3) \cdot \ell_2  s_2  (1/p_2 + 1/p_3) = 3/4 + p_1/(12p_3)$ which can be smaller than $1$.  However in this case $p_2 \neq 1$ since $v_2$ cannot be rational, which also implies that $p_3 \neq 1$ by the relation $2p_3 = p_1 + p_2$.  Therefore, imposing a prong of order one when reducing the coefficients of RMB graphs in the minimal strata makes sense.
Finally we show that an extra contribution from multiple top vertices (i.e., the $v^\top$-term) is necessary in <ref>. Consider a three-level graph where $u_1, \ldots, u_n$ are trivial vertices on the top level, $v$ is a hyperelliptic vertex (of age zero) on the middle level, and $w$ is a trivial vertex on the bottom level. Suppose each $u_i$ joins $v$ by a short edge of prong $p_1$, $v$ joins $w$ by a short edge of prong $p_2$, and each $u_i$ joins $w$ by a long edge of prong $p_3$, where $p_1 + p_2 = 2p_3$ and $p_1, p_2, p_3$ are pairwise relatively prime.  The GRC imposes zero residue for each polar edge of $v$, hence the age of $v$ can be zero.  Moreover, every edge in every undegeneration of this graph is NCT.  Take $s_1 = s_2 = 1/(2p_3)$ which satisfies the half-integer requirement along every short edge and the integer requirement along every long edge.  The top level passage has $2n$ edges and the bottom level passage has $n+1$ edges. Then we obtain $c(\bftau) = 1/(2n) \cdot \ell_1 s_1 (n/p_1 + n/p_3) + 1/(n+1) \cdot \ell_2 s_2  (1/p_2 + n/p_3)  \sim  3/4 < 1$ for $n \gg 0$ and $p_3 \gg p_2$. Hence in this case an extra contribution from $v^{\top}$ is needed.

§ PULLBACK CLASSES AND THE CANONICAL CLASS

In this section we recall basic properties of the tautological ring of
the moduli space of multi-scale differentials and express some divisor classes
in terms of standard generators. We apply this to the formula for the canonical
bundle and also compute the classes of divisors of Brill–Noether type that are
pulled back from the moduli space of curves.
The divisorial part of the tautological ring is generated by
the $\psi$-classes and boundary classes. It contains standard tautological divisor
classes $\xi, \lambda_1, \kappa_1$, which are all proportional in the strata interior
but can differ along the boundary (see e.g., <cit.>).
For their conversion we define rational numbers

κ_μ∑̄_ m_i≠-1 m_i(m_i+2)/m_i+1
2̄g - 2 + s + ∑_m_i≠-1 m_i/m_i+1
for any signature $\mu = (m_1, \ldots, m_n)$, where $s$ is the number of entries equal
to $-1$ (i.e. the number of simple poles).
The values  $\kappa^\bot := \kappa_{\mu_{\Gamma}^{\bot}}$ and $\kappa^\top :=
\kappa_{\mu_{\Gamma}^{\top}}$ are similarly defined for
the bottom and top level strata of $\Gamma$, including the edges
as legs. In particular

$$ \kappa^\bot + \kappa^\top \= \kappa_\mu\,.$$

This constant $\kappa_\mu$ previously
appeared (with an additional factor $\tfrac1{12}$) in <cit.> for relating sums of
Lyapunov exponents and (area) Siegel-Veech constants in the strata of holomorphic
differentials.  Our definition here includes meromorphic signatures as well. We also
define the $\psi$-class over simple poles to be

$$\psi_{-1} \= \sum_{ m_i = -1} \psi_i\,.$$

The main conversion result of divisor classes we need is the following relation,
whose proof is given in <ref>.
In the tautological ring we have
the relation

κ_μξψ̄_-1 + 12 λ_1 - [D_h]
- ∑_Γ∈_1 ℓ_Γ[D_Γ] .

The canonical class of the coarse moduli space follows from the canonical class of the stack computed in <cit.>, using the above conversion formula and taking into account the branching behavior at the boundary. Recall
from <ref> the definition of graphs $\Gamma$ of type HTB, HBT and HBB (and their prime versions for meromorphic strata) as well as the corresponding ramification divisors $D_\Gamma^{\rm H}$. Recall also that $N$ denotes the dimension of the (unprojectivized) strata.

Let $\mu$ be a holomorphic signature not of type $(2m,2g-2-2m)$ (and the hyperelliptic component is excluded if $\mu = (2g-2)$),
or a meromorphic signature not of type $(2m_1, 2m_2, 2g-2-2m_1 - 2m_2)$ (and the hyperelliptic component is excluded if $\mu = (2g-2+2m, -2m)$).
Then the class of the canonical bundle of the coarse moduli space
$\bP\MScoarse$  is given by

κ_μ/N    _̧1(K_)
ψ̄_-1 + 12 λ_1 - (1 + κ_μ/N)[D_h]

- ∑_Γ∈_1 (ℓ_Γκ_μ_Γ^ - κ_μ/N (ℓ_ΓN_Γ^-1)
) [D_Γ]
- κ_μ/N ∑_Γ  is   HTB   orHBT  or  HBB
in $\CH^1(\bP\MScoarse) = \CH^1(\bP\LMS)$, where for meromorphic signatures the last
sum is for graphs of type HTB', HBT' and HBB'.

We remark that for $\mu$ of holomorphic type $(2m,2g-2-2m)$ or of meromorphic type
$(2m_1, 2m_2, 2g-2-2m_1 - 2m_2)$ the above expressions have to be modified
by the class of the ramification divisor in the interior described by
The proof of <ref> together with the variants for
connected components of the strata is given in <ref>.

§.§ Relations among tautological classes

We denote by
R^∙()  ⊂  ^∙()
the tautological ring, being the smallest subring
that contains the $\psi$-classes, that is closed under the push-forward of level-wise clutching and forgetting a marked point, and moreover that contains the class of the horizontal divisor $D_h$ (including the class of each component if $D_h$ is reducible).[A version of (small) tautological ring without $D_h$ was also considered in <cit.> for the purpose of running diffstrata.]
Note that $D_h$ is irreducible in each holomorphic stratum (component), but in general it can be reducible for the meromorphic strata.
The tautological ring contains all boundary strata classes and
standard tautological classes such as the $\kappa$-classes. Let $\pi\colon \cX \to \ol{B}$ be the universal family with $s_i\colon \ol{B}
\to \cX$ the universal sections, $S_i \subset \cX$ their images and $\omega_\pi$ the relative cotangent bundle.
Define the Miller–Morita–Mumford class $\kappa_1 = \pi_{*} (\c_1 (\omega_{\pi})^2)$.
We give a closed expression for $\kappa_1$ as follows.

The class $\kappa_1$ can be expressed in terms of the standard generators
of $R^\bullet(\bP\LMS)$ as
κ_1 -̄ψ_-1 + κ_μξ+ ∑_Γ∈_1 ℓ_Γ(- ∑_e∈E(Γ) 1/p_e) [D_Γ]
where $\psi_{-1}$ is the sum of $\psi$-classes associated to marked simple poles.

We remark that the above expression can be converted to the
Arbarello–Cornalba $\kappa$-class via the relation
κ_1^AC  π̄_* (_̧1 (ω_π(∑_i=1^n S_i))^2)  κ̄_1 + ψ
where $\psi = \sum_{i=1}^n \psi_i$ is the total $\psi$-class.
The prescribed vanishing of the differentials in the universal family along the sections $S_i$ implies that

$$ \c_1 (\omega_{\pi}) \= \pi^{*}\xi + \sum_{i=1}^n m_i S_i
+ \sum_{\Gamma\in \LG_1} \ell_\Gamma [\cX_\Gamma^{\bot}]\,$$

where $\cX_\Gamma^{\bot}$ is the vertical vanishing divisor over the locus
with level graph $\Gamma$. We compute
κ_1    π̄_*  (( π^*ξ+ ∑_i=1^n m_i S_i
+ ∑_Γ∈_1 ℓ_Γ[_Γ^])^2)

-̄∑_i=1^n m_i^2 ψ_i +  π_* ((∑_Γ∈_1 ℓ_Γ[_Γ^])^2) + (4g-4) ξ+ 2∑_i=1^n ∑_Γ∈_i_1 m_i ℓ_Γ[D_Γ]

(̄4g-4) ξ-∑_i=1^n m_i^2 ψ_i +  2∑_i=1^n ∑_Γ∈_i_1 m_i ℓ_Γ[D_Γ]  + ∑_Γ∈_1 ℓ_Γ^2 π_* ([_Γ^]^2)
where ${}_i{\LG_1}$ means the $i$-th marked point is in lower level.
Next we evaluate
$\pi_{*}([\cX_\Gamma^{\bot}]^2)$. Since $[\cX_\Gamma^{\bot}] + [\cX_\Gamma^{\top}] = \pi^{*} [D_\Gamma]$, it suffices to evaluate $\pi_{*} ([\cX_\Gamma^{\bot}]\cdot [\cX_\Gamma^{\top}])$, which is a class supported on $D_\Gamma$ with suitable multiplicity. Geometrically $\cX_\Gamma^{\bot}$ and $\cX_\Gamma^{\top}$ intersect along the (vertical) edges of $\Gamma$. For $e \in E(\Gamma)$, to figure out its contribution to the multiplicity, it suffices to take a general one-parameter family $C$ crossing through $D_\Gamma$.  The local singularity type of $\cX|_C$ at $e$ is $\ell_\Gamma / p_e$, i.e., locally the corresponding node is defined by
$xy = t^{\ell_\Gamma / p_e}$ where $t$ is the base parameter. Hence the local contribution is $p_e / \ell_\Gamma$ (as the reciprocal of the exponent of $t$). We conclude that

$$\pi_{*}([\cX_\Gamma^{\bot}]^2) \=  - \pi_{*}([\cX_\Gamma^{\bot}]\cdot [\cX_\Gamma^{\top}])
\=  - \frac{1}{\ell_\Gamma}\Big(\sum_{e\in E(\Gamma)} p_e\Big)[D_\Gamma]\,. $$

Moreover by the relation at the beginning we have

ξ(̄m_i+1)ψ_i - ∑_Γ∈_i_1 ℓ_Γ[D_Γ]
(see e.g., <cit.>) and consequently

$$ (2g-2) \xi \= \sum_{i=1}^n (m_i^2 + m_i) \psi_i -  \sum_{i=1}^n \sum_{\Gamma\in {}_i{\LG_1}} m_i \ell_\Gamma [D_\Gamma] \,.$$

In particular, $\psi_i$ can be converted to $\xi$ with boundary classes as long as $m_i \neq -1$, and $\xi = - \sum_{\Gamma\in _i\LG_1} \ell_\Gamma [D_\Gamma]$ for $m_i = -1$. It follows that

\begin{align*}
\kappa_1 + \psi_{-1}
&\=  (2g-2+s)\xi + \sum_{m_i\neq -1} m_i \psi_i + \sum_{\Gamma\in \LG_1} \ell_\Gamma
\Big( \sum_{i \in L(\Gamma)\atop m_i\neq -1} m_i  - \sum_{e\in E(\Gamma)} p_e\Big) [D_\Gamma] \\
&\=  \kappa_\mu \xi + \sum_{ m_i\neq -1} \frac{m_i}{m_i+1} \sum_{\Gamma\in _i\LG_1} \ell_\Gamma [D_\Gamma] + \sum_{\Gamma\in \LG_1} \ell_\Gamma
\Big( \sum_{i \in L(\Gamma)\atop m_i\neq -1} m_i  - \sum_{e\in E(\Gamma)} p_e\Big) [D_\Gamma] \\
&\= \kappa_\mu \xi + \sum_{\Gamma \in \LG_1} \ell_\Gamma \Big( \sum_{i\in L(\Gamma)\atop m_i\neq -1} \frac{m_i^2+2m_i}{m_i+1} -   \sum_{e\in E(\Gamma)} p_e    \Big)[D_\Gamma] \\
&\=  \kappa_\mu \xi + \sum_{\Gamma \in \LG_1} \ell_\Gamma \Big(\kbot - \sum_{e\in E(\Gamma)} \frac{1}{p_e}\Big) [D_\Gamma]\,
\end{align*}

as claimed, where $L(\Gamma)$ denotes the lower level of $\Gamma$ and in the last step we used
that a vertical edge $e$ with pole order $-p_e - 1$ contributes $-p_e + 1/p_e$ in $\kbot$.

Next we compute the first Chern class of the Hodge bundle $\lambda_1$ as shown in <ref>. For that purpose we need to pull back boundary divisor classes from the moduli space of curves to the moduli space of multi-scale differentials.
Let $\delta_{[n]}$ be the total boundary divisor class in $\barmoduli[{g,n}]$. Denote by $f_{[n]}\colon \bP\LMS \to \barmoduli[g,n]$ the natural map remembering the underlying pointed stable curves only.

The pullback of $\delta_{[n]}$ to the moduli space of multi-scale differentials has divisor
f_[n]^*(δ_[n]) [̄D_h] + ∑_Γ∈_1 ℓ_Γ(∑_e∈E(Γ)
1/p_e)[D_Γ] .

The local equation of the universal curve over $\barmoduli[{g,n}]$ is $xy=t$ for the node defining each boundary divisor. In the family over $\bP\LMS$ the local equation is
$xy=t$ for $D_h$ by the horizontal plumbing <cit.>
and $xy = t^{\ell_\Gamma / p_e}$ for each (vertical) edge $e$ in
a graph $\Gamma\in \LG_1$ by Equation (12.8) in loc. cit.
We thus obtain the sum on the right-hand side of
the desired equation and each summand contributes $\ell_\Gamma / p_e$.

Recall the well-known relation $12\lambda_1 = \kappa_1 + \delta_{[n]}$ on $\barmoduli[{g,n}]$ (see e.g. <cit.>). Combining it with
<ref> and <ref> we thus conclude the desired formula.

For convenience of later calculations we combine <ref>  and <ref> to get

ψ_i %̄s̄/̄%̄s̄1κ_μ(m_i +1) (
ψ_-1 + 12 λ_1 - [D_h] -∑_Γ∈_1 (κ_μ_Γ^ - δ_i, κ_μ) ℓ_Γ[D_Γ]
if $m_i \neq -1$, where we define the 'Kronecker' symbol $\delta_{i,\bot}$ to be 1 if
the $i$-th leg is on bottom level and zero otherwise.

§.§ The canonical bundle formula

Our goal here is to prove the canonical bundle formula in
<ref>. We continue to use $\mathbf{D}_\Gamma$,
$\mathbf{D}_h$, etc, for the reduced boundary divisors of the coarse moduli space $\bP\MScoarse$, to
distinguish from the divisors $D_\Gamma$, $D_h$, etc, in the stack $\bP\LMS$. We use ${\bf D}$ and $D$ to denote respectively the total boundary divisors.
We will express the canonical classes in standard generators in $\CH^1(\bP\LMS)$, both for the stack and the
coarse moduli space, so that even for the latter no boldface objects appear
in the final formulas. We start with:

The class of the log canonical bundle of the smooth Deligne–Mumford stack $\bP\LMS$ is
given by
κ_μ/N _̧1(Ω^d_(logD)) ψ̄_-1 +
12 λ_1 - [D_h] + ∑_Γ∈_1 ℓ_Γ(κ_μN_Γ^/N -
κ_μ_Γ^) [D_Γ]
in $\CH^1(\bP\LMS)$, where $d = \dim \bP\LMS$ and $N_\Gamma^{\bot}$ is the dimension of the unprojectivized bottom level stratum in $D_\Gamma$.

This follows from <cit.> and <ref>.

Next we pass to the coarse moduli space, still in the logarithmic context:

Let $\mu$ be a holomorphic signature not of type $(2m, 2g-2-2m)$ (and the hyperelliptic component is excluded if $\mu = (2g-2)$),
or a meromorphic signature not of type $(2m_1, 2m_2, 2g-2-2m_1 - 2m_2)$ (and the hyperelliptic component is excluded if $\mu = (2g-2+2m, -2m)$). Then
the class of the reflexive log canonical bundle of the coarse moduli space
$\bP\MScoarse$  is given by
κ_μ/N _̧1(Ω^[d]_(log𝐃)) ψ̄_-1 +
12 λ_1 - [D_h] + ∑_Γ∈_1 ℓ_Γ(κ_μN_Γ^/N - κ_μ_Γ^ ) [D_Γ]
in $\CH^1(\bP\MScoarse) = \CH^1(\bP\LMS)$, where $d = \dim \bP\MScoarse$.

By the assumption <ref> says that the map
$\varphi\colon \bP\LMS \to \bP\MScoarse$ is unramified in the interior.
For any linear combination of boundary divisors $\sum a_i D_i$ in the stack, suppose $\varphi$ is ramified with order $e_i$ at $D_i$ and with image $\mathbf{D}_i$ in the coarse moduli space. Then the ramification
formula passing from the stack to the coarse moduli space (e.g. <cit.>) gives an equality of $\bQ$-Cartier
φ^*(K_ + ∑_i e_i - 1 + a_i/e_i 𝐃_i)
K̄_ + ∑_i a_i D_i .
Setting all $a_i = 1$, i.e., taking the combination $D_h + \sum_\Gamma D_\Gamma$, <ref>
thus follows from the above
together with <ref>.  Next we can set $a_i=0$
for unramified boundary divisors and $a_i = -1$ for ramified boundary divisors (of order two). Then
<ref> follows from
<ref> and <ref>.

Next we explain how to apply the above formulas to each connected component of a disconnected stratum.
Recall that the connected components of strata of holomorphic differentials have
been classified by Kontsevich–Zorich (<cit.>). For special signatures these connected components are distinguished by the parity of the spin
structure, odd or even, and by consisting entirely of hyperelliptic differentials.
We denote these components by an upper index odd,
even or hyp respectively. Note that if a holomorphic stratum has three components, then the hyperelliptic component has a fixed spin parity
depending on $g$. We emphasize that the superscript odd or even excludes
the hyperelliptic component with that spin parity. The smooth
compactification $\bP\LMS$ still separates these components and we distinguish
them by the same superscripts. In this case we also add the same superscript to
decompose a boundary divisor $D_\Gamma$ or $D_h$, e.g.,
D_ΓD̄_Γ^ ∪D_Γ^ ∪D_Γ^
with   D_Γ^∙ ⊂  ^∙as a disjoint union, where $\bullet \in \{\hyp, \odd, \even\}$. Similar
decompositions occur if there are only two components, odd and even,
or hyperelliptic and non-hyperelliptic.
All the steps of the proof of <cit.> can be performed on
each connected component separately. We thus deduce that the component-wise
version of <ref>

κ_μ/N _̧1(Ω^d_^∙(logD)) ψ̄_-1 +
12 λ_1 - [D_h^∙] + ∑_Γ∈_1 ℓ_Γ(κ_μN_Γ^/N
- κ_μ_Γ^) [D_Γ^∙]
holds with $\bullet \in \{\hyp, \odd, \even\}$ for holomorphic signatures $\mu$. As before this formula
can be converted to

κ_μ/N    _̧1(K_^∙)
ψ̄_-1 + 12 λ_1 - (1 + κ_μ/N)[D_h^∙]

+ ∑_Γ∈_1 (κ_μ/N (ℓ_ΓN_Γ^-1)
- ℓ_Γκ_μ_Γ^ ) [D_Γ^∙]
- κ_μ/N ∑_Γ  is   HTB  or HBT  or  HBB
in $\CH^1(\bP\MScoarse) = \CH^1(\bP\LMS)$ for the (non-logarithmic) canonical
bundle. Here $D^{H,\bullet}_\Gamma$ is the (possibly empty) component of $D^{\rm H}_\Gamma$
in the component indicated by $\bullet \in \{\hyp, \odd, \even\}$. We remark that $D_\Gamma^{\bullet}$ and $D^{H,\bullet}_\Gamma$ can be further reducible due to connected components of the strata in each level of $\Gamma$ and also due to the equivalence classes of prong-matchings of the multi-scale differentials they encode. However for our purpose we do not need to classify their irreducible components.
The connected components of strata of meromorphic differentials have been classified by Boissy (<cit.>). These connected components are similarly distinguished by the spin and hyperelliptic structures, with the only exception in genus one where the distinction is given by divisors of the gcd of the entries in $\mu$ (also called the rotation or torsion numbers, see <cit.> and <cit.>). We can analogously add the corresponding superscripts in order to apply <ref> and <ref> to each connected component of a meromorphic stratum.

§.§ Divisors of Brill–Noether type

Recall that two kinds of effective divisors in $\barmoduli[g]$ were used by
Harris and Mumford in <cit.> and <cit.>
which depend on the parity of $g$. We proceed similarly. First for $g$ odd,
consider the admissible cover compactification of the locus

BN_g     {̄ X ∈[g] : X   has a   ^1_k},

{̄ X ∈[g] : there is a cover
πX →^1,   (π) = k} .
This is a classical Brill–Noether divisor by the description
using linear series. We normalize the class of the Brill–Noether divisor computed
in <cit.> to be

$$ [\BN_g] \=  6 \lambda_1 - \frac{g+1}{g+3} \delta_{\irr} -
\sum_{i=1}^{[g/2]} \frac{6 i (g-i)}{g+3} \delta_i \quad \in \CH^1(\barmoduli[g])\,.$$

We define the pullback of the Brill–Noether divisor $\BN_\mu$
(as a $\bQ$-divisor) to be the total transform
$[\BN_\mu] = f^*[\BN_g]$ where $f\colon \bP\LMS \to\barmoduli[g]$ is the forgetful map. To state the class
of the pullback Brill–Noether divisor we use the following notation.
For an edge $e$ in a level graph $\Gamma$ we write $e \mapsto  \Delta_i$ if
contracting all edges of $\Gamma$ but $e$ results in a graph of compact type
parametrized by the boundary divisor $\Delta_i$ in $\barmoduli[g]$. We write $e \mapsto
\Delta_{\irr}$ if the edge is non-separating, equivalently if the contraction
results in a graph parameterizing irreducible one-nodal
Let $g\geq 3$ be odd.
If the stratum $\bP\LMS$ is connected then the class of the Brill–Noether divisor
in  $\bP\LMS$

$$ [\BN_\mu] \= 6\lambda_1 - \frac{g+1}{g+3} [D_h] - \sum_{\Gamma\in \LG_1}
b_\Gamma [D_\Gamma]\, $$

$$b_\Gamma \= \ell_\Gamma \Big(\sum_{i=1}^{[g/2]}\sum_{e\in E(\Gamma) \atop e \mapsto \Delta_i} \frac{6i(g-i)}{(g+3) p_e} +
\sum_{e\in E(\Gamma) \atop e \mapsto \Delta_{\irr}} \frac{g+1}{(g+3) p_e} \Big)\, $$

is an effective divisor class.
If the stratum is disconnected and $\bullet \in \{\odd, \even\}$ denotes
a non-hyperelliptic component then similarly
[_μ^∙] 6̄λ_1 - g+1/g+3 [D_h^∙]
- ∑_Γ∈_1 b_Γ[D_Γ^∙]
is an effective divisor class.

It follows from <cit.> that for odd $g$ and any
non-hyperelliptic connected component of a stratum $\bP\LMS$ not every curve
parameterized therein admits a ${\frakg}^1_{k}$ for $k = (g+1)/2$.[Strictly speaking <cit.> only considered the strata of holomorphic differentials. However the same argument works for the meromorphic case as well by merging all zeros and poles and specializing to the minimal strata, with the exception for $\mu = (2g-2+m, -m)$ with $m > 1$ where the zero and pole cannot be merged due to the GRC. For the exceptional case one can still argue as in loc. cit. by taking a Brill–Noether general curve in $\bP\Omega\cM_{g-1}(2g-4)^{{\rm nonhyp}}$ union an elliptic tail in $\bP\Omega\cM_{1}(2-2g, 2g+2-m, -m)$. ] This implies that
the pullback of the Brill–Noether divisor to such strata (components) is an effective divisor.
To compute ($c$ times) its class we can perform the same local computation as in the proof of

We remark that for special $\Gamma$, e.g., if the underlying curves parameterized by a vertex of $\Gamma$ have gonality much smaller than expected, then the total transform $\BN_\mu$ can contain the corresponding boundary divisor $D_\Gamma$, which can be subtracted from $\BN_\mu$ to make the remaining effective divisor class more extremal.
Second, to cover $g$ even we need two other types of effective divisors.
For $g$ even Harris used in <cit.> the closure of

_g    {̄ X ∈[g] : there is a cover
πX →^1,   (π) = (g+2)/2,

$\pi$ has a point of multiplicity three }
where multiplicity being $m$ means locally the cover is given by $x \mapsto x^m$, i.e., being of ramification order $m-1$. We normalize the class of the Hurwitz divisor computed in loc. cit. to be

$$  [\Hur_g] \=  6 \lambda_1 - \frac{3g^2 + 12g -6}{(g+8)(3g-1)}
\delta_{\irr} - \sum_{i=1}^{g/2} \frac{6 i (g-i)(3g+4)}
{(g+8)(3g-1)} \delta_i\,. $$

Similarly to the case above we let $[\Hur_\mu] = f^*[\Hur_g]$ be the
pullback Hurwitz divisor class.

For even $g \geq 6$, the Hurwitz divisor $\Hur_\mu$ is an effective divisor in every connected $\bP\LMS$ and in every non-hyperelliptic component of disconnected $\bP\LMS$. For $g=4$, $\Hur_\mu$ is an effective divisor in every connected $\bP\LMS$ and in every odd spin  component of disconnected $\bP\LMS$.
Moreover, the class of the Hurwitz divisor in a connected
$\bP\LMS$ is

$$ [\Hur_\mu] \= 6\lambda_1 -  \frac{3g^2 + 12g -6}{(g+8)(3g-1)} [D_h] - \sum_{\Gamma\in \LG_1}
h_\Gamma [D_\Gamma]\, $$

$$h_\Gamma \= \ell_\Gamma \Big(\sum_{i=1}^{g/2}\sum_{e\in E(\Gamma) \atop e \mapsto \Delta_i} \frac{6 i (g-i)(3g+4)}{(g+8)(3g-1) p_e} +
\sum_{e\in E(\Gamma) \atop e \mapsto \Delta_{\irr}} \frac{3g^2 + 12g -6}{(g+8)(3g-1) p_e} \Big)\, $$

and the same expression holds for the spin components of $\bP\LMS$ with $D_\Gamma$ decorated by $\bullet \in \{ \odd, \even\}$ respectively.

The divisor class calculation is the same as in the proof of <ref>. In order to prove that no strata (components) are  contained in $\Hur_\mu$ in the claimed range, by merging zeros and poles it suffices to prove it for the minimal strata, i.e., for $(2g-2)^{\odd}$ with even $g \geq 4$ and for $(2g-2)^{\even}$ with even $g\geq 6$ (with the only exception for $\mu = (2g-2+m, -m)$ with $m > 1$ since in this case the zero and pole cannot be merged due to the GRC, which we will treat separately at the end). We will exhibit a (boundary) point in each case that is not contained in $\Hur_\mu$.