\section{Residual characters and the criterion for recovering the algebra}
\label{sec:weights-and-parameters}

We continue under the assumption $\charac k=2$, with $A_0=k[\mathcal P]$ ($\mathcal P=\ray A_0$)
and $\C=\gr_\JJ\ind(\modu A)\simeq\ind(\modu A_0)$. In this section we compute
the universal group of additive weights on the ray category $\mathcal P$, extend
these weights to the arrows of $\C$ so that every mesh relation is
homogeneous, and use the first-order module
lifting of Section~\ref{sec:positive-deformations} to embed the
positive-degree Hochschild cohomology $\HH^2(A_0)_{>0}$ into the mesh
cokernel $\bigoplus_{m\geq2}\operatorname{coker}D_m(\C)$. The
weight lattice $L=W(\mathcal P)$ separates the cohomology classes attached to
distinct residual components even when they share the same categorical
degree.

By Lemma~\ref{lem:unit-mesh-coefficients}, we assume from now on that
all mesh coefficients of $\C$ equal one. The same quiver and
unit-coefficient relations define an $\F_2$-linear category
$\mathcal C^{(2)}$ with $k\otimes_{\F_2}\mathcal C^{(2)}\cong\C$;
we use it in Section~\ref{sec:integral-deformations}.

\subsection{The weight group of a ray category}

We now establish an integral counterpart of the scalar weight trivialization
of Lemma~\ref{lem:scalar-object-values}. This result holds in arbitrary
characteristic (compare the fundamental-group and coefficient analysis in
Bretscher and Gabriel \cite[Sections~2.11--2.12]{BretscherGabriel}) and will
also be used in Part~\ref{part:characteristic-two} to lift modulo-two
weights on smaller weighted multiplication tables.

\begin{definition}
\label{def:scalar-weight-group}
For any finite category $\mathcal R$ with zero morphisms, its \emph{weight group} is the abelian
group
\[
 W(\mathcal R)=\Z\langle[r]:r\ne0,1\rangle/
       \langle[rs]-[r]-[s]:rs\ne0\rangle,
\]
where the classes of identity morphisms are set to zero. Thus homomorphisms
$W(\mathcal R)\to M$ into an abelian group $M$ correspond bijectively to additive
weights on the nonzero morphisms of $\mathcal R$ (and, for $M=k^*$ written
multiplicatively, $\Hom(W(\mathcal R),k^*)$ is the group of scalar
weights $a(rs)=a(r)a(s)$ for $rs\ne0$).
\end{definition}

\begin{proposition}
\label{prop:integral-ray-weights}
Let $A_0=k[\mathcal P]$ ($\mathcal P=\ray A_0$) for a connected basic representation-finite
algebra $A$ with $n$ primitive idempotents, let $G=\pi_1(K)$ be the free deck
group of Subsection~\ref{subsec:universal-cover}, and let $\mathcal L$ be the
category of lifted projectives. Let
$I_{\mathcal L}\subset\Z^{(\operatorname{ob}\mathcal L)}$ denote the subgroup
of formal integer combinations of objects of $\mathcal L$ whose coefficients
sum to zero. Write $\mathbf e_x$ for the formal basis vector at $x$.
\begin{enumerate}[label={\upshape(\arabic*)}]
\item Sending the nonzero morphism line $\mathcal L(x,y)$ to the difference
$\mathbf e_y-\mathbf e_x\in I_{\mathcal L}$ induces an isomorphism
$W(\mathcal P)\cong(I_{\mathcal L})_G$ with the group of $G$-coinvariants of
$I_{\mathcal L}$. Under this identification, the coinvariant class
$\theta_g=[\mathbf e_{gx}-\mathbf e_x]$ is independent of $x\in\operatorname{ob}\mathcal L$,
and the map $g\mapsto\theta_g$ embeds the abelianization $G_{\mathrm{ab}}$
as a direct summand of $W(\mathcal P)$ with free complement $\Z^{n-1}$:
\[
                       W(\mathcal P)\cong\Z^{n-1}\oplus G_{\mathrm{ab}}.
\]
In particular, since $G=\pi_1(K)$ is finitely generated, $W(\mathcal P)$ is a free
abelian group of finite rank.
\item Every additive weight on $\mathcal L$ with values in an abelian group
is a difference of object values.
\end{enumerate}
\end{proposition}
\begin{proof}
For a thin category $T$, finite or infinite, use the presentation of
Definition~\ref{def:scalar-weight-group} on its nonzero morphism lines to
define $W(T)$. First let $T$ be finite, directed and representation-finite.
If $s$ is a source of $T$ and
$T'=T\setminus\{s\}$, let $R$ be the representation-finite poset incidence
category obtained from $T'$ and $\rad T(s,-)$ as in
Lemma~\ref{lem:source-basis-extension}. Because the tautological weight
$r\mapsto[r]\in W(T')$ is additive on $R$, Lemma~\ref{lem:poset-weights}
provides elements $u_y\in W(T')$ for $y\in R$, normalized to $u_{y_D}=0$ at a
chosen base object $y_D$ in each connected component $D\in\pi_0(R)$, such
that $[b_{yz}]=u_z-u_y$ for every nonzero product $f_yb_{yz}=f_z$, where
$f_y$ denotes the nonzero morphism line $T(s,y)$. Since
the relations in $W(T)$ involving the new generators $[f_y]$ are precisely
$[f_y]+[b_{yz}]=[f_z]$, substituting $[f_y]=\mathbf u_D+u_y$, with one new generator
$\mathbf u_D$ for each component $D$, gives a direct-sum
decomposition
\[
 W(T)\cong W(T')\oplus\Z^{\pi_0(R)},
\]
and the dual decomposition holds when adjoining a sink. Passing to the direct
limit along the finite or countably infinite exhaustion $(\mathcal L_m)$
of Lemma~\ref{lem:cover-exhaustion} shows that $W(\mathcal L)$ is a free
abelian group.

Now consider the $G$-equivariant homomorphism
$\partial:W(\mathcal L)\to I_{\mathcal L}$ sending $[r:x\to y]$ to
$\mathbf e_y-\mathbf e_x$. Because $\mathcal L$ is connected, $\partial$ is surjective. We
claim that $\partial$ is also injective. Since $W(\mathcal L)$ is free
abelian and $k^*$ is infinite (so for any nonzero integer $m$ there exists
$a\in k^*$ with $a^m\ne1$), any nonzero element
$z\in\ker\partial$ would be detected by some group homomorphism
$\varphi:W(\mathcal L)\to k^*$; however, by
Lemma~\ref{lem:scalar-object-values}, every scalar weight
$\varphi\in\Hom(W(\mathcal L),k^*)=\operatorname{Scal}(\mathcal L)$ has the
form $\varphi([r:x\to y])=\zeta_y/\zeta_x$ and therefore factors through
$\partial$, forcing $\varphi(z)=1$. Thus $\partial:W(\mathcal L)\xrightarrow{\sim}I_{\mathcal L}$
is a $G$-equivariant isomorphism, which immediately implies (2).

Because the nonzero rays of $\mathcal P$ and their nonzero products are the $G$-orbits
of the nonzero morphism lines of $\mathcal L$ and their nonzero products,
$W(\mathcal P)$ is the group of $G$-coinvariants $W(\mathcal L)_G\cong(I_{\mathcal L})_G$.
Fix reference lifts $x_0,x_1,\ldots,x_{n-1}\in\operatorname{ob}\mathcal L$ of
the $n$ projective vertices of $A_0$. In $(I_{\mathcal L})_G$, the relation
$\mathbf e_{gy}-\mathbf e_x=(\mathbf e_{gy}-\mathbf e_{gx})+(\mathbf e_{gx}-\mathbf e_x)\equiv(\mathbf e_y-\mathbf e_x)+(\mathbf e_{gx}-\mathbf e_x)$ shows
that $\theta_g=[\mathbf e_{gx}-\mathbf e_x]$ is independent of $x$ and satisfies
$\theta_{gh}=\theta_g+\theta_h$. Thus $(I_{\mathcal L})_G$ is generated by
the classes $z_v=[\mathbf e_{x_v}-\mathbf e_{x_0}]$ ($1\leq v\leq n-1$) and $\theta_g$
($g\in G$), and the homomorphism $(I_{\mathcal L})_G\to\Z^{n-1}\oplus G_{\mathrm{ab}}$
induced by sending the object $gx_v$ to $(\mathbf e_v,[g])$, where $\mathbf e_1,\ldots,\mathbf e_{n-1}$
are the standard basis vectors of $\Z^{n-1}$ and $\mathbf e_0=0$, provides
the inverse isomorphism, proving (1).
\end{proof}

\subsection{Integer weights on rays and meshes}

We use integer weights on $\mathcal P$ and on $\C$ to refine the categorical grading
and separate first-order deformations attached to different periodic
components.

\begin{definition}
\label{def:integer-weights}
Let $A_0=k[\mathcal P]$ ($\mathcal P=\ray A_0$) and $\C=\gr_\JJ\ind(\modu A)$.
\begin{enumerate}[label={\upshape(\arabic*)}]
\item An \emph{integer ray weight} is a function $h:\mathcal P\setminus\{0\}\to\Z$
that vanishes on identity rays and satisfies $h(rs)=h(r)+h(s)$ whenever
$rs\ne0$. We write $H_{\Z}=\Hom_{\Z}(W(\mathcal P),\Z)$ for the group of integer ray
weights, set $L=W(\mathcal P)$ (called the \emph{weight lattice} of $A_0$), and write
$\ell(r)=[r]\in L$ for the weight class of a nonzero ray $r\in \mathcal P$.
\item An \emph{integer mesh weight} on $\C$ is a pair $(w,\nu)$ assigning an
integer $w(a)\in\Z$ to each arrow $a$ of $\Gamma_A$ and an integer
$\nu_v\in\Z$ to each nonprojective vertex $v$ of $\Gamma_A$ such that
\[
                   w(\sigma a)+w(a)=\nu_v\qquad\text{for every arrow }a:u\to v.
\]
We write $H^{\mathrm{mesh}}_{\Z}$ for the abelian group of integer mesh
weights on $\C$.
\end{enumerate}
\end{definition}
An integer mesh weight is precisely a closed integer edge weight on $K$:
assign $w(a)$ to each arrow edge and $\nu_v$ to the translation edge $t_v$.
The closedness condition around its mesh triangles is the equality in
Definition~\ref{def:integer-weights}(2).


Note that the categorical degree function $\deg_\JJ$ of Lemma~\ref{lem:ray-degree}
is an element of $H_{\Z}$. Because $L=W(\mathcal P)$ is free abelian of finite rank by
Proposition~\ref{prop:integral-ray-weights}(1), biduality identifies $L$
with $\Hom_{\Z}(H_{\Z},\Z)$ via $\chi(h)=h(\chi)$ for $\chi\in L$ and
$h\in H_{\Z}$. We call elements of $L$ \emph{characters} with respect
to this identification. The algebra $A_0=k[\mathcal P]$ is naturally $L$-graded by placing each
ray $r\in \mathcal P$ in degree $\ell(r)\in L$; a Hochschild $2$-cochain sending a
pair of rays $(r,s)$ to a multiple of $u$ shifts $L$-degree by
$\ell(u)-\ell(r)-\ell(s)\in L$. Thus the $L$-grading refines the categorical
degree $\deg_\JJ$: evaluating a character on $\deg_\JJ\in H_\Z$ recovers its
categorical radical length, whereas evaluating on any $h\in H_\Z$ yields the
corresponding integer ray weight. For a cochain, \emph{internal degree}
refers to the shift in categorical degree and \emph{$L$-weight} to the shift
in $L$-degree. As the next lemma shows, integer mesh weights extend these
gradings compatibly to all of $\C$.

\begin{lemma}
\label{lem:integer-weight-lifting}
For $A_0=k[\mathcal P]$ and $\C=\gr_\JJ\ind(\modu A)$, assigning to each path in $\C$ the
sum of the weights of its arrows under a mesh weight $w\in H^{\mathrm{mesh}}_{\Z}$
induces a well-defined grading on $\C$ and restricts on the projective
corners to an integer ray weight in $H_{\Z}$. This restriction fits into a
short exact sequence of abelian groups
\[
 0\longrightarrow
 \{(\zeta_X)\in\Z^{\operatorname{ob}\C}:\zeta_P=0
                         \text{ for all projective }P\}
 \longrightarrow H^{\mathrm{mesh}}_{\Z}\longrightarrow H_{\Z}
 \longrightarrow0,
\]
where the left-hand map sends a tuple $(\zeta_X)$ to the coboundary mesh
weight $w(a)=\zeta_{\operatorname{target}(a)}-\zeta_{\operatorname{source}(a)}$
and $\nu_v=\zeta_v-\zeta_{\tau v}$.
\end{lemma}
\begin{proof}
For any $w\in H^{\mathrm{mesh}}_{\Z}$, every term $(\sigma a)a$ of the mesh
relation at a nonprojective vertex $v$ has total weight $\nu_v$, so the mesh
ideal of $\C$ is homogeneous with respect to path weight. Because each
nonzero homogeneous component $A_0(x,y)_d=\C_d(P_y,P_x)$ is one-dimensional
by Lemma~\ref{lem:ray-degree}, all paths representing a given ray $r\in \mathcal P$
have the same $w$-weight, and Proposition~\ref{prop:projective-ray-product}(2)
shows that this weight is additive on nonzero ray products. Thus restriction
defines a homomorphism $H^{\mathrm{mesh}}_{\Z}\to H_{\Z}$.

To see that restriction is surjective, pull any $h\in H_{\Z}$ back to an
additive weight on the category $\mathcal L$ of lifted projectives. By
Proposition~\ref{prop:integral-ray-weights}(2), there exist integers $b_x$
for $x\in\operatorname{ob}\mathcal L$ such that $h(r)=b_y-b_x$ for every
nonzero morphism $r\in\mathcal L(x,y)$, and by $G$-invariance the difference
$\gamma(g)=b_x-b_{gx}$ is independent of $x\in\operatorname{ob}\mathcal L$
and homomorphic in $g\in G$. On the lifted projective vertices of
$\widetilde\C$, set $\zeta_{P_x}=-b_x$ (the sign change reflecting that
$\mathcal L$ is the opposite category), and extend $\zeta$ to all vertices of
$\widetilde\C$ by choosing arbitrary integers on one representative of each
$G$-orbit of nonprojective vertices and setting
$\zeta_{g\widetilde X}=\zeta_{\widetilde X}+\gamma(g)$. Then the differences
of $\zeta$ along lifted arrows and translation edges are $G$-invariant and
therefore descend to an integer mesh weight on $\C$ whose restriction to $\mathcal P$
is $\zeta_{P_x}-\zeta_{P_y}=b_y-b_x=h(r)$.

Finally, if $w\in H^{\mathrm{mesh}}_{\Z}$ restricts to zero on $\mathcal P$, pull $w$
back to a closed integer edge weight on the simply connected complex
$\widetilde K$ to write $w$ as the difference of vertex integers
$\zeta_{\widetilde X}$. Because $w$ vanishes on all morphisms between lifted
projectives and $\mathcal L$ is connected, $\zeta$ is constant on all lifted
projective vertices; subtracting this constant makes $\zeta_{P_x}=0$ for all
$x$. Since $\zeta_{g\widetilde X}-\zeta_{\widetilde X}$ is independent of
$\widetilde X$ and vanishes on lifted projectives, $\zeta$ is $G$-invariant
and descends to a vertex assignment $(\zeta_X)$ on $\C$ vanishing at the
projective vertices. Because every connected component of $\Gamma_A$ contains
at least one projective vertex, this coboundary map is injective.
\end{proof}

For each residual component $i\in I$ and its representative path
$p_i:\tau v_i\to v_i$ of length $d_i=2s_i+2$ from
Proposition~\ref{prop:residual-correction-spaces}(2), define the linear
functional $\chi_i$ on $H^{\mathrm{mesh}}_{\Z}$ by
\begin{equation}
 \chi_i(w)=w(p_i)-\nu_{v_i}.
 \label{eq:parameter-character}
\end{equation}
Because $p_i$ and every term of the mesh relation at $v_i$ share the same
endpoints $\tau v_i$ and $v_i$, any vertex coboundary $(\zeta_X)$ shifts
$w(p_i)$ and $\nu_{v_i}$ by the same amount $\zeta_{v_i}-\zeta_{\tau v_i}$.
By Lemma~\ref{lem:integer-weight-lifting}, $\chi_i(w)$ therefore depends only
on the restricted ray weight in $H_{\Z}$ and defines an element
$\chi_i\in\Hom_{\Z}(H_{\Z},\Z)=L$, which we call the \emph{residual
character} of the $i$th residual component.

More generally, each mesh weight $w\in H^{\mathrm{mesh}}_{\Z}$ grades
$V_{m+1}(\C)$ by assigning weight $w(p)-\nu_v$ to a path $p:\tau v\to v$ in
the summand $\C_{m+1}(\tau v,v)$, and grades $U_m(\C)$ by assigning weight $w(p)-w(a)$ to a path $p$
parallel to an arrow $a$. The mesh derivative $D_m$ of
\eqref{eq:mesh-derivative} preserves this grading, so
$\operatorname{coker}D_m(\C)$ becomes an $L$-graded space in which the line
$k[p_i]$ is homogeneous of $L$-degree $\chi_i$ (and of positive internal
degree $\chi_i(\deg_\JJ)=d_i-2=2s_i>0$).

To construct weights that distinguish the residual components, we first
extend closed edge weights from the periodic subcomplexes of $K$.

\begin{lemma}
\label{lem:periodic-weight-extension}
Every closed edge weight on the union of the periodic subcomplexes of $K$
extends to a closed edge weight on $K$.
\end{lemma}
\begin{proof}
As in the first paragraph of the proof of Lemma~\ref{lem:mesh-walk-free}, we
may collapse all triangles outside the periodic subcomplexes one by one along
finite $\sigma$-chains. Assign weight zero to all nonperiodic edges that
remain after these collapses, and then restore the collapsed triangles in
reverse order: since each restored triangle has a unique free edge that was
removed at that step, its weight is uniquely determined by requiring the
boundary sum around the triangle to vanish. This extends the given periodic
weight to a closed edge weight on all of $K$.
\end{proof}

The residual characters are integrally independent: more precisely, they
extend to a basis of the weight lattice.

\begin{proposition}
\label{prop:primitive-parameter-characters}
Assume $\charac k=2$. In the weight lattice $L=W(\mathcal P)$ of
Definition~\ref{def:integer-weights}, the residual characters
$\{\chi_i\}_{i\in I}$ defined by \eqref{eq:parameter-character} extend to a
$\Z$-basis of $L$.
\end{proposition}
\begin{proof}
For each $i\in I$, identify the $i$th residual component with
$\Z\Delta/\langle\tau^{s_i}\rho\rangle$ as in
Proposition~\ref{prop:residual-correction-spaces}(1), with $\Delta$ oriented
away from the root fixed by $\rho$. On the vertices $(x,j)$ of $\Z\Delta$,
set
\[
                   F_i(x,j)=\left\lfloor j/s_i\right\rfloor,
\]
and assign to every arrow and translation edge of $\Z\Delta$ the difference
of $F_i$ between its target and source. Because the deck generator
$(x,j)\mapsto(\rho x,j-s_i)$ subtracts $1$ from $F_i$ at every vertex, these
edge differences are invariant under $\langle\tau^{s_i}\rho\rangle$ and
descend to a closed integer edge weight on the $i$th residual component.
Assigning weight zero to all other periodic components of $\Gamma_A$ and
extending to a closed edge weight on the entire mesh complex $K$ via
Lemma~\ref{lem:periodic-weight-extension} produces an integer mesh weight
$w_i\in H^{\mathrm{mesh}}_{\Z}$ on $\C$; let $h_i\in H_{\Z}=\Hom_{\Z}(L,\Z)$
be its restriction to $\mathcal P$.

Lift the path $p_i:\tau v_i\to v_i$ of length $2s_i+2$ to $\Z\Delta$ starting
at a vertex $(x,r)$ over $\tau v_i$. Its endpoint has the form
$(\rho^m x,r+1-ms_i)$ for some $m\in\Z$ and lies at the same distance from
the root of $\Delta$ as $x$. Since outward arrows in $\Z\Delta$ preserve the
second coordinate while inward arrows increase it by $1$, the path $p_i$ of
length $2s_i+2$ traverses an equal number $s_i+1$ of outward and inward
arrows and therefore ends at second coordinate $r+s_i+1$. Subtracting the
weight of the translation edge $t_{v_i}$ from $(x,r)$ to $(x,r+1)$ gives
\[
 h_i(\chi_i)=\chi_i(w_i)
 =\left\lfloor\frac{r+s_i+1}{s_i}\right\rfloor
       -\left\lfloor\frac{r+1}{s_i}\right\rfloor=1,
\]
whereas $h_i(\chi_j)=\chi_j(w_i)=0$ for $j\ne i$ because $w_i$ vanishes on
the $j$th periodic component. Consequently, the homomorphism
$(h_i)_{i\in I}:L\to\Z^I$ is a split surjection with section sending the
$i$th standard basis vector to $\chi_i$, so
$L=\bigoplus_{i\in I}\Z\chi_i\oplus\ker(h_i)_{i\in I}$. Since $\ker(h_i)_{i\in I}$
is a subgroup of the finitely generated free abelian group $L$, it is free
abelian, and any $\Z$-basis of it completes $\{\chi_i\}_{i\in I}$ to a
$\Z$-basis of $L$.
\end{proof}

\subsection{Hochschild cohomology and first-order mesh deformations}
\label{subsec:hochschild-restriction}

We now relate the Hochschild cohomology of $A_0$ to the mesh derivative
cokernel by lifting the degree-normalized indecomposable $A_0$-modules to
first-order deformations of $A_0$ over $k[\varepsilon]/(\varepsilon^2)$.
The comparison with the projective corner is a graded application of
the Cartan--Eilenberg spectral sequences recalled by Buchweitz and Liu
\cite[Proposition~3.9]{BuchweitzLiu}; the required vanishing in
nonnegative internal degrees is Lemma~\ref{lem:positive-ext-vanishing}.
The module lifts below express the degree-two comparison as a map to
mesh errors and retain the $L$-grading.

Because the degree-zero vertex subalgebra $S=\prod_x ke_x$ is separable over
$k$, the inclusion of the normalized $S$-relative cochain complex
$C^\bullet_S(A_0,A_0)$ of \eqref{eq:normalized-relative-cochains} into the
ordinary Hochschild cochain complex is a quasi-isomorphism
\cite[Theorem~1.2]{GerstenhaberSchack}. Both the internal categorical degree
and the $L$-grading on $A_0=k[\mathcal P]$ induce gradings on $\HH^2(A_0)$; we write
$\HH^2(A_0)_q$ for the component of internal degree $q$,
$\HH^2(A_0)_{>0}=\bigoplus_{q>0}\HH^2(A_0)_q$, and $\HH^2(A_0)_\chi$ for the
component of $L$-weight $\chi\in L$. Under this comparison, a Hochschild
two-cocycle of internal degree $q$ corresponds to arrow corrections of path
degree $m=q+1$ and mesh relation errors of path degree $d=q+2$, as summarized
below together with the specialization to the $i$th residual component:
\[
\begin{array}{c|c|c}
\text{Hochschild internal degree}&\text{arrow correction degree}&
\text{mesh error path degree}\\\hline
q&m=q+1&d=q+2\\
2s_i&m=d_i-1=2s_i+1&d_i=2s_i+2
\end{array}
\]

\begin{proposition}
\label{prop:hochschild-mesh-restriction}
Let $A_0$ be the standard form of Construction~\ref{con:projective-algebra}
and $\C=\gr_\JJ\ind(\modu A)$. For every internal degree $q>0$, lifting
degree-normalized modules to first-order deformations induces an $L$-graded
injection
\[
 \Psi:\HH^2(A_0)_q\hookrightarrow\operatorname{coker}D_{q+1}(\C).
\]
In particular, when $\charac k=2$, there is a subset $I_0\subseteq I$ of the
residual components of Proposition~\ref{prop:residual-correction-spaces}
such that
\[
 \HH^2(A_0)_{>0}=\bigoplus_{i\in I_0}\HH^2(A_0)_{\chi_i},
 \qquad \dim_k\HH^2(A_0)_{\chi_i}=1\quad(i\in I_0).
\]
\end{proposition}
\begin{proof}
Write $m_0$ for the multiplication of $A_0$. Represent a cohomology class in $\HH^2(A_0)_q$ by a normalized $S$-relative
$2$-cocycle $\mu\in C^2_S(A_0,A_0)_q$, and form the first-order deformation
$A_{0,\mu}=(A_0\otimes_k k[\varepsilon]/(\varepsilon^2),m_0+\varepsilon\mu)$
with $\deg\varepsilon=-q$. By Lemma~\ref{lem:positive-ext-vanishing}(2) and
Lemma~\ref{lem:square-zero-selected-lifts}, the degree-normalized
indecomposable $A_0$-modules $M_1,\ldots,M_h$ lift to graded
$A_{0,\mu}$-modules $M_i^\mu$ (with $M_i^\mu=e_x A_{0,\mu}$ at the projective
vertices), and each arrow $a$ of $\Gamma_A$ lifts to a degree-one morphism
$\tilde a$ in the Hom category $\C_\mu$ of the lifted modules. Because the
mesh relation $r_v=\sum_{a:u\to v}(\sigma a)a$ vanishes in $\C$, its evaluation
on the lifted arrows $\tilde a$ in $\C_\mu$ vanishes modulo $\varepsilon$; by
the freeness of the lifted Hom spaces over $k[\varepsilon]/(\varepsilon^2)$
(Lemma~\ref{lem:square-zero-selected-lifts}(2)), this mesh error has the form
$\varepsilon z_v$ for a unique element $z_v\in\C_{q+2}(\tau v,v)$ (since
$\deg\varepsilon=-q$ and $\deg r_v=2$). Modifying the arrow lifts by
$\tilde a\mapsto\tilde a+\varepsilon f_a$ with $f_a\in\C_{q+1}$ changes the
tuple $z=(z_v)_v$ by $D_{q+1}(f)$.

By Lemma~\ref{lem:square-zero-selected-lifts}(1), any other graded lifts of
the modules $M_i$ are isomorphic to $M_i^\mu$ via degree-zero isomorphisms
reducing to the identity modulo $\varepsilon$. Transporting the lifted arrows
along these isomorphisms conjugates each mesh error $\varepsilon z_v$ by
endpoint automorphisms congruent to the identity modulo $\varepsilon$, which
leaves the coefficient $z_v$ unchanged, while any further adjustment of the
lifted arrows changes $z$ by an element of $\im D_{q+1}$. Moreover, if
$\mu'=\mu+dh$, then
\[
  1+\varepsilon h:A_{0,\mu'}\xrightarrow{\sim}A_{0,\mu}
\]
is an algebra isomorphism because $(dh)(a,b)=a h(b)-h(ab)+h(a)b$, and pulling
the lifted modules and arrows back along $1+\varepsilon h$ yields lifts for
$\mu'$ with the same reduced mesh errors $z_v$. The assignment
$[\mu]\mapsto\Psi([\mu])=[z]$ is therefore well-defined.

Because the first-order module-action and arrow-lifting equations in
Lemma~\ref{lem:square-zero-selected-lifts} are linear in $\mu$ and in the
first-order corrections, and any product of two $\varepsilon$-corrections
vanishes modulo $\varepsilon^2$, the map $\Psi$ is $k$-linear. To see that
$\Psi$ is injective, suppose $\Psi([\mu])=0$. Then we can adjust the arrow
lifts $\tilde a$ by $\varepsilon f_a$ so that every mesh relation $r_v(\tilde a)$
vanishes identically in $\C_\mu$. The assignment $a\mapsto\tilde a$ then
defines a graded $k[\varepsilon]/(\varepsilon^2)$-linear functor
$\C\otimes_k k[\varepsilon]/(\varepsilon^2)\to\C_\mu$ that reduces to the
identity modulo $\varepsilon$. By Lemma~\ref{lem:square-zero-selected-lifts}(2),
the morphism spaces of $\C_\mu$ are free over $k[\varepsilon]/(\varepsilon^2)$
with bases lifting those of $\C$, so by Nakayama's lemma over the local ring
$k[\varepsilon]/(\varepsilon^2)$ this functor is an isomorphism on every Hom
space. Restricting to the projective vertices $e_x A_{0,\mu}$ yields a
graded algebra isomorphism $A_0\otimes_k k[\varepsilon]/(\varepsilon^2)\xrightarrow{\sim}A_{0,\mu}$
reducing to the identity modulo $\varepsilon$, which means that $[\mu]=0$ in
$\HH^2(A_0)_q$.

Finally, choosing an $L$-homogeneous cocycle $\mu$ of weight $\chi\in L$ and
lifting a $\Z$-basis of $H_\Z$ to mesh weights on $\C$ via
Lemma~\ref{lem:integer-weight-lifting}, Lemma~\ref{lem:square-zero-selected-lifts}(3)
shows that when $\varepsilon$ is assigned $L$-weight $-\chi$, the lifted
modules and arrows can be chosen $L$-homogeneous, so the resulting mesh
error $z_v$ has weight $\chi$ after subtracting the mesh weight $\nu_v$. Thus $\Psi$ preserves
$L$-weights. Changing the chosen mesh-weight lifts adds differences of weights attached to objects by Lemma~\ref{lem:integer-weight-lifting}; these cancel between
a path $p:\tau v\to v$ and the mesh weight $\nu_v$, so the cokernel weights are unchanged.
Since the characters $\chi_i$ are distinct by
Proposition~\ref{prop:primitive-parameter-characters}, injectivity of $\Psi$
and Proposition~\ref{prop:residual-correction-spaces}(2) give the stated
decomposition of $\HH^2(A_0)_{>0}$ in characteristic two.
\end{proof}

Here $I$ indexes all nonzero mesh cokernel lines $[p_i]$, whereas
$I_0\subseteq I$ indexes those lines that lie in the image of $\Psi$; our
construction of the deformation family requires only the components in $I_0$.
In the classification of BGRS \cite[Sections~9.2--9.7]{BGRS}, nonstandard
deformations arise from type-$D$ configurations, and in the selfinjective
case Riedtmann \cite[Theorem, p.~232]{RiedtmannDn} places the nonstandard
alternatives in tree class $D_{3m}$ in characteristic two.