\section{Hochschild representatives from the integers modulo four}
\label{sec:integral-deformations}

We now prove Proposition~\ref{prop:integral-bockstein} and combine it
with Theorem~\ref{thm:integral-partial-lifting} to obtain cocycles supported
on zero products. Retain $A^{(2)}=\F_2[\mathcal P]$ and
$A^{(4)}=\mathcal O[\mathcal P]$, where $\mathcal O=\Z/4$.
All cochains are normalized with respect to the vertex idempotents and
homogeneous for the categorical degree and the weight lattice $L$.
For the Hochschild Bockstein construction, see
\cite[Sections~4--5]{GerstenhaberSchack} and
\cite[Section~2]{Linckelmann}.

\subsection{The model over the two-element field and its module lifts}

The all-ones mesh relations of Section~\ref{sec:weights-and-parameters} define a graded $\F_2$-linear category $\mathcal C^{(2)}$ satisfying $k\otimes_{\F_2}\mathcal C^{(2)}=\C$. Its full subcategory on the projective objects $P_x$ has opposite endomorphism algebra $\End(\bigoplus_x P_x)^{\op}=A^{(2)}$, since each nonzero homogeneous component of $A^{(2)}(x,y)$ is one-dimensional over $\F_2$ and every nonzero product of basis rays has coefficient~$1$. For each object $X_i$ of $\mathcal C^{(2)}$, define the graded right $A^{(2)}$-module $M_i^{(2)}$ by
\[
  M_i^{(2)}e_x=\mathcal C^{(2)}(P_x,X_i).
\]
Evaluation on the projective objects identifies $\mathcal C^{(2)}$ with the full graded homomorphism category of the modules $M_i^{(2)}$, because it does so after the faithfully flat base change $-\otimes_{\F_2}k$. Since tensoring a finitely generated projective resolution with $k$ commutes with graded $\Ext$, Lemma~\ref{lem:positive-ext-vanishing} implies that
\begin{equation}
  \Ext^n_{A^{(2)}}(M_i^{(2)},M_j^{(2)})_d=0\qquad(n\geq1,\ d\geq0).
  \label{eq:prime-field-ext-vanishing}
\end{equation}
When $X_i=P_x$ is a projective object, $M_i^{(2)}=e_xA^{(2)}$ is the corresponding indecomposable projective $A^{(2)}$-module.

\begin{proposition}
\label{prop:fixed-integral-companion}
The graded $A^{(2)}$-modules $M_i^{(2)}$ lift to graded $A^{(4)}$-modules $M_i^{(4)}$ that are free over $\mathcal O=\Z/4$, and their full graded homomorphism category $E_4$ has the following properties:
\begin{enumerate}[label={\upshape(\arabic*)}]
\item Each morphism space of $E_4$ is a free $\mathcal O$-module, and $E_4/2E_4=\mathcal C^{(2)}$.
\item At the projective objects $X_i=P_x$, we have $M_i^{(4)}=e_xA^{(4)}$, so the projective corner of $E_4$ has opposite endomorphism algebra $\End(\bigoplus_x e_xA^{(4)})^{\op}=A^{(4)}$.
\item The category $E_4$ admits a quadratic scalar mesh presentation over $\mathcal O$ whose coefficients are units reducing modulo~$2$ to the all-ones mesh presentation of $\mathcal C^{(2)}$.
\end{enumerate}
Moreover, the lifts may be chosen to preserve compatible $L$-gradings
on the modules and morphisms.
\end{proposition}
\begin{proof}
Choose arbitrary degree-preserving lifts of the action matrices of $A^{(2)}$ on $M_i^{(2)}$ to free graded $\mathcal O$-modules. Their associativity defect is divisible by~$2$; dividing it by~$2$ and reducing modulo~$2$ produces a degree-zero cocycle representing a class in $\Ext^2_{A^{(2)}}(M_i^{(2)},M_i^{(2)})_0$, which vanishes by~\eqref{eq:prime-field-ext-vanishing}. Subtracting $2$ times a bounding one-cochain cancels the defect modulo~$4$ (since $(2\mathcal O)^2=0$), yielding an associative graded $A^{(4)}$-module structure $M_i^{(4)}$. The same bar-resolution calculation using the vanishing of $\Ext^1$ in nonnegative degrees shows that $M_i^{(4)}$ is unique up to isomorphism and that every homogeneous morphism of nonnegative degree between the modules $M_i^{(2)}$ lifts to $E_4$. At the projective objects $X_i=P_x$, we take $M_i^{(4)}=e_xA^{(4)}$ to be the regular projective $A^{(4)}$-modules, which gives~(2).

To verify~(1), fix any pair of objects $X_i,X_j$ and lift an $\F_2$-basis $\bar f_1,\ldots,\bar f_r$ of $\mathcal C^{(2)}(X_i,X_j)_d$ ($d\geq0$) to homogeneous $A^{(4)}$-module homomorphisms $f_1,\ldots,f_r\in E_4(X_i,X_j)_d$. Any $g\in E_4(X_i,X_j)_d$ that reduces to zero modulo~$2$ has matrix entries in $2\mathcal O$; dividing by~$2$ gives an $A^{(2)}$-module homomorphism of degree~$d$, which is an $\F_2$-linear combination of $\bar f_1,\ldots,\bar f_r$, so $f_1,\ldots,f_r$ span $E_4(X_i,X_j)_d$ over $\mathcal O$. Likewise, if $\sum_{m=1}^r a_mf_m=0$ with $a_m\in\mathcal O$, reducing modulo~$2$ shows that $a_m=2b_m$ for all $m$, and dividing by~$2$ and reducing modulo~$2$ again shows that every $b_m$ is even, so $a_m=0$ in $\mathcal O$. The same two-step reduction shows that $E_4(X_i,X_j)_d=0$ for $d<0$.

For~(3), Nakayama's lemma implies that the degree-one morphisms generate $E_4$ as an $\mathcal O$-linear category. Because $(E_4)_2$ is a free $\mathcal O$-module, the composition map from length-two paths onto $(E_4)_2$ splits, so its kernel is a free $\mathcal O$-module direct summand. Lifting the mesh relations of $\mathcal C^{(2)}$ to a basis of this kernel defines a quadratic $\mathcal O$-linear category $\widetilde E_4$ equipped with a graded surjection $\widetilde E_4\to E_4$ that reduces modulo~$2$ to the identity on $\mathcal C^{(2)}$. Each graded piece $N=(\widetilde E_4)_d$ is finitely generated over $\mathcal O$.
For $d$ above the top degree of $\mathcal C^{(2)}$, its reduction modulo~$2$
is zero, so $N=2N=4N=0$. Thus $\widetilde E_4$ is finite over $\mathcal O$. Since $E_4$ is free over $\mathcal O$, the surjection $\widetilde E_4\to E_4$ splits as a map of $\mathcal O$-modules, and its kernel vanishes by Nakayama's lemma because it reduces to zero modulo~$2$. Finally, because each arrow $\tau X\to Y$ in a mesh of $\mathcal C^{(2)}$ has a unique mate $Y\to X$ and the irreducible morphism spaces are one-dimensional, every length-two path between $\tau X$ and $X$ is a mesh path, so the lifted relations are quadratic scalar mesh relations with unit coefficients in $\mathcal O^\times$.

Finally, lifting a $\Z$-basis of $H_\Z$ to mesh weights via
Lemma~\ref{lem:integer-weight-lifting} equips the modules $M_i^{(2)}$ with
compatible $L$-gradings. Because the relative bar differentials preserve
$L$-weights, each bounding cochain above may be projected onto its component
of the required $L$-weight as in Lemma~\ref{lem:square-zero-selected-lifts}(3),
so that all lifted actions, morphisms, and mesh relations preserve the
$L$-grading.
\end{proof}

\subsection{Comparison with the mesh derivative}

The comparison with mesh errors uses the following integral version of
Proposition~\ref{prop:residual-correction-spaces}. Its proof is in
Appendix~\ref{app:integral-corrections}.

\begin{lemma}
\label{lem:integral-mesh-cokernels}
Let $\mathcal O=\Z/4$, and let $E_4$ be the finite free mesh category over $\mathcal O$ of Proposition~\ref{prop:fixed-integral-companion}, with mod-$2$ reduction $\mathcal C^{(2)}$ over $\F_2$. Write $D_m(E_4)$ for its mesh derivative with respect to the unit coefficients of that presentation. With $I$ and $\chi_i$ as in Proposition~\ref{prop:residual-correction-spaces} and~\eqref{eq:parameter-character}, there is a degree- and weight-preserving isomorphism
\[
  \bigoplus_{m\geq2}\operatorname{coker}D_m(E_4)
  \cong\bigoplus_{i\in I}(\mathcal O/2\mathcal O)_{\chi_i},
\]
where the summand indexed by $i\in I$ lies in $\operatorname{coker}D_{d_i-1}(E_4)$ ($d_i=2s_i+2$) and has internal degree $d_i-2$.
\end{lemma}

By Lemma~\ref{lem:integral-mesh-cokernels}, the groups
$\mathcal Q_m=\operatorname{coker}D_m(E_4)$ satisfy
\begin{equation}
  2\mathcal Q_{q+1}=0\qquad(q>0).
  \label{eq:mod-four-mesh-annihilator}
\end{equation}
Here $q$ is the Hochschild internal degree and $q+1$ is the corresponding
arrow-correction degree, as in Section~\ref{sec:weights-and-parameters}.
For any cochain $\bar\mu$ over $\F_2$, we write $2\widetilde\mu$ for twice an arbitrary coefficient lift $\widetilde\mu$ of $\bar\mu$ to $\mathcal O=\Z/4$; since $2\mathcal O\cong\F_2$, the cochain $2\widetilde\mu$ depends only on $\bar\mu$.

\begin{lemma}
\label{lem:mod-four-comparison}
Let $\bar\mu$ be a normalized Hochschild two-cocycle of $A^{(2)}$ of internal degree $q>0$. Then $2\widetilde\mu$ is a coboundary in the normalized Hochschild complex of $A^{(4)}$. The same statement holds within each weight component $\chi\in L$ with $\chi(\deg_\JJ)=q>0$.
\end{lemma}
\begin{proof}
Write $m$ for the multiplication of $A^{(4)}$. Set $R=\mathcal O[\varepsilon]/(\varepsilon^2)$ with $\deg(\varepsilon)=-q$, and equip $B=A^{(4)}\otimes_{\mathcal O}R$ with the deformed multiplication $m+2\varepsilon\widetilde\mu$. Its associator is $2\varepsilon\,d\widetilde\mu=0$ because $d\bar\mu=0$ over $\F_2$, so $B$ is an associative graded $R$-algebra.

Over the reduced ring $\F_2[\varepsilon]/(\varepsilon^2)$, we may apply Lemma~\ref{lem:square-zero-selected-lifts} to the first-order deformation $m+\varepsilon\bar\mu$ of $A^{(2)}$ and the modules $M_i^{(2)}$, since the required $\Ext^2$ and $\Ext^1$ groups vanish by~\eqref{eq:prime-field-ext-vanishing}. Let $\bar\nu_i$ be the resulting first-order corrections to the module actions and let $\bar g_a$ be the corrections to the generating arrow morphisms. On the free $R$-modules $M_i^{(4)}\otimes_{\mathcal O}R$, write $\alpha_i^{(4)}$ for the original action on $M_i^{(4)}$. We define deformed actions $\alpha_i^{(4)}+2\varepsilon\widetilde\nu_i$ and deformed arrow maps $a^{(4)}+2\varepsilon\widetilde g_a$. Because every correction term carries a factor of $2\varepsilon$ and $(2\varepsilon)^2=0$, the module-action and module-homomorphism conditions depend only on the reductions $\bar\nu_i$ and $\bar g_a$ modulo~$2$ and are therefore satisfied over $R$. At the projective objects $P_x$, we take the lifted modules to be the regular projective $B$-modules $e_xB$, whose action corrections match the deformed multiplication $m+2\varepsilon\widetilde\mu$ of $B$.

Now evaluate the quadratic scalar mesh relations of $E_4$ on the deformed arrow maps $a^{(4)}+2\varepsilon\widetilde g_a$. The constant terms vanish in $E_4$, and because the unit coefficients of the mesh relations in $E_4$ reduce to~$1$ modulo~$2$, the remaining error at each mesh ending at $X$ is $2\varepsilon\widetilde z_X$, where $\bar z_X$ is the mesh error of the modulo-two lift. Choose a homogeneous morphism $z_X\in E_4(\tau X,X)_{q+2}$ lifting $\bar z_X$; since the underlying representation spaces are free over $\mathcal O$, evaluating $2\varepsilon z_X$ produces precisely the error $2\varepsilon\widetilde z_X$. By~\eqref{eq:mod-four-mesh-annihilator}, the class of $2z=(2z_X)$ vanishes in $\mathcal Q_{q+1}=\operatorname{coker}D_{q+1}(E_4)$, so there exist homogeneous $A^{(4)}$-module morphisms $f_a\in E_4(\operatorname{source}a,\operatorname{target}a)_{q+1}$ satisfying
\[
  D_{q+1}(f)=2z.
\]
Replacing the deformed arrow maps by $a^{(4)}+2\varepsilon\widetilde g_a-\varepsilon f_a$ preserves the $B$-module homomorphism property (since $f_a$ is $A^{(4)}$-linear and $\varepsilon^2=0$) and subtracts $\varepsilon D_{q+1}(f)=2\varepsilon z$ from the mesh errors, so all mesh errors vanish. The corrected arrow maps therefore satisfy all relations of $E_4$ and define an $R$-linear functor from $E_4\otimes_{\mathcal O}R$ to the category of lifted $B$-modules.

Restricting this functor to the projective objects $P_x$ and passing to opposite endomorphism algebras via Proposition~\ref{prop:fixed-integral-companion}(2) yields an $R$-algebra homomorphism
\[
  A^{(4)}\otimes_{\mathcal O}R\longrightarrow B
\]
that reduces to the identity modulo $\varepsilon$ and fixes the vertex idempotents. On the underlying free $R$-module it has the form $\id+\varepsilon h$ for a normalized one-cochain $h$ of $A^{(4)}$, and multiplicativity $(\id+\varepsilon h)(ab)=(\id+\varepsilon h)(a)\cdot_B(\id+\varepsilon h)(b)$ expands to
\[
  h(ab)=h(a)b+ah(b)+2\widetilde\mu(a,b),
\]
so $2\widetilde\mu=-dh$ is a coboundary. Giving $\varepsilon$ weight $-\chi$ keeps the entire construction homogeneous of weight zero, so $h$ may be taken in weight $\chi$.
\end{proof}


\begin{proof}[Proof of Proposition~\ref{prop:integral-bockstein}]
Put $S_2=\prod_x\F_2e_x$ and $S_4=\prod_x\mathcal Oe_x$.
The short exact sequence of normalized relative Hochschild cochain complexes
\[
  0\longrightarrow C^*_{S_2}(A^{(2)},A^{(2)})\xrightarrow{\,2\,}C^*_{S_4}(A^{(4)},A^{(4)})\longrightarrow C^*_{S_2}(A^{(2)},A^{(2)})\longrightarrow0
\]
induces a long exact sequence in cohomology. By Lemma~\ref{lem:mod-four-comparison}, multiplication by~$2$ from $\HH^2(A^{(2)})_q$ to $\HH^2(A^{(4)})_q$ is zero for every $q>0$, so the connecting homomorphism $\beta\colon\HH^1(A^{(2)})_q\to\HH^2(A^{(2)})_q$ is surjective. By the standard definition of the connecting homomorphism, $\beta([\bar\delta])$ is represented by $(d\delta)/2$ for any cochain lift $\delta\in C^1_{S_4}(A^{(4)},A^{(4)})$, and all maps respect the weight grading by $L$.
\end{proof}

\subsection{A coefficient criterion for supported representatives}

Before combining Proposition~\ref{prop:integral-bockstein} with Theorem~\ref{thm:integral-partial-lifting}, we use the following lemma to produce cocycle representatives supported on a prescribed subset of basis elements.

\begin{lemma}
\label{lem:supported-bockstein}
Let $p$ be a prime, let $C^*$ be a cochain complex of free $\Z/p^2$-modules, and set $\overline C^*=C^*/pC^*$. Let $\mathcal Z\subseteq C^2$ be a submodule spanned by a subset of a fixed $\Z/p^2$-basis of $C^2$, and assume that:
\begin{enumerate}[label={\upshape(\arabic*)}]
\item the map $H^2(\overline C^*)\xrightarrow{p}H^2(C^*)$ induced by the inclusion $\Z/p\hookrightarrow\Z/p^2$ is zero;
\item every one-cocycle $\bar\delta\in Z^1(\overline C^*)$ lifts to a one-cochain $\delta\in C^1$ satisfying $d\delta\in \mathcal Z$.
\end{enumerate}
Then every cohomology class in $H^2(\overline C^*)$ is represented by a two-cocycle lying in the subspace $\mathcal Z/p\mathcal Z\subseteq\overline C^2$.
\end{lemma}
\begin{proof}
By hypothesis~(1) and the long exact cohomology sequence for $0\to\overline C^*\xrightarrow{p}C^*\to\overline C^*\to0$, the Bockstein connecting map $H^1(\overline C^*)\to H^2(\overline C^*)$ is surjective. Given any class in $H^2(\overline C^*)$, choose a preimage represented by $\bar\delta\in Z^1(\overline C^*)$ and lift $\bar\delta$ to $\delta\in C^1$ with $d\delta\in \mathcal Z$ via~(2). Since $d\bar\delta=0$ in $\overline C^2$, the coboundary $d\delta$ lies in $\mathcal Z\cap pC^2=p\mathcal Z$ (as $\mathcal Z$ is spanned by a subset of a basis of $C^2$). Dividing $d\delta$ by $p$ therefore yields a representative of the given cohomology class in $\mathcal Z/p\mathcal Z$.
\end{proof}

\subsection{Moving the representative onto zero products}

We apply Lemma~\ref{lem:supported-bockstein} to the Hochschild complex
and the coefficient lifts just constructed.

\begin{corollary}
\label{cor:zero-supported-coverage}
Assume that Proposition~\ref{prop:table-weight-lifting} holds for all weighted multiplication tables with fewer than $\dim A_0$ nonzero elements, as in Theorem~\ref{thm:integral-partial-lifting}. Then for each $i\in I_0$, the one-dimensional weight space $\HH^2(A_0)_{\chi_i}$ has a nonzero cocycle representative supported exclusively on composable pairs of rays $(r,s)$ satisfying $rs=0$ and $r\circledast s\ne0$, where $\circledast$ is the stem product of Definition~\ref{def:stem-product}, and its value on each such pair $(r,s)$ lies in the one-dimensional endpoint socle $U_{r\circledast s}$. Moreover, this cocycle may be chosen over $\F_2$ and extended by scalars to $k$.
\end{corollary}
\begin{proof}
Fix $i\in I_0$ and work first over $\F_2$ with $A^{(2)}=\F_2[\mathcal P]$ and $A^{(4)}=\mathcal O[\mathcal P]$. By Proposition~\ref{prop:primitive-parameter-characters}, the residual character $\chi_i\in L$ is part of a $\Z$-basis of $L$ and satisfies $\chi_i(\deg_\JJ)=2s_i>0$, so $\chi_i$ is positive and primitive as required by Setting~\ref{set:fixed-character}. Let $C^*$ be the weight-$\chi_i$ subcomplex of the normalized Hochschild complex of $A^{(4)}$, so that $\overline C^*$ is the weight-$\chi_i$ subcomplex for $A^{(2)}$, and let $\mathcal Z\subseteq C^2$ be the free $\mathcal O$-submodule of cochains that vanish on all pairs $(r,s)$ with $rs\ne0$ or $r\circledast s=0$ and take values in the free line $\mathcal O(r\circledast s)$ when $rs=0$ and $r\circledast s\ne0$.

Condition~(1) of Lemma~\ref{lem:supported-bockstein} is equivalent,
by the long exact coefficient sequence, to the Bockstein surjectivity
proved in Proposition~\ref{prop:integral-bockstein}. For condition~(2), any derivation $\bar\delta\in Z^1(\overline C^*)$ of weight $\chi_i$ acts on shiftable rays by $\bar\delta(r)=\bar c_r r^+$ and vanishes on identities and unshiftable rays, where the coefficients $(\bar c_r)$ satisfy the nonzero-product Leibniz equations~\eqref{eq:partial-leibniz-group} modulo~$2$. By Theorem~\ref{thm:integral-partial-lifting}, $(\bar c_r)$ lifts to a solution $(c_r)$ over $\mathcal O=\Z/4$, so the lifted one-cochain $\delta(r)=c_r r^+\in C^1$ satisfies $(d\delta)(r,s)=0$ whenever $rs\ne0$. On any pair $(r,s)$ with $rs=0$, we have $\delta(rs)=0$, and any outer successor substitution $r^+=\widehat q_xr$ or $s^+=s\widehat q_z$ contributes $\widehat q_x(rs)=0$ or $(rs)\widehat q_z=0$ to $(d\delta)(r,s)$. Thus the only possible nonzero contributions to $(d\delta)(r,s)=c_r r^+s+c_s rs^+$ come from middle actions $r^+=r\widehat q_y$ or $s^+=\widehat q_ys$, both of which evaluate to $r\widehat q_ys$. By Corollary~\ref{cor:active-zero-pairs} and Proposition~\ref{prop:intrinsic-stem-product}, $r\widehat q_ys$ vanishes if $r\circledast s=0$ and, if $r\circledast s\ne0$, equals $r\circledast s$ when $y$ is supporting and zero otherwise. Hence $d\delta\in\mathcal Z$, verifying condition~(2) of Lemma~\ref{lem:supported-bockstein}.

By Lemma~\ref{lem:supported-bockstein}, the space
$\HH^2(A^{(2)})_{\chi_i}$ has a nonzero cocycle representative in $\mathcal Z/2\mathcal Z$.
Tensoring this representative with $k$ over $\F_2$ gives the required
generator of $\HH^2(A_0)_{\chi_i}$.
\end{proof}