\subsection{Existence of multiplicative bases in every characteristic}
\label{subsec:main-theorems}

We now assemble the preceding results to prove the inductive step for Proposition~\ref{prop:table-weight-lifting} and deduce our main theorems. We first show that if Proposition~\ref{prop:table-weight-lifting} holds for all representation-finite weighted multiplication tables with fewer than $\dim A$ nonzero elements, then every basic connected representation-finite algebra $A$ of dimension $\dim A$ in characteristic two satisfies all three conclusions of Proposition~\ref{prop:dimension-induction}.

\begin{lemma}
\label{lem:recovery-with-lifting}
Assume that $\charac k=2$ and let $A$ be a basic connected representation-finite algebra over $k$. If Proposition~\ref{prop:table-weight-lifting} holds for all weighted multiplication tables with fewer than $\dim A$ nonzero elements, then all three conclusions of Proposition~\ref{prop:dimension-induction} hold for $A$.
\end{lemma}
\begin{proof}
Since $\dim A_0=\dim A$, the hypothesis of Theorem~\ref{thm:integral-partial-lifting} holds for the weighted multiplication tables $T_\chi$. For each $i\in I_0$, Corollary~\ref{cor:zero-supported-coverage} therefore provides a cocycle generator of the one-dimensional weight space $\HH^2(A_0)_{\chi_i}$ supported on the pairs in $Z$. By Proposition~\ref{prop:indicator-cocycles}(2), this cocycle is a $k$-linear combination of indicator cocycles $\mu_E$ of weight $\chi_i$. Consequently, for each $i\in I_0$ we can choose a single equivalence class $E_i\in Z/{\sim}$ of weight $\chi_i$ whose indicator cocycle $\mu_i:=\mu_{E_i}$ represents a nonzero cohomology class in $\HH^2(A_0)_{\chi_i}$.

Because the characters $\chi_i$
($i\in I_0$) are pairwise distinct by Proposition~\ref{prop:primitive-parameter-characters}, the classes $E_i$ are distinct. Specializing the family of Proposition~\ref{prop:zero-product-family} by setting $T_{E_i}=t_i$ for $i\in I_0$ and $T_E=0$ for all other classes $E\notin\{E_i:i\in I_0\}$ produces the polynomial family of multiplications
\begin{equation}
  m_t=m+\sum_{i\in I_0}t_i\mu_i
  \label{eq:selected-binary-family}
\end{equation}
over $k[t_i:i\in I_0]$. By Proposition~\ref{prop:zero-product-family}, $m_t$ is associative, every scalar fibre has ray category $\mathcal P$, and every binary fibre ($t_i\in\{0,1\}$) has the nonzero rays of $\mathcal P$ as a normed multiplicative basis. Together with our choice of $\mu_i$ as a basis of $\HH^2(A_0)_{>0}=\bigoplus_{i\in I_0}\HH^2(A_0)_{\chi_i}$, all hypotheses of Proposition~\ref{prop:family-criterion} are satisfied. Hence there exists a binary specialization $A_t$ ($t_i\in\{0,1\}$) and a filtered algebra isomorphism $A_t\xrightarrow{\sim}A$ whose associated graded isomorphism $\gr A_t\xrightarrow{\sim}\gr A$ rescales each homogeneous line of $A_0$. Transporting the normed multiplicative ray basis of $A_t$ across this isomorphism gives a normed multiplicative ray basis of $A$, proving conclusion~(1) of Proposition~\ref{prop:dimension-induction}.

Furthermore, because $\ray A_t=\mathcal P=\ray A_0$ by Proposition~\ref{prop:zero-product-family}(3) and the filtered isomorphism $A_t\xrightarrow{\sim}A$ induces a diagonal rescaling on $\gr A_t=A_0=\gr A$, the initial-form bijection of Lemma~\ref{lem:ray-degree} identifies the ray multiplication in $\ray A$ with that in $\ray A_0$, proving conclusion~(2). Finally, $W(\ray A)\cong W(\mathcal P)$ is free abelian by Proposition~\ref{prop:integral-ray-weights}, proving conclusion~(3).
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:table-weight-lifting}]
We proceed by strong induction on the number $n=\dim k[T]$ of nonzero elements of $T$. The base case $n=0$ is vacuous, and if $T$ is disconnected, it decomposes as a disjoint union of smaller weighted multiplication tables whose linearizations are representation-finite direct factors of $k[T]$, so the claim follows componentwise.

Assume now that $T$ is connected with $n$ nonzero elements, and set $B=k[T]$. Because nonidentity elements of $T$ have strictly positive degree $\partial(\ell(r))>0$, they span a nilpotent ideal of $B$ whose quotient is $\prod_x k\cdot 1_x$; thus $B$ is a basic connected representation-finite algebra of dimension $n$.

By Lemma~\ref{lem:table-rays}, $T\cong\ray B$.

If $\charac k\ne2$, Corollary~\ref{cor:away-from-two-basis} identifies $\ray B$ with $\ray B_0$, and if $\charac k=2$, the strong induction hypothesis and Lemma~\ref{lem:recovery-with-lifting} applied to $B$ likewise identify $\ray B$ with $\ray B_0$. By Proposition~\ref{prop:integral-ray-weights}, the weight group $W(\ray B_0)$ is free abelian. Since $W(T)\cong W(\ray B)\cong W(\ray B_0)$ is free abelian, every homomorphism $W(T)\to\Z/2$ lifts to a homomorphism $W(T)\to\Z$, completing the induction.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:dimension-induction}]
Decomposing $A$ into connected components, we may assume that $A$ is connected. When $\charac k=2$, Proposition~\ref{prop:table-weight-lifting} implies the hypothesis of Lemma~\ref{lem:recovery-with-lifting} for every dimension smaller than $\dim A$, which gives~(1), (2), and~(3). When $\charac k\ne2$, Corollary~\ref{cor:away-from-two-basis} gives~(1) and~(2), and Proposition~\ref{prop:integral-ray-weights} gives~(3).
\end{proof}

We can now state the basis conclusion, including the nonbasic case.

\begin{theorem}[{\cite[Sections~1.2 and~1.12]{BGRS}}]
\label{thm:final-basis}
Let $A$ be a finite-dimensional representation-finite algebra over an algebraically closed field $k$.
\begin{enumerate}[label={\upshape(\arabic*)}]
\item The algebra $A$ admits a normed multiplicative basis (Definition~\ref{def:normed-multiplicative-basis}).
\item If $A$ is basic, this basis can be chosen to consist of one representative from each nonzero ray of $A$, with the primitive idempotents representing the identity rays.
\item If $A$ is basic and $\charac k\ne2$, these representatives may be chosen so that they induce an algebra isomorphism $k[\ray A]\xrightarrow{\sim}A$.
\end{enumerate}
\end{theorem}
\begin{proof}
When $A$ is basic, Proposition~\ref{prop:dimension-induction}(1) gives~(1) and~(2), and Corollary~\ref{cor:away-from-two-basis} gives~(3). When $A$ is nonbasic, Lemma~\ref{lem:nonbasic-basis} extends the normed multiplicative basis from its basic corner to $A$.
\end{proof}

Theorem~B was proved in Corollary~\ref{cor:away-from-two-basis}.
The next theorem identifies the standard form of Bretscher and
Gabriel~\cite[Section~3]{BretscherGabriel} with the linearization
$k[\ray A]$ in every characteristic, as in~\cite[Section~9]{BGRS}, and
records the freeness of its ray weight group. To state the result for an
abstract finite category $\mathcal R$ with zero morphisms, we say that
$\mathcal R$ is a \emph{finite ray category} when it can be recovered from
its own linearization $k[\mathcal R]$ via Definition~\ref{def:ray-linearization}(1).

\begin{definition}
\label{def:finite-ray-category}
A finite skeletal category with zero morphisms $\mathcal R$ is called a \emph{finite ray
category} if the identity morphisms of $\mathcal R$ form a complete set of primitive
orthogonal idempotents in $k[\mathcal R]$, the ray product \eqref{eq:ray-product} on
$k[\mathcal R]$ is independent of the choice of representatives, and the map
$r\mapsto[r]$ identifies $\mathcal R$ with $\ray(k[\mathcal R])$.
\end{definition}

For the standard form $A_0$, the primitive idempotents, the representative
independence of Lemma~\ref{lem:ray-product-independent}, the product
description of Proposition~\ref{prop:projective-ray-product}, and the
isomorphism $A_0\cong k[\mathcal P]$ of
Proposition~\ref{prop:scalar-normalization} show that $\mathcal P=\ray A_0$
is a finite ray category; by Proposition~\ref{prop:dimension-induction}(2),
the same holds for $\ray A$ for every basic representation-finite algebra $A$.

\begin{theorem}
\label{thm:final-standardness}
Let $A$ be a basic representation-finite algebra over an algebraically closed field $k$, and form $k(\Gamma_A)$ using the mesh coefficients from the almost split sequences of $\modu A$. In every characteristic:
\begin{enumerate}[label={\upshape(\arabic*)}]
\item Taking initial forms identifies $\ray A$ with $\ray A_0$. In particular, $\ray A$ is an associative ray category, $A_0\cong k[\ray A]$ is representation-finite, and
\[
  \add k(\Gamma_A)\simeq\modu k[\ray A].
\]
\item The algebra $A$ is standard if and only if $A\cong k[\ray A]$.
\item For every finite ray category $\mathcal R$ whose linearization $k[\mathcal R]$ is representation-finite, the weight group $W(\mathcal R)$ is free abelian and
\[
  k(\Gamma_{k[\mathcal R]})\simeq\ind(\modu k[\mathcal R]),
\]
with each vertex sent to the corresponding indecomposable module.
\end{enumerate}
\end{theorem}
\begin{proof}
Parts~(1) and~(2) follow immediately from Proposition~\ref{prop:dimension-induction}(2), Proposition~\ref{prop:scalar-normalization}, and Corollary~\ref{cor:graded-standard-form}.

For~(3), we first check that $B=k[\mathcal R]$ is basic. By Definition~\ref{def:finite-ray-category}, the identity elements $e_x=1_x$ form a complete set of orthogonal primitive idempotents in $B$. If two distinct vertices $x\ne y$ gave isomorphic projective modules $e_xB\cong e_yB$, there would exist elements $f\in e_xBe_y$ and $g\in e_yBe_x$ satisfying $fg=e_x$ and $gf=e_y$. Conjugation by $f$ and $g$ would identify the local rings $R_x$ and $R_y$ and their radicals, ensuring that $M(f,g)\subseteq\rad R_x$ and $M(g,f)\subseteq\rad R_y$. The ray products in $\ray B\cong \mathcal R$ would then satisfy $[f][g]=[e_x]$ and $[g][f]=[e_y]$, making $x$ and $y$ isomorphic in $\mathcal R$ and contradicting the skeletality of $\mathcal R$ (Definition~\ref{def:finite-ray-category}). Thus $k[\mathcal R]$ is basic. Applying~(1) and~(2) together with Proposition~\ref{prop:dimension-induction}(3) to each connected component of $k[\mathcal R]$ (and noting that $\ray(k[\mathcal R])\cong \mathcal R$ by Definition~\ref{def:finite-ray-category}, so $(k[\mathcal R])_0\cong k[\mathcal R]$) shows that $W(\mathcal R)$ is free abelian and $k(\Gamma_{k[\mathcal R]})\simeq\ind(\modu k[\mathcal R])$.
\end{proof}