\subsection{Successor coefficients and primitive reverses} \label{sec:primitive-ray-equations} Whereas the preceding restrictions identify which zero products can acquire socle-valued corrections, we now express the Leibniz condition \eqref{eq:nonzero-leibniz} in coordinates for a one-cochain of fixed weight $\chi$, obtaining the linear system that Section~\ref{sec:integral-partial-lifts} will solve. For each ray $r$, its \emph{successor} $r^+$ specifies the unique basis element in which a homogeneous one-cochain of weight $\chi$ can take a nonzero value on $r$, and the \emph{endpoint type} of $r$ determines which terms of the Leibniz equation on a nonzero product $rs$ survive. We also analyze how local radical generators factor through return partners, which couples the coefficients of outgoing and incoming rays. \begin{definition} \label{def:primitive-characters} Evaluating characters in $L$ on the categorical degree $\deg_\JJ\in H_\Z$, we say that a character $\chi\in L$ is \emph{positive} if $\chi(\deg_\JJ)>0$, and \emph{primitive} if it cannot be written as $n\eta$ for an integer $n>1$ and a character $\eta\in L$. \end{definition} By Proposition~\ref{prop:primitive-parameter-characters} and the identity $\chi_i(\deg_\JJ)=2s_i$, every residual character $\chi_i$ is both positive and primitive. \begin{setting} \label{set:fixed-character} For the remainder of this section and Section~\ref{sec:integral-partial-lifts}, we fix a positive primitive character $\chi\in L$. For each vertex $x$, we define \[ \widehat q_x=\begin{cases} q_x, & \text{if }q_x\ne0\text{ and }\ell(q_x)=\chi,\\ 0, & \text{otherwise}. \end{cases} \] Thus $q_x$ continues to denote the actual homogeneous generator of $\rad R_x$ from Setting~\ref{set:projective-companion}, whereas $\widehat q_x$ retains $q_x$ only when its weight equals the fixed character $\chi$. \end{setting} \begin{definition} \label{def:successor-types} Work with $A_0=k[\mathcal P]$ and a positive primitive character $\chi$ as in Settings~\ref{set:projective-companion} and~\ref{set:fixed-character}. \begin{enumerate}[label={\upshape(\arabic*)}] \item A vertex $x$ is \emph{supporting} (for $\chi$) if $\widehat q_x\ne0$, that is, if $q_x\ne0$ and $\ell(q_x)=\chi$. More generally, $x$ has \emph{primitive local weight} if $q_x\ne0$ and $\ell(q_x)$ is primitive in $L$. \item A nonidentity ray $r\colon x\to y$ is \emph{shiftable} if there is a ray $r^+\colon x\to y$ of weight $\ell(r)+\chi$, which we call the \emph{successor} of $r$. \item The \emph{endpoint type} of a shiftable ray $r\colon x\to y$ is classified by which of the endpoint actions $\widehat q_xr$ and $r\widehat q_y$ are nonzero: \[ \begin{array}{c|c|l} \text{Name} & \text{Symbol} & \text{Defining condition} \\ \hline \text{left-sided} & \mathsf L & \widehat q_xr=r^+\ne0,\ r\widehat q_y=0 \\ \text{right-sided} & \mathsf R & \widehat q_xr=0,\ r\widehat q_y=r^+\ne0 \\ \text{two-sided} & \mathsf B & \widehat q_xr=r\widehat q_y=r^+\ne0. \end{array} \] We say that $r$ is \emph{one-sided} if its endpoint type is $\mathsf L$ or $\mathsf R$. \end{enumerate} \end{definition} The three endpoint types exhaust all shiftable rays. \begin{lemma} \label{lem:successor-trichotomy} In Settings~\ref{set:projective-companion} and~\ref{set:fixed-character}, every shiftable ray $r\colon x\to y$ satisfies $r^+=\widehat q_xr$ or $r^+=r\widehat q_y$. Thus it has exactly one of the types in Definition~\ref{def:successor-types}(3). \end{lemma} \begin{proof} The corner bimodule $A_0(x,y)$ is cyclic over at least one of its endpoint rings $R_x$ or $R_y$, so its rays are successive powers of that endpoint's radical generator applied to the generating ray of $A_0(x,y)$. The weight difference $\ell(r^+)-\ell(r)=\chi$ must therefore be a positive integer multiple of the weight of that radical generator. Because $\chi$ is primitive, the multiple is necessarily~$1$, which means that $r^+$ is obtained from $r$ by a single multiplication by $\widehat q_x$ on the left or by $\widehat q_y$ on the right (and $r^+$ is unique because each weight space in $A_0(x,y)$ is at most one-dimensional). \end{proof} Notice the distinction between cyclicity in Definition~\ref{def:generating-rays}, which is a property of the entire corner bimodule $A_0(x,y)$ over $R_x$ or $R_y$, and the endpoint type $\mathsf L,\mathsf R,\mathsf B$, which records whether the action of weight $\chi$ on a specific shiftable ray $r$ remains nonzero on the left, on the right, or on both sides. For example, if $A_0(x,y)$ is a three-dimensional corner bimodule generated on the left at a supporting vertex $x$, its rays are $f$, $\widehat q_xf$, and $\widehat q_x^2f$, which lie in endpoint layers $0$, $1$, and $2$, respectively. The rays $f$ and $\widehat q_xf$ are shiftable, with successors $\widehat q_xf$ and $\widehat q_x^2f$, whereas the socle ray $\widehat q_x^2f$ is unshiftable. If the right action of $\widehat q_y$ satisfies $f\widehat q_y=\widehat q_xf$, then both $f$ and $\widehat q_xf$ have type $\mathsf B$; if instead $f\widehat q_y=0$, then both have type $\mathsf L$. Over a coefficient ring $R$, any normalized $R$-linear one-cochain $\delta$ of weight $\chi$ vanishes on the primitive idempotents $1_x$ and on all unshiftable rays, and is determined on shiftable nonidentity rays $r$ by coefficients $c_r\in R$ via $\delta(r)=c_r r^+$. On a nonzero product $rs=t$ of nonidentity rays, the Leibniz equation $\delta(rs)=\delta(r)s+r\delta(s)$ compares $c_t t^+$ with $c_r r^+s+c_s rs^+$. We refer to the products $r^+s$ and $rs^+$ as the \emph{successor substitutions} of $rs$; each substitution either equals $t^+$ or vanishes. Before tabulating these equations, we note a basic consequence of primitive local weight that will be used throughout our analysis. \begin{lemma} \label{lem:primitive-action-generation} In Setting~\ref{set:projective-companion}, let $f\colon x\to y$ be a generating ray, and suppose that $\ell(q_x)$ is primitive. If $q_xf\ne0$, then $A_0(x,y)=R_xf$. If in addition $A_0(x,y)$ is cyclic at $y$, then $\ell(q_y)=\ell(q_x)$ and $q_xf=fq_y\ne0$. The dual assertions hold when $\ell(q_y)$ is primitive and $fq_y\ne0$. \end{lemma} \begin{proof} If $A_0(x,y)$ is cyclic at $y$, then $q_xf\ne0$ must take the form $q_xf=fq_y^a$ for some integer $a\geq1$. Comparing weights gives $\ell(q_x)=a\ell(q_y)$, and the primitivity of $\ell(q_x)$ implies $a=1$. Iterating the relation $q_xf=fq_y$ then shows that $R_xf=fR_y=A_0(x,y)$. If $A_0(x,y)$ is not cyclic at $y$, then Lemma~\ref{lem:endpoint-chains}(3) immediately implies that $A_0(x,y)=R_xf$. The second assertion follows by the same weight comparison, and the dual statements follow by passing to the opposite algebra. \end{proof} \begin{lemma} \label{lem:successor-substitutions} In Settings~\ref{set:projective-companion} and~\ref{set:fixed-character}, let $rs=t$ be a nonzero product of nonidentity rays. If $t$ is unshiftable, then both successor substitutions $r^+s$ and $rs^+$ vanish (whenever defined), so $\delta(rs)=\delta(r)s+r\delta(s)$ imposes no condition on the coefficients $c_r,c_s$. If $t$ is shiftable, the possible endpoint types of $(r,s)$ and the resulting Leibniz equations for $c_t$ are given by the following table (where the entry ``any'' allows either a shiftable or an unshiftable ray): \begin{equation} \label{eq:product-type-table} \begin{array}{c|c|c} \text{Type of }t=rs & \text{Types of }(r,s) & \text{Leibniz equation} \\ \hline \mathsf L & (\mathsf L,\text{any}) & c_t=c_r \\ \mathsf L & (\mathsf B,\mathsf L) & c_t=c_r+c_s \\ \mathsf R & (\text{any},\mathsf R) & c_t=c_s \\ \mathsf R & (\mathsf R,\mathsf B) & c_t=c_r+c_s \\ \mathsf B & (\mathsf B,\mathsf B)\text{ or }(\mathsf L,\mathsf R) & c_t=c_r+c_s. \end{array} \end{equation} \end{lemma} \begin{proof} Let $rs=t\ne0$ with $r\colon x\to y$ and $s\colon y\to z$. If $t$ is unshiftable, then $A_0(x,z)$ has no ray of weight $\ell(t)+\chi$, so neither $r^+s$ nor $rs^+$ can be nonzero. Assume now that $t$ is shiftable. Suppose first that $t$ has type $\mathsf L$, so that $t^+=\widehat q_xrs\ne0$ while $rs\widehat q_z=0$. Since $\widehat q_xrs\ne0$, we have $\widehat q_xr\ne0$, so $r$ is shiftable of type $\mathsf L$ or $\mathsf B$ and $r^+s=\widehat q_xrs=t^+$. \begin{itemize} \item If $r$ has type $\mathsf L$, then $r\widehat q_y=0$. If $s$ is shiftable, its successor $s^+$ equals either $\widehat q_ys$ or $s\widehat q_z$; in either case $rs^+$ equals $r\widehat q_ys=0$ or $rs\widehat q_z=0$. Thus only the first substitution $r^+s=t^+$ is nonzero, yielding $c_t=c_r$. \item If $r$ has type $\mathsf B$, then $r\widehat q_y=\widehat q_xr$, so $r\widehat q_ys=\widehat q_xrs=t^+\ne0$. Hence $\widehat q_ys\ne0$, which means $s$ is shiftable with a nonzero left action, while $s\widehat q_z$ cannot equal $\widehat q_ys$ because $rs\widehat q_z=0\ne r\widehat q_ys$. Thus $s$ has type $\mathsf L$ and $rs^+=r\widehat q_ys=t^+$, giving $c_t=c_r+c_s$. \end{itemize} The two rows for $t$ of type $\mathsf R$ follow symmetrically by passing to the opposite algebra. Finally, suppose that $t$ has type $\mathsf B$, so that $t^+=\widehat q_xrs=rs\widehat q_z\ne0$. Then $\widehat q_xr\ne0$ and $s\widehat q_z\ne0$, so $r\in\{\mathsf L,\mathsf B\}$ and $s\in\{\mathsf R,\mathsf B\}$. The mixed pairs $(\mathsf L,\mathsf B)$ and $(\mathsf B,\mathsf R)$ cannot occur: for $(\mathsf L,\mathsf B)$ we would have $t^+=rs\widehat q_z=r\widehat q_ys=0$, and for $(\mathsf B,\mathsf R)$ we would have $t^+=\widehat q_xrs=r\widehat q_ys=0$. In the remaining cases $(\mathsf L,\mathsf R)$ and $(\mathsf B,\mathsf B)$, both $r^+s=\widehat q_xrs$ and $rs^+=rs\widehat q_z$ equal $t^+$, which gives $c_t=c_r+c_s$. \end{proof} We now turn to factorizations of a primitive local generator $q_x$ through other vertices. For a factorization $q_x=fh$ through a vertex $y\ne x$, if $\ell(q_x)=\chi$ and $q_x^2\ne0$, Lemma~\ref{lem:successor-substitutions} yields a linear relation linking $c_f$, $c_h$, and $c_{q_x}$. We first describe the corner subcategories formed by the vertices through which such factorizations pass. \begin{definition} \label{def:primitive-return-corner} Using the local radical generators $q_x$ of Setting~\ref{set:projective-companion} (independently of the fixed character $\chi$): \begin{enumerate}[label={\upshape(\arabic*)}] \item A \emph{primitive reverse} of a ray $f\colon x\to y$ is a ray $h\colon y\to x$ satisfying $fh=q_x\ne0$ and $hf=q_y\ne0$ (neither $f$ nor $h$ is required to be shiftable). \item A \emph{return partner} of $x$ is a vertex $y\ne x$ for which there exists a generating ray $f\colon x\to y$ that admits a primitive reverse and satisfies $q_xf=fq_y\ne0$. \item Writing $\mathcal V_x$ for the set consisting of $x$ together with all of its return partners, the \emph{primitive return corner} $\mathcal K_x$ is the full corner subcategory of $A_0$ on $\mathcal V_x$. \end{enumerate} \end{definition} \begin{lemma} \label{lem:primitive-cyclic-corners} Work in Setting~\ref{set:projective-companion}, with primitive reverses and return corners as in Definition~\ref{def:primitive-return-corner}. \begin{enumerate}[label={\upshape(\arabic*)}] \item Let $f\colon x\to y$ ($x\ne y$) be a generating ray whose corner bimodule $A_0(x,y)$ is cyclic at both endpoints with $q_xf=fq_y\ne0$, and suppose that $\ell(q_x)$ is primitive. If $fA_0(y,x)\ne0$, then the generating ray $h$ of $A_0(y,x)$ satisfies \[ fh=q_x,\qquad hf=q_y. \] \item Let $x$ be a vertex of primitive local weight. The generating rays $f\colon x\to y$ ($y\ne x$) that admit a primitive reverse and satisfy $q_xf=fq_y\ne0$ are linearly ordered by factorization, and the primitive return corner $\mathcal K_x$ is a basic cyclic Nakayama algebra. \item As $x$ ranges over all vertices of primitive local weight with $N_x\geq3$, the vertex sets $\mathcal V_x$ are pairwise disjoint or equal, and every factorization $q_x=uv$ into nonidentity rays $u\colon x\to y$ and $v\colon y\to x$ has its intermediate vertex $y$ in $\mathcal K_x$. \end{enumerate} \end{lemma} \begin{proof} \textit{(1)} Because $A_0(x,y)=fR_y$ is cyclic at $y$, multiplying the generating ray $h$ of $A_0(y,x)$ by powers of $q_y$ or $q_x$ simply multiplies $fh$ by powers of $q_x$. The assumption $fA_0(y,x)\ne0$ therefore ensures that $fh=q_x^a\ne0$ for some integer $a\geq1$. Since $h$ generates the corner, no nonzero return through $y$ has a smaller exponent. If $a\geq2$, then the faithful projective support at $x$ contains the four-point subposet with lower elements $q_x,f$ and upper elements $q_x^a,q_xf$. All four comparisons from $\{q_x,f\}$ to $\{q_x^a,q_xf\}$ hold because $q_xf=fq_y$ and $fh=q_x^a$. On the other hand, $q_x$ and $f$ are incomparable (since $f$ is generating and $a\geq2$ is minimal), and by left cancellation $q_x^a$ and $q_xf$ are likewise incomparable, while neither upper element can precede a lower element by degree. Thus $\{q_x,f,q_x^a,q_xf\}$ forms a four-point crown, contradicting Lemma~\ref{lem:dimension-infinity}. Hence $a=1$ and $fh=q_x$. Since $f(hf)=(fh)f=q_xf=fq_y\ne0$, we also have $hf=q_y$. \textit{(2)} By Definition~\ref{def:primitive-return-corner} and Lemma~\ref{lem:primitive-action-generation}, the targets of the specified generating rays $f\colon x\to y$ are precisely the return partners of $x$, and each associated corner bimodule $A_0(x,y)$ is cyclic at both endpoints. Given two such rays $f_i\colon x\to y_i$ and $f_j\colon x\to y_j$, their primitive reverses show that both $f_i$ and $f_j$ precede both $q_xf_i$ and $q_xf_j$ in the faithful projective support at $x$. By left cancellation of $q_x$, $f_i$ and $f_j$ are comparable if and only if $q_xf_i$ and $q_xf_j$ are comparable; moreover, no upper element $q_xf_i$ can precede a lower element $f_j$ because a generating ray $f_j$ cannot factor through $q_x$ (Lemma~\ref{lem:generating-paths}). If $f_i$ and $f_j$ were incomparable, the four rays $\{f_i,f_j,q_xf_i,q_xf_j\}$ would therefore form a four-point crown. Consequently, these outgoing generating rays form a chain under factorization. Since their categorical degrees lie strictly between $0$ and $\deg_\JJ(q_x)$ and no two distinct rays from $x$ to the same partner have the same weight, the return partners of $x$ may be listed in strictly increasing factorization order as $y_1,\ldots,y_t$. For $1\leq i0$, take the consecutive cycle arrows to be $f_1$, $h_{12},\ldots,h_{t-1,t}$, and $h_t$. By left cancellation in $f_ih_{ij}=f_j$, the composite of these arrows along the segment from $y_i$ to $y_j$ equals $h_{ij}$, the complementary segment from $y_j$ to $y_i$ equals $h_jf_i=h_{ji}$, the segments from and to $x$ equal $f_i$ and $h_i$, and one full turn based at any vertex $v\in\mathcal V_x$ equals $q_v$. The resulting algebra homomorphism from the path algebra of the oriented cycle \[ x\longrightarrow y_1\longrightarrow\cdots\longrightarrow y_t \longrightarrow x \] to $\mathcal K_x$ is surjective because the corner bases above consist precisely of the nonzero images of cycle paths. Since paths sharing the same endpoints differ by a number of full turns, their nonzero images have distinct categorical degrees and are linearly independent; thus the kernel is spanned by paths mapping to zero, and $\mathcal K_x$ is a basic cyclic Nakayama algebra (when $t=0$, the same holds for $\mathcal K_x=R_x=k[q_x]/(q_x^{N_x})$ on a single loop). \textit{(3)} Let $x_0$ be any vertex of primitive local weight $\xi=\ell(q_{x_0})$, and let $x_1,x_2\in \mathcal V_{x_0}\setminus\{x_0\}$ be two distinct return partners of $x_0$, reached by generating rays $u\colon x_0\to x_1$ and $v\colon x_0\to x_2$ with primitive reverses $\bar u,\bar v$. By the chain property in~(2), we may assume after swapping $x_1$ and $x_2$ if necessary that $v=uw$ for a ray $w\colon x_1\to x_2$. Setting $h=\bar v u\colon x_2\to x_1$, we have \[ hw=\bar v uw=\bar v v=q_{x_2},\qquad u(w h)=(uw)\bar v u=v\bar v u=q_{x_0} u\ne0. \] Since $q_{x_0}$, $q_{x_1}$, and $q_{x_2}$ all have weight $\xi$, the nonzero ray $w h\in R_{x_1}$ has weight $\xi$ and therefore equals $q_{x_1}$. Neither $w$ nor $h$ can be a proper ray, since pulling out a local generator from $w h=q_{x_1}$ or $hw=q_{x_2}$ would factor $1_{x_1}$ or $1_{x_2}$ through a distinct vertex. By associativity, \[ q_{x_1}w=(w h)w=w(hw)=w q_{x_2},\qquad q_{x_2}h=(hw)h=h(w h)=hq_{x_1}. \] If either $N_{x_1}\geq3$ or $N_{x_2}\geq3$, then $q_{x_1}^2=w q_{x_2}h$ or $q_{x_2}^2=hq_{x_1}w$ is nonzero, so $q_{x_1}w=w q_{x_2}\ne0$ and $q_{x_2}h=hq_{x_1}\ne0$; hence $x_1$ and $x_2$ are mutual return partners whenever $\max(N_{x_1},N_{x_2})\geq3$. Likewise, if $N_{x_0}\geq3$, then $u\bar u q_{x_0}=q_{x_0}^2\ne0$ implies $q_{x_1}\bar u=\bar u q_{x_0}\ne0$, so $x_0\in \mathcal V_{x_1}$; if both $N_{x_0}\geq3$ and $N_{x_1}\geq3$, the argument above gives $\mathcal V_{x_1}\subseteq \mathcal V_{x_0}$, and interchanging $x_0$ and $x_1$ gives $\mathcal V_{x_0}\subseteq \mathcal V_{x_1}$. Now suppose that $N_x\geq3$ and $N_{x'}\geq3$, and let $z\in \mathcal V_x\cap \mathcal V_{x'}$. Then $z$ has the same primitive local weight as $x$ and $x'$. If $N_z\geq3$, the equality just proved gives $\mathcal V_x=\mathcal V_z=\mathcal V_{x'}$. If instead $N_z=2$, then since $N_x\geq3$ and $N_{x'}\geq3$, the reverse rays from $z$ to $x$ and to $x'$ show as above that $x,x'\in \mathcal V_z$; applying the preceding paragraph to the partners $x,x'$ of $z$ shows that $x$ and $x'$ are mutual return partners, and hence $\mathcal V_x=\mathcal V_{x'}$. Thus the sets $\mathcal V_x$ ($N_x\geq3$) are pairwise disjoint or equal. Finally, suppose that $q_x=uv$ is a factorization of $q_x$ into nonidentity rays $u\colon x\to y$ and $v\colon y\to x$ at a vertex $x$ of primitive local weight $\xi$ with $N_x\geq3$ (so $q_x^2\ne0$). Then $u(vu)=(uv)u=q_xu\ne0$ (since $q_x^2=q_xuv\ne0$), so $vu=q_y^j\ne0$ for some $j\geq1$. Comparing weights gives $\xi=j\ell(q_y)$, so primitivity of $\xi$ implies $j=1$ and $vu=q_y$. Because $q_x^2=uvuv\ne0$, we have $q_xu=uq_y\ne0$ and $vq_x=q_yv\ne0$, and both $u$ and $v$ are generating rays because $uv=q_x$ cannot contain an extra local radical factor. Thus $y\in \mathcal V_x$, so the factorization passes through $\mathcal K_x$. \end{proof} The next chain property is used in the proof of Proposition~\ref{prop:proper-ray-products}(1) and~(4) in Appendix~\ref{app:proper-products}: it compares generating rays that leave a primitive return corner. \begin{lemma} \label{lem:external-hom-chain} In Setting~\ref{set:projective-companion}, let $x$ be a vertex of primitive local weight with $N_x\geq3$. Then the generating rays $f\colon x\to y$ satisfying $q_xf\ne0$ and $y\notin \mathcal V_x$ are linearly ordered by factorization. The dual statement holds for incoming generating rays $g\colon y\to x$ with $gq_x\ne0$ and $y\notin \mathcal V_x$. \end{lemma} \begin{proof} Suppose for a contradiction that $f\colon x\to y$ and $h\colon x\to z$ are two incomparable generating rays with $q_xf\ne0$, $q_xh\ne0$, and $y,z\notin \mathcal V_x$. In the faithful projective support at $x$, consider the five rays \[ 1_x,\quad q_x,\quad q_x^2,\quad q_xf,\quad q_xh. \] Clearly $1_x=Latex,scale=.9,every node/.style={font=\small}] \draw[rounded corners] (-.65,-.7) rectangle (5.15,1.65); \node at (4.6,1.3) {$\mathcal K_x$}; \node (x) at (0,0) {$x$}; \node (v1) at (1.5,1) {$v_1$}; \node (dots) at (3,1) {$\cdots$}; \node (vt) at (4.5,0) {$v_t$}; \draw[->] (x)--(v1); \draw[->] (v1)--(dots); \draw[->] (dots)--(vt); \draw[->] (vt) to[bend left=14] (x); \node at (2.3,.3) {one cycle gives $\widehat q_x$}; \node (y) at (0,-2.1) {$y\notin \mathcal V_x$}; \draw[->] (x) to[bend right=28] node[left] {${f_0}\ (\mathsf L)$} (y); \draw[->] (y) to[bend right=28] node[right] {$l\ (\mathsf R)$} (x); \node[align=left] at (3,-1.8) {${f_0}l=\widehat q_x^2$,\quad $l{f_0}=0$\\$\widehat q_x^3\ne0$,\quad $\widehat q_x^4=0$}; \end{tikzpicture} \caption{A primitive return corner $\mathcal K_x$ together with a nondeep external return $({f_0},l)$ through $y\notin \mathcal V_x$. The cycle arrows depict the cycle in Lemma~\ref{lem:primitive-cyclic-corners}(2), whereas ${f_0}$ and $l$ are generating rays that may be composite paths in the quiver of $A_0$. The relations $l{f_0}=0$ and $\widehat q_x^4=0$ (so $N_x=4$) are established in Lemma~\ref{lem:nondeep-two-object-return}.} \label{fig:primitive-corner-return} \end{figure} The next proposition, proved in Appendix~\ref{app:proper-products}, catalogs all proper shiftable products of generating rays and describes how nondeep external returns interact with other rays at $x$. These restrictions determine how the coefficient of a product depends on the coefficients of its generating factors. We use them to extend a solution of the Leibniz equations from two-sided rays to one-sided rays in Theorem~\ref{thm:integral-partial-lifting}. \begin{proposition} \label{prop:proper-ray-products} Work in Settings~\ref{set:projective-companion} and~\ref{set:fixed-character}, with endpoint layers as in Definition~\ref{def:generating-rays}, endpoint types as in Definition~\ref{def:successor-types}, and primitive return corners $\mathcal K_x$ as in Definition~\ref{def:primitive-return-corner}. \begin{enumerate}[label={\upshape(\arabic*)}] \item If a generating ray $f\colon x\to y$ has type $\mathsf L$ (respectively, $\mathsf R$), then $\dim A_0(x,y)\leq3$, with equality implying $N_x=3$ (respectively, $N_y=3$). Moreover, whenever a nonzero product $rs$ and its first factor $r$ both have type $\mathsf L$, they have the same endpoint layer; the dual statement holds when $rs$ and $s$ both have type $\mathsf R$. \item If $r\colon x\to y$ of type $\mathsf B$ and $s\colon y\to z$ of type $\mathsf L$ are generating rays whose product $rs$ is proper and shiftable, then $rs$ has endpoint layer~$1$, the source $x$ has local length $N_x=3$ ($\widehat q_x^2\ne0$ and $\widehat q_x^3=0$), and $r$ admits a primitive reverse (so $y\in \mathcal V_x$). The dual statement holds for generating rays of types $(\mathsf R,\mathsf B)$. \item Let $r\colon x\to y$ of type $\mathsf L$ and $s\colon y\to z$ of type $\mathsf R$ be generating rays whose product $rs$ is proper and shiftable. Then $rs$ has endpoint layer~$2$: \begin{enumerate}[label={\upshape(\alph*)}] \item If $x=z$, so that $rs=\widehat q_x^2\in R_x$, then $(r,s)$ is a nondeep external return at $x$ (Definition~\ref{def:nondeep-return}). \item If $x\ne z$, write $rs=\widehat q_x^2f=f\widehat q_z^2$ where $f\colon x\to z$ is the generating ray of $A_0(x,z)$. Then at least one of the factorizations \[ r=fa,\quad as=\widehat q_z^2\qquad\text{or}\qquad s=bf,\quad rb=\widehat q_x^2 \] holds, where $(a,s)$ (respectively, $(r,b)$) is a nondeep external return at $z$ (respectively, at $x$) and $a$ (respectively, $b$) is a generating one-sided ray. \end{enumerate} \item If $x$ admits a nondeep external return, then there is no generating two-sided ray $f\colon x\to z$ with $z\notin \mathcal V_x$ and $\widehat q_xf\ne0$, and dually no generating two-sided ray $g\colon z\to x$ with $z\notin \mathcal V_x$ and $g\widehat q_x\ne0$. \end{enumerate} \end{proposition} For convenience, the table below summarizes the local ray terminology introduced in this section and its role in the coefficient equations. Note that the endpoint-type symbols $\mathsf L,\mathsf R,\mathsf B$ label the three shiftable ray actions, independently of the weight lattice $L$ and of the auxiliary algebra $B_\chi$ constructed in the next section. \begin{center} \small \begin{tabularx}{\textwidth}{@{}lX@{}} \toprule Data & Terms and their roles \\\midrule Corner bimodule & Generating/proper rays, cyclicity and endpoint layers (Definition~\ref{def:generating-rays}) describe its basis and endpoint actions. \\ Products & The stem product (Definition~\ref{def:stem-product}) records multiplication of endpoint subbimodules, including when the ray product is zero. \\ Fixed weight $\chi$ & Supporting vertices, shiftable rays and successors (Definition~\ref{def:successor-types}) specify where a cochain of weight $\chi$ can be nonzero. \\ Leibniz equations & The endpoint types $\mathsf L,\mathsf R,\mathsf B$ determine which terms survive in~\eqref{eq:partial-leibniz-group}. \\ Returns & Primitive reverses and return partners describe products equal to a local generator; nondeep external returns have value $\widehat q_x^2$ with $\widehat q_x^3\ne0$ (Definition~\ref{def:nondeep-return}). \\ Exclusions & Homogeneous categories and faithful supports realize factorizations in representation-finite categories; four-point crowns contradict Lemma~\ref{lem:dimension-infinity}. \\ \bottomrule \end{tabularx} \end{center}