Separating Latent-Source Arrows with Cumulants

The latent source variables $S_0,\ldots,S_{m+1}$ are mutually independent with respect to $\mu$: $$S_0\perp\cdots\perp S_{m+1}.$$

With forward loading vectors $u_0=(1,\gamma)$, $u_j=(1,\rho_j)$ for $1\le j\le m$, and $u_{m+1}=(0,1)$, the bivariate observed outcome vector admits the forward latent linear representation $$(X,Y)^\top=\sum_{j=0}^{m+1} u_j S_j .$$

With reverse loading vectors $v_0=(1,0)$, $v_j=(\sigma_j,1)$ for $1\le j\le m$, and $v_{m+1}=(\delta,1)$, the bivariate observed outcome vector admits the reverse latent linear representation $$(X,Y)^\top=\sum_{j=0}^{m+1} v_j S_j .$$

Let $m\geq 1$, put $n=m+2$ and $K=2m+2=2n-2$. Use the support-annihilator and differential-operator conventions of Definition. There exist sets $$U_m^{\mathrm{right}}\subseteq \Theta^{\mathrm{right},\circ}_{m,K}, \qquad U_m^{\mathrm{left}}\subseteq \Theta^{\mathrm{left},\circ}_{m,K}$$ such that:

• (Generic loci.) Each of $U_m^{\mathrm{right}}$ and $U_m^{\mathrm{left}}$ is relatively Zariski open and relatively Zariski dense in the corresponding complex parameter space $\Theta^{\mathrm{right}}_{m,K}$ or $\Theta^{\mathrm{left}}_{m,K}$ of Definition P-3, and is contained in the corresponding generic parameter locus of Definition P-7.
• (Open neighborhoods meeting feasibility.) There are Euclidean-open sets $O^{\mathrm{right}},O^{\mathrm{left}}$ of real parameter space such that $$O^{\mathrm{right}}\cap F^{\mathrm{right}}_{m,K}\neq\varnothing, \qquad O^{\mathrm{left}}\cap F^{\mathrm{left}}_{m,K}\neq\varnothing,$$ where the feasible regions are those of Definition P-11; use the coordinatewise-complexification convention of Definition. Then $$\left(O^{\mathrm{right}}\cap F^{\mathrm{right}}_{m,K}\right)_{\mathbb C} \subseteq U_m^{\mathrm{right}}, \qquad \left(O^{\mathrm{left}}\cap F^{\mathrm{left}}_{m,K}\right)_{\mathbb C} \subseteq U_m^{\mathrm{left}}.$$
• (Forward recovery and separation.) For every $\theta\in U_m^{\mathrm{right}}$, $$R^{\mathrm{left}}_{m,K}\!\left(\Phi^{\mathrm{right}}_{m,K}(\theta)\right)=\varnothing.$$ Moreover, there is a nonzero squarefree polynomial $Q\in\mathbb C[z]$ such that $Q(0)\neq0$ and the multiset of roots of $Q$ is $\{\gamma,\rho_1,\ldots,\rho_m\}$. Every generic forward parameter $\theta''$ satisfying $$\Phi^{\mathrm{right}}_{m,K}(\theta'')= \Phi^{\mathrm{right}}_{m,K}(\theta)$$ has the same multiset $\{\gamma,\rho_1,\ldots,\rho_m\}$. If $$f_r(x,y)=\sum_{a=0}^{r}\binom{r}{a}t_{r,a}x^{r-a}y^a, \qquad t=\Phi^{\mathrm{right}}_{m,K}(\theta),$$ let $D$ be the forward loading-direction support. Then $$Q_D\neq0,\qquad x\mid Q_D,\qquad y\nmid Q_D,$$ and, for every homogeneous binary form $q$ of degree $n$, $$\bigl[q(\partial)f_{n+k}=0\ \text{for every }0\leq k\leq m\bigr] \quad\Longleftrightarrow\quad \exists c\in\mathbb C:\ q=cQ_D.$$
• (Reverse recovery and separation.) For every $\eta\in U_m^{\mathrm{left}}$, $$R^{\mathrm{right}}_{m,K}\!\left(\Phi^{\mathrm{left}}_{m,K}(\eta)\right)=\varnothing.$$ Moreover, there is a nonzero squarefree polynomial $Q\in\mathbb C[z]$ such that $Q(0)\neq0$ and the multiset of roots of $Q$ is $\{\delta,\sigma_1,\ldots,\sigma_m\}$. Every generic reverse parameter $\eta''$ satisfying $$\Phi^{\mathrm{left}}_{m,K}(\eta'')= \Phi^{\mathrm{left}}_{m,K}(\eta)$$ has the same multiset $\{\delta,\sigma_1,\ldots,\sigma_m\}$. If $$f_r(x,y)=\sum_{a=0}^{r}\binom{r}{a}t_{r,a}x^{r-a}y^a, \qquad t=\Phi^{\mathrm{left}}_{m,K}(\eta),$$ let $D$ be the reverse loading-direction support. Then $$Q_D\neq0,\qquad y\mid Q_D,\qquad x\nmid Q_D,$$ and, for every homogeneous binary form $q$ of degree $n$, $$\bigl[q(\partial)f_{n+k}=0\ \text{for every }0\leq k\leq m\bigr] \quad\Longleftrightarrow\quad \exists c\in\mathbb C:\ q=cQ_D.$$

For every $m\in\mathbb N$ with $m\ge 1$, set $K=2m+2$. There exist complex loci $U^{\mathrm{right}}_m\subseteq \Theta^{\mathrm{right},\circ}_{m,K}$ and $U^{\mathrm{left}}_m\subseteq \Theta^{\mathrm{left},\circ}_{m,K}$, with the generic loci as in Definition P-7, such that:

• (Relative genericity.) Each of $U^{\mathrm{right}}_m$ and $U^{\mathrm{left}}_m$ is relatively Zariski open and relatively Zariski dense in the parameter spaces $\Theta^{\mathrm{right}}_{m,K}$ and $\Theta^{\mathrm{left}}_{m,K}$ of Definition P-4 and Definition P-5, respectively.
• (Real incidence.) There is a Euclidean-open set of real forward parameters whose intersection with $F^{\mathrm{right}}_{m,K}$ from Definition P-11 is nonempty, and every parameter in this intersection has coordinatewise complexification contained in $U^{\mathrm{right}}_m$. Likewise, there is a Euclidean-open set of real reverse parameters whose intersection with $F^{\mathrm{left}}_{m,K}$ is nonempty, and every parameter in this intersection has coordinatewise complexification contained in $U^{\mathrm{left}}_m$.
• (Forward generic fibers.) For every $\theta=(\gamma,\rho,c)\in U^{\mathrm{right}}_m$, every forward parameter in the forward fiber $$R^{\mathrm{right}}_{m,K}\!\left(\Phi^{\mathrm{right}}_{m,K}(\theta)\right)$$ of Definition P-8 and Definition P-4 has the same unordered loading-slope multiset $$\{\gamma,\rho_1,\ldots,\rho_m\}$$ as $\theta$. The opposite-arrow fiber $$R^{\mathrm{left}}_{m,K}\!\left(\Phi^{\mathrm{right}}_{m,K}(\theta)\right)$$ of Definition P-8 (formed with the reverse map of Definition P-5) is empty. For every latent index $i=1,\ldots,m$, there is a generic forward parameter $\theta'\in\Theta^{\mathrm{right},\circ}_{m,K}$ obtained by swapping the full pairs $(\gamma,(c_{0r})_{r=2}^K)$ and $(\rho_i,(c_{ir})_{r=2}^K)$, leaving every other coordinate fixed, such that $$\Phi^{\mathrm{right}}_{m,K}(\theta') = \Phi^{\mathrm{right}}_{m,K}(\theta),$$ but $\theta'$ is not in the admissible $G_m$-orbit of $\theta$ from Definition P-9. If $m\ge 2$, this forward fiber has exact relative Zariski dimension $$\frac{m(m-1)}{2}.$$
• (Reverse generic fibers.) For every $\eta=(\delta,\sigma,d)\in U^{\mathrm{left}}_m$, every reverse parameter in the reverse fiber $$R^{\mathrm{left}}_{m,K}\!\left(\Phi^{\mathrm{left}}_{m,K}(\eta)\right)$$ has the same unordered loading-slope multiset $$\{\delta,\sigma_1,\ldots,\sigma_m\}$$ as $\eta$. The opposite-arrow fiber $$R^{\mathrm{right}}_{m,K}\!\left(\Phi^{\mathrm{left}}_{m,K}(\eta)\right)$$ is empty. For every latent index $i=1,\ldots,m$, there is a generic reverse parameter $\eta'\in\Theta^{\mathrm{left},\circ}_{m,K}$ obtained by swapping the full pairs $(\delta,(d_{m+1,r})_{r=2}^K)$ and $(\sigma_i,(d_{ir})_{r=2}^K)$, leaving every other coordinate fixed, such that $$\Phi^{\mathrm{left}}_{m,K}(\eta') = \Phi^{\mathrm{left}}_{m,K}(\eta),$$ but $\eta'$ is not in the admissible $G_m$-orbit of $\eta$ from Definition P-9. If $m\ge 2$, this reverse fiber has exact relative Zariski dimension $$\frac{m(m-1)}{2}.$$

For every $m\geq 1$, at $K=2m+2$, let $\overline E_m$, $H^{\mathrm{right}}_m$, and $H^{\mathrm{left}}_m$ be as in Definition P-10, and let $C^{\mathrm{right}}_{m,K}$ and $C^{\mathrm{left}}_{m,K}$ be as in Definition P-6. Then:

• (Codimension.) The exceptional closure $\overline E_m$ has codimension exactly $1$ in each of $C^{\mathrm{right}}_{m,K}$ and $C^{\mathrm{left}}_{m,K}$. Moreover, if $m\geq 2$, then $\overline E_m$ does not have codimension $m$ in either $C^{\mathrm{right}}_{m,K}$ or $C^{\mathrm{left}}_{m,K}$.
• (Forward preimage.) For every $\theta\in\Theta^{\mathrm{right}}_{m,K}$, $$\theta\in H^{\mathrm{right}}_m \quad\Longleftrightarrow\quad \theta\in\Theta^{\mathrm{right},\circ}_{m,K} \ \text{and}\ R^{\mathrm{left}}_{m,K}\!\left(\Phi^{\mathrm{right}}_{m,K}(\theta)\right)\neq\varnothing .$$
• (Reverse preimage.) For every $\eta\in\Theta^{\mathrm{left}}_{m,K}$, $$\eta\in H^{\mathrm{left}}_m \quad\Longleftrightarrow\quad \eta\in\Theta^{\mathrm{left},\circ}_{m,K} \ \text{and}\ R^{\mathrm{right}}_{m,K}\!\left(\Phi^{\mathrm{left}}_{m,K}(\eta)\right)\neq\varnothing .$$
• (Low-dimensional specializations.) If $m=1$, then $$E_1 = \left\{ t:\ \exists p\text{ satisfying the system }A_1,\ \Phi^{\mathrm{right}}_{1,4}(p^{\mathrm{right}})=t \right\}.$$ If $m=2$, then $$E_2 = \left\{ t:\ \exists p\text{ satisfying the system }A_2,\ \Phi^{\mathrm{right}}_{2,6}(p^{\mathrm{right}})=t \right\},$$ where $A_1,A_2$ are the compatibility instances of Definition P-16.

The theorem-local symbols $E^{\mathrm{right}}$ and $E^{\mathrm{left}}$ below are real parameter-space exceptional subsets and are unrelated to the cumulant-space compatibility locus $E_m$. For every $m\in\mathbb N$ with $3\le m$, at truncation order $L=2m+1$:

• (Forward generic separation.) There is a proper real algebraic subset of the real forward parameter space whose complement meets $F^{\mathrm{right}}_{m,L}$; choose one such subset and denote it locally in this theorem by $E^{\mathrm{right}}$. For every $\theta\in F^{\mathrm{right}}_{m,L}\setminus E^{\mathrm{right}}$ there is no $\eta\in F^{\mathrm{left}}_{m,L}$ such that $$\Phi^{\mathrm{right}}_{m,L}(\theta)=\Phi^{\mathrm{left}}_{m,L}(\eta).$$
• (Reverse generic separation.) There is a proper real algebraic subset of the real reverse parameter space whose complement meets $F^{\mathrm{left}}_{m,L}$; choose one such subset and denote it locally in this theorem by $E^{\mathrm{left}}$. For every $\eta\in F^{\mathrm{left}}_{m,L}\setminus E^{\mathrm{left}}$ there is no $\theta\in F^{\mathrm{right}}_{m,L}$ such that $$\Phi^{\mathrm{left}}_{m,L}(\eta)=\Phi^{\mathrm{right}}_{m,L}(\theta).$$

Consequently, for the information order $K^\star(m)$ of Definition P-12, $$K^\star(m)\le 2m+1 \qquad\text{and}\qquad K^\star(m)\ne 2m+2 .$$