Note that $\rpoly_{G}^3(\prob,\ldots, \prob)$ as a polynomial in $\prob$ has degree at most six. Next, we figure out the exact coefficients since this would be useful in our arguments:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Lemma}\label{lem:qE3-exp}
%When we expand $\poly_{G}^3(\vct{X})$ out and assign all exponents $e \geq 1$ a value of $1$, we have the following result,
Hence $\rpoly_{G}^3(\vct{X})$ has degree six. Note that the monomial $\prod_{\ell=1}^{3}X_{i_\ell}X_{j_\ell}$ will contribute to the coefficient of $\prob^\nu$ in $\rpoly_{G}^3(\vct{X})$, where $\nu$ is the number of distinct variables in the monomial.
%Rather than list all the expressions in full detail, let us make some observations regarding the sum.
\textsc{case 1:}$e_1= e_2= e_3$ (all edges are the same). When we have that $e_1= e_2= e_3$, then the monomial corresponds to $\numocc{G}{\ed}$. There are exactly $\numedge$ such triples, each with a $\prob^2$ factor in $\rpoly_{G}^3\left(\prob,\ldots, \prob\right)$.
\textsc{case 2:} This case occurs when there are two distinct edges of the three, call them $e$ and $e'$. When there are two distinct edges, there is then the occurence when $2$ variables in the triple $(e_1, e_2, e_3)$ are bound to $e$. There are three combinations for this occurrence in $\poly_{G}^3(\vct{X})$. Analogusly, there are three such occurrences in $\poly_{G}^3(\vct{X})$ when there is only one occurrence of $e$, i.e. $2$ of the variables in $(e_1, e_2, e_3)$ are $e'$. %Again, there are three combinations for this.
This implies that all $3+3=6$ combinations of two distinct edges $e$ and $e'$ contribute to the same monomial in $\rpoly_{G}^3$. % consist of the same monomial in $\rpoly$, i.e. $(e_1, e_1, e_2)$ is the same as $(e_2, e_1, e_2)$.
Since $e\ne e'$, this case produces the following edge patterns: $\twopath, \twodis$, which contribute $6\prob^3$ and $6\prob^4$ respectively to $\rpoly_{G}^3\left(\prob,\ldots, \prob\right)$.
\textsc{case 3:} All $e_1,e_2$ and $e_3$ are distinct. For this case, we have $3!=6$ permutations of $(e_1, e_2, e_3)$, each of which contribute to the same monomial in the \textsc{SMB} representation of $\poly_{G}^3(\vct{X})$. This case consists of the following edge patterns: $\tri, \oneint, \threepath, \twopathdis, \threedis$, which contribute $6\prob^3, 6\prob^4, 6\prob^4, 6\prob^5$ and $6\prob^6$ respectively to $\rpoly_{G}^3\left(\prob,\ldots, \prob\right)$.
\qed
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Since $\prob$ is fixed, \Cref{lem:qE3-exp} gives us one linear equation in $\numocc{G}{\tri}$ and $\numocc{G}{\threedis}$ (we can handle the other counts due to equations (\ref{eq:1e})-(\ref{eq:3p-3tri})). However, we need to generate one more independent linear equation in these two variables. Towards this end we generate another graph related to $G$:
For $\ell\geq1$, let graph $\graph{\ell}$ be a graph generated from an arbitrary graph $G$, by replacing every edge $e$ of $G$ with a $\ell$-path, such that all inner vertexes of an $\ell$-path replacement edge are disjoint from all other vertexes.\footnote{Note that $G\equiv\graph{1}$.}% of any other $\ell$-path replacement edge. % in the sense that they only intersect at the original intersection endpoints as seen in $\graph{1}$.
We will prove \Cref{th:single-p-hard} by the following reduction:
\begin{Theorem}\label{th:single-p}
Fix $\prob\in(0,1)$. Let $G$ be a graph on $\numedge$ edges.
If we can compute $\rpoly_{G}^3(\prob,\dots,\prob)$ exactly in $T(\numedge)$ time, then we can exactly compute $\numocc{G}{\tri}$%count the number of triangles, 3-paths, and 3-matchings in $G$
in $O\inparen{T(\numedge)+\numedge}$ time.
\end{Theorem}
For clarity, we repeat the notion of $\numocc{G}{H}$ to mean the count of subgraphs in $G$ isomorphic to $H$.
The following lemmas relate these counts in $\graph{2}$ to $\graph{1}$ ($G$). The lemmas are used to prove \Cref{lem:lin-sys}.
Finally, the following result immediately implies \Cref{th:single-p}:
\begin{Lemma}\label{lem:lin-sys}
Fix $\prob\in(0,1)$. Given $\rpoly_{\graph{\ell}}^3(\prob,\dots,\prob)$ for $\ell\in[2]$, we can compute in $O(m)$ time a vector $\vct{b}\in\mathbb{R}^3$ such that
\[\begin{pmatrix}
1 - 3p & -(3\prob^2 - \prob^3)\\
10(3\prob^2 - \prob^3) & 10(3\prob^2 - \prob^3)
\end{pmatrix}
\cdot
\begin{pmatrix}
\numocc{G}{\tri}]\\
\numocc{G}{\threedis}
\end{pmatrix}
=\vct{b},
\]
allowing us to compute $\numocc{G}{\tri}$ and $\numocc{G}{\threedis}$ in $O(1)$ time.
We can compute $\graph{2}$ from $\graph{1}$ in $O(m)$ time. Additionally, if in time $O(T(m))$, we have $\rpoly_{\graph{\ell}}^3(\prob,\dots,\prob)$ for $\ell\in[2]$, then the theorem follows by \Cref{lem:lin-sys}.
\qed
\end{proof}
In other words, if \Cref{th:single-p} holds, then so must \Cref{th:single-p-hard}.