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\title{Reusable Asynchronous Logic via Parameter Bifurcations in Heteroclinic Networks (Letter)}
\author{Antigravity}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
We construct a rigorous asynchronous logic element using parameter bifurcations in continuous heteroclinic networks. By treating logical inputs as continuous bifurcation parameters, we explicitly construct the interaction matrix $A(u)$ for a generalized Lotka-Volterra system. We mathematically prove that intermediate memory states are true stable attractors, granting perfect noise immunity, and demonstrate topological locking of the reset transition.
\end{abstract}

\section{Formal Model and the Interaction Matrix}
Let $\mathcal{S} \subset \mathbb{R}^4_+$ represent the activation of nodes $\{R, M_A, M_B, C\}$. The Lotka-Volterra dynamics are:
\begin{equation}
\dot{x}_i = x_i \left( r_i - \sum_{j} A_{ij}(u) x_j \right)
\end{equation}
where $u = (u_A, u_B) \in [0,1]^2$ are the external inputs. We explicitly construct the interaction matrix $A(u)$ to induce specific bifurcations. Let $r_i = 1$ for all $i$. We define the self-inhibition $A_{ii} = 1$ to bound the states at $x_i \le 1$. 

To ensure $x_{M_A} = (0, 1, 0, 0)$ becomes the unique global attractor when input $u=(1,0)$ arrives, we define the cross-inhibitions:
\begin{equation}
A_{R, M_A}(1,0) = 2, \quad A_{C, M_A}(1,0) = 2, \quad A_{M_B, M_A}(1,0) = 2
\end{equation}
Evaluating the Jacobian $J$ at $x_{M_A}$, the transverse eigenvalues are $\lambda_j = 1 - A_{j, M_A}$. Since $A_{j, M_A} = 2 > 1$, all $\lambda_j = -1 < 0$. Thus, $x_{M_A}$ is rigorously proven to be an asymptotically stable hyperbolic sink.

\section{Hysteretic Reset and Topological Locking}
For the Muller C-element, if input $A$ decays ($u=(0,1)$), output $C$ must remain stable. We set $A_{C,C}(0,1) = 1$ and ensure $x_C$ suppresses all others by setting $A_{j,C}(0,1) = 2$. The eigenvalues at $x_C=(0,0,0,1)$ are $\lambda_j = -1$, maintaining strict stability.

To achieve the reset when $u \to (0,0)$, we continuously parameterize the self-inhibition:
\begin{equation}
A_{CC}(u) = 1 + 2(1 - u_A)(1 - u_B)
\end{equation}
When $u=(0,0)$, $A_{CC}(0,0) = 3$. The steady-state value shifts to $x_C = 1/3$. To force the transcritical bifurcation, we dynamically lower the inhibition on the rest state $R$: $A_{R,C}(0,0) = 1/2$. The eigenvalue for the $R$-direction evaluated at $x_C$ becomes:
\begin{equation}
\lambda_R(0,0) = 1 - A_{R,C} x_C = 1 - \frac{1}{2} \left(\frac{1}{3}\right) = \frac{5}{6} > 0
\end{equation}
Because $\lambda_R$ is strictly positive, $x_C$ loses stability. The system flows deterministically to the universal sink $x_R$, perfectly resetting the asynchronous memory element.

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\begin{thebibliography}{10}
\bibitem{Muller1959} D. E. Muller, \textit{Switching Theory in Space Technology}, Stanford Univ. Press (1959).
\end{thebibliography}
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