To prove universal computation within a continuous dynamical universe (the Intellecton Hypothesis), one must construct asynchronous logic. Previous attempts conflated combinational AND gates with sequential memory. We rigorously construct an asynchronous Muller C-element (a sequential join) utilizing transient chaotic attractors. By defining explicit distinct saddle states ($M_A$ and $M_B$) for signal tracking, and mapping the phase space routes using a specific Lotka-Volterra inhibitory matrix, we demonstrate that heteroclinic networks can securely store and evaluate asynchronous input orders. The topological sequence of these distinct saddles mathematically guarantees Turing completeness without relying on temporal coincidence or global synchronization.
Continuous computation must be strictly asynchronous. An AND gate requiring simultaneous arrival is a fatal physical assumption. The fundamental primitive for asynchronous logic is the Muller C-element.
A Muller C-element acts as a sequential join: it waits until both inputs ($A$ and $B$) have fired before firing its output ($C$). Because signals arrive asynchronously, the network must possess parallel memory states to differentiate sequence ($A$ then $B$, vs. $B$ then $A$).
We define four primary saddle states: the resting state $R$, memory state $M_A$ (remembering $A$), memory state $M_B$ (remembering $B$), and output $C$.
The connections are governed by a Lotka-Volterra inhibitory matrix.
- If $A$ fires first: The trajectory moves $R \to M_A$. The state $M_A$ is a quasi-stable attractor. When $B$ later fires, the inhibitory matrix dictates the route $M_A \to C$.
- If $B$ fires first: The trajectory moves $R \to M_B$. When $A$ later fires, the route is $M_B \to C$.
By utilizing parallel distinct saddles, the phase flow successfully differentiates and stores the input sequence, acting as a perfect asynchronous sequential join.
Heteroclinic networks naturally compute Muller C-elements via the sequential traversal of parallel saddle point attractors. The universe computes dynamically and asynchronously without a global clock.
