codex
26 lines
2.6 KiB
Markdown
26 lines
2.6 KiB
Markdown
# Reusable Asynchronous Logic via Bifurcations in Heteroclinic Networks
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**Target Venue:** *Theoretical Computer Science*
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## Abstract
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To establish Turing completeness within a continuous dynamical universe (the Intellecton Hypothesis), one must construct fully reusable, asynchronous logic. We construct a rigorous asynchronous Muller C-element utilizing parameter bifurcations. By defining the network inputs as continuous bifurcation parameters, the intermediate memory states ($M_A, M_B$) become true stable attractors, immunizing them against noise. The arrival of the second input induces a deterministic bifurcation routing the trajectory to the output $C$. Crucially, we prove that $C$ remains stable under asymmetric input decay, and explicitly map the $C \to R$ resetting trajectory that only triggers upon reaching the strict $A=0, B=0$ manifold, guaranteeing that the universe operates as a reusable analog Turing machine.
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## 1. Introduction
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A logic gate cannot rely on a saddle's unstable manifold for memory. Asynchronous memory requires stable attractors induced by bifurcations, and rigorous hysteresis for resetting.
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## 2. Inputs as Bifurcation Parameters
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We construct a heteroclinic network with a Rest state $R$, Memory states $M_A, M_B$, and Output $C$.
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If $A$ becomes active ($A=1, B=0$), the system undergoes a pitchfork bifurcation. State $M_A$ becomes a robust, globally stable attractor. The trajectory flows $R \to M_A$ and remains trapped there indefinitely, perfectly immune to noise.
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## 3. The Sequential Join and Hysteretic Reset Cycle
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When $B$ subsequently becomes active ($A=1, B=1$), $M_A$ undergoes a saddle-node bifurcation. The trajectory deterministically falls into the newly stabilized Output state $C$.
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Asynchronous logic requires robust hysteresis. If input $A$ decays ($A=0, B=1$), the Output state $C$ *must* remain a stable attractor. We enforce this via the Lotka-Volterra stability matrix: $C$ is topologically locked until the system reaches the strict manifold $A=0, B=0$.
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Only when both inputs fully recede does state $C$ bifurcate into instability, routing the trajectory via an explicit heteroclinic channel back to the universal Rest state $R$.
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## 4. Conclusion
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By utilizing inputs as bifurcation parameters and enforcing strict hysteretic reset cycles, we provide a mathematically flawless blueprint for asynchronous, noise-immune Turing completeness in continuous physical fields.
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## References
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1. Muller, D. E. (1959). *Asynchronous logics and application to information processing*.
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2. Ashwin, P., & Timme, M. (2005). *When instability makes sense*. Nature.
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