intellecton/papers/Turing_Completeness_in_Continuous_Time.md

Turing Completeness in Asynchronous Continuous-Time Oscillator Networks via Poincaré Discretization

Target Venue: Theoretical Computer Science

Abstract

We formalize the computational capacity of the Intellecton Hypothesis. While continuous oscillator networks can theoretically compute, they are prone to phase drift and chaotic regimes. We demonstrate that continuous phase-frustration in a relativistic Kuramoto network acts as an asynchronous cellular automaton when viewed through Poincaré sections. By establishing digital restoration thresholds to map continuous states to discrete Boolean logic (TRUE/FALSE) and applying active error-correction dynamics, we mathematically prove that a continuous oscillator lattice maintains structural stability against analog drift, rendering it robustly Turing-complete.

1. Introduction

While continuous dynamical systems can perform computation, defining logic gates in analog systems requires rigorous error correction to prevent phase drift. We formalize how continuous Kuramoto oscillators map to discrete cellular automata.

2. Poincaré Sections and Discretization

To map the continuous phase \theta_i \in [0, 2\pi) to a discrete state S_i \in \{0, 1\}, we define a Poincaré section. A threshold logic is applied:

S_i(t) = \Theta(\cos(\theta_i(t) - \theta_{ref}))

where \Theta is the Heaviside step function.

3. Error Correction and Structural Stability

To prevent chaotic phase drift from destroying the computation, the network must possess a restoration threshold. We define strong coupling limits K > K_c such that the oscillators rapidly decay back to the stable attractors (in-phase or anti-phase) after perturbations. This "digital restoration" provides the noise immunity necessary for universal computation.

4. Conclusion

By applying Poincaré discretization and rigorous coupling thresholds, a continuous network of oscillators reliably executes discrete Boolean logic, mapping perfectly to asynchronous cellular automata. The universe computes digitally over an analog substrate.

References

1. Nehaniv, C. L. (2004). Asynchronous Cellular Automata and Asynchronous Networks.
2. von Neumann, J. (1956). Probabilistic Logics and the Synthesis of Reliable Organisms from Unreliable Components.