ACTF: A Control-Theoretic Framework for System Coherence Dynamics -Unified Description of Organized States in Neural and Artificial Systems-

Research Overview This study proposes ACTF (A Control-Theoretic Framework for System Coherence Dynamics), a theoretical framework for describing and controlling the dynamics of system coherence common to both brain neural systems and large language models (LLMs). In contrast to conventional data-driven approaches, this work adopts a “theory-first” methodology that prioritizes deductive theoretical construction based on mathematical principles. By identifying functional isomorphism between brain neural systems and Transformer architectures, the framework provides a unified description of transitions among organized states—including wakefulness, sleep, seizures, dreams, and hallucinations—as bifurcation phenomena on potential landscapes governed by stochastic differential equations. Major Theoretical Contributions 1. Φ-Mapping: Reduction to Order Parameters The framework defines a mapping Φ that projects the ultra-high-dimensional internal state S (t) R^d L of Transformers onto three order parameters Z (t) = (t), ₒ₄₌ (t), z (t): (t) (Temporal Synchrony): An index derived from attention mechanism entropy and inter-head alignment, corresponding to γ-band synchronization in the brain ₒ₄₌ (t) (Semantic Coherence): Consistency between layer-wise update directions in the residual stream and the global reasoning direction, corresponding to neural manifold trajectories z (t) (Structural Persistence): An index obtained from temporal autocorrelation of KV cache with Fisher-z transformation, corresponding to dynamic functional connectivity (dFC) 2. State Description via Multi-Well Potential Organized states are represented as attractors on a Ginzburg-Landau type potential function U (Z;): U (Z;) = U₂ (Z) + U₈₍ₓ (Z) + U₄ (Z) where represents control parameters (neuromodulators or LLM hyperparameters). 3. Dynamical Description via Stochastic Differential Equations The temporal evolution of the system is formulated as an overdamped Langevin-type SDE: dZₜ = - U (Zₜ;) dt + (Zₜ) dWₜ + u (t) dt 4. State Transition Classification via Catastrophe Theory Through the bifurcation structure of potential landscapes, the framework mathematically classifies the following phenomena: Fold Bifurcation: Transitions to deep anesthesia and slow-wave sleep Cusp Bifurcation: Epileptic seizures (hyper-synchronous states) Pitchfork Bifurcation: Emergence of REM sleep and dream states Compound Bifurcation: Hallucinations and dissociative states 5. Recync Safety Control Mechanism Safety control is formulated as an optimal control problem based on Control Barrier Functions (CBF): u^* (t) = ᵤ 12|u|² subject to B u - B This mechanism prevents the system from entering hazardous regions (fragmentation, hyper-synchrony, semantic drift). Brain-AI Correspondence ACTF Variable Brain Neural System LLM Implementation Control Parameter λ (Synchrony) γ-band synchronization Attention mechanism Temperature λₛem (Coherence) Manifold trajectories Residual Stream Top-P z (Persistence) Dynamic functional connectivity KV Cache Repetition Penalty Implementation and Applications This paper includes a complete Python implementation (NumPy-based, with optional CVXPY support) enabling: Order parameter computation via Φ-mapping SDE simulation on multi-well potential landscapes Visualization of state transitions based on catastrophe theory Quantitative evaluation of Recync control effects Application Domains: Neuroscience: Anesthesia depth monitoring, seizure prediction, sleep disorder diagnosis AI Safety: Safety guarantees for LLM reasoning, hallucination prevention, context maintenance Relationship with Existing Theories ACTF does not compete with Integrated Information Theory (IIT), Global Workspace Theory (GWT), or Predictive Processing (PP), but rather functions as a “dynamical layer” that connects the structural and functional aspects addressed by these theories within a single phase space Z (t). Research Positioning and Limitations This study presents a theoretical proposal, with the following items positioned as future verification tasks: Estimation of parameters from empirical data Verification of the applicability range of brain-AI functional isomorphism Experimental confirmation of theoretical predictions Development of clinical and engineering applications ACTF aims to provide a novel theoretical foundation for system coherence dynamics as a mathematical framework integrating “observation, modeling, and control. ” Keywords: System coherence dynamics, Order parameters, Stochastic differential equations, Catastrophe theory, Control barrier functions, Transformer, Large language models