Inference Is a Controlled Dynamical System

It Doesn't Need a Second Brain. It Needs a Gyroscope.

There is a prevailing "architectural incompleteness" narrative in AI today: the belief that the fundamental flaws of LLMs (hallucinations, overconfidence, long-horizon collapse) stem from the lack of a "System-2" module for slow deliberation. This view calls for architectural reconstruction.

We agree with the diagnostic (LLMs suffer from uncontrolled inference dynamics), but we reject the prescription (we don't need a second brain).

In RN-003, we proved that inference can be controlled (Controllability). In RN-004, we prove that inference can be controlled continuously, in real-time (Execution).

We do not modify parameters. We do not change architectures. We do not retrain. We simply inject a tiny perturbation driven by a PD controller into the residual stream—and that is enough to stabilize the trajectory and break death loops. The missing ingredient isn't a second brain; it's a gyroscope attached to the existing one.

Dynamic Operator Mixing (DOM)

Autoregressive generation is the dynamical evolution of hidden states. We introduce a residual stream control injection:

H_{t+1} = f(H_t, x_t) + u_t

The control input u_t mixes a stabilization operator (to counter divergence) and an exploration operator (to counter rigid collapse). The mixing ratio is driven by real-time trajectory coherence (STR).

The physical intuition:

• When the system gets stuck in a loop (STR spikes), the controller injects exploration.
• When the system diverges into nonsense (STR collapses), the controller forces stabilization.

The Experimental Matrix: A Unified Failure Theory

Through three core experiments, we prove that various cognitive defects of LLMs are actually different manifestations of Trajectory Instability.

1. Trajectory Drift & Long-Horizon Collapse

In long-chain reasoning (e.g., ≥50 steps), standard models eventually collapse, accompanied by violent STR spikes. Static control also fails due to accumulated error. With closed-loop DOM, STR remains bounded and accuracy stays stable.

Conclusion: Uncontrolled inference accumulates instability over time. Control is a necessity for stability.

2. Stability Without Friction

The optimal control isn't constant intervention. We implemented an Adaptive Band (Deadzone) Controller. The system evolves freely within a safety band (~80% of the time, control friction is 0), with sparse corrections only upon boundary crossing. This crushed the collapse rate from 52% down to 20%.

Conclusion: Proper control means "defending the topological boundary," not micromanaging every step.

3. The Ultimate Decoupling: Dynamics vs. Semantics

We injected severe structural conflict noise at the end of a generation chain. The baseline model collapsed and transitioned to a new state. The DOM-equipped model perfectly recovered its STR stability.

But when we looked at the semantic output: the model had completely switched from solving math to writing Python code.

Conclusion: You can force it to stay conscious, but you cannot force what it thinks about.

Control guarantees dynamical stability, but it does not guarantee the preservation of semantic intent. Stability and semantics are decoupled. We precisely characterize what control cannot do.

The Deeper Implication

RN-004 is not just a new algorithm; it's a paradigm shift: Inference is a controlled dynamical system.

The root of cognitive failure is dynamical loss of control. The solution is closed-loop feedback.

This note draws on theoretical results from "Continuous Control of Inference Dynamics" (Haelio Tang, 2026). Full experimental protocol, phase diagrams, and source code available on request.