When Your Own Output Becomes Your Training Data: Noise-to-Meaning Loops and a Formal RSI Trigger

TL;DR

The paper formalizes Noise-to-Meaning Recursive Self-Improvement (N2M-RSI), a minimal loop where an agent’s own outputs become inputs via a noise-to-meaning operator $\$ and drive the context $C$ to evolve. It proves two core results: (i) no non-trivial fixed point exists under injectivity and positive-entropy noise, and (ii) surpassing an information-integration threshold $\Gamma$ with a monotone update leads to unbounded growth, with explicit measures $\Omega$ (e.g., compression gain, Fisher information, $\Phi$) that trigger divergence. The framework unifies self-prompting, AutoML, and RSI theories, while detailing multi-agent amplification, threshold dynamics, and practical safety levers to prevent runaway behavior. It also discusses empirical validations, practical considerations like compute costs, and potential pathologies, offering structured avenues for both theoretical and safety-focused future work. Overall, N2M-RSI provides a rigorous lens for understanding and bounding strongly self-referential AI systems through formal thresholds and information-theoretic measures.

Abstract

We present Noise-to-Meaning Recursive Self-Improvement (N2M-RSI), a minimal formal model showing that once an AI agent feeds its own outputs back as inputs and crosses an explicit information-integration threshold, its internal complexity will grow without bound under our assumptions. The framework unifies earlier ideas on self-prompting large language models, Gödelian self-reference, and AutoML, yet remains implementation-agnostic. The model furthermore scales naturally to interacting swarms of agents, hinting at super-linear effects once communication among instances is permitted. For safety reasons, we omit system-specific implementation details and release only a brief, model-agnostic toy prototype in Appendix C.

Figures (4)

• Figure 1: Schematic loop; arrows denote information flow. Minimal N2M--RSI loop (see Eq. \ref{['eq:loop']}). Self-generated noise $N$ is passed to the noise-to-meaning operator $\Psi$, yielding meaning $M$ that updates context $C$, which in turn influences the next iteration of $\Psi$.
• Figure 2: Conceptual map situating N2M--RSI among adjacent research themes.
• Figure 3: Empirical self‑feedback loop on Llama‑3‑8B (10 iterations). Injective sampling ($T=1.0$) diverges linearly whereas deterministic sampling ($T=0$) plateaus after two iterations, corroborating Theorem \ref{['thm:diverge']} and Proposition \ref{['prop:deterministic']}.
• Figure 4: Phase portrait illustrating Theorem \ref{['thm:diverge']}: once $\|C\|>\Gamma$, every step adds a positive increment $\Delta$, forcing divergence.

Theorems & Definitions (66)

• Definition 1: Noise-to-Meaning Operator
• Definition 2: $\varepsilon$‑injectivity
• Lemma 1: Fixed‑point absence under $\varepsilon$‑injectivity
• proof : Sketch
• Lemma 2: Positive lower bound on $\Gamma$ for compression gain
• proof
• Definition 3: Recursive Loop
• Definition 4: $\delta$‑Monotone Update Operator
• Theorem 1: No Non-trivial Fixed Point
• proof