An Empirical Invariant for Transformer Scaling: Towards an Information Incompleteness Hypothesis

The phenomenon of performance plateaus in large language models (LLMs) despite increasing computational resources reveals a fundamental limit analogous to Gödel's incompleteness theorems and Heisenberg's uncertainty principle. We propose a quantitative framework for Information Incompleteness for Complex Systems (LIICS), establishing that any system attempting to model a whole of which it is a part faces an irreducible computational boundary.

Domain scope: This study focuses exclusively on Transformer-based large language models; generalization to other architectures requires empirical validation. Through meta-analysis of scaling laws data from GPT-3, Chinchilla, PaLM, and LLaMA, we derive a quantitative formulation: G_S(C) = Ψ·(N·D)/(L·E·H·V), where the dimensionless invariant G_S(C) → 1 indicates the efficiency limit.

We derive and validate an architecture-specific efficiency coefficient Ψ_LL M ≈ 1.27 × 10⁻¹¹ for Transformer architectures, exhibiting stability across multiple state-of-the-art models. Statistical significance is limited by sample size (n=3 compute-optimal models); this invariant provides a predictive tool for resource allocation and suggests a hypothesis connecting scaling limits to information-processing constraints.