EOS 1.0-3.0: Experimental Validation of Coherence Phase Transitions in Large Language Models

Version 1.2 (August 2026) — reference audit. Version 1.1 corrected one erroneous bibliographic entry and committed the author to resolving every remaining entry in the reference list against its source. That audit is complete. Of 25 remaining entries, 20 were confirmed as cited and 4 were found defective and are corrected here: one entry combined the author list of one paper with the title and identifier of another (corrected to Dohmatob, Feng, and Kempe, NeurIPS 2024); one carried a nonexistent title and cited a mouse-cortex study in support of a claim about artificial networks (replaced with Aitken, Garrett, Olsen, and Mihalas, PLoS Computational Biology 2022); one did not exist as cited and is removed (its claim is now supported by Liu Ziyin, arXiv:2309.16932, 2023, alongside the thesis already cited); one carried the wrong venue (corrected to Neural Computation 16(7), 2004). No claims, methods, data, figures, or conclusions are altered; in two cases the source supporting a sentence changes, and the sentence does not. All defects share the signature of language-model-generated references that were not independently resolved, which the author discloses. The accompanying Version Note and Erratum documents each correction. The reference-list audit committed to in Version 1.1 is closed with this release. Earlier versions remain available under their DOIs. Abstract. Large language models (LLMs) exhibit complex dynamics that resemble high-dimensional physical systems, yet no unified framework connects representation lifecycle stability to the thermodynamic formalisms that describe analogous physical phenomena. This paper introduces EOS (Emergent Order Systems), a three-part framework that formalizes how representation structure changes under perturbation. EOS 1.0 defines coherence as a bounded stability metric inversely related to representation drift. EOS 2.0 models energy injection—operationalized through gradient-driven fluctuations—and predicts how accumulating energy degrades coherence. EOS 3.0 extends this relationship to identify critical behavior, demonstrating that LLMs exhibit ordered, transitional, and disordered regimes analogous to second-order phase transitions in statistical physics. To evaluate the framework, we monitor the coherence evolution of a GPT-2–class model over a controlled training trajectory, measuring drift and coherence at each step. The results reveal smooth coherence decay in low-energy regimes and a reproducible catastrophic collapse at a critical threshold, consistent with the predictions of EOS 3.0. The observed coherence–energy trajectory parallels classical order-parameter behavior in systems described by Landau mean-field theory and the Ising model. By unifying representation drift, optimization dynamics, and critical phenomena into a single operational framework, EOS provides a practical means of assessing stability in neural networks and offers a foundation for a broader physics-based theory of learning-system longevity.