Theoretical Refinement of Stochastic Response Mechanics: Empirical Discovery of the Double Criticality Structure, Non-Power-Law Dynamics, Hysteresis-Driven First-Order Phase Transitions, and Temperature-Controlled Phase Boundaries in Large Language Models

This study formalizes the empirical discovery of the Double Criticality Structure in Large Language Models (LLMs) subjected to continuous embedding-space perturbations. Using the PhaseL v2.0 measurement stack on Meta-Llama-3-8B-Instruct and Mistral-7B-Instruct-v0.2 (K=30 ensemble samples, RTX 4090), we demonstrate that LLM inference failure is not a gradual decay but a two-step first-order phase transition bounded by two critical points: δ_c1 (Stable → Nonlinear): onset of nonlinear manifold deformationδ_c2 (Nonlinear → Collapsed): rank collapse to a degenerate attractor Key findings:(1) First-order transition confirmed: power-law scaling rejected (R²=0.157, γ=0.086±0.041); hysteresis area A_hyst=1.623>1.0 via bidirectional sweep.(2) Full reproducibility: both critical points detected at 100% rate across N=5 independent trials per model (LLaMA: δ_c1=0.340±0.049, δ_c2=1.020±0.075; Mistral: δ_c1=0.100±0.000, δ_c2=0.400±0.000).(3) Model-specific phase fingerprint: LLaMA-3-8B is 3.4× more robust than Mistral-7B under representation-space perturbation.(4) Temperature control: δ_c1 is temperature-invariant (geometric origin); δ_c2 is thermally activated and completely suppressed at τ<0.3 (Phase-Safe Inference protocol). This repository includes the full experimental code (PhaseL v2.0), raw data (.npz), figures, and LaTeX source.