Phase-Dependent Scaling and Topological Stability in Noisy Dicke Model Simulations: An N=100 Analysis

We report the systematic characterization of a first-order phase transition in collective intensity within open quantum systems. Using the Lindblad master equation formalism on NVIDIA A100 GPU hardware, we map the scaling behavior of the Dicke Model across noise amplitudes (gamma_phi) from 0.0001 to 100 for system sizes up to N=100. Our high-resolution simulations reveal three distinct topological regimes: Coherent Phase (alpha approx 0): A stable integrated core that demonstrates topological resilience, maintaining intensity levels even as system size increases. Transition Regime: A sharp first-order phase transition occurring between gamma_phi approx 5.0 and 10.0. Fragmented Phase: A functional collapse characterized by the 'Critical Scaling Gap'—a 1.91x reduction in steady-state intensity compared to the coherent phase at the N=100 limit. While earlier observations suggested a universal '-1.36 scaling law,' this $N=100$ study identifies that value as a transient scaling state. We demonstrate that both the coherent and fragmented phases eventually reach stable intensity plateaus, establishing the 1.91x Intensity Ratio as a quantitative signature of the phase boundary. This repository provides the full QuTiP simulation framework, raw data for the phase diagram (Run A vs. Run B), and the revised manuscript establishing these phase-dependent scaling plateaus as a foundational metric for quantum biology, anesthetic modeling, and AI consensus protocols. Note: The 1.91x number may vary slightly; however, the topological phase integration protocol remains valid. "Note on Scaling Constants: Earlier drafts and associated works in this research program may refer to a 'Universal -1.36 Scaling Law.' High-resolution N=100 simulations (this work, V3.0) have since identified that the -1.36 exponent represents a transient scaling state. The finalized metric for stable phase-dependent integration is defined herein as the 1.91x Critical Scaling Gap. Readers should treat all prior '-1.36' references as precursors to the 1.91x plateau-based framework."