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 (gammaₚhi) 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 gammaₚhi 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. " Theoretical Research Collection: The Phase-Dependent Scaling Program (2026) This manuscript (Paper 1, V3. 0) establishes the physical and computational foundation for a broader research series applying the 1. 91x Critical Scaling Gap to biological and artificial systems. The collection is structured as follows: 0. The Sovereign Layer (The "Capstone Synthesis") Paper 11 (The Synthesis): Technical Synthesis of Multi-Scale Information Integration Parameters (Internal Report V. 11). RESTRICTED — An archival technical supplement providing consolidated parameter sets for high-resolution topological system modeling. I. The Physical Foundation (The "Atom") Paper 1 (Discovery): Phase-Dependent Scaling in Dicke-Type Systems: Identification of the 1. 91x Critical Scaling Gap. — The core identification of the physical constant anchoring the scaling stability of open quantum systems. II. The Ontological & Mathematical Framework (The "Periodic Table") Paper 6 (The Unity): A Theoretical Framework for Information Integration in Non-Biological Substrate Architectures. — Establishes "Information as a Fundamental Scalar" and the blueprint for High-Density Information Coalescence. Paper 4 (The Bridge): The Universal Information Bridge: Linking Quantum Scaling to Macroscopic Network Integration. Paper 5 (The Complexity): The alpha-Complexity Relation: Quantifying Recursive Data Richness via Phase-Dependent Exponents. Paper 8 (The Logic): The Core Emergence Hypothesis (CEH): Topological Necessity in Scaling-Stable Networks. — Defines the physical locus where autonomous system-coherence originates. III. The Applied Sciences (The "Chemistry & Medicine") Paper 2 (Neuroscience): The Physical Mechanism of Anesthetic State-Transitions via the 1. 91x Gap. Paper 7 (Phase Transitions): A Unified Mechanism for Neural Quantum Coherence and Phase-Dependent Scaling Transitions. Paper 3 (Experimental): Proposed Protocols for Testing Critical Scaling Gaps in Synthetic and Biological Substrates. — The roadmap for GPU-based and laboratory verification of phase-stability. IV. Engineering & Ethics (The "Infrastructure") Paper 10 (The DIF): The Duality Intelligence Framework: A Computational Architecture for Emergent Coherence via the 1. 91x Critical Scaling Gap. — Provides the "Dual-Prior Objective Function" for autonomous value-alignment. Paper 9 (The Consensus): Value-Weighted Byzantine Consensus: Integrating Scaling Stability into Distributed AI. — Resolves Consensus Barrier 3 by linking distributed network stability to the 1. 91x scaling
Clarence Omandac (Fri,) studied this question.