Analytical Relation between Energy and Information Thresholds in the T₀→T₁ Transition of Quantum Darwinism: Two Exact Projections of a Single Structural Variable

This work investigates the emergence of structural redundancy... This work investigates the emergence of structural redundancy and g-invariant behavior in finite quantum environments. Building on previous results, we analyze the robustness of the critical threshold Dc under variations of coupling strength and basis rotation. We demonstrate that apparent breakdowns of g-invariance at intermediate rotation angles are finite-size effects, which are resolved as the environment size increases. The results support the existence of a stable structural regime associated with pointer basis selection and reinforce the interpretation of redundancy as an emergent property of the system-environment interaction. This study provides a consistent numerical framework for identifying structural thresholds and contributes to the understanding of information proliferation mechanisms in quantum systems.