Structural Energy–Information Ordering at the T₀→T₁ Transition: Emergence of a Critical Energy Threshold Ec (N)

We present a structural framework for the emergence of classical reality from quantum systems based on a dimension-driven transition in redundancy. We derive an exact analytical condition for the critical redundancy threshold Dc (N), defined by a transcendental equation independent of the coupling strength. This establishes that the transition from non-redundant (T₀) to redundant (T₁) regimes is governed by system dimensionality rather than dynamical parameters. We introduce a critical environmental energy threshold Ec (N), numerically shown to be invariant under variations of the coupling strength (CV < 2%) for dephasing-type (σᵦ) interactions. This demonstrates that the onset of redundancy occurs only after a finite amount of energy has flowed into the environment. The results establish a structural ordering: energy flow precedes stable information redundancy. In the thermodynamic limit, Ec (N) converges toward Dc (N), suggesting a unified threshold governing both energy and information at large system size. We further show that this behavior is specific to dephasing interactions. Under σₓ coupling, the invariance of Ec (N) breaks down, indicating a fundamentally different dynamical structure. These findings provide a new perspective on the quantum-to-classical transition, identifying dimension and energy thresholds as the primary drivers of classical emergence.