Structural Robustness of the T0→T1 Transition: Dimensional Truncation and Coupling Basis

The structural robustness of the informational and energetic thresholds associated with the T0→T1 transition in the Spin-Boson model is investigated along two independent axes. (i) The local environmental Hilbert space is extended from d = 2 to d = 3. (ii) The system–environment coupling is continuously rotated from pure dephasing (σz) toward bit-flip (σx) via a rotation angle φ. All results are obtained via exact numerical simulation. The decoherence threshold Dc(N) is robust under dimensional truncation: shifts below 4% for N = 5 and below 1.5% for N = 6, with g-invariance preserved. The energetic threshold Ec(N) acquires a systematic upward correction of 6–12% for N = 5 and 7–11% for N = 6 at d = 3, while g-invariance is maintained. Under coupling basis rotation, g-invariance of Dc breaks down for N = 5 above φ ≈ 56°, but is restored for N = 6 up to φ = 60° before collapsing at φ = 75°. Simulations for N = 7 show the same regime pattern as N = 6 at both tested angles, consistent with saturation of φc ≈ 65–70° at N ≥ 6. These results indicate that α*(N) remains the structural reference for both thresholds: Dc stays robust under truncation, while Ec preserves the same structural trend but acquires a systematic upward correction at d = 3. The pointer basis associated with σz coupling is structurally privileged within this framework.