Operational Structure of Entanglement Time: Numerical Evidence for Quadratic Scaling with Modular Flow

We provide rigorous numerical verification of the modular time definition through Tomita–Takesaki theory and establish a scaling relation between entanglement time τₑnt = dS/dt and modular time τₘod = √Var (K_ρ) in Heisenberg XXX spin chains. Extensive simulations for system sizes N = 6–20 and systematic robustness checks reveal that τₑnt ∝ τₘod^γ with exponent γ∞ = 2. 07 ± 0. 04 for N ≥ 14, consistent with diffusion-like growth of entanglement in modular time. We verify that the modular flow satisfies all four axioms of a Tomita–Takesaki automorphism group to numerical precision ~10⁻¹⁵. Furthermore, we demonstrate that emergent time is intensive (non-additive): τ (2N) / (2τ (N) ) = 0. 76 ± 0. 17 ≠ 1. These results provide strong numerical evidence that different notions of time in quantum many-body systems are related by a scale-invariant law, with potential implications for emergent gravity.