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 τ_ent = dS/dt and modular time τ_mod = √Var(K_ρ) in Heisenberg XXX spin chains. Extensive simulations for system sizes N = 6–20 and systematic robustness checks reveal that τ_ent ∝ τ_mod^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.