Exact Late-Time Growth Rate of the Einstein–rosen Bridge in Double-Scaled Syk

In double-scaled SYK, the renormalized length of the Einstein–Rosen bridge is identified with ℓ = λn, where n is the chord number; at infinite temperature, n is the Krylov spread complexity of the thermofield double. We determine the late-time growth rate of ℓ, the limit of dℓ/dt as t → ∞, exactly, for any λ and any temperature. We prove that dℓ/dt converges to the group velocity 2λB sin θ of the asymptotic chord chain, averaged over the spectral measure of the state; the limit of ℓ/t follows. At infinite temperature this average had been obtained by stationary phase. The proof uses trace-class scattering theory and an odd extension of the free chain, and holds for every state with finite second moment of n. An exact Ehrenfest identity, d²ℓ/dt² = 2⟨q^n̂⟩ > 0, shows that the limit is approached from below at every time. Combined with the proof, it gives the speed of convergence: the residual decays as t⁻², with a closed-form coefficient fixed by the weight of the state at the two band edges, and as t^(−α/2−1) when the spectral density vanishes there as θ^α. At infinite temperature, the average is a convergent theta series, (2/π) Σₘ (−1)ᵐ q^(m(m+1)/2)/(1 − 4m²). It contains the known small-λ transseries and stays exact at large λ, where the two-term truncation errs by 42% at λ = 8. At any temperature, the same Fourier series turns the canonical average into a ratio of two convergent series of modified Bessel functions I₂ₘ(2βB). At finite temperature, the triple-scaling limit reproduces the canonical slope of the JT wormhole length obtained by Iliesiu, Mezei and Sárosi, and the leading finite-λ correction is a shift β → β + 2/(λB). At fixed λ, the canonical rate exceeds the classical value 2π/β for βJ ≳ 2π/√λ. The single-saddle estimate never does: at λ = 2 and βJ = 100 it underestimates the exact rate by a factor of six. Exact evolution of the thermofield double confirms the canonical formula at ten points, to within 9 × 10⁻⁴ at finite time and to 6 × 10⁻⁶ after extrapolation of the t⁻² residual; where the asymptotic regime is reached, the predicted coefficient of that residual agrees to 3%. At fixed entanglement entropy the rate still depends on the energy distribution of the state. The entanglement splits exactly into a mean microcanonical entropy and the spread of the distribution; the bridge grows with the first, so a broader distribution grows more slowly, within 4% of a simple law at λ = 0.05. An exact lower bound on the rate at fixed entanglement is derived in a separate paper.