IFF_BEC_Prediction

Standard Hawking radiation is exactly thermal: the two-mode frequency cross-correlator ⟨ a†ωaω′⟩ vanishes for ω ≠ ω′. We show that retaining the vacuum entanglement term Ivac(BH:rad) — which is discarded by normal-ordering in standard quantum field theory — generates a non-vanishing off-diagonal correlator from t= 0. We prove that the jump operator Lvac ∝ a†ωaω′ is an eigenvector of the Hessian of the relaxation functional Γ[ρ] = D(ρ ‖ σ), with eigenvalueλ_vac(ω,ω') = (1 + n̄_ω)(1 + n̄_ω') · [exp(βω) + exp(βω')]where n̄k = [exp(βωk) − 1]⁻¹ is the Bose-Einstein occupation number and β = 2π/κ is the inverse Hawking temperature. This identifies Lvac as a well-defined Lindblad jump operator arising from the gradient-flow dynamics — a derivable consequence of vacuum entanglement, not an assumption. The proof holds exactly for any N-mode bosonic system at finite UV cutoff, which is precisely the physical regime of a Bose-Einstein condensate (BEC) analogue black hole. The resulting correlator takes the formG(ω, ω′; t) = (1 − t/t_H)^{2/3} · exp(−κt/2π) · √(n̄_ω n̄_ω') · (2π/κ) · sinc²(π(ω−ω′)/κ)with dephasing timescale τ₁ = 2π/κ and power-law amplitude (1 − t/tH)²/³. The prediction is qualitatively distinct from standard Hawking radiation (correlator identically zero), the island formula (discontinuous onset at the Page time), and loop quantum gravity (Planck-scale suppressed). It is testable in BEC analogue systems via the time-drift ratio R = G(2τ₁)/G(τ₁) = exp(−1) ≈ 0.368, which eliminates all time-stationary backgrounds by construction.