The black hole information paradox has challenged theoretical physics for 50 years, presenting an apparent conflict between quantum mechanics (unitarity) and general relativity (information destruction at event horizons). We propose an operational reformulation: rather than asking "is information destroyed?", we ask "how is information distributed between accessible (C) and transferred (γ) components?". We prove that for thermal quantum channels modeling horizon radiation, the conservation law C + γ = 1 + O(1/d) holds, where d is the Hilbert space dimension. Numerical simulations across 384,000 independent trajectories at 30 qubits (~10⁹ dimensions) confirm this relation with precision σ ~ 10⁻¹². The framework JADE naturally connects to the ISLAND formula, Hayden-Preskill protocol, and extends to cosmological horizons in de Sitter space. Our results suggest that information conservation in horizons is computationally accessible. Code and data available at: jocsanlaguna.com/jade