This paper derives a lower bound on how long an evaporating black hole must live if it eventually returns the information that fell into it. Working only with quantities measured far from the hole, the authors combine energy conservation with a requirement that the final quantum state be pure (that is, information is recovered) to show that the final purification stage cannot be arbitrarily fast. For an initial black hole mass M0 they find a minimum purification time that grows like M0^4 divided by ħ^{3/2}. With one extra assumption — that a Planck-mass black hole is metastable — the required time can become astronomically large, growing exponentially with the square of the initial mass (equivalently with its initial area).

The argument works in an ‘‘asymptotically semiclassical’’ framework. This means the authors assume quantum-gravity effects stay confined to the region near the hole and that observers far away can be treated with ordinary quantum field theory on a background spacetime. They use a recent expression for the radiative energy flux seen at future null infinity (the place where outgoing radiation is collected). In their spherically symmetric model the flux is ⟨F_rad⟩(u)=α ħ k(u)^2, where k(u) is a function called the redshift exponent that encodes the correlations of the outgoing field, and α is a constant set by the field content (for N massless scalar fields α = N/(48π)). The paper also uses a recent formula for the renormalized entanglement entropy of the radiation at null infinity: S_rad(u)=4πα ∫_{-∞}^u k(u') du'. These two relations tie the energy carried away and the quantum entropy of the radiation to the same function k(u), allowing the authors to place constraints without having to solve the unknown quantum-gravity dynamics in the interior.

Using these tools the authors split the evaporation into three phases. Phase A is the familiar semiclassical Hawking evaporation, during which the black hole loses mass and emits nearly thermal radiation; its duration scales roughly like M0^3/ħ (they quote an estimate τ_A ≈ 163 α M0^3/ħ). Phase B is a possible short quiescent period when the hole reaches Planckian mass and the radiative flux could vanish. Phase C is the purification phase, when the radiation must become correlated in such a way that the combined state at infinity becomes pure. By asking how much energy is available at infinity to create the correlations needed for purification, and using the flux and entropy relations above, they prove a lower bound on the duration of this purification phase that scales as M0^4/ħ^{3/2}.