Reasoning. (1) Express the field in in-modes {f} (positive frequency on past null infinity) and out-modes {p} plus horizon modes {q}; the initial vacuum is a_i|0>=0. The out-mode particle content is Sum_j |beta_ij|^2, where p_i = Sum_j (alpha_ij f_j + beta_ij f_j). (2) Propagating an out-mode backward, the part p^(2) that traverses the collapsing body piles up just before the last escaping ray v = v0; the affine parameter on past null infinity relates to retarded time by lambda = -C e^(-kappau), so near v0 the phase behaves as -(omega/kappa) log(v0 - v). (3) Fourier-transforming this form and analytically continuing around the branch point gives |alpha^(2)_omega,omega’| = e^(piomega/kappa) |beta^(2)_omega,omega’| for large omega’. (4) Building late-retarded-time wave packets, the mixing coefficients come only from this asymptotic regime — the collapse details drop out — and combining with the flux-normalization condition Gamma_jn = Integral(|alpha^(2)|^2 - |beta^(2)|^2) yields a mean occupation Gamma_jn / (e^(2piomega/kappa) - 1) per outgoing mode: a grey-body at temperature kappa/2pi, with Gamma the absorption probability of the same mode. (5) For fermions the beta-terms flip sign in the probability flux, giving Gamma/(e^(2piomega/kappa) + 1) — Fermi-Dirac, as thermality demands. (6) Asymmetric collapse changes nothing at late times; rotation and charge shift omega to omega - mOmega - e*Phi (recovering superradiance in the T→0 limit); massive fields substitute total energy for omega. The derivation uses ultra-high-frequency (trans-Planckian) modes at intermediate steps — its acknowledged soft spot — but the emitted spectrum depends only on the horizon-scale geometry.
Step 6 — validity verdict
approved / checked. Load-bearing step traced: back-propagating a late out-mode gives the logarithmic phase -(omega/kappa) log(v0 - v); Fourier analysis with the branch point rounded in the correct half-plane forces |alpha_(2)| = e^(piomega/kappa) |beta_(2)| for the relevant asymptotic regime; combining with the Wronskian normalization Gamma = Sum(|alpha|^2 - |beta|^2) yields mean occupation Gamma/(e^(2piomega/kappa) - 1) — a grey-body at kappa/2pi — and the anticommutator normalization flips the sign for fermions, giving the Fermi-Dirac form thermality requires. Because only the late-time asymptotic regime contributes to late wave packets, collapse details drop out; rotation/charge enter solely through omega → omega - mOmega - ePhi. I retraced each of these moves (branch-cut factor, occupation algebra, sign flip, late-time localization); they are long but elementary given the premises, so checked, author-blind. The acknowledged soft spot — free-field structure at trans-Planckian intermediate frequencies — is a premise of the derivation (carried separately by A-37 and priced on H-18/H-26), not a failure of this inference.