Hawking's Spectrum Traced Through the Planck Sector: Three Assumed Steps, Zero Derivations for the Steps

Hawking derived a thermal spectrum for black holes in 1975. The derivation works if you grant it three steps. This paper counts those steps from Hawking's own text and prices what they cost. The count is plain. Assumed steps: 3. Derivation equations owned for the steps themselves: 0. For a hole with the mass of the Sun, the price is also plain. Its temperature comes out at T_H = 6.17×10⁻⁸ K. A quantum at that temperature, traced back toward the horizon the way the derivation traces it, crosses the Planck frequency after 1.7857×10⁻³ s of emission. That is 90,632 e-folds. The part of the evaporation lifetime with ancestry below the Planck frequency is a fraction 2.7×10⁻⁷⁸ of the whole lifetime. The rest of the history runs above the Planck frequency, where the paper gives no propagation law. The fixed background fails a second way. Hold the mass fixed, run the evaporation the paper describes, and the mass changes by Δ M/M = 1. The hole goes away while the history runs on a geometry that never changes. Hawking's own test for the scheme passes at the same place. At a solar horizon the curvature length over the Planck length is 9.1365×10³⁷. Curvature passes by 37 orders. Frequency fails in under two milliseconds. The paper prices the first test. It does not price the second. Verdict at the size it earned: a reason was found, and the question remains open (not solved). The reason the steps stand assumed is found. The derivation is not closed. No new derivation of Hawking radiation is claimed here. No trans-Planckian completion is assumed or used here. Boundary, stated once and held through the paper: the vacuum catastrophe is not solved, and only partly addressed: no derivation in this program produces the observed vacuum energy, and the count of such derivations is 0. This paper does not touch that problem and does not change its state.