Joint Consistency of the Optical Depth and the 21-cm Trough: Escape Fraction in the Bounce Frame

Two constraints on the end of the cosmic dawn era are often used separately: the microwave-background optical depth, which integrates the ionisation history, and the absorption trough, which timestamps its beginning. Using the stated cosmology and the brightness-temperature relations described in the Methods, the calculation gives the following headline values: f esc = 0.16; f esc standard = 0.10; tau model = 0.058; tau planck = 0.054. The comparison values taken from the published record — including the EDGES and SARAS reports, HERA limits and the Planck optical depth where relevant — are used exactly as published, and the bounce contribution enters only through the named timing, heating or shear inputs. The result is a computed prediction for the delayed-onset history, with its sensitivity to each input stated in the Discussion. This study is simulated and computed from stated assumptions; it is not a measurement.