The Lyman-alpha Background Required for Full Spin Coupling by Redshift Twenty

Full coupling of the spin temperature to the gas temperature by a given redshift requires a minimum Lyman-alpha background intensity. Using the stated cosmology and the brightness-temperature relations described in the Methods, the calculation gives the following headline values: J alpha 21 = 0.80; z = 20; flux ph cm2 s Hz sr = 1.0e-21 units; coupling x = 1.44. 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.