Recovering the Temperature-Density Relation of the Intergalactic Medium from a Line-Width Cutoff

This study asks whether the temperature-density relation can be recovered from the lower edge of the line-width distribution with simple arithmetic. Using published Lyman-alpha forest measurements as stated inputs, and a bounce frame in which the early universe turned around at a density of 9.72 x 10^54 grams per cubic centimetre, the relevant quantities are computed directly and compared with the precision of the data. The headline outcome is as follows: T0_K_recovered = 9212; gamma_recovered = 1.33; bounce_T0_shift_K = 1.6298057522712044e-79; result = T0 and gamma recovered from envelope; bounce shift effectively zero (negative). The inversion returns a temperature at mean density near nine thousand kelvin and a slope parameter near 1. All values are computed from the stated assumptions, not measured, and the computation is supplied so that every number can be reproduced.