Variance and Shape of the Transmitted-Flux Probability Distribution as a Forest Diagnostic

This study asks what the full distribution of transmitted flux, beyond its mean, reveals about the absorbing gas. 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: mean_flux_from_pdf = 0.436; variance = 0.0866; skewness = 0.272; bounce_variance_shift = 1.532185137777778e-84; result = PDF moments recovered; bounce shift negligible (negative). The distribution returns a mean flux near 0. All values are computed from the stated assumptions, not measured, and the computation is supplied so that every number can be reproduced.