A Density Bound Cannot Set a Rotation Angle: The Missing Dimensionless Input, Quantified

This paper examines whether any theory whose new content is a single density bound can, on dimensional grounds alone, predict a polarisation rotation angle, and how large a Chern-Simons-type coefficient would have to be if the observed angle were produced over the bound's own characteristic time rather than over the Hubble path. The calculation is performed from stated assumptions and published polarisation measurements, using the deposited script so that every quoted number can be reproduced. The headline outputs are: t A s = 1.24e-24; required k over one tA eV = 2.78e+06; required k over cmb path eV = 2.41e-36; ratio = 1.2e+42; angle from rho alone = not determined - dimensionless input missing. The treatment separates what the published measurements supply from what a theoretical framework must additionally provide; where a required quantity is not supplied, that absence is reported plainly as a negative result rather than replaced by an assumed value. The study is computed from stated assumptions and published inputs; it is not a new measurement. SIMULATED, NOT MEASURED in the sense that all derived quantities are calculated, not observed.