The Miscalibration Degeneracy: Instrument Angle Error, Foreground Self-Calibration, and the Reported Rotation

This paper examines how an error in the assumed orientation of a polarisation detector enters the measured rotation, why it is exactly degenerate with a uniform cosmic angle at map level, and what the common practice of self-calibrating on foregrounds does to the quantity that remains. 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: true deg = 0.30; bias at alpha0p3 = 0.30; observed at alpha0p3 = 0.60; degeneracy = exact at map level for a uniform angle. 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.