Preserve direction when bounding a rotation
The current propagation method uses vector-length bounds in several places. They are conservative, but can assign uncertainty to every coordinate even when the geometry keeps one coordinate nearly fixed. This is a possible source of unnecessary height uncertainty, not yet a demonstrated explanation of the full prediction widths.
A simple component now has an exact bound: rotating a vertical unit vector around its own axis cannot change its vertical component. With a small sideways tilt, the change in that component is quadratic in the tilt, even if rotation around the vertical axis is large.
The implementation uses rotation-vector coordinates, not Euler angles. It passes 12 local checks, including 384 independent numerical rotations, known-axis, exact rational and large-angle cases. It has not been connected to the aircraft propagation code. All existing studies and their bounds remain unchanged.
Mathematical bound
Let u be a coordinate unit vector, w a rotation vector in radians, and
R = exp([w]×). Put q = |w × u|². Rodrigues' formula gives
u · R u = 1 − q (1 − cos |w|) / |w|².
The limiting value at zero is understood. Since the ratio lies between zero and
one half, the component along u is between max(−1, 1 − q/2) and 1.
Also |R u − u|² = 2(1 − u · R u) ≤ min(4, q). The squared transverse component
is at most min(1, q). For a supplied box, bound q by the sum of squared maximum
absolute values of the two perpendicular rotation-vector coordinates. All
arithmetic in the component implementation uses exact rational numbers.
For example, perpendicular coordinates bounded by 0.01 and 0.005 radians imply a parallel-component loss of at most 0.0000625. The parallel coordinate may range over ±20 radians without changing that bound. This is a mathematical example, not an inferred aircraft calibration.
Next qualification
Derive a source-bound action on a general force vector and a known installation rotation. Preserve transverse force, reference-center departures, calibration errors and numerical margins. Test it against independent known motions before replacing any norm-based operation in a separately frozen propagation candidate. A successful isolated component cannot qualify a full prediction method.