Full contraction analysis of the composed slice map

Derive a full contraction analysis for the composed CORAM slice map, including the effects of correlated or uneven expert updates and the manifold means over the rotation, spectral-shift, and Stiefel factors.

Background

CORAM uses an amplification factor to compensate for the contraction produced by averaging expert-specific factors on curved spaces. Proposition 1 establishes a $\sqrt{N}$ amplification scale only under idealized assumptions of pairwise orthogonal updates with equal norms, while the empirically measured $c_{\mathrm{RMS}}$ is intended to account for correlated or uneven updates.

The paper reports empirical values for the restoration strength $\kappa$, including a high-dispersion value calibrated on only one base model. A general theoretical characterization of the contraction induced by the complete composed slice transformation remains unresolved.