Control all branches associated with the inverse-matrix transforms

Characterize and control all possible branches of the mathcal{R}_1 and mathcal{R}_2 transforms associated with the inverse of a large random matrix vb A, rather than relying on a single potentially available branch when determining the spectral boundary of vb Avb B.

Background

To obtain spectral-boundary formulas for products vb Avb B, the paper relates the transforms of vb A{-1} to those of vb A. The authors derive a formula that can provide one branch of the relevant transform for vb A{-1}, but they explicitly acknowledge that the full branch structure is not controlled. This leaves unresolved whether the proposed construction captures every branch needed to determine all spectral boundaries.

References

We use the word ``can'', since we currently have no control over all the possible branches associated with \vb A{-1}.

— Spectral properties of deterministic matrices multiplied by rotationally invariant random non-Hermitian ensembles  (2609.03618 - Bousseyroux et al., 3 Sep 2026) in Section 4, immediately before Lemma 1 (the lemma labeled \ref{propinverse})