Chirotope factorization for strongly simply‑embedded CTLNs

Determine whether the chirotope of the oriented matroid associated with a combinatorial threshold‑linear network that has a strongly simply‑embedded partition admits a factorization analogous to the proven factorization of the Cramer's determinants s_i. If such a factorization exists, derive explicit formulas and structural conditions; otherwise, provide counterexamples and bounds.

Background

The work develops determinant factorizations (for s_i) under simply‑embedded and strongly simply‑embedded structures and connects these to fixed point characterization. In the chirotope section, the authors seek to extend these ideas to the oriented-matroid level, asking whether the chirotope itself can be similarly factorized for strongly simply‑embedded partitions.

A positive result would unify combinatorial geometry (oriented matroids) with CTLN structural theorems, potentially enabling complete combinatorial descriptions of fixed points and attractors via chirotope signatures.