SYNCHRONY IN GENE REGULATORY NETWORKS

The paper characterizes synchrony and lifts for SUM and MULT GRN models using block row-sum conditions (SUM) and multiplicity row-sums plus a per-row product-of-weights condition (MULT). Specifically: Proposition 3.4 gives the SUM synchrony criterion via constant block row-sums, and Proposition 3.6 shows the restricted dynamics are the SUM model on the quotient with entries q± determined by those sums . For MULT, Proposition 3.12 requires (i) constant block sums of multiplicities and (ii) constancy of the product of nonzero weights per row; Proposition 3.14 then fixes the quotient multiplicities and constrains the product of the quotient weights across each row . The lift characterizations in Theorems 4.1 and 4.2 match these: SUM requires blocks with constant row-sums; MULT requires block multiplicity row-sums and a per-row product condition on nonzero entries of the concatenated block row (Q+...Q−) to equal the product of the q’s . The candidate solution re-derives exactly these equivalences, using the same limit-based separation of act/rep guaranteed by the Hill-like assumptions (2.6)–(2.7) . Minor differences are expository (e.g., not emphasizing the non-uniqueness of Q in the MULT quotient noted in Remark 3.15) but do not affect correctness .