Positive steady states of a class of power law systems with independent decompositions

The paper’s Theorem 3.1 asserts that for PL-RDK systems with an independent decomposition, if δ = Σi δi and the augmented kinetic-order matrices satisfy T̂ = ⊕i T̂i (rank additivity of the augmented T-matrices), then E+(N,K) ≠ ∅ iff each E+(Ni,Ki) ≠ ∅. The proof reduces the “gluing” step to solving a stacked linear system in ln-variables using T̂-independence and a Farkas-type corollary, yielding a common positive steady state; the factor-map/Laplacian formulation makes Y·L·θ(x)=0 once θi(x) are scaled versions of viable vi for each subnetwork . The candidate solution develops the same core argument in a slightly different formalism: it writes the SFRF in a reactant-lumped monomial form f(x)=M exp(T⊤ ln x), shows that each subnetwork admits μi with Mi μi=0 and ln μi∈Im(Ti⊤), and uses T̂-independence to solve Ti⊤w=ln μi+ui ei so that x=e^w is a common positive steady state. This is the same “matching ln-exponents modulo the all-ones direction per subnetwork” mechanism the paper uses (with γi=e^{ui}). One minor observation: although the theorem assumes δ = Σi δi, the proof’s nontrivial direction uses only T̂-independence; the deficiency-sum condition appears redundant in the proof as written. Nonetheless, both proofs are correct and essentially the same in structure (different notational frameworks) .