Finite chain-growth bookkeeping ($X_n=S/C$)

Stipulate a feed of exactly $M_0=40$ monomer molecules, four successfully initiated linear chains, one polymerization-active site per chain, no chain transfer in chain polymerization, no loss of incorporated units, and no further chain-growth initiation. Count only initiated chains in $C$; exclude unreacted monomers. Assume each consumed monomer molecule contributes exactly one monomeric unit. With incorporated feed fraction $q=S/M_0$, the number-average degree of polymerization is the counting identityThis is conditional conservation bookkeeping, not a universal empirical law. Immediately after chain-growth initiation, the four lengths are $(1,1,1,1)$, so $(C,A,S)=(4,4,4)$ and $q=0.1$. Stipulate twelve subsequent chain-growth propagation events, three on each chain: the lengths become $(4,4,4,4)$, giving $(C,A,S)=(4,4,16)$, $q=0.4$ and $X_n=4$. Sixteen units have been incorporated, while twenty-four free monomer molecules remain. The chain-length assignment is deliberately chosen, not a prediction of equal growth in an experiment.