Generic polar divisors and flag residues for root-system zeta functions

Let $\Phi$ be an irreducible crystallographic root system and let $Z_\Phi(\mathbf{s})$ be the untwisted Komori-Matsumoto-Tsumura zeta function of $\Phi$, with one complex exponent for each positive coroot; its diagonal specialization is the single-variable Witten zeta function. For a nonempty set $S$ of simple nodes and an integer $\ell\ge 0$, let $H_{S,\ell}$ be the hyperplane on which the exponents of the roots meeting $S$ sum to $|S|-\ell$. We prove that every proper-support hyperplane $H_{S,\ell}$ is a genuine polar divisor at a generic point, while exact homogeneity leaves only the unshifted full-support divisor; each generic residue is an explicit finite Taylor-jet sum of projective periods and polynomially weighted zeta functions of complementary subsystems. On the maximal support wonderful model, boundary contributions are indexed by strict decorated flags. We derive recursive flag residues, component-mass gamma factors, and an incidence-complete formula for the Laurent coefficients on any transverse slice: the pole order equals the largest order with nonzero aggregate coefficient, not the largest order of an individual flag. The general formulas recover the classical $A_2$ and $A_3$ singular data, Zhao's Euler-Zagier residue formulas and the Akiyama-Egami-Tanigawa lists, and the $C_2$ and $G_2$ residue functions. For $B_3$ and $C_3$ we derive the carrier geometry and lower-rank factorizations of the positive diagonal residues, identify the three-term cancellation at $s=1/8$, and show that negative half-integers are the only possible double-pole locations, recovering the known $B_3$ double coefficient at $s=-1/2$.