Active Scalar Completion of Cyclic Observables for Nonselfadjoint Root Pencils

Commuting scalar markers retain the content of cyclic products in a nonselfadjoint root pencil but not their order. For an alphabet of size Q, the multilinear response of degree r loses exactly binom(Q,r)((r-1)!-1) independent principal-part coordinates. This loss persists for the complete determinant germ: two distinct pencils in a fixed calibrated architecture may have identical scalar determinant data and different ordered cyclic responses. Each missing cyclic word is recovered by a finite scalar state extension whose square-free determinant coefficient is the prescribed trace word. Given one pole anchor of order ν, words of length at most R recover the inverse pencil modulo t^((R-1)ν). This exponent is optimal, and the inverse limit recovers the fixed-frame pencil. For a cyclic root module O/(t^m), the reconstructed Laurent data determine a minimal measured basis of the monic deformation space and the associated analytic control germ. The finite constructions admit scalar heteroclinic realizations for Pollicott–Ruelle networks, where the compiled coefficients also have a primitive-orbit interpretation.