A unified archimedean tail certification for real primitive Dirichlet characters

Groskin (arXiv:2607.02828) proves that, past the Galerkin band, the omitted archimedean tail of the truncated Weil quadratic form for zeta is a strictly totally positive Cauchy–Stieltjes increment, yielding an explicit two-sided finite-cutoff certification rule. We show that the two properties of the archimedean density used in that proof — pointwise positivity, and continuity ensuring positive measure on every open subinterval — are exactly the properties needed, and that both hold for primitive real Dirichlet characters of either parity, so the same positivity and total-positivity conclusions follow. The only new ingredient is an exact digamma-reflection identity relating the odd archimedean multiplier to the even (zeta) one. This is an extension of an existing certification framework, not a new positivity mechanism; the exact numerical tail-budget constant of Groskin's Corollary 3.3(iii) is not claimed to transfer unchanged, only its O(log T/T) order.