Stieltjes constants without closed form II: the mirror principle, CM atoms and the exact range m | 24

Part II of a study of the Stieltjes constants through the Laurent expansions of quadratic zeta functions at s = 1. Part I split each coefficient into a Herglotz part and an elementary part. It found non-classical polylogarithms (Nielsen S₃,₂) in the elementary part from the level of γ₃ on. Here we show that this content is an artifact of the decomposition. For imaginary fields, the elementary part is a Gamma-type term minus a “boundary triangle” integral. The Herglotz part carries exactly the opposite triangle term plus the Fourier part of the Eisenstein series at the CM point, so the polylogarithms cancel in ζ_K (mirror principle). For real fields, three natural decompositions of ζ_Q(√5) carry three different polylogarithmic contents. The Γ-invariant Kronecker–Hecke form carries none. The canonical “atoms” are therefore the values at CM points, or the integrals over closed geodesics, of the Laurent coefficients of the real-analytic Eisenstein series. Genus characters reduce real atoms to CM atoms at all orders (Kronecker–Pell at all orders). As a result, every generalized Stieltjes constant γₙ(r/m) with m | 24 is explicit in elementary constants plus Bessel towers at CM points, verified for n ≤ 16–20 at 50–60 digits. A parity argument shows that m | 24 is exactly the range of imaginary-quadratic CM data.