Stieltjes constants without closed form: higher Kronecker limit formulas, Herglotz functions and the classical frontier

The Stieltjes constants γₙ are the Laurent coefficients of the Riemann zeta function at s = 1. Apart from Euler’s constant, no closed form is known for any of them. This paper does not look for closed forms. It studies how these constants combine inside the Laurent expansions of zeta functions of quadratic fields, and climbs a ladder of “higher Kronecker limit formulas” one level at a time. At level 1 (the level of γ₁) we give an explicit formula built from: a second-order Herglotz function G; a new two-slope function R₂; polylogarithms up to Li₃. The formula holds as an identity for arbitrary real slopes. It yields L''(1, χ_D) for every quadratic field. It also yields L'(1, ψ) for non-abelian class-group characters, which give the Euler–Kronecker constants and first Stieltjes constants of non-Galois cubic, quintic and septic fields. Extensive PSLQ searches show that these level-1 values are not linear or bilinear in level-0 data (units, eta-quotients, Chowla–Selberg periods). An exact Euler–Maclaurin identity explains the structure at every level: Herglotz functions arise as Laurent coefficients of a “Herglotz zeta function”; two-slope defects arise from off-diagonal terms; the whole elementary part is a single incomplete beta integral, whose coefficients are Nielsen polylogarithms. As a consequence, the formulas are classical (expressible with Liₙ) up to level 2. At level 3, the level of γ₃, the non-classical Nielsen polylogarithm S₃,₂ appears. A Bloch-group invariant shows that this contribution does not cancel in the field zeta functions. For imaginary fields of class number one, the invariant equals −2 π∧π̄, where π is a root of Euler’s polynomial x² − x + c (for example c = 41 when D = −163). For real fields it involves the fundamental unit. Every statement is labelled as proved, proved up to routine computations, or numerically verified. The paper includes tables for independent self-verification, computed at 20–160 digits with independent methods.