Abstract
Kramers-Kronig (KK) relations are usually invoked for causal response functions, but their precise status for non-Markovian quantum memory kernels is less explicit. We separate three Laplace-domain objects: the Nakajima-Zwanzig memory kernel K(z), the reduced-state transform σ(z), and the force-fit effective kernel K eff(z). Under a real-axis spectral-representation hypothesis for the projected generator, with a coupling-weighted spectral density in L¹ ∩ L^p, we show that K(z) belongs to the operator-valued Hardy space H^p+ and obeys KK or subtracted KK relations. This gives a Hardy-space consistency criterion for CPTP reduced dynamics, a passivity-analyticity compatibility statement for passive bosonic baths, and a finite-truncation Carleman diagnostic for moment-based kernel reconstructions. In contrast, σ(z) is analytic in the upper half-plane for any initial system-bath state, including correlated states, because microscopic unitarity gives |σ(t)| ≤ 1. Apparent acausality can therefore enter only through the force-fit object: in scalar channels, uncancelled zeros of σ(z) can generate upper-half-plane poles of K_ eff(z). Numerically, we verify the full matrix-valued KK relation for an extracted 4 × 4 Jaynes-Cummings memory kernel. The measured integrated relative residual, 3.8%, lies below the calibrated noise floor of the circular FFT-Hilbert protocol, about 5%, and is therefore consistent with exact KK within numerical accuracy. We also present Born-order and correlated-state diagnostics showing how discarded inhomogeneous terms can contaminate force-fit kernels without violating microscopic causality.