To address staining variability in digital pathology, we formulate normalization as a variational inverse problem balancing structural fidelity and chromatic alignment. We propose Screened Poisson Normalization (SPN), a convex model in CIE L∗a∗b∗ space governed by the modified Helmholtz equation. By coupling a first-order gradient consistency term with a zeroth-order chromatic anchor, SPN yields a screened reconstruction controlled by a single screening parameter λ. Beyond establishing well-posedness in H1(Ω) and Lipschitz stability, we derive a property crucial for gigapixel wholeslide analysis: the screening term induces an intrinsic localization length ℓ ∼ λ−1/2, which leads to exponential decay of boundary-induced errors. This theoretical insight provides a principled justification for a scalable, non-overlapping subdomain implementation; when combined with an efficient DCT-based spectral solver, it enables artifact-free processing without elaborate tiling heuristics or boundary post-processing. Extensive experiments on multi-center datasets further demonstrate that SPN achieves robust chromatic consistency while faithfully preserving diagnostically relevant microstructural detail.