Decoupling of spatial scales in breast pathology reveals fractal-like nuclear organization emergent from tissue spatial architecture

Quantitative biophysical signatures of nuclear spatial reorganisation across breast carcinoma progression remain insufficiently characterised. We apply two complementary fractal descriptors, Correlation dimension (Dc) and Minkowski dimension (Dm), to 4276 regions of interest across seven breast tissue subtypes from the BRACS dataset, validating observed dimensions against systematically constructed null spatial models to distinguish genuine structural organisation from geometric irregularity. All subtypes significantly exceed the complete spatial randomness baseline, confirming universal departure from random nuclear arrangement. The observed scaling is characterised as statistically monofractal within a bounded pre-fractal range. Invasive carcinoma uniquely fails to exceed the clustered null in Dc while simultaneously showing the weakest Dm null deviation, a dual convergence toward stochastic baselines consistent with the progressive removal of architectural constraints. Flat epithelial atypia exhibits a unique directional dissociation with the lowest Dc across all subtypes combined with high Dm null deviation, a co-occurrence not observed in any other subtype and geometrically consistent with decoupled nuclear spatial organisation at the centroid distribution and boundary morphology scales. Interpreted within a percolation-theoretic framework, the non-monotonic null deviation trajectory maps onto qualitative regime transitions, providing a physically grounded explanation for the observed discrimination profile across pathological transitions. These findings position fractal-like nuclear architecture as a potential descriptor for pre-malignant transitional states.