New Framework for Interfacial Statistics: Exact <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> -Point Correlations of Gaussian Level Sets

We derive exact analytical expressions for higher-order correlations of Gaussian level-set interfaces, establishing a direct link between bulk field statistics and interface geometry. This framework enables efficient reconstruction of disordered media, detailed modeling of Gaussian foams-structured, double-thresholded interfaces with tunable morphology-and analysis of memory-enhanced anisotropy reflecting directional persistence in the surface structure. These results open new possibilities for characterizing complex transport, guiding stochastic reconstructions, designing materials with desired properties, quantifying the information content of correlation functions, and modeling directional processes on irregular boundaries.