Operator-Specific Information Retention in Spectral Membrane Analysis

Abstract

Membrane analysis in cryo-electron tomography necessarily proceeds through a sequence of representation changes, including meshing, graph construction, differential-operator construction, clipping, simplification, and spectral truncation. These transformations are not information-neutral. This paper introduces a comparative operator-response framework for spectral membrane analysis, asking how explicit representation operators induce differential response across global operator-derived summaries and local empirical-distribution summaries, rather than asking only whether a fixed descriptor remains stable under perturbation. The framework partitions descriptor observables into structural and distributional channels — formalized as a distinction between coupling-sensitive operator summaries and empirical local-field distribution summaries — and uses retention, deformation, and disagreement functionals to quantify response under matched interventions. It is instantiated using normalized graph algebraic connectivity, normalized cotangent second eigenvalue, degree-distribution entropy, and curvature-distribution entropy. Experiments on five synthetic membrane families together with a small real-membrane proof-of-concept set from EMPIAR-11370 show three recurring patterns: baseline morphology already separates topology-like and geometry-aware channels in constricted geometries; radial clipping produces morphology-dependent structural responses larger than distributional shifts; and graph–cotangent disagreement under transformation is largest where morphology makes the representation most sensitive to operator choice and can itself be diagnostically informative. A random-subsample decimation study reveals clean, transition, and fragmentation-stress regimes, while a topology-preserving edge-collapse control shows that this regime structure is pipeline-specific. Spectral truncation provides qualified support for low-mode concentration in the narrower sense tested here. The contribution is methodological: the paper does not propose a new general theory of descriptor robustness, but a framework for measuring operator-conditioned channel asymmetry and graph–cotangent disagreement under representation change.

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