Oxi-Shapes: Tropical geometric analysis of bounded redox proteomic state spaces

Redox proteomics generates bounded biochemical measurements that are categorically mismatched to conventional linear algebraic formalisms. This work introduces Oxi-Shapes, a tropical geometric framework for the measurement-native analysis of bounded redox proteomic data. Oxi-Shapes represents cysteine oxidation as a scalar field over a discrete lattice, enabling global and site-wise analysis without rescaling, interpolation, or kinetic assumptions. At the global level, the framework yields internal redox entropy, lattice curvature, and derived energy functionals that characterise the geometric structure of the redox proteome. At the site level, Oxi-Shapes defines a bounded change space that makes explicit hard geometric constraints on admissible redox transitions and enables a normalised signed representation of site-wise change as a fraction of available redox freedom. Applied to an ageing mouse brain dataset, Oxi-Shapes reveals that a small decrease in mean oxidation arises from a profound redistribution of site-wise redox states, with thousands of residues shifting toward the reduced absorbing boundary. These results demonstrate that categorically correct algebraic representations expose structure in proteomic data that is inaccessible to mean-centric or unbounded analyses.