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.
Chemically reversible cysteine oxidation redox regulates biological processes by posttranslationally modifying protein structure and function 1 . Advances in redox proteomics have enabled large-scale quantification of cysteine oxidation across thousands of sites, tissues, and conditions, providing detailed snapshots of redox state across the proteome [2][3][4] . However, the analytical frameworks commonly used to interpret these data often rely on algebraic assumptions that are mismatched to the structure of the underlying biochemical state space.
Cysteine oxidation measurements are intrinsically bounded to the unit interval and defined over discrete, invariant residue identities 5 . They report biochemical state occupancies rather than unconstrained continuous variation. Bounded biochemical state spaces with absorbing boundaries and many-to-one observation are categorically incompatible with linear and Euclidean representations. In such spaces, translation invariance, global invertibility, and symmetry do not hold, and equal-magnitude numerical changes do not encode equal physical meaning [6][7][8] . Consequently, common operations such as linear projection, Euclidean distance, or variance maximisation can generate invalid states, obscure boundary-induced structure, and irretrievably conflate distinct biochemical histories. The consequence of this algebraic mismatch is a loss of access to physically meaningful information about the order-disorder, entropy, and energy of the measured redox states. This work introduces Oxi-Shapes, a tropical geometry-based framework for analysing redox proteomic data that is categorically matched to bounded biochemical state spaces. In Oxi-Shapes, invariant cysteine sites define a discrete lattice, and measured oxidation occupancies are represented as a bounded scalar field over this lattice. Structure is therefore induced directly by experimental measurements rather than imposed by an external embedding. Within this representation, entropy corresponds to geometric volume, order-disorder emerges naturally along the bounded oxidation axis, and curvature and energetic structure can be computed using well-defined Dirichlet and Morse formulations, with fully reduced states constituting the ground configuration 9 . Without invoking dynamics or mechanisms, Oxi-Shapes establishes what can be physically inferred from bounded biochemical snapshots once representation is correct. The Oxi-Shapes framework can be applied to any bounded biochemical state space that is categorically matched to its algebra, including other posttranslational modifications such as protein phosphorylation 10 .
The analyses presented in this work proceed from a fixed mathematical representation of bounded biochemical state spaces. The algebraic structure and geometric properties of the Oxi-Shapes representation are derived formally in the Supplemental Notes, independent of a given empirical dataset. Implementation details, including data preprocessing and construction of the discrete lattice representation, are described in Methods and applied here to a foundational redox proteomic dataset (Oxi-Mouse 11 ).
The results are organised around geometric objects that exist on bounded biochemical state spaces rather than around analytical procedures. These objects fall into two complementary classes. Local geometric objects describe site-wise structure and admissible state transitions within the bounded oxidation interval, capturing capacity, feasibility, and directionality of redox change at individual residues. Global geometric objects describe high-dimensional organisation of the redox state across the proteome, capturing ensemble-level orderdisorder, entropy, and energy.
For each object, this work establishes what can and cannot be inferred under many-to-one observation and then apply it to experimental redox proteomic data. This structure ensures that each result reflects a necessary consequence of categorically correct representation rather than an outcome of optimisation, modelling, or algorithmic choice.
As a measurement-native framework, Oxi-Shapes operates on datasets in which cysteine oxidation is quantified as bounded occupancies in the [0,1] state space. To represent these data as a high-dimensional object, each cysteine residue is assigned a unique and invariant position in a discrete lattice index set, providing a fixed correspondence between measured sites and lattice elements across conditions. Measured oxidation values are then realised as bounded scalar values along a third (oxidation) axis over this lattice (Supplemental Note 1), yielding a discrete tropical representation in which structure is induced directly by the measured biochemical state rather than by imposed relational or embedding assumptions.
In Oxi-Shapes, each cysteine site therefore carries an intrinsic state variable-its oxidation occupancy-that is mutable across conditions but conserved at the level of information: site ident
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