A tropical geometry for bounded biochemical state spaces.

Many biochemical measurements define state spaces that are bounded, absorbing, and generically noninvertible yet they are routinely analyzed using categorically mismatched linear and Euclidean frameworks. This categorical mismatch can irretrievably obscure biological structure. This work formalizes the minimal algebraic properties imposed by bounded biochemical state spaces, demonstrating that a bounded tropical semiring provides a natural category match by replacing additive linear structure with order-based, piecewise-linear operations that encode dominance, saturation, and path-dependence without contradiction. To instantiate this framework, measurement-native tropical geometric analysis was applied to cysteine redox proteomic data (Oxi-Shapes). Oxi-Shapes represents cysteine oxidation occupancies as a bounded scalar field over a discrete lattice of invariant residue identities, enabling analysis of both ensemble-level organization and site-wise admissible transitions. As demonstrated on empirical cysteine redox proteomic data, categorically correct algebraic structures exposed geometric features that remained inaccessible under unbounded linear representations.