A Geometric and Category-Theoretic Theory of Viability: How Sequential Adaptations Induce Path-Dependent Risk

Adaptive systems survive fluctuatingenvironments by adjusting internal strategies. Sequences ofindividually harmless adjustments can push a system into failurewhen environmental shifts occur in a non-commuting order. Wedevelop the geometric and category-theoretic framework definingand quantifying this phenomenon: \emph{viability-weightedcurvature}, the viability loss a closed loop accumulatesthrough policy holonomy. The theory has four pillars.(i)~The \emph{SAVGS architecture} unifies control base, Fisher--Raopolicy bundle, viability margin, maintenance graph, and$2$-categorical boundary span into one stratified bundle.(ii)~The $2$-category $\mathbf{StCon}(B)$ of stratified connectionscarries a lax-functorial gluing theorem and a piecewise-holonomyformula, with boundary resets at their true order for transversalwall-crossing loops in \emph{pairs}. Each stratum carries theFisher-minimal transport law (KKT projection) as connection, andthe small-loop theorem bounds endpoint erosion by theviability-weighted curvature.(iii)~A single composition theorem types seven bridges as optics,with per-optic Lipschitz constants and a Banach contraction of theKrasnoselskii--Mann-averaged update; the projected CPTP contractionsettles the Zeno self-reference.(iv)~The filtered-colimit construction of RAF sets is proved at Setlevel, verified at scale; the $\infty$-categorical extension inhomotopy type theory carries marked proof-sketch status. Thevalidation battery isrobust across six axes: carbon, oxygen, nitrogen, phosphate, and ironsupply plus non-medium maintenance stress leave labels invariant,re-stratifying only at regime switches andnitrogen-source substitution. The association isinvariant under canonical flux selection, the declared tie-breakclosing its near-degeneracy boundary. The application paperdevelops the atomic curvature measure and its genome-scalevalidation