Calculus of intelligence: A topos-monadic framework for agentic workflows

In this paper, we propose “Calculus of Intelligence” (COIN), a mathematical framework that formalizes agentic tasks as typed free-monadic task spaces inside a Grothendieck topos, and studies them as a calculus: differentiation via task-space decomposition, differentials as the resulting local task spaces, and integration via monadic composition together with sheaf-theoretic compatibility of declared overlaps. COIN treats a task as a valid-plan space rather than as a single output: the user intent is progressively elaborated into a typed task space, and each valid decomposition presents a sound subspace of plans obtainable from local solutions whose declared overlaps agree and whose composition is certified.