Cybernetics of AI Companies

Models an AI company as an open, resource-dependent cybernetic system whose sensing, learning, deployment, and governance close multiple feedback loops. It builds a formal language from contextual ∞-topoi, higher groupoids, categories, and lenses to record states, admissibility certificates, irreversible interfaces, and coherent witnessed equivalences; identity types and univalence capture what implementations count as the same while preserving directional operations. Concrete organizational results include certified update preservation, descent conditions, limits on what observation and regulation can determine, a formal panoptic hypothesis relating trace coverage to observation/intervention/governance inequalities, and a distinction between learning dynamics and mere semantic equivalence. The scope explicitly treats these as mathematically specified models rather than empirical inevitabilities about markets or surveillance.

At the differentiable level the work derives an exact optimizer-dependent neural tangent kernel along gradient flow and introduces parameter jets, a neural-tangent hierarchy, and an input-jet tangent kernel to record higher-order optimization behavior; it documents jet-algebra limits (e.g., differentiation lowers finite jet order) and shows no fixed kernel suffices for all dynamics. A central theorem gives a unique global holonomic-jet factorization: every extensional optimizer on smooth objectives factors through holonomic jets while retaining runtime and oracle state, with a separate locality condition for pointwise factorization. Overall this combines homotopy-type ideas with jets and control-theoretic interfaces into a research program for organizational cybernetics of AI firms.