The Cognition Substrate Theorem (CST)

The definition of cognition remains fragmented across disciplines, often conflatedwith behavioral heuristics or raw processing power. This paper presents the Cog-nition Substrate Theorem (CST), a formal framework defining the necessary andsufficient conditions for a system to sustain a valid cognitive worldline. We definethe cognitive substrate as a 6-tuple C = ⟨T, S, F, J, ⊕, Φ⟩, treating cognitive.validity not as a scalar output, but as a structural invariant: a Justified EpistemicTrajectory. We establish three primary metatheoretic results: (1) the necessityof each operator to prevent structural collapse (e.g., into stochasticity or infiniteregress), (2) the sufficiency of the 6-tuple for sustaining a non-Markovian trajec-tory, and (3) the minimality of these requirements, establishing the 6-tuple as theIrreducible Intersection of structural necessity. The theorem posits that validityis a property of Structural Orchestration (Φ), functionally independent of thecomputational capacity of the inference engine (F ). Crucially, these theoreticalclaims are supported by machine-checked formalizations in three independentproof assistants—Lean 4, Coq, and Isabelle/HOL—providing constructive proofsof existence and establishing axiomatic independence via a hybrid strategy ofstructural ablation (for components) and logical non-formation (for the proto-col). The CST thus provides a mathematically verified, implementation-agnosticfoundation for defining, comparing, and constructing biological, artificial, andinstitutional cognitive substrates.