Pairwise Independence of Representation, Classification, and Composition in Finite Extensional Magmas

Abstract

Nontrivial combinatory algebras with S and K must be infinite. Associativity is incompatible with combining a classifier and a retraction pair in a finite extensional magma. These obstructions exclude several standard settings from the finite extensional framework studied here, most notably nontrivial finite S+K-style combinatory algebras and associative structures (semigroups, monoids, groups, rings) carrying both a classifier and a retraction pair. What algebraic structure exists in the remaining landscape: finite, non-associative, total? We identify three properties of finite extensional 2-pointed magmas: self-representation (R), the classifier dichotomy (D), and the Internal Composition Property (H). We prove they are pairwise independent. Lean-verified finite counterexamples at sizes 4 through 10 establish all six non-implications, four with provably tight bounds. The minimum coexistence witness has N = 5, which is optimal: ICP requires 3 pairwise distinct core elements, so N \ge 5. The three-category decomposition induced by D is an isomorphism invariant, and the ICP is logically equivalent to the standard Compose+Inert axioms. All results are formalized in Lean 4 with zero sorry.