Pairwise Independence of Representation, Classification, and Composition in Finite Extensional Magmas
For researchers in algebraic logic and combinatory logic, this clarifies the minimal algebraic structure needed for certain properties in finite settings, but the results are incremental and highly specialized.
The paper proves that three properties (self-representation, classifier dichotomy, and internal composition property) are pairwise independent in finite extensional 2-pointed magmas, providing six Lean-verified counterexamples of sizes 5–10 and showing the minimum coexistence witness size is 5.
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. Six Lean-verified finite counterexamples at sizes 5 through 10 establish all six non-implications. The minimum coexistence witness has N=5, which is optimal: ICP requires 3 pairwise distinct core elements, so N >= 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 Lean4 with zero sorry.