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

Stefano Palmieri

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. 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.

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. 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.

Paper Structure

This paper contains 56 sections, 13 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Definitions
  4. Universal Theorems
  5. Theorems from D alone
  6. Theorems from R+D
  7. The K-infinity obstruction
  8. Decomposition Invariance
  9. ICP Characterization
  10. Enrichments (not used in this paper).
  11. Independence
  12. $R \not\Rightarrow D$ ($N\!=\!8$, the Countermodel).
  13. $R \not\Rightarrow H$ ($N\!=\!6$, structural).
  14. $D \not\Rightarrow H$ ($N\!=\!10$).
  15. $H \not\Rightarrow D$ ($N\!=\!10$).
  16. ...and 41 more sections

Key Result

theorem 1

In any extensional 2-pointed magma satisfying D, every element is a zero, a classifier, or a non-classifier. The classes are pairwise disjoint: in particular, $C \cap N = \emptyset$ (no element is partially classifying).

Figures (1)

  • Figure 1: Full independence structure. All six pairwise non-implications are proved by Lean-verified finite counterexamples at the indicated sizes. No capability implies any other.

Theorems & Definitions (31)

  • definition 1: Extensional 2-Pointed Magma
  • definition 2: Faithful Retract Magma
  • Remark 1: Retraction pair convention
  • definition 3: Classifier Dichotomy
  • definition 4: Internal Composition Property
  • definition 5: Capability R
  • definition 6: Capability D
  • definition 7: Capability H
  • theorem 1: Three-Category Decomposition
  • theorem 2: Asymmetry
  • ...and 21 more