Published January 3, 2026 | Version 1.0
Journal article Open
- 1. McGill University
Description
Classification systems (type systems, ontologies, taxonomies, schemas) operate over fixed sets of classification axes. We prove this architectural choice has unavoidable consequences: any fixed-axis system is incomplete for some domain. This is not a limitation of specific implementations; it is an information-theoretic impossibility.
The Core Theorems All proofs in Lean 4 (2700+ lines, 142+ theorems, 0 sorry):
Fixed Axis Incompleteness: For any axis set A and any axis a ∉ A, there exists a domain D that A cannot serve. The information required to answer D’s queries does not exist in A.
Parameterized Immunity: For any domain D, there exists an axis set A_D that is comple…
Published January 3, 2026 | Version 1.0
Journal article Open
- 1. McGill University
Description
Classification systems (type systems, ontologies, taxonomies, schemas) operate over fixed sets of classification axes. We prove this architectural choice has unavoidable consequences: any fixed-axis system is incomplete for some domain. This is not a limitation of specific implementations; it is an information-theoretic impossibility.
The Core Theorems All proofs in Lean 4 (2700+ lines, 142+ theorems, 0 sorry):
Fixed Axis Incompleteness: For any axis set A and any axis a ∉ A, there exists a domain D that A cannot serve. The information required to answer D’s queries does not exist in A.
Parameterized Immunity: For any domain D, there exists an axis set A_D that is complete for D. This set is computable: A_D = ⋃_{q ∈ D} requires(q).
The Asymmetry: Fixed systems guarantee failure for some domain. Parameterized systems guarantee success for all domains. One dominates the other absolutely.
Uniqueness: For any domain D, all minimal complete axis sets have equal cardinality. "Dimension" is well-defined for classification problems.
Minimality ⇒ Orthogonality: Every minimal complete axis set is orthogonal. Orthogonality is not imposed; it is derived from minimality.
The Prescriptive Force. These are not design recommendations. They are mathematical necessities:
Given any domain D, the required axes are computable, not chosen. Missing axes cause impossibility, not difficulty. No implementation overcomes a missing axis. The choice of axis-parameterization is forced by the requirement of domain-agnosticism. Application to Type Systems. We instantiate the framework to programming language type systems, proving:
(B, S) (Bases and Namespace) is the unique minimal complete representation of class semantics (B, S, H) extends this for hierarchical configuration systems (adding a Scope axis) B-inclusive systems strictly dominate B-exclusive systems when B ≠ ∅ (in type system terminology: nominal typing dominates structural typing) Systems using only {S} require Ω(n) error localization; systems using {B, S} achieve O(1) The Broader Claim. The impossibility theorems apply to any classification system with fixed dimensions, not only type systems. Biological taxonomies, library classification schemes, database schemas, knowledge graphs: all are subject to the same constraints. A fixed set of axes guarantees domains that cannot be served.
Corollary (Incoherence of Preference). Claiming "classification system design is a matter of preference" while accepting the uniqueness theorem instantiates P ∧ ¬P. Uniqueness entails ¬∃ alternatives; preference presupposes ∃ alternatives. The mathematics admits no choice.
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Additional details
2026-01-03
Initial publication