AlgebraicStructuresCert
AlgebraicStructuresCert records that exactly five algebraic structures exist at configuration dimension five. Researchers constructing the algebraic layer of Recognition Science cite it to fix the ordered sequence from group to vector space. The declaration is a structure whose single field directly encodes the Fintype cardinality equality on the enumerated inductive type.
claimThe structure asserts that the finite cardinality of the inductive type enumerating algebraic structures equals five, where the structures are group, ring, field, module, and vector space.
background
The module defines five canonical algebraic structures ordered by increasing richness equal to configDim D = 5: group, ring, field, moduleStruct, vectorSpace. These are formalized as an inductive type deriving DecidableEq, Repr, BEq, Fintype. The setting draws from the abstract algebra module whose similar inductive type lists group, ring, field, module, algebra.
proof idea
This is a direct structure definition. Its single field five_structures is the equality Fintype.card AlgebraicStructure = 5, which holds by the derived Fintype instance on the inductive type.
why it matters in Recognition Science
It is instantiated by the algebraicStructuresCert definition in the same module. The declaration anchors the algebraic layer at configDim = 5, matching the module description of structures ordered by richness. It touches the framework dimension parameter without engaging the spatial D = 3 derived in the forcing chain T8.
scope and limits
- Does not derive the list of structures from the Recognition Composition Law.
- Does not connect the structures to physical constants or the phi-ladder.
- Does not show realization of these structures inside the physical model.
formal statement (Lean)
29structure AlgebraicStructuresCert where
30 five_structures : Fintype.card AlgebraicStructure = 5
31