algebraicStructureCount
plain-language theorem explainer
The theorem establishes that the recognition lattice Q₃ supports exactly five canonical algebraic structures. A researcher deriving abstract algebra from the Recognition Science forcing chain would cite this count to confirm the match with configuration dimension D=5. The proof is a direct decision procedure on the finite inductive type enumerating the structures.
Claim. The set of canonical algebraic structures on the recognition lattice Q₃ has cardinality five: group, ring, field, module, and algebra. Thus $ |S| = 5 $ where $ S = $ {group, ring, field, module, algebra}.
background
The module Abstract Algebra from RS states that the recognition lattice Q₃ carries the structure of the abelian group (ℤ/2)³ with order 8 = 2³ = 2^D for D=3. The inductive type enumerates five structures (group, ring, field, module, algebra) that arise when the configuration dimension equals 5. This count is consistent with the upstream inductive definitions of the same structures in the AlgebraicStructuresFromConfigDim module.
proof idea
The proof is a one-line wrapper that applies the decide tactic. The tactic evaluates the cardinality directly from the Fintype instance derived on the inductive type with five constructors.
why it matters
This result supplies the five_structures field to the abstractAlgebraCert definition that packages the algebraic facts for Q₃. It realizes the module claim that five structures arise for configDim D=5, aligning with Recognition Science landmarks T7 (eight-tick octave) and T8 (D=3). The downstream certificate uses the count to certify the full algebraic structure of the lattice.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.