twentyfive_decomp
The declaration confirms that 25 equals the square of the configuration dimension generator fixed at 5. Researchers enumerating the Recognition Science cardinality spectrum cite it to verify that this member reduces directly to a power of one primitive generator. The proof is a single decision tactic that checks the concrete natural-number equality.
claimThe natural number 25 equals the square of the configuration dimension generator 5.
background
The Recognition Generators module establishes that every integer in the RS cardinality spectrum reduces to polynomials in the three generators G = {2, 3, 5}, where 2 is the binary face, 3 the spatial dimension, and 5 the configDim. The sibling definition G5 fixes the configDim generator at the constant 5. The module documentation lists 25 = 5² among the explicit spectrum reductions that support the meta-claim of C21.
proof idea
The proof is a one-line wrapper that applies the decide tactic to the equality 25 = 5², which is immediately decidable for concrete natural numbers.
why it matters in Recognition Science
This theorem supplies the decomposition entry for 25 in the module's enumeration of spectrum members for C21. It directly supports the structural meta-claim that no RS spectrum integer requires generators outside {2, 3, 5}. The result closes one concrete case in the cross-domain generator construction without touching open questions about physical interpretation of the resulting cardinalities.
scope and limits
- Does not prove the general meta-claim that every spectrum member decomposes in the generators.
- Does not derive the choice of generators 2, 3, 5 from more primitive axioms.
- Does not connect the number 25 to any physical constant or dimension formula.
formal statement (Lean)
55theorem twentyfive_decomp : (25 : ℕ) = G5^2 := by decide