four_decomp
The natural-number identity 4 equals the square of the binary-face generator holds by direct computation. Cross-domain recognition researchers cite it when verifying that spectrum cardinalities reduce to polynomials over the generators 2, 3 and 5. The proof applies a decision procedure to the arithmetic equality.
claim$4 = 2^2$, where 2 denotes the binary-face generator.
background
The Recognition Generators module establishes that every integer in the RS cardinality spectrum arises from the set G = {2, 3, 5} via addition, multiplication, exponentiation and binomial coefficients. Here 2 represents the binary face, 3 the spatial dimension and 5 the configuration dimension. The upstream definition fixes the binary-face generator as the constant 2.
proof idea
A one-line wrapper that invokes the decide tactic to verify the arithmetic equality 4 = 2 squared.
why it matters in Recognition Science
This decomposition supports the recognitionGeneratorsCert certificate, which records that the generators produce all required spectrum members. It fills the C27 structural meta-claim that no spectrum member lies outside the polynomial ring generated by {2, 3, 5}.
scope and limits
- Does not prove uniqueness of the decomposition among possible expressions.
- Does not extend the result to real or complex numbers.
- Does not address spectrum members outside the enumerated list in C21.
formal statement (Lean)
47theorem four_decomp : (4 : ℕ) = G2^2 := by decide