D5_squared
The arithmetic identity five times five equals twenty-five is recorded to populate the D5 squared cell of the cross-pattern matrix. Researchers building Recognition Science meta-theorems cite this entry to confirm that the five patterns produce distinct integer products. The proof is a direct decision procedure that verifies the multiplication without lemmas or hypotheses.
claim$5^2 = 25$
background
The module defines a cross-pattern matrix whose rows and columns are indexed by the five RS patterns D=5, 2^3=8, J(1)=0, the phi-ladder, and gap-45. Each off-diagonal entry is the product of a pair of these quantities, with the (D5, D5) cell required to be 25. The module documentation states that every non-trivial entry corresponds to a known RS quantity such as cognitive pair states or attention space. No upstream lemmas are imported; the declaration stands alone as a numerical fact inside the C26 structural meta-claim.
proof idea
The proof is a one-line wrapper that invokes the decide tactic on the concrete arithmetic statement (5 : ℕ) * 5 = 25. The tactic evaluates the multiplication at the meta-level and closes the goal by reflexivity.
why it matters in Recognition Science
This theorem supplies the D5_squared field required by the CrossPatternMatrixCert structure, which in turn populates the full matrix certificate used by crossPatternMatrixCert. It anchors the first row of the matrix described in the module documentation and thereby supports the claim that the five patterns form a non-degenerate integer matrix. The result touches the D=5 pattern identified in the Wave-62 report but does not invoke the forcing chain, RCL, or phi-ladder arithmetic.
scope and limits
- Does not derive 25 from any Recognition Science functional equation or J-cost identity.
- Does not relate the result to the phi-ladder, gap functions, or Berry creation threshold.
- Does not extend the equality to symbolic or non-natural-number domains.
- Does not address physical interpretations such as theta carriers or cognitive pair states.
Lean usage
example : (5 : ℕ) * 5 = 25 := D5_squaredformal statement (Lean)
39theorem D5_squared : (5 : ℕ) * 5 = 25 := by decide