gap_squared
plain-language theorem explainer
45 squared equals 2025 supplies the bottom-right cell of the cross-pattern matrix assembled from five Recognition Science patterns. Cross-domain meta-theorem work cites this arithmetic fact when building the certification record for the full matrix. The verification applies a direct decision procedure to the equality.
Claim. $45^2 = 2025$
background
The module defines a 5-by-5 cross-product matrix whose rows and columns are the patterns D=5, 2^3=8, J=0, the phi-ladder, and gap-45. The supplied matrix table lists the entry at gap-45 crossed with itself as 2025 and labels it the full-turn ceiling. This theorem populates that single integer cell; the module states that every non-trivial entry matches a known RS quantity such as 360 = 2^3 * 45.
proof idea
The proof is a term-mode application of the decide tactic that evaluates the concrete arithmetic equality 45 * 45 = 2025.
why it matters
The result fills the gap_squared field inside the CrossPatternMatrixCert structure, which assembles all five pattern products into a single certificate. It thereby closes the 2025 cell of the Wave-64 meta-claim and supplies the numerical anchor for the full-turn ceiling relation 2^3 * 45 = 360. The framework treats this integer as the square of the gap-45 pattern that appears in the eight-tick octave construction.
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