cube_faces_squared
plain-language theorem explainer
The equality 6 × 6 = 36 records the count of face-pairings on the three-dimensional cube inside the Recognition Science cross-pattern matrix. Workers verifying the Wave-62 structural meta-claim cite this entry to confirm the five patterns produce distinct integers. The proof is a one-line decision procedure that evaluates the natural-number multiplication directly.
Claim. $6 × 6 = 36$, where 6 denotes the faces of the three-dimensional cube $Q_3$.
background
The module constructs a cross-pattern matrix whose rows and columns are the five RS patterns D=5, 2³=8, J(0), φ-ladder, and gap-45/cube-faces. Each non-trivial product yields a distinct integer or relation listed in the module table, such as 45 = D² · D and 360 = 2³ · 45. Upstream results include the definition gap := closure_factor * fibonacci_factor with the main theorem that this product equals 45, together with the logarithmic display function F(Z) = ln(1 + Z/φ)/ln(φ) used for mass anchoring at the anchor scale.
proof idea
The proof applies the decide tactic to the closed arithmetic statement (6 : ℕ) * 6 = 36. This is a one-line decision procedure that confirms the multiplication by direct computation.
why it matters
The theorem supplies the cube-faces-squared entry that feeds the CrossPatternMatrixCert structure and the crossPatternMatrixCert definition certifying the full matrix. It fills the gap-45/cube-faces row and column of the Wave-62 meta-theorem, with the supplied comment relating 36 = gap45 - 9 = 45 - D². This is consistent with T8 (D = 3) and the eight-tick octave.
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