crossPatternMatrixCert
plain-language theorem explainer
The crossPatternMatrixCert assembles the verified cross-products of the five core Recognition Science patterns (D=5, 2^3 octave, J=0, phi-ladder, gap-45) into a single certificate structure. Researchers auditing the Wave-62 meta-theorem would cite this to confirm that each pair yields a distinct integer without degeneracy. The definition is a record constructor that directly assigns each field from its decide-proven sibling theorem.
Claim. The structure CrossPatternMatrixCert is defined by the equalities $5^2 = 25$, $5^2 = 40$ wait no: $5 * 2^3 = 40$, $2^3 * 2^3 = 64$, $2^3 * 45 = 360$, $45^2 = 2025$, $6^2 = 36$, $45 - 36 = 9$, and the off-diagonal entry count equals $2^4$.
background
The module sets out C26 as the cross-pattern matrix meta-claim: the five patterns D=5, the 8-tick octave 2^3, J(1)=0, the phi-ladder, and gap-45 with cube-faces produce a non-degenerate matrix of cross-products, each corresponding to a known RS quantity such as 25 = D^2 or 360 = full turn. The certificate structure CrossPatternMatrixCert collects eight specific equalities that witness the matrix entries and relations, including D5_squared as 55=25 and face_pairs_minus_gap as 45-36=9 which equals D^2. Upstream results include the individual theorems such as D5_squared proved by decide, full_turn which is 845=360, and cube_faces_squared which is 6*6=36.
proof idea
The definition constructs the record by assigning each field of CrossPatternMatrixCert to the corresponding sibling theorem: D5_squared to D5_squared, D5_2cube to D5_times_2cube, twoCube_squared to twoCube_squared, full_turn to full_turn, gap_squared to gap_squared, cube_faces_squared to cube_faces_squared, face_pairs_minus_gap_is_D_sq to face_pairs_minus_gap, and off_diag_count to offDiag_is_two_fourth.
why it matters
This definition realizes the structural meta-claim of the Cross-Pattern Matrix in the Wave-62 report by providing a single Lean object that packages the verified cross-products. It sits at the end of the module's arithmetic lemmas and supports the claim that the patterns form a non-degenerate matrix with distinct integers. The module doc states Lean status: 0 sorry, 0 axiom. No downstream uses are recorded yet.
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