crossPatternMatrixCert
This definition assembles the verified arithmetic identities among the five Recognition Science patterns (D=5, 2^3, J=0, phi, gap-45) into a single certificate record. A researcher citing the Wave-64 meta-theorem would reference it to treat the cross-product matrix as a closed, non-degenerate object whose entries match known RS quantities. The construction is a direct record literal that plugs in the decide lemmas for each field.
claimThe cross-pattern matrix certificate is the structure with fields $5^2=25$, $5*2^3=40$, $2^3*2^3=64$, $2^3*45=360$, $45^2=2025$, $6^2=36$, $45-6^2=9$, and off-diagonal entry count equal to $2^4$.
background
The CrossPatternMatrixCert structure collects the arithmetic identities realizing the cross-domain matrix of the module documentation. The five patterns are D=5 (spatial dimension), 2^3=8 (octave tick count from T7), J=0 (J-cost at unity), the golden ratio phi, and gap-45 (Berry creation threshold scaled by D^2). Upstream results include the individual theorems such as D5_squared proving 55=25 by decision and full_turn establishing 845=360 as the full angular turn.
proof idea
The definition is a one-line record constructor that directly assigns each field of CrossPatternMatrixCert to the corresponding theorem: D5_squared to the decide proof of 25, D5_2cube to the product 40, twoCube_squared to 64, full_turn to 360, gap_squared to 2025, cube_faces_squared to 36, face_pairs_minus_gap to 9, and offDiag_is_two_fourth to the off-diagonal count.
why it matters in Recognition Science
This definition realizes the structural meta-claim of the Cross-Pattern Matrix by packaging the verified entries for the Wave-64 meta-theorem. It closes the scaffolding for cross-domain relations among D=5, the eight-tick octave, and the gap-45 structure, aligning with T7 and T8 landmarks through the explicit 2^3 and 5 factors. The module states zero sorry and zero axiom, confirming the matrix entries are distinct and match known RS quantities such as 25 for cognitive pair states and 360 for full turn.
scope and limits
- Does not prove uniqueness of matrix entries beyond the eight listed equalities.
- Does not derive physical interpretations such as 5 phi as a theta carrier frequency.
- Does not extend the matrix to patterns outside the five listed in the module documentation.
formal statement (Lean)
111def crossPatternMatrixCert : CrossPatternMatrixCert where
112 D5_squared := D5_squared
proof body
Definition body.
113 D5_2cube := D5_times_2cube
114 twoCube_squared := twoCube_squared
115 full_turn := full_turn
116 gap_squared := gap_squared
117 cube_faces_squared := cube_faces_squared
118 face_pairs_minus_gap_is_D_sq := face_pairs_minus_gap
119 off_diag_count := offDiag_is_two_fourth
120
121end IndisputableMonolith.CrossDomain.CrossPatternMatrix