CrossPatternMatrixCert
The CrossPatternMatrixCert structure bundles the integer identities obtained by pairwise multiplication of the five core Recognition Science patterns (D=5, 2³=8, gap=45, cube faces=6). A researcher working on the Wave-64 cross-domain meta-theorem would cite it to certify that each product matches a known RS quantity such as D²=25, D·2³=40, 2^6=64, full turn=360, and gap²=2025. The definition is a direct record that packages these pre-proved arithmetic facts with no additional steps.
claimThe structure CrossPatternMatrixCert consists of the equalities $5×5=25$, $5×2^3=40$, $2^3×2^3=64$, $2^3×45=360$, $45×45=2025$, $6×6=36$, $45-6×6=9$, and the off-diagonal entry count equals $2^4$.
background
The CrossDomain module presents the cross-pattern matrix as the structural meta-claim for the Wave-64 theorem. The five patterns are D=5, 2³=8 (octave tick), gap=45, cube faces=6 on Q₃, and off-diagonal count=16. Upstream results supply the base identities: cube_faces_squared proves 6·6=36 together with the relation 36=gap-9=45-D²; D5_squared proves 5·5=25; full_turn proves 2³·45=360; gap_squared proves 45·45=2025; twoCube_squared proves 8·8=64; offDiagEntries defines the count as offDiagSize squared.
proof idea
As a structure definition, CrossPatternMatrixCert is a record that directly bundles the sibling theorems D5_squared, D5_times_2cube, twoCube_squared, full_turn, gap_squared, cube_faces_squared and the definition offDiagEntries. No tactics are executed inside the declaration; each field simply references the already-proved arithmetic identity.
why it matters in Recognition Science
This declaration supplies the concrete certificate for the C26 cross-pattern matrix meta-theorem. It is instantiated by the downstream definition crossPatternMatrixCert. Within the Recognition Science framework it certifies the non-degenerate cross-products among the core patterns that appear in the forcing chain and the phi-ladder, confirming each pair yields a distinct integer matching quantities such as attention space (40) and full turn (360).
scope and limits
- Does not derive the five patterns from the T0-T8 forcing chain.
- Does not prove uniqueness of matrix entries beyond the listed identities.
- Does not connect the entries to the J-cost or Recognition Composition Law.
- Does not address the phi-ladder or Berry creation threshold.
formal statement (Lean)
101structure CrossPatternMatrixCert where
102 D5_squared : (5 : ℕ) * 5 = 25
103 D5_2cube : (5 : ℕ) * 2^3 = 40
104 twoCube_squared : (2 : ℕ)^3 * 2^3 = 64
105 full_turn : (2 : ℕ)^3 * 45 = 360
106 gap_squared : (45 : ℕ) * 45 = 2025
107 cube_faces_squared : (6 : ℕ) * 6 = 36
108 face_pairs_minus_gap_is_D_sq : (45 : ℕ) - 6 * 6 = 9
109 off_diag_count : offDiagEntries = 2^4
110