offDiagEntries_eq
The cross-pattern matrix built from the five Recognition Science patterns has exactly 16 off-diagonal entries. Researchers citing the Wave-62 meta-theorems use this count to establish that the matrix of pattern products is non-degenerate. The equality is obtained by direct numerical evaluation of the squared off-diagonal size.
claimThe number of off-diagonal entries in the 5-by-5 cross-pattern matrix of Recognition Science patterns equals 16.
background
The Cross-Pattern Matrix module defines a structural array whose rows and columns are indexed by the five patterns D=5, 2^3=8, J=0, the phi-ladder, and gap-45/cube-faces. Each non-trivial product is listed explicitly, for example 25 = D^2 for cognitive pair states, 40 = D · 2^3 for attention space, 64 = 2^6 for the double cube, 5 phi for the theta carrier, and 360 = 2^3 · 45 for the full turn. The off-diagonal entry count is obtained by squaring the off-diagonal size, which evaluates to 16.
proof idea
The proof is a one-line wrapper that applies the decide tactic to the definition of the off-diagonal entry count as the square of the off-diagonal size.
why it matters in Recognition Science
This result supplies the numerical base for the C26 meta-claim that the five patterns form a non-degenerate matrix of cross-products. It confirms the count of 16 positions whose values include 5 phi, 8 phi, 45 phi, and 360, aligning with the eight-tick octave and phi-ladder landmarks. No downstream theorems are recorded, so the declaration functions as a terminal fact within the cross-domain layer.
scope and limits
- Does not enumerate which matrix positions count as off-diagonal.
- Does not derive the numerical values of those entries.
- Does not extend the construction to pattern sets larger than five.
- Does not assign physical units or observational tests to the entries.
formal statement (Lean)
96theorem offDiagEntries_eq : offDiagEntries = 16 := by decide
proof body
Term-mode proof.
97
98/-- 16 = 2⁴ (the off-diagonal entry count is a power of 2). -/