twoCube_squared
The arithmetic identity eight times eight equals sixty-four certifies the double-cube entry in the cross-pattern matrix. Wave-64 meta-theorem authors cite this result to confirm the 2^6 term arising from the eight-tick octave squared. The proof proceeds by direct evaluation via a decision procedure on natural numbers.
claim$2^3 · 2^3 = 64$
background
The module establishes a structural meta-claim that five Recognition Science patterns (D=5, 2³=8, J=0, φ, gap-45) form a non-degenerate matrix of cross-products. Each pair yields a distinct integer or relation, with the (2³, 2³) slot defined as 64 and interpreted as 2^6 (DFT × DFT, double cube). The eight-tick octave appears here as the period-2^3 factor from the forcing chain.
proof idea
The proof is a one-line wrapper that invokes the decide tactic to verify the natural-number equality by direct computation.
why it matters in Recognition Science
This result supplies the twoCube_squared field inside the CrossPatternMatrixCert structure, which certifies the full wave-64 matrix. It anchors the 64 = 2^6 entry that links the eight-tick octave to itself, consistent with T7 in the forcing chain. The matrix as a whole demonstrates that all non-trivial pattern products remain distinct integers.
scope and limits
- Does not extend the equality beyond natural numbers.
- Does not supply a symbolic or inductive proof.
- Does not address floating-point or approximate arithmetic.
- Does not connect the identity to physical units or constants.
Lean usage
def cert : CrossPatternMatrixCert := { D5_squared := D5_squared, D5_2cube := D5_times_2cube, twoCube_squared := twoCube_squared, full_turn := full_turn, gap_squared := gap_squared }formal statement (Lean)
43theorem twoCube_squared : (2 : ℕ)^3 * 2^3 = 64 := by decide