cube_sq_plus_cube
The arithmetic identity (2³)² + 2³ = 72 is recorded as a supporting fact inside the cross-pattern matrix. Researchers verifying the Wave-64 meta-theorem entries for the cube pattern would cite it when checking RS bounds. The proof is a single decision procedure that evaluates the natural-number expression directly.
claim$(2^{3})^{2} + 2^{3} = 72$
background
The module constructs a 5-by-5 cross-product matrix whose rows and columns are indexed by the five core RS patterns: D=5, 2³=8, J(1)=0, the phi-ladder, and gap-45/cube-faces. Each off-diagonal or derived entry is required to match a known RS quantity such as 64 = 2⁶ or 360 = 2³ · 45. The present identity supplies the derived value 72 that appears in certain bounds involving the cube pattern. Upstream definitions of configDim supply the integer 5 (or 3) that fixes the spatial ledger dimension in the surrounding matrix construction.
proof idea
One-line wrapper that applies the decide tactic to confirm the numerical equality on natural numbers.
why it matters in Recognition Science
The declaration supplies one concrete matrix entry required by the C26 meta-theorem that every non-trivial cross-product equals a recognized RS quantity. It therefore participates in the structural claim that the five patterns generate a non-degenerate integer matrix. The cube factor 2³ directly instantiates the eight-tick octave (T7) of the forcing chain; the identity therefore anchors the arithmetic layer that later feeds mass-ladder and alpha-band calculations.
scope and limits
- Does not claim the identity holds for real or complex arguments.
- Does not derive any deeper RS relation beyond the arithmetic fact.
- Does not reference or constrain the value of configDim.
- Does not assert uniqueness among possible matrix-derived bounds.
formal statement (Lean)
69theorem cube_sq_plus_cube : (2 : ℕ)^3 * 2^3 + 2^3 = 72 := by decide
proof body
Term-mode proof.
70
71/-- D × cube faces = 30 (configDim × cube-face count). -/