D_times_cube_faces
Arithmetic identity five times six equals thirty is recorded for the configuration dimension times cube faces entry. Researchers assembling the cross-product matrix in the Wave-64 meta-theorem cite this when tabulating integer relations among D, the cube, and gap quantities. The proof reduces to a single decision procedure on natural numbers.
claimThe product of the configuration dimension $5$ and the cube face count $6$ equals $30$.
background
The Cross-Pattern Matrix module organizes five RS patterns into a non-degenerate matrix of cross-products: configuration dimension D=5, eight-tick cube 2³=8, J-uniqueness at zero, phi-ladder, and gap-45 with cube faces. Each pair produces a distinct integer or relation, with D times cube faces listed as thirty in the supplied table. The module imports the hypothesis structure from PhiLadderLattice.that, described as a structure that can be assumed in downstream theorems and discharged when analytic infrastructure becomes available, specifically for phi-ladder Poisson summation on rapidly decreasing functions.
proof idea
The proof is a one-line wrapper that applies the decide tactic to confirm the natural-number equality.
why it matters in Recognition Science
This theorem supplies the D times cube faces cell of thirty in the C26 cross-pattern matrix meta-claim. It contributes to the structural assertion that the five patterns yield distinct entries tied to quantities such as D²=25 and full-turn 360. The result closes a basic arithmetic step in the matrix without engaging open questions in the phi-ladder summation.
scope and limits
- Does not derive the equality from Recognition Science axioms or forcing chain steps.
- Does not extend to real numbers or other numeric types.
- Does not reference J-cost, defectDist, or phi-ladder summation details.
- Does not compute any other matrix entry or cross-product.
formal statement (Lean)
72theorem D_times_cube_faces : (5 : ℕ) * 6 = 30 := by decide
proof body
Term-mode proof.
73
74/-- D² × cube faces = 150 — a quantity that exceeds gap45 by exactly D²·D - 1·45 = 5·D = 25. -/