cube_faces_squared
The equality 6 times 6 equals 36 encodes the face-pairing count on the three-cube within the Recognition Science cross-pattern matrix of five core patterns. Researchers verifying the Wave-62 meta-theorem would cite this to anchor the gap-45 row entries. The proof is a one-line decision procedure that confirms the natural-number arithmetic directly.
claim$6$ times $6$ equals $36$, where $6$ is the number of faces of the three-dimensional cube $Q_3$ and the product counts face pairings in the cross-pattern matrix.
background
The CrossPatternMatrix module constructs a non-degenerate matrix of cross-products among the five RS patterns D=5, the eight-tick octave $2^3$, J(1)=0, the golden ratio phi, and gap-45. The module document tabulates the entries and states that each non-trivial product matches a known RS quantity such as 25 = D squared for cognitive pair states and 45 = D squared times D for the gap itself.
proof idea
The proof is a one-line wrapper that applies the decide tactic to verify the natural-number equality directly.
why it matters in Recognition Science
This theorem supplies a verified entry that feeds the CrossPatternMatrixCert structure, which assembles the full matrix for the Wave-62 cross-domain meta-theorem. It links to the gap-45 structure via the relation 36 = 45 minus 9 and to D=3 spatial dimensions from the unified forcing chain T0-T8. The parent result crossPatternMatrixCert uses these equalities to demonstrate matrix non-degeneracy.
scope and limits
- Does not derive 36 from the gap definition as closure factor times Fibonacci factor.
- Does not incorporate phi-ladder or golden-ratio relations.
- Does not address physical interpretations such as theta carriers or attention space.
- Does not extend to other matrix entries or higher-dimensional analogs.
formal statement (Lean)
78theorem cube_faces_squared : (6 : ℕ) * 6 = 36 := by decide
proof body
Term-mode proof.
79
80/-- 36 + 8 = 44 = 2 · gap45 - 46 (relation between face-pairs and the
81 gap-45 structure: 36 = gap45 - 9 = 45 - 9 = 45 - D²). -/