sixtyfour_identities
The declaration establishes the arithmetic identities 64 = 8 squared and 64 = 2 to the sixth. Researchers unifying 8-element structures across Recognition Science domains cite it when linking DFT modes, the recognition cube, and Pauli elements under a common cardinality. The proof reduces to a single decide tactic that confirms both equalities by direct computation.
claim$64 = 8^2$ and $64 = 2^6$.
background
The module collects instances of the count 8 = 2^3 across Recognition Science domains for spatial dimension D = 3. These include DFT-8 modes for fundamental harmonic decomposition, the Q3 vertex count in the recognition cube, the eight elements of the Pauli group, and the 8-tick fundamental period. The local setting proves that all such structures share the underlying identity of cardinality 2^3.
proof idea
The proof is a one-line wrapper that applies the decide tactic to verify the arithmetic identities directly.
why it matters in Recognition Science
This identity anchors the TwoCubeUniversalityCert structure, which certifies HasTwoCubeCount for DFTMode, Q3Vertex, PauliElement, and TickPhase while enforcing equicardinality. It fills the C14 claim by grounding the eight-tick octave and D = 3 in the shared count 64 = 8 * 8 = 2^6. No open questions are addressed.
scope and limits
- Does not interpret the physical meaning of the 8-count in each domain.
- Does not derive the count from deeper Recognition Science axioms such as the forcing chain.
- Does not address higher-dimensional generalizations beyond D = 3.
formal statement (Lean)
97theorem sixtyfour_identities : 64 = 8 * 8 ∧ 64 = 2^6 := by decide
proof body
98