two_cube_pair_64
plain-language theorem explainer
The theorem shows that the Cartesian product of any two finite types each of cardinality 8 has cardinality 64. Cross-domain researchers cite it when combining distinct 2³-cube instances such as DFT modes and Q3 vertices. The proof is a one-line wrapper that unfolds the HasTwoCubeCount assumptions and applies the standard product cardinality identity.
Claim. If $A$ and $B$ are finite types with $|A|=8$ and $|B|=8$, then $|A×B|=64$.
background
The module C14 collects instances of the count 8=2³ across Recognition Science domains for spatial dimension D=3, including DFT-8 modes, Q3 vertices, Pauli elements, and 8-tick periods. HasTwoCubeCount is the predicate asserting that a finite type T satisfies Fintype.card T = 2^3. The local setting is the structural identity that all listed domains share the maximal-periodic 2³-cube count.
proof idea
The proof is a one-line wrapper. It unfolds HasTwoCubeCount at the hypotheses to obtain the equalities card A = 8 and card B = 8, then applies simp with Fintype.card_prod to reduce the product cardinality directly to 64.
why it matters
This declaration supports the parent theorem dft_q3_product, which concludes Fintype.card (DFTMode × Q3Vertex) = 64. It fills the cross-domain product step in the C14 universality claim for the 2³-cube, linking to the eight-tick octave (T7) and D=3 (T8) in the forcing chain. The module maintains zero sorry and zero axiom for these identities.
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