cubeFaces_eq
plain-language theorem explainer
The equality states that the unit cube face count equals the product of the binary face factor and the spatial dimension. Cross-domain cardinality analyses in Recognition Science cite this identity when verifying that spectrum members decompose into the primitives 2 and 3. The proof is a one-line decision procedure that evaluates the concrete arithmetic on the supplied definitions.
Claim. The number of faces on the unit cube equals twice the spatial dimension: $6 = 2 × 3$.
background
In the CrossDomain.CardinalitySpectrum module the local setting is the RS cardinality spectrum whose members are generated from the small cube-generators {2, 3}, the config dimension 5, and gap45. Dspatial is the definition Dspatial : ℕ := 3. twoFace is the definition twoFace : ℕ := 2 (binary face count). cubeFaces is the sibling definition cubeFaces : ℕ := 6 (explicitly 2 * Dspatial). Upstream results supply the same cubeFaces value in ParadigmShiftLattice and FreudenthalTriangulationCert, each carrying the doc-comment “Six cube faces of Q₃” or “Unit cube face count.”
proof idea
The proof is a one-line term proof that invokes the decide tactic. decide evaluates the equality 6 = 2 * 3 directly from the concrete natural-number definitions of cubeFaces, twoFace and Dspatial.
why it matters
This theorem supplies the cubeFaces_as_D field inside the cardinalitySpectrumCert definition, confirming that the cube face count belongs to the structured spectrum. It aligns with the T8 forcing step that sets D = 3 spatial dimensions and supports the Freudenthal triangulation certificate by providing the matching face-count identity. The result closes a verification step showing the spectrum is generated by RS primitives rather than chosen arbitrarily.
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