cubeFaceUniversalityCert
plain-language theorem explainer
CubeFaceUniversalityCert assembles the HasCubeFaceCount proofs for quarks, leptons, cortical layers, Braak stages, and robotic degrees of freedom into one structure, together with the identities 6 = 2 × 3 and the Euler characteristic of the 3-cube. Researchers unifying spatial-dimension signatures across Recognition Science domains cite this certificate. The definition is a direct record construction that supplies each field from a pre-proved decide-based theorem.
Claim. The structure CubeFaceUniversalityCert certifies HasCubeFaceCount(Quark), HasCubeFaceCount(Lepton), HasCubeFaceCount(CorticalLayer), HasCubeFaceCount(BraakStage), HasCubeFaceCount(RoboticDOF), the equality $6 = 2 · 3$, the Euler relation $8 - 12 + 6 = 2$, the sum of quark and lepton cardinalities equaling 12, and $6^3 = 216$.
background
Recognition Science asserts that the spatial dimension D = 3 appears as the face count of the 3-cube Q₃ across domains. The structure CubeFaceUniversalityCert collects HasCubeFaceCount instances for the five listed domains plus the supporting arithmetic facts. Upstream theorems such as quark_has_6, lepton_has_6, cortical_has_6, and braak_has_6 each unfold HasCubeFaceCount and decide the cardinality. The identity cube_face_identity states that 6 equals 2 times 3, reflecting binary face states times spatial dimension, while q3_euler confirms V - E + F = 2 for the 3-cube.
proof idea
The definition constructs an instance of CubeFaceUniversalityCert by direct field assignment: quark_6 receives quark_has_6, lepton_6 receives lepton_has_6, cortical_6 receives cortical_has_6, braak_6 receives braak_has_6, robotic_6 receives robotic_has_6, six_equals_2_D receives cube_face_identity, q3_euler receives q3_euler, fermion_count receives quark_lepton_sum, and six_cubed receives six_cubed. Each referenced theorem is proved by unfolding HasCubeFaceCount and applying decide.
why it matters
This definition supplies the concrete certificate for the cube-face universality claim in the CrossDomain module, unifying the recurrence of the number 6 across particle physics, neuroscience, and robotics under the Recognition Science framework. It instantiates the structural claim that 6 = D_cube · D_spatial with D_spatial = 3 from the module documentation. The certificate stands as the terminal object for this cross-domain enumeration and would be referenced by any higher-level theorem asserting the universality of the 3-cube face count.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.