robotic_has_6
The theorem establishes that the RoboticDOF type satisfies HasCubeFaceCount by having cardinality exactly 6. Cross-domain researchers cite it when building the aggregated certificate that unifies the count 6 across quarks, leptons, cortical layers, Braak stages, and robotic axes. The proof is a one-line term that unfolds the cardinality predicate and invokes decide on the derived Fintype instance.
claimThe robotic degrees of freedom type has cube-face count: $|RoboticDOF| = 6$.
background
HasCubeFaceCount is the proposition that a finite type T satisfies Fintype.card T = 6. RoboticDOF is the inductive type whose six constructors enumerate the three translational axes x, y, z and the three rotational axes rollAxis, pitchAxis, yawAxis, deriving Fintype, DecidableEq, and Repr. The module sets this instance inside the C15 structural claim that the number 6 appears universally as the face-level enumeration for spatial dimension D = 3, expressed as 6 = 2 · 3 where the factor 2 is the binary state count of a cube face. Upstream results include the spatial-dimension definition dim := D from the lepton-generations module and the identity functors from ObserverForcing and VantageCategory that anchor the forcing chain.
proof idea
The proof is a term-mode one-liner. It unfolds HasCubeFaceCount to reduce the goal to Fintype.card RoboticDOF = 6, then applies the decide tactic, which succeeds because RoboticDOF derives Fintype and is enumerated by exactly six constructors.
why it matters in Recognition Science
This theorem supplies the robotic instance to cubeFaceUniversalityCert, the definition that packages all six-domain instances into a single certificate. It directly fills the C15 claim in the module documentation that 6 = D_cube · D_spatial for D = 3. In the Recognition Science framework it reinforces the spatial dimension D = 3 forced at T8 of the unified forcing chain and the eight-tick octave, supplying an engineered-system example that parallels the six leptons and six quarks.
scope and limits
- Does not derive the six robotic axes from the Recognition Composition Law or J-cost minimization.
- Does not map the axes onto any specific step of the T0-T8 forcing chain.
- Does not prove the count 6 for continuous manifolds or non-discrete structures.
- Does not address physical realizability or dynamics of the robotic degrees of freedom.
Lean usage
example : HasCubeFaceCount RoboticDOF := robotic_has_6formal statement (Lean)
59theorem robotic_has_6 : HasCubeFaceCount RoboticDOF := by
proof body
Term-mode proof.
60 unfold HasCubeFaceCount; decide
61
62/-- The cube-face identity: 6 = 2 × 3 = face-binary × spatial-dim. -/