generations_eq_dimension
The number of opposite face pairs on the three-dimensional cube equals three, fixing the count of particle generations. Researchers deriving Standard Model structure from the forced dimension D=3 would cite this equality. The proof is a direct term application of the three_generations lemma.
claimThe number of pairs of opposite faces on the three-dimensional cube equals three, which sets the fermion generation count to three.
background
The Spectral Emergence module starts from the forced spatial dimension D=3 (T8) to construct the binary cube Q₃ with eight vertices. The face_pairs definition counts D pairs of opposite faces on a D-dimensional cube, so face_pairs D := D. Upstream results include the ParticleGenerations.face_pairs definition and the three_generations lemma that specializes it at D=3. This sits inside the self-consistency loop T8 (D=3) → Q₃ → face-pair count → three generations, alongside the |Aut(Q₃)|=48 fermion states and φ-ladder masses.
proof idea
The proof is a one-line term wrapper that directly invokes the three_generations lemma from the ParticleGenerations module.
why it matters in Recognition Science
This theorem supplies the explicit link from cube geometry to the three generations required by the module's derivation of SU(3)×SU(2)×U(1) content and the 24 chiral flavors. It fills the step D=3 → 3 face pairs → 3 generations in the T8-forced chain. The result supports the claim that |Aut(Q₃)|=48 matches the Standard Model fermion state count with zero free parameters.
scope and limits
- Does not derive the generation count without assuming the cube model from D=3.
- Does not compute masses or mixing parameters on the φ-ladder.
- Does not address why D=3 is selected over other dimensions beyond the listed requirements.
- Does not extend the equality to non-cube geometries or higher-dimensional analogs.
formal statement (Lean)
210theorem generations_eq_dimension : face_pairs 3 = 3 := three_generations
proof body
Term-mode proof.
211
212/-! ## Part 4: Fermion Census — The 24 and 48 Theorems
213
214The total number of chiral fermion flavors is D × 2^D = 24.
215The total fermionic state count is |Aut(Q₃)| = 48. -/
216
217/-- The chiral fermion flavor count: each sector contributes a number
218 of flavors determined by matter_dim × generations × chiralities_per_sector. -/