no_fourth_generation
The theorem shows that a three-dimensional cube has three pairs of opposite faces, so the ledger cannot accommodate four fermion generations. Physicists deriving the fermion spectrum from spatial dimension would cite this to close the three-generation count. The proof is a one-line numerical normalization that unfolds the face-pairs definition to the numeral 3.
claimIn three spatial dimensions the number of opposite face pairs on the cube is not four: $3 ≠ 4$.
background
The module formalizes P-001: three generations of fermions. It starts from D = 3 spatial dimensions (forced in DimensionForcing) and the geometry of the 3-cube Q₃. The cube possesses exactly three pairs of opposite faces; each pair is identified with one generation in the ledger's mode structure. The local definition states face_pairs(D : ℕ) := D, so the count equals the dimension itself. An upstream definition in SpectralEmergence instead gives face_pairs(D) := D(D-1)/2, but the present theorem uses the simpler count that directly matches the cube's three axes.
proof idea
This is a one-line wrapper that applies norm_num to the face_pairs definition. The tactic unfolds face_pairs 3 to the numeral 3 and reduces the inequality 3 ≠ 4 to a decidable arithmetic fact.
why it matters in Recognition Science
The result supplies the foundation-level step for the no_fourth_generation theorems in MassHierarchy and ThreeGenerations. Those theorems tie the absence of a fourth generation to the eight-tick octave (T7) and the forced value D = 3 (T8). It completes the P-001 registry item by showing that the cube geometry admits only three face-pairs and therefore only three generations; the framework predicts neither two nor four.
scope and limits
- Does not derive D = 3 from the forcing chain.
- Does not compute CKM mixing angles between generations.
- Does not address mass splittings inside each generation.
- Does not exclude non-standard-model fourth-generation states.
formal statement (Lean)
59theorem no_fourth_generation :
60 face_pairs 3 ≠ 4 := by
proof body
Term-mode proof.
61 norm_num [face_pairs]
62
63/-- For D = 3, there cannot be 2 face-pairs. -/