face_pairs_at_D3
plain-language theorem explainer
The result shows that a cube in three dimensions has exactly three pairs of opposite faces. This grounds the three fermion generations in Recognition Science via the forced spatial dimension. The proof is immediate by reflexivity from the definition of the face-pair count.
Claim. A cube in three spatial dimensions possesses exactly three pairs of opposite faces.
background
The ParticleGenerations module formalizes P-001 on three fermion generations. It starts from the dimension-forcing result that spatial dimensions equal three and considers the geometry of a three-dimensional cube. Each pair of opposite faces maps to one independent coherence mode in the ledger structure, producing three generations total.
proof idea
The proof is a one-line term wrapper that applies reflexivity. It reduces directly to the definitional equality that the face-pair count equals the dimension value when the argument is three.
why it matters
This declaration completes the P-001 resolution by confirming the face-pair count for D=3. It connects to the framework landmark that D=3 is forced by the eight-tick octave and linking structure, yielding the unique three-generation outcome. The module notes that this count distinguishes the framework from two or four generations; downstream siblings such as three_generations_from_dimension and no_fourth_generation build on the same geometric step.
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