ckm_dimension
plain-language theorem explainer
The CKM quark-mixing matrix is forced to be exactly 3×3: spatial dimension three yields three opposite-face pairs on the cube, hence three fermion generations. Anyone deriving CKM structure from Q₃ geometry cites this dimension count. The proof is pure reflexivity on the definition that the face-pair count equals the dimension.
Claim. The number of pairs of opposite faces on a cube in spatial dimension $D = 3$ equals $3$. Consequently the Cabibbo-Kobayashi-Maskawa matrix is exactly $3 \times 3$, matching the three fermion generations.
background
Recognition Science forces spatial dimension $D = 3$ at step T8 of the unified forcing chain. On a $D$-dimensional cube, opposite faces come in exactly $D$ pairs; that count is identified with the number of fermion generations. Upstream, the face-pair count is defined as the identity on dimension and documented as: "Number of pairs of opposite faces on a $D$-dimensional cube. For a cube, opposite faces come in pairs: $D$ pairs total."
This module builds the CKM matrix from Q₃ hypercube geometry, generation torsion ${0, 11, 17}$, and Gray-code chirality. Mass eigenstates carry those torsion levels; weak eigenstates are labeled by the three even-sign-flip SU(2) generators. Both bases therefore live on a three-dimensional generation space, so every overlap $V_{ij}$ is an entry of a $3 \times 3$ matrix.
proof idea
Term-mode one-liner: reflexivity. Opposite-face pairs on a $D$-cube are defined to equal $D$ itself, so the case $D = 3$ is definitional equality. No lemmas are invoked.
why it matters
Records the structural reason the CKM matrix is $3 \times 3$ inside Recognition Science: $D = 3$ (forcing-chain T8) implies three opposite-face pairs, hence three generations. The module treats $V$ as the overlap between mass and weak eigenstates on that three-generation space, with amplitudes suppressed by $\varphi^{-|\Delta\tau|}$ from torsion gaps.
Sibling results (torsion gaps, flip weights, Wolfenstein-style suppressions) all presuppose a $3 \times 3$ index set. No downstream theorem currently depends on this fact in the dependency graph, but every CKM-from-cube formula is indexed by the three generations it certifies. It closes the dimension-counting step that links cube geometry to the observed three-family structure of quark mixing.
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