bit2_flips_two
plain-language theorem explainer
Along the canonical 3-bit Gray-code walk on the cube, bit 2 flips exactly twice in one 8-tick cycle. Anyone deriving generation-3 coupling, CKM flip weights, or the chirality certificate cites this count. The proof is a one-line native decision over the explicit flip list.
Claim. If $\mathrm{bitFlipCount}(b)$ counts how many times bit $b\in\{0,1,2\}$ flips along the directed 8-step Gray-code cycle on $Q_3$, then $\mathrm{bitFlipCount}(2)=2$.
background
The module treats the canonical Gray-code cycle on the 3-cube $Q_3$ as the geometric carrier of the eight-tick recognition octave (T7). The walk visits all eight vertices once; successive edges flip exactly one coordinate bit. The sequence of flipped bits is $[0,1,0,2,0,1,0,2]$.
bitFlipCount b is the multiplicity of bit $b$ in that list: how often axis $b$ is toggled during one full period. Axes are identified with particle generations via the face-winding / ParticleGenerations correspondence, so unequal flip counts mean unequal kinematic drive per generation.
J-cost remains symmetric under $x\leftrightarrow 1/x$ (CPT), but the directed walk breaks $S_3$ axis permutation. The flip counts $[4,2,2]$ produce the asymmetry vector $\Delta=(4/3,-2/3,-2/3)$ that measures chirality.
proof idea
One-line computational proof: native_decide evaluates bitFlipCount 2 against the concrete List.ofFn flippedBit and checks equality with 2. No algebraic lemmas are required; the finite 8-element list is decided by the kernel.
why it matters
Closes the third coordinate of the flip-count triple used everywhere chirality or flavor mixing is certified. Downstream: chiralityCert packages it with the bit-0 and bit-1 counts; gen3_flips is a direct alias identifying generation 3 with axis 2; mixingData and the CKM structure certificate (flipWeight_sum, flipWeight_values, ckmStructureCert) normalize these counts to weights $1/2,1/4,1/4$; berry_gen3 uses the same count for the generation-3 Berry phase. Together with the 4:2:2 split this is the kinematic source of CKM/PMNS misalignment while CPT (J-symmetry) stays intact. Landmark link: eight-tick octave (T7) plus directed Gray walk on $D=3$ (T8).
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