bit0_flips_four
plain-language theorem explainer
In the canonical 3-bit Gray-code cycle on the cube, axis 0 flips four times per eight-tick period. Anyone deriving generation-dependent coupling, CKM structure, or the η_B rung from chirality cites this count. The proof is a one-line native decision on the finite flip list.
Claim. If $\mathrm{bitFlipCount}(b)$ is the number of times bit $b\in\{0,1,2\}$ flips along the directed 8-tick Gray-code walk on $Q_3$, then $\mathrm{bitFlipCount}(0)=4$.
background
The module establishes that the canonical 3-bit Gray-code cycle on the cube $Q_3$ is chiral: the directed walk distinguishes clockwise from counterclockwise face traversal. That chirality is the Recognition Science origin of CP violation, while $J(x)=J(1/x)$ keeps CPT intact.
The eight-tick recognition operator walks the Gray path $[0,1,3,2,6,7,5,4]$, flipping bits in the pattern $[0,1,0,2,0,1,0,2]$. The helper bitFlipCount simply counts occurrences of each axis in that flip list. Face-pairs correspond to particle generations, so unequal flip counts mean unequal generation coupling over one octave (the T7 eight-tick period).
The flip-asymmetry vector is then $\Delta=(4-8/3,2-8/3,2-8/3)=(4/3,-2/3,-2/3)$, whose nonzero norm breaks $S_3$ axis permutation symmetry.
proof idea
One-line computational proof: native_decide evaluates bitFlipCount 0 on the finite list List.ofFn flippedBit and checks equality with 4. No algebraic lemmas are required beyond the definition of the flip-count.
why it matters
This is the first coordinate of the flip triple $[4,2,2]$ that feeds the whole chirality stack. Downstream it is packaged into chiralityCert (as flipCounts), re-exported as gen1_flips (generation 1 sees four flips), and supplied to mixingData for weak-basis mixing angles.
In cosmology it is the left factor of the structural identity chirality_product_equals_gap_minus_one: $\mathrm{bitFlipCount}(0)\times|\mathrm{torsionGap}(0,1)|=4\times 11=44$, matching $\mathrm{dimensionGap}(D)-A$ at $D=3$, and thereby fixing eta_B_rung_from_chirality = -44. The same count underwrites CKM structural certificates (ckmStructureCert, flipWeight_sum, flipWeight_values).
Framework landmarks: T7 eight-tick octave, T8 $D=3$, and the geometric origin of flavor mixing / CP violation while CPT is preserved by $J$-symmetry.
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