twoThreePrimeMixedDirection

In the primitive recognition calculus, rational comparison data live in RatioOrbit: an integer numerator over a nonzero distinction-nat denominator (K4.7). Reciprocal on ratio orbits is total and sends zero to zero, matching the usual map on $\mathbb{Q}$ (K4.12). The dual-recognition symmetry on positive reals is the involution $\iota(x)=x^{-1}$.

The module builds native cost uniqueness tests by comparing characters and costs on prime directions and their composites. The two-prime and three-prime directions are the canonical orbits of the primes $2$ and $3$. A two-adic axis twist flips the $2$-branch while leaving other prime branches fixed; at the composite $2\cdot 3$ that forced mixed image is exactly $3/2$.

One-line definitional construction: multiply the reciprocal of the two-prime direction by the three-prime direction inside the ratio-orbit monoid. No tactics or lemmas beyond the constructors RatioOrbit.mul, RatioOrbit.recip, and the two named prime directions.

This orbit is the concrete mixed image that a two-adic axis twist character must hit at the composite $2\cdot 3$. Downstream theorems use it to show the twist character sends that composite to $3/2$ (cross-equality), that the mixed direction is not cross-equal to the plain composite $2\cdot 3$, and that two-adic generated native costs and twist characters fail prime-pair product cost consistency and calibration. Those failures feed the refutation of the native cost uniqueness target: uniqueness fails once two-adic axis twists are admitted. In the broader forcing chain this is local scaffolding for J-cost uniqueness (T5) and the recognition composition law, by ruling out non-canonical cost factors on small prime composites.