twoThreePrimeCompositeDirection
plain-language theorem explainer
Names the rational orbit for the composite direction 2·3, built as the product of the canonical two-prime and three-prime directions. Anyone working the two-adic obstruction or native-cost uniqueness cites it as the first mixed composite test point. The body is a one-line multiplicative construction on RatioOrbit.
Claim. Let the two-prime and three-prime directions be the canonical rational orbits associated to the primes $2$ and $3$. The two-three prime composite direction is the product orbit $2\cdot 3$ in the space of rational orbits (integer numerator over a nonzero orbit denominator).
background
In the Primitive Recognition Calculus, a RatioOrbit is a display of a rational as an integer numerator over a nonzero orbit denominator (K4.7). Prime directions are the canonical RatioOrbit values attached to prime orbits; twoPrimeDirection and threePrimeDirection are those for $2$ and $3$, the latter flagged as the canonical non-two prime for the first concrete two-adic mixed-composite test.
The surrounding module develops native-cost uniqueness and two-adic axis-twist characters on ratio orbits. Composite directions are needed because local orientation and cost-consistency conditions must be checked not only on primes but on mixed products such as $2\cdot 3$, where two-adic twisting can fail to preserve prime-pair product structure.
proof idea
Pure definitional abbreviation: apply RatioOrbit.mul to twoPrimeDirection and threePrimeDirection. No lemmas or tactics; the value is exactly the multiplicative composite of those two prime directions.
why it matters
This is the first mixed composite direction in the two-adic obstruction story. Downstream it appears in the local-orientation predicate (image of $2\cdot 3$ must be the composite or its reciprocal), in the image theorem for two-adic axis-twist characters on that mixed point, and in several refutations: native-cost uniqueness target, prime-calibration forcing nonunit local orientation, and failure of prime-pair product cost consistency for the two-adic axis-twist character. It is the concrete test orbit that separates admissible native costs from two-adically twisted impostors in the PRC uniqueness chain.
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