prc_single_prime_calibration_insufficient

In the Primitive Recognition Calculus (PRC), costs live on ratio orbits of the rational carrier, compared by cross-equivalence: two orbits match when cross-scaled numerators and denominators balance as signed orbits. A PRC ratio character is a map on orbits that preserves the unit, multiplies, and sends reciprocals to reciprocals, all up to that relation. It is the discrete stand-in for a d'Alembert factorization factor of a native cost.

The module studies whether the native cost $J$ is forced on this discrete carrier by prime-axis calibrations alone. Each prime orbit is an independent free generator of the multiplicative group of positive rationals (up to units and signs in the orbit picture). Orientation of an axis means whether the character acts as the identity or as inversion on that prime direction.

Upstream, cross-equivalence is transitive and symmetric, and prime directions are never cross-equivalent to their own reciprocals. Those facts underwrite the contradiction step below.

Term-mode existence: instantiate $\chi$ as the three-adic axis twist character $x \mapsto x \cdot 3^{-2 v_3(x)}$. Feed the three packaged lemmas that it is a PRC ratio character and that it acts as the identity on the prime-2 direction.

The remaining goal is non-identity on the prime-3 direction. Assume for contradiction that the twist fixes the 3-direction up to cross-equivalence. Symmetry plus the packaged fact that the twist sends the 3-direction to its reciprocal then yields, by transitivity, that the 3-direction is cross-equivalent to its own reciprocal. That contradicts the lemma that a prime orbit is never cross-equivalent to its reciprocal. Hence the character cannot fix the 3-axis.

This is the complementary-axis headline blocker for $\delta$-native cost non-forcing. Its sibling flips 2 and fixes the odds; this one fixes 2 and flips 3. Downstream, both are collapsed into a single base-parameterized theorem: every prime axis is an independent orientation freedom, so no finite (indeed no proper) set of prime calibrations forces $J$ on the rational carrier.

In the Recognition forcing chain this pins why T5 $J$-uniqueness needs the continuous completion. The continuous law that forces $J$ escapes the obstruction only because its calibration is a second-derivative condition at the unit, constraining a full neighborhood at once. The discrete carrier has one free sign per prime; the continuous carrier does not. The forward repair is therefore to derive that single neighborhood-level calibration from the cost of one $\delta$ act, not to pile more prime pins.

It also sits next to the $\lambda=2$ cost family member used on the completion side as the matching continuous non-forcing witness (wrong curvature at the unit).