twoAdicGeneratedNativeCost_not_strengthened_hypotheses

In the Primitive Recognition Calculus, a native cost is a map $F$ on ratio orbits meant to realize the RS cost functional. The ordinary native-cost interface packages RCL-type identities, normalization, and calibration probes. After a two-adic no-go, the module strengthens that interface: keep those fields, and add prime-pair product calibration at the cost level (not only on characters or traces).

The two-adic generated native cost is the concrete counter-model under study: on the unit orbit it is zero; elsewhere it is the cost induced by the two-adic axis-twist character. Upstream work already shows this $F$ fails prime-pair product calibration. The strengthened hypothesis structure is exactly ordinary native hypotheses plus that prime-pair product field, so any cost missing the latter cannot inhabit the strengthened interface.

One-line projection proof. Assume the strengthened native-cost hypotheses for the two-adic generated cost. Project to the prime_pair_product_cost field to obtain prime-pair product calibration. Discharge the goal by the prior theorem that this same cost is not prime-pair product calibrated. No new arithmetic is done here.

This seals the two-adic generated cost out of the post-no-go strengthened interface. In the Recognition forcing picture, native-cost uniqueness feeds the J-uniqueness landmark (T5): the cost must be the standard $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$), not a two-adic twist. The declaration records that strengthening by prime-pair product calibration is enough to exclude this family, so later uniqueness theorems can assume the strengthened package without re-checking the two-adic model.

No downstream consumers are wired yet (used_by is empty); the lemma is a local closure step inside PRC native-cost uniqueness, sitting beside the direct prime-pair product no-go for the same cost.