PRCPrimeCalibrationForcesPrimePairProductCostConsistencyTarget_refuted
plain-language theorem explainer
Prime-direction calibration of a ratio character does not force cost consistency on products of two native prime directions. The product-calibration target is therefore false. Anyone tracking branch-rigidity blockers in the primitive recognition calculus cites this. The proof transfers the claim through an equivalence to the already-refuted identity-branch uniformity target.
Claim. It is false that every ratio character $\chi$ which is prime-direction calibrated has the property that the induced cost on the product of any two native prime directions satisfies the cross-equality identity. In other words, prime calibration need not propagate to pairwise prime products.
background
In the primitive recognition calculus, a ratio character $\chi$ assigns orbit data on ratio orbits. From $\chi$ one builds an induced cost via costFromCharacter. Prime-direction calibration asks that this cost match the native cost on every native prime direction.
The product-calibration target strengthens that demand: once $\chi$ is a ratio character and is prime-direction calibrated, the induced cost on the product of any two native prime directions must satisfy the cross-equality identity. The module doc frames this as the natural composite surface; its $2\cdot p$ mixed-orientation instance is the current branch-rigidity blocker.
Upstream, identity-branch uniformity (same calibration forcing a uniform identity branch) is already refuted by an explicit two-adic axis-twist ratio character. An equivalence theorem identifies the product-calibration target with that identity-branch uniformity target.
proof idea
One-line transfer proof. Assume the product-calibration target. Apply the forward direction of the equivalence
PRCPrimeCalibrationForcesPrimePairProductCostConsistencyTarget_iff_prime_identity_branch_uniformity
to obtain identity-branch uniformity. Discharge the goal by the prior theorem that identity-branch uniformity is false (itself obtained from the constructed two-adic axis-twist ratio character). No new analytic work is done here.
why it matters
This closes the product-calibration surface as a viable forcing route. Downstream it is the engine for three further refutations: the two-prime-identity forcing target, the two-prime mixed-composite cost-consistency target, and the full prime-calibration propagation target. Each of those proofs simply rewrites its hypothesis into product-calibration and applies this lemma.
It also appears in the universal-foundation conditional certificate chain, marking that prime-calibration propagation cannot be assumed as an unconditional axiom. In Recognition Science terms this is a negative result inside the native-cost uniqueness program: J-uniqueness (T5) and the Recognition Composition Law still stand, but naive prime-to-composite calibration transfer is blocked by branch non-rigidity. The open path is whatever residual hypotheses survive after these refuted targets are stripped out.
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