PRCPrimeCalibrationForcesPrimeIdentityForcesTwoPrimeIdentityTarget_refuted
plain-language theorem explainer
The one-sided claim that prime-direction calibration forces the orbit-2 prime axis onto the identity branch whenever any calibrated prime axis is on identity is false. Native-cost uniqueness and foundation-certificate work cite this as a closed negative. The proof is a short transport: an already-proved equivalence moves the claim onto the pair-product cost-consistency target, which is already refuted.
Claim. It is false that every ratio-orbit character $\chi$ that is a PRC ratio character and is prime-direction calibrated must place the orbit-$2$ prime axis on the identity branch whenever it places any calibrated prime axis on the identity branch.
background
In the Primitive Recognition Calculus, ratio-orbit characters $\chi$ assign each ratio orbit a branch (identity or reciprocal). A character is prime-direction calibrated when its values on prime axes obey the native calibration constraints used to pin the cost. The orbit-$2$ axis is the distinguished binary axis tied to the eight-tick / period-$2^3$ structure of the forcing chain.
The target being denied asserts a one-sided forcing law: identity on any calibrated prime axis would force identity on the orbit-$2$ axis. An equivalent formulation is the prime-pair product cost-consistency target (pair products of prime axes must match the native cost on the product orbit). That equivalence is already proved in-module, and the pair-product target is already refuted.
Local setting is native cost uniqueness for PRC: which branch-uniformity and calibration constraints can actually hold for characters that reproduce the J-cost / doubled-trace structure.
proof idea
Assume the one-sided identity-forces-two target. Apply the right-to-left direction of the proved equivalence
PRCPrimeCalibrationForcesPrimePairProductCostConsistencyTarget_iff_prime_identity_forces_two
to obtain the pair-product cost-consistency target. Discharge by the already-proved refutation
PRCPrimeCalibrationForcesPrimePairProductCostConsistencyTarget_refuted.
Pure transport across an iff; no new analytic work.
why it matters
Closes a distinguished-axis forcing route that would have rigidified prime calibration toward a unique native cost character. Downstream it feeds the two-sided identity iff-two refutation, both reciprocal-branch one-sided refutations, and the native-cost uniqueness blocker certificate. That blocker is part of the conditional universal-foundation certificate stack, so the negative result is load-bearing for what uniqueness can still claim.
In framework terms this sits under PRC native-cost uniqueness rather than T5 J-uniqueness itself: it shows that prime calibration alone does not force the orbit-2 identity branch from other prime identities, so cost uniqueness cannot ride on that one-sided law. The reciprocal analogues are discharged by the same lemma via further equivalences.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.