PRCPrimeCalibrationForcesPrimeIdentityIffTwoPrimeIdentityTarget_refuted
plain-language theorem explainer
The identity-iff-two prime-calibration target is false: prime-direction calibration need not force identity orientation on every prime axis exactly when it does so on the distinguished orbit-2 axis. Native-cost uniqueness and foundation-certificate work cite this refutation. The proof is a short reduction through an equivalence to the already-refuted one-sided identity-forces-two target.
Claim. It is not the case that every ratio character $\chi$ that is prime-direction calibrated satisfies: $\chi$ has identity orientation on an arbitrary prime axis if and only if it has identity orientation on the distinguished orbit-$2$ prime axis.
background
In the Primitive Recognition Calculus, ratio characters are maps on ratio orbits that encode multiplicative orientation data used to build native costs. Prime-direction calibration restricts how such a character may act along prime axes. The distinguished orbit-$2$ axis is the preferred reference among those primes.
The identity-iff-two target asserts a biconditional rigidity: under prime calibration, identity orientation on any prime axis holds exactly when identity holds on orbit $2$. A one-sided companion target only requires that identity on any calibrated prime axis force identity on orbit $2$. An in-module equivalence identifies the biconditional target with that one-sided target.
The local module develops uniqueness and obstruction results for native costs built from characters and doubled traces, including a chain of calibration targets and their refutations used by blocker certificates.
proof idea
Assume the biconditional target. Apply the forward direction of the in-module equivalence that identifies it with the one-sided identity-forces-two target, obtaining that one-sided target as a hypothesis. Discharge the goal by the already-proved refutation of the one-sided target. The argument is a pure logical reduction: no new analytic or arithmetic work.
why it matters
This closes the biconditional prime-identity calibration target as false, matching the one-sided refutation already in the chain. Downstream, the two-prime-branch-controls-primes target is refuted by reducing to this result. The native-cost uniqueness blocker certificate and the universal-foundation conditional certificate both sit above this obstruction layer: they record which factorization and calibration targets are proved versus refuted so that uniqueness claims stay honest.
In the broader Recognition forcing picture, native-cost uniqueness is the bridge from abstract ratio characters to the unique $J$-cost (T5) and the self-similar $\varphi$ fixed point (T6). Refuting over-strong calibration identities prevents false uniqueness routes and keeps the certificate stack aligned with what the calculus actually forces.
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