PRCPrimeCalibrationForcesPrimeIdentityTraceCoherenceTarget_refuted
plain-language theorem explainer
Prime calibration does not force identity orientation to propagate coherently across prime axes for every ratio character. Native-cost uniqueness and foundation-certificate work cite this as a closed negative. The proof is a short transfer: the target is equivalent to branch-uniformity, already refuted by an explicit two-adic axis twist.
Claim. It is false that every ratio-orbit character $\chi$ that is prime-direction calibrated is automatically prime-identity trace-coherent. Equivalently: prime calibration alone does not force identity orientation to propagate across all prime axes.
background
In the Primitive Recognition Calculus, a ratio character is a map on ratio orbits obeying the multiplicative structure used to build native cost. Prime-direction calibration means the character is normalized on the distinguished prime axes so that each axis carries a preferred orientation. Trace coherence then asks that identity orientation on one calibrated prime axis force the same identity branch on every other native prime axis.
The module studies whether those calibration hypotheses pin down a unique native cost (via doubled-trace and d'Alembert structure linked to the J-cost). The trace-coherence target is the exact remaining demand that "prime calibration must make identity orientation propagate across prime axes."
Upstream, that target is proved equivalent to a branch-uniformity target (every identity-oriented native prime axis puts all native prime axes on the identity branch). Branch-uniformity is already refuted by a concrete two-adic axis-twist ratio character.
proof idea
Assume the trace-coherence target. Apply the right-to-left direction of the proved equivalence with branch-uniformity to obtain the branch-uniformity target. Discharge the assumption by the existing refutation of branch-uniformity, which comes from the constructed two-adic axis-twist ratio character. No new counterexample is built here; the argument is pure target transport along the iff.
why it matters
This closes one of the remaining positive forcing claims in the native-cost uniqueness ledger: calibration does not buy free identity-trace coherence. Downstream, the same refutation is reused to kill the common-trace-extension, comparable-trace, and trace-transport variants (each reduced to this target by an iff). It also feeds the native-cost uniqueness blocker certificate and, through that stack, the conditional universal-foundation certificate.
In the broader Recognition chain, unique native cost is what should recover the J-cost of T5 and the RCL. A refuted coherence bridge marks a genuine obstruction: uniqueness cannot be finished by prime calibration alone and must route around this gap or strengthen hypotheses.
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