PRCCharacterTwoPrimeReciprocalRespectsTraceConnected_of_forces
plain-language theorem explainer
If a ratio-orbit character that is reciprocal on the distinguished orbit-2 prime axis is forced reciprocal on every native prime axis, then it is already reciprocal on every prime axis reachable by a finite δ-trace from orbit 2. Anyone equating the global and trace-connected reciprocal normal forms cites this. The proof is a one-line specialization that drops the connectivity hypothesis.
Claim. Let $\chi$ map ratio orbits to ratio orbits. Suppose that whenever $\chi$ sends the orbit-$2$ prime axis to its reciprocal class, it sends every native prime axis to its reciprocal class. Then, for every prime axis that is $\delta$-trace-connected to the orbit-$2$ prime axis, the same implication holds: reciprocal orientation at orbit $2$ forces reciprocal orientation at that prime.
background
In the primitive recognition calculus, ratio orbits are rational displays (signed numerator over a nonzero distinction-nat denominator). A character $\chi$ is a map on these orbits. Reciprocal orientation means $\chi$ lands in the same cross-equivalence class as the orbit reciprocal.
Two normal forms package positive reciprocal-branch transport from the distinguished orbit-$2$ prime axis. The strong form says reciprocal orientation at orbit $2$ forces reciprocal orientation at every native prime axis. The weaker form restricts the conclusion to primes that admit a finite $\delta$-trace connection from the orbit-$2$ axis.
This module develops native-cost uniqueness for PRC characters. The two forms are the global and path-restricted versions of the same reciprocal transport law used later in calibration and blocker certificates.
proof idea
Term-mode one-liner. Introduce a prime $p$, its primality witness, the unused trace-connection hypothesis, and the reciprocal hypothesis at orbit $2$. Apply the strong forcing hypothesis to that reciprocal fact and to $p$. Connectivity is never used: the universal quantifier over all primes already covers the trace-connected subclass.
why it matters
Closes one half of the equivalence between the global and trace-connected reciprocal normal forms (PRCCharacterTwoPrimeReciprocalRespectsTraceConnected_iff_forces). Downstream, the calibration target PRCPrimeCalibrationForcesTwoPrimeReciprocalTraceConnectedTarget_of_forces applies this direction to pass from a forces-style calibration hypothesis to the trace-connected target. The native-cost uniqueness blocker certificate also depends on this cluster.
In the Recognition forcing chain this sits inside the foundation layer that isolates the unique native cost before J-uniqueness (T5) and the self-similar fixed point $\phi$ (T6). It does not itself force $J$ or $\phi$; it only equates two presentations of reciprocal branch transport on prime axes.
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