PRCPrimeCalibrationForcesPrimeIdentityCanonicalAddTraceTarget_of_branch_uniformity
plain-language theorem explainer
If prime calibration forces identity-branch uniformity on native prime axes, then it also forces identity orientation to transport along canonical additive orbit-position traces. Native-cost uniqueness and the universal-foundation certificate cite this implication when collapsing two calibration targets. The argument is a one-line reduction: instantiate the target hypothesis and apply the character-level branch-uniformity lemma.
Claim. Assume that every ratio character $\chi$ that is prime-direction calibrated places all native prime axes on the identity branch whenever one identity-oriented native prime axis is present. Then every such $\chi$ also preserves identity orientation under the concrete finite common-extension traces $\mathrm{orbitPositionTrace}(p+r)$ for native primes $p,r$.
background
In the Primitive Recognition Calculus, a ratio character $\chi$ is a map on ratio orbits that encodes how recognition cost orients multiplicative structure. Prime-direction calibration asks that native prime axes sit in a preferred orientation relative to the identity branch. Two calibration targets are compared here.
The branch-uniformity target is trace-free: once one native prime axis is identity-oriented, every native prime axis must lie on the identity branch. The canonical-add-trace target is stronger in appearance: identity orientation must transport through the concrete finite common extension given by the orbit-position trace of $p+r$. Upstream, the character-level lemma already shows that branch uniformity alone yields respect for those canonical additive traces; the structural connectivity of the prime-axis trace graph is treated as already established.
This module sits in the native-cost uniqueness development: the goal is to force the native cost functional (built from doubled traces and d'Alembert-type identities tied to the J-cost) down to a unique calibrated form.
proof idea
Term-mode wrapper. Introduce a ratio character $\chi$ together with the hypotheses that it is a PRC ratio character and is prime-direction calibrated. Apply the assumed branch-uniformity target at $\chi$ to obtain identity-branch uniformity for that character. Feed the resulting uniformity hypothesis into the upstream lemma that any branch-uniform character respects canonical additive traces. No further arithmetic or connectivity argument is needed at this layer.
why it matters
This is one direction of the equivalence between the branch-uniformity target and the canonical-add-trace target. Downstream, that equivalence is packaged as an iff, so either formulation may be used when discharging prime-calibration obligations in the native-cost uniqueness blocker certificate. The same target feeds the conditional universal-foundation certificate, which assembles kernel, ordered-field, and trace-logic passes into a single foundation bundle.
In the broader Recognition Science chain, native-cost uniqueness is the PRC-side route toward forcing the J-cost $J(x)=(x+x^{-1})/2-1$ (T5) as the unique admissible cost, before $\phi$ and the eight-tick structure appear. Collapsing two seemingly different identity-transport targets removes a bookkeeping fork: one may prove the easier-looking uniformity statement and inherit the concrete $p+r$ trace-transport form for free.
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