PRCPrimeCalibrationForcesPrimeIdentityCanonicalAddTraceTarget_of_common_trace_extension
plain-language theorem explainer
If prime calibration forces identity orientation to respect some common finite δ-trace extension, then it forces transport through the concrete canonical extension orbitPositionTrace(p+r). Native-cost uniqueness and the universal-foundation certificate cite this implication. The proof is a one-line pointwise wrapper of the character-level common-to-canonical lemma.
Claim. Assume that every prime-direction-calibrated ratio character $\chi$ has identity orientation that respects some explicitly witnessed common finite $\delta$-trace extension. Then every such $\chi$ has identity orientation that transports through the concrete finite common extension given by the orbit-position trace of $p+r$.
background
In the Primitive Recognition Calculus, ratio characters $\chi$ act on ratio orbits and encode orientation data used to build native costs. Prime-direction calibration is the constraint that $\chi$ is aligned on prime generators. Two related targets package what that calibration should force for identity orientation under finite $\delta$-trace transport.
The common-trace-extension target is the sharper existential form: identity orientation must respect some witnessed common finite $\delta$-trace extension. The canonical-add-trace target specializes the witness to the concrete finite extension orbitPositionTrace (p + r). Upstream, the character-level lemma already shows that any $\chi$ that respects a common extension automatically respects that canonical add-trace witness (by instantiating the extension as orbitPositionTrace (p + r)).
This declaration lifts that character-level implication to the quantified target propositions used by the native-cost uniqueness certificate stack.
proof idea
Term-mode one-line wrapper. Introduce a ratio character $\chi$ together with the ratio-character and prime-calibration hypotheses. Apply the common-trace target hypothesis at $(\chi, h\chi, h\mathrm{prime})$ to obtain the character-level common-extension property, then feed that into PRCCharacterPrimeIdentityRespectsCanonicalAddTrace_of_common_trace_extension, which specializes the common witness to orbitPositionTrace (p + r).
why it matters
Closes one direction of the equivalence between the common-trace-extension target and the canonical-add-trace target (..._iff_common_trace_extension). That equivalence keeps the native-cost uniqueness blocker certificate free to phrase prime-identity transport either existentially or via the concrete $p+r$ witness without changing logical strength.
Downstream it feeds prc_native_cost_uniqueness_blocker_certificate and, through the foundation stack, prc_universal_foundation_conditional_certificate. In Recognition Science terms this is bookkeeping inside the PRC native-cost uniqueness path that underwrites the J-cost forcing chain (T5 uniqueness of $J(x)=(x+x^{-1})/2-1$), not a new physical law by itself.
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