PRCCharacterPrimeIdentityRespectsTraceConnected_of_common_trace_extension
plain-language theorem explainer
Identity orientation of a ratio-orbit character transports along any prime-axis trace connection once it transports under a shared finite δ-trace extension. Native-cost uniqueness and prime-calibration transport arguments cite this implication to pass from the explicit common-extension hypothesis to the connected form. The proof unpacks the connection witness into a common Trace and applies the stronger hypothesis.
Claim. Let $\chi$ be a map on ratio orbits. Suppose that whenever two prime-axis position traces extend into a common finite $\delta$-trace $T$, identity orientation of $\chi$ on the first prime direction forces identity orientation on the second. Then $\chi$ respects prime-axis trace connection: identity orientation transports along any finite $\delta$-trace component relating two prime axes.
background
In the Primitive Recognition Calculus, a ratio orbit is an integer-numerator display over a nonzero distinction-nat denominator. Characters act on these orbits; the native-cost uniqueness program asks when such a character is forced to the J-cost shape.
Prime axes are prime distinction-nats with their orbit directions. A finite $\delta$-trace records a chain of recognition steps. Two prime axes are trace-connected when some common Trace extends both of their orbit-position traces. The connected transport rule says identity orientation (cross-equality of $\chi$ on a prime direction with that direction itself) moves along any such connection.
The common-trace-extension rule is the more explicit sibling: the same transport conclusion, but quantified over an explicit shared extension $T$ rather than over the connection predicate. This module builds the uniqueness ladder from doubled-trace d'Alembert hypotheses up through prime-identity transport.
proof idea
One-line unpacking of the connection witness. Introduce the two primes, the connection hypothesis, and the identity-orientation hypothesis on the first prime. Destructure the connection as an existential: some Trace $T$ extending both orbit-position traces. Feed those data, together with the identity hypothesis, into the common-trace-extension assumption. The goal is exactly the identity orientation on the second prime.
why it matters
This is the bridge from the explicit common-extension transport hypothesis to the connected form used throughout the native-cost uniqueness stack. Downstream, the canonical-add-trace specialization applies it after reducing canonical add-trace respect to common-extension respect. The prime-calibration target theorem uses it to turn a common-extension calibration hypothesis into full prime-identity trace-transport. Both feed the native-cost uniqueness blocker certificate, which packages zero-calibrated factorization and related refutations.
In the broader Recognition Science forcing chain, native cost uniqueness pins the J-cost (T5: $J(x)=(x+x^{-1})/2-1$) as the unique character compatible with the Recognition Composition Law and prime-axis calibration. Closing transport of identity orientation along δ-traces is a necessary step before that uniqueness can be certified.
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