PRCPrimeCalibrationForcesPrimeIdentityTraceCoherenceTarget_of_branch_uniformity
plain-language theorem explainer
Prime calibration that forces identity-branch uniformity on native prime axes already forces identity orientation to propagate along prime-axis traces. Anyone closing native-cost uniqueness or the universal-foundation certificate cites this reduction. The proof is a short quantifier chase: apply the branch-uniformity target, then the pointwise branch-to-trace lemma.
Claim. If prime calibration forces every identity-oriented native prime axis of a ratio character $\chi$ to place all native prime axes on the identity branch, then prime calibration also forces identity orientation of $\chi$ to propagate across prime axes along the prime-axis trace graph.
background
In the Primitive Recognition Calculus, a ratio character $\chi$ is a map on ratio orbits that encodes how multiplicative structure is read as cost. Prime-direction calibration means $\chi$ is aligned on native prime axes. Two residual targets remain for native-cost uniqueness: branch uniformity (every identity-oriented native prime axis puts all native prime axes on the identity branch) and trace coherence (identity orientation propagates along prime-axis trace connections).
The module already proves the structural connectivity of the prime-axis trace graph. The remaining work is logical: show that the calibration hypotheses force the identity branch to be coherent, not merely locally chosen. Upstream, branch uniformity of a single character immediately yields its trace coherence by specializing the uniformity statement along any prime-axis connection.
This declaration lifts that pointwise implication to the quantified calibration targets: the universal branch-uniformity target implies the universal trace-coherence target.
proof idea
Term-mode quantifier chase. Introduce a ratio character $\chi$ together with the character and prime-calibration hypotheses. Apply the assumed branch-uniformity target at $\chi$ to obtain identity-branch uniformity for that character. Feed the result into the upstream lemma that any branch-uniform character is already trace-coherent. No extra arithmetic or connectivity argument is needed here; connectivity is already discharged above.
why it matters
Native-cost uniqueness in PRC needs identity orientation to travel across prime axes once calibration is fixed. This lemma shows the remaining trace-coherence target is no stronger than the trace-free branch-uniformity target, so the two can be treated interchangeably.
It is one direction of the in-module equivalence between those targets, and it feeds the native-cost uniqueness blocker certificate and the conditional universal-foundation certificate. In the broader Recognition chain this sits under cost uniqueness for the J-cost forced at T5, before phi, the eight-tick octave, and $D=3$ are installed. It does not itself force J; it clears a calibration-coherence blocker on the path to a unique native cost.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.