PRCPrimeCalibrationForcesTwoPrimeBranchControlsPrimesTarget_iff_prime_identity_iff_two
plain-language theorem explainer
Equivalence of two blocker formulations under prime calibration in the primitive recognition calculus: the orbit-2 branch controlling every native prime branch is the same demand as identity orientation holding on every prime axis exactly when it holds on the orbit-2 axis. Cited by anyone tracking native-cost uniqueness blockers and their refutations. Proof is the pair of already-proved one-way implications packaged as a biconditional.
Claim. The following are equivalent. (i) For every ratio character that is prime-direction calibrated, the branch chosen at the distinguished orbit $2$ controls every native prime branch. (ii) For every such character, identity orientation is forced on an arbitrary prime axis if and only if it is forced on the orbit-$2$ prime axis.
background
In the primitive recognition calculus, ratio characters are maps on ratio orbits that encode admissible multiplicative structure for native cost. Prime-direction calibration restricts how those characters may orient prime axes. Two blocker targets package the same obstruction in different normal forms.
The distinguished-prime form asserts that, after prime calibration, the branch chosen at orbit $2$ controls every native prime branch. The identity-iff-two form asserts that identity orientation is forced on any prime axis exactly when it is forced on the orbit-$2$ axis. Both are universal statements over ratio characters satisfying the character and calibration hypotheses.
The surrounding module develops native-cost uniqueness via doubled-trace and d'Alembert structure on these characters. The two one-sided implications between the targets are already available upstream; this declaration only identifies them.
proof idea
Term-mode biconditional: the forward direction is the existing theorem that two-prime-branch control implies the identity-iff-two target; the reverse is the existing theorem that the identity-iff-two target implies two-prime-branch control. No new case analysis. Each one-way proof reduces, after introducing a calibrated character, to a local orientation lemma already proved for that character (local two-prime-branch control, respectively local prime-identity-iff-two), then applies the corresponding character-level conversion lemma.
why it matters
Native-cost uniqueness in Recognition Science needs a clean inventory of calibration blockers: which prime-axis constraints are forced, which are independent, and which collapse. This iff shows the distinguished-orbit-$2$ control form and the identity-iff-two form are the same obstruction, so refuting one refutes the other.
Downstream, the two-prime-branch target is refuted by transporting the identity-iff-two refutation across this equivalence. The same linkage feeds the native-cost uniqueness blocker certificate and, at one remove, the conditional universal-foundation certificate. In the forcing chain this sits in the foundation layer that pins the unique J-cost and the admissible character calculus before constants and the phi-ladder are read off.
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