PRCCharacterPrimeFloorOrbitIdentityContractsSuccessorStep
plain-language theorem explainer
A ratio-orbit character satisfies backward identity transport above the unit floor when identity orientation at the successor of any non-unit distinction level forces identity orientation at that level. Native-cost uniqueness and prime-floor coherence arguments cite this as the contracting half of successor transport. The declaration is a pure Prop definition: a universal quantification over distinction indices, with no proof body.
Claim. For a map $\chi$ from ratio orbits to ratio orbits: for every nonzero distinction index $p$ that is not a unit, if $\chi$ has identity orbit-direction orientation at the successor of $p$, then $\chi$ has identity orbit-direction orientation at $p$.
background
In the Primitive Recognition Calculus, ratio orbits are the equivalence classes of positive ratios under the reciprocal identification used by the J-cost. A character $\chi$ is a self-map of ratio orbits encoding orientation (identity versus reciprocal flip) along each orbit. Distinction indices (DistinctionNat) grade the discrete ladder generated by the one-step $\delta$-extension of traces; the unit marks the self-reciprocal floor orbit (the J-cost minimum at ratio $1$).
Identity orbit-direction orientation means $\chi$ fixes the orbit rather than sending it to its reciprocal twin. The local setting is native-cost uniqueness: one wants characters whose doubled-trace values match a unique native cost functional, which requires coherent orientation along prime-floor paths.
Upstream structure supplies the successor on the distinction ladder (one more generator application), the reciprocal automorphism on events and costs, and the identity recognition event at cost zero. The present Prop isolates the backward half of one-step identity transport once the path sits strictly above that unit floor.
proof idea
Definitional packaging only. The body is the quantified implication
$\forall p\neq 0,; \neg\mathrm{unit}(p),; \mathrm{IdOri}(\chi,\mathrm{succ},p)\Rightarrow\mathrm{IdOri}(\chi,p)$.
No tactics or lemmas fire here. Downstream proofs discharge the Prop by introducing $p$, the non-zero and non-unit hypotheses, and the successor-identity assumption, then applying local orientation and no-adjacent-mixed-orientation hypotheses on $\chi$.
why it matters
This is the contracting half of the corrected prime-floor successor-transport rule. Its sibling conjunction with the extending half yields the full successor-transport Prop, documented as "the exact layer needed for prime-to-prime trace coherence."
It is discharged from local non-unit orientation plus no-adjacent-mixed-orientation, and is the conclusion demanded by the prime-calibration target that every ratio character with prime-direction calibration should satisfy this contraction. That target is itself recorded (and separately refuted for a two-adic axis-twist character) inside the Pass-25 native-cost uniqueness blocker certificate, which splits the still-open uniqueness problem into exact Lean obligations. In the broader RS chain this sits under character-to-cost matching toward J-uniqueness (T5) and the Recognition Composition Law, not yet at the forced constants.
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