native_cost_recip_symmetric
plain-language theorem explainer
Native recognition cost on ratio orbits is unchanged by taking the reciprocal orbit: cost(q^{-1}) equals cost(q) as rationals. Anyone building the native reciprocal-generator split certificate cites this. The proof is a four-rewrite term chain through the jq representation, reciprocal-to-rational, and J-inversion.
Claim. For every ratio orbit $q$, the native cost of the reciprocal orbit equals the native cost of $q$ when both are read as rationals: $\mathrm{cost}(q^{-1})=\mathrm{cost}(q)$.
background
In the Primitive Recognition Calculus structural ledger, positive ratios are tracked up to the orbit data RatioOrbit, and the native cost on an orbit is the rational value of the character cost jq (the discrete J-cost pulled back to the orbit). The classical continuous cost is $J(x)=(x+x^{-1})/2-1$, which is visibly invariant under $x\mapsto x^{-1}$.
Reciprocal structure appears throughout the foundation stack: the CostAlgebra reciprocal automorphism, LedgerForcing's reciprocal event (swap source/target and invert the ratio), and ObserverForcing's event cost as J-cost. Here the same symmetry is stated natively on ratio orbits rather than on real ratios.
The local module packages structural facts about that native cost (nonnegativity, zeros, closed-form identities) before assembling generator-split certificates.
proof idea
Term-mode proof by a single rewrite chain. Unfold native cost on both sides via jq_onRatioOrbit, so the goal becomes equality of jq values. Apply RatioOrbit.recip_toRat to turn the reciprocal orbit into the reciprocal rational, then finish with jq_inv, the algebraic identity that the discrete J-character is invariant under inversion. No case splits or induction.
why it matters
This is the cost-symmetry leg of nativeReciprocalGeneratorSplit_holds, which assembles the NativeReciprocalGeneratorSplit structure: involutivity of reciprocal, cost symmetry, unit as the unique zero-cost fixed point, and the no-native-golden-scale purchase obstruction.
In the broader framework it is the orbit-level shadow of T5 J-uniqueness and the Recognition Composition Law: $J(x)=J(x^{-1})$ is forced once cost is the unique symmetric comparator. Without this equality the reciprocal generator could not be treated as cost-neutral structure in the native ledger, and the split certificate would not close.
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