PRCNativeCostCharacterFactorizationTarget_of_doubled_trace_coherent_root
plain-language theorem explainer
Assuming every native doubled trace admits a coherent multiplicative root character, every admissible PRC-native RCL cost factors through a ratio character. Anyone closing discrete d'Alembert uniqueness for native costs would cite this reduction. The proof is a pure composition: coherent-root yields character-trace lift, which yields factorization.
Claim. If every map $T$ on ratio orbits satisfying the native doubled-trace hypotheses admits a multiplicative ratio character $\chi$ with $\chi(q)+\chi(q)^{-1}$ cross-equal to $T(q)$, then every admissible PRC-native RCL cost $F$ factors as $F(q)$ cross-equal to the cost built from some ratio character $\chi$.
background
In the Primitive Recognition Calculus, costs live on ratio orbits. A ratio character is a multiplicative map $\chi$ on those orbits. The native cost built from a character is the discrete analogue of the J-cost trace: essentially $\chi+\chi^{-1}$ packaged as a cost (the same algebraic shape as $J(x)=(x+x^{-1})/2-1$ from the forcing chain T5).
The Recognition Composition Law forces a d'Alembert-type functional equation on admissible costs. The first exact uniqueness blocker is factorization: every native cost $F$ meeting the PRC native-cost hypotheses should equal the cost-from-character of some ratio character.
A weaker intermediate asks only that the doubled trace of $F$ lift to a character (trace-lift target). The coherent-root target is still weaker on the surface: it asks that any map $T$ obeying doubled-trace hypotheses be realized as the character trace $\chi+\chi^{-1}$. This theorem packages the chain from that root hypothesis up to full factorization.
proof idea
Term-mode composition of two already-proved bridges. First apply the lemma that turns the coherent-root hypothesis into the character-trace-lift target (build the doubled trace of $F$, feed the native-cost hypotheses into the doubled-trace hypotheses, then extract $\chi$). Then apply the one-step lemma that turns any trace-lift witness into factorization: once $\chi$ matches the doubled trace of $F$, the cost-from-character cross-equality follows from the character-trace-matches-cost comparison lemma used inside that bridge.
why it matters
This is the top of the local reduction tower for the first exact blocker in PRC native-cost uniqueness: discrete d'Alembert factorization of every admissible native RCL cost through a ratio character. That blocker is the discrete counterpart of J-uniqueness (forcing-chain T5) and of the Recognition Composition Law identity $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$.
No downstream consumers are wired yet in the graph; the declaration exists to discharge factorization once the coherent-root target is proved or assumed. Closing that root (existence of multiplicative $\chi$ realizing any native doubled trace) would finish this uniqueness step and feed the broader PRC native-cost uniqueness program toward the unique J-shaped cost.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.