rclCombiner_ledgerLinearResponse

This module isolates the remaining algebraic condition for Phase 3: derive the T4-to-T5 bridge from the recognition ledger instead of taking it as an analytic input. The free ledger already proves unconditional additivity of ledger cost. If a two-variable combiner also has ledger-linear response in its second argument, the rightAffine field of the d'Alembert factorization gate follows; together with symmetry, the boundary law, and the unit diagonal, the gate theorem forces the RCL polynomial.

Ledger-linear response packages five properties for a combiner $P:\mathbb{R}\to\mathbb{R}\to\mathbb{R}$: symmetry $P(u,v)=P(v,u)$; zero boundary $P(u,0)=2u$; unit diagonal $P(1,1)=6$; right-response (affine dependence on the second argument governed by the unit-mass increment); and free-ledger additivity, which pins the bridge to the existing free-ledger theorem rather than leaving it as prose.

The canonical RCL combiner is $P(u,v)=2uv+2u+2v$, the bilinear form appearing in the Recognition Composition Law $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$. Upstream, that same polynomial is already known to satisfy the full factorization-associativity gate.

Instance construction for ledger-linear response on the canonical RCL combiner. Symmetry, zero boundary, and unit diagonal are projected directly from the existing gate theorem for that combiner. Right-response is proved by unfolding the combiner definition and closing with ring: the identity $2uv+2u+2v=((2u+2)-2u)v+2u$ is pure algebra. Free-ledger additivity is a one-line appeal to the free-ledger cost-additivity theorem on defect ledgers $\Gamma+\Delta$.

This declaration is the concrete witness that the RCL polynomial meets the ledger-side hypothesis of the T4-to-T5 bridge. Module intent is explicit: free-ledger additivity is already unconditional; once a combiner has ledger-linear response, right-affine response follows, and the factorization gate forces the RCL form. That gate is the algebraic engine behind T5 J-uniqueness ($J(x)=(x+x^{-1})/2-1$), so closing the ledger-to-combiner link keeps the forcing chain ledger-native rather than analytic.

No downstream consumers are wired yet in the dependency graph; the theorem stands as the Phase-3 closure point that later factorization or mass-ladder developments can invoke when they need the combiner to speak ledger language. It does not itself re-prove J-uniqueness or the eight-tick structure; it only certifies that the canonical combiner is admissible under the ledger-linear interface.