(1) Plain English statement
The declaration branch_selection asserts: if a polynomial combiner of the form P(u, v) = 2u + 2v + c·u·v must be a coupling combiner (i.e., not separately additive in its two arguments), then the coefficient c cannot be zero.
(2) Role in Recognition Science
Recognition Science derives the cost functional J from a functional equation whose solutions split into a bilinear family (c eq 0, representative J(x) = (x + x^{-1})/2 − 1) and an additive family (c = 0, representative ½(ln x)^{2}). The theorem supplies the structural reason the additive branch is excluded once composition consistency is strengthened to require genuine coupling between arguments. This isolates the bilinear branch at the operator level.
(3) Reading the formal statement
The Lean statement is:
theorem branch_selection (c : ℝ)
(hCoupling : IsCouplingCombiner (RCLCombiner c)) :
c
eq 0 :=
(RCLCombiner_isCoupling_iff c).mp hCoupling
It takes a real parameter c and a hypothesis that the RCL combiner for that c satisfies IsCouplingCombiner; the conclusion is c eq 0. The proof is a one-line application of the equivalence RCLCombiner_isCoupling_iff.
(4) Visible dependencies and certificates
The theorem depends only on definitions and lemmas inside the same module: IsCouplingCombiner, RCLCombiner, interactionDefect, and the equivalence RCLCombiner_isCoupling_iff. A certificate structure BranchSelectionCert packages the same facts together with the contrapositive additive_branch_not_coupling.
(5) What the declaration does not prove
It does not derive the functional equation itself, does not calibrate the residual α-family freedom, does not connect to spacetime emergence or physical constants, and does not address the companion paper's higher-derivative or action-functional conditions.