Explain the Lean theorem `RCLCombiner_isCoupling_iff` in module `IndisputableMonolith.Foundation.BranchSelection`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

(1) Plain English Statement

The theorem RCLCombiner_isCoupling_iff asserts that the specific polynomial combiner RCLCombiner c (defined as P(u, v) = 2u + 2v + c·u·v) qualifies as a "coupling combiner"—one whose output cannot be expressed as separate functions of each input alone—if and only if the real parameter c is nonzero.

(2) Relevance to Recognition Science

In RS, the Recognition Composition Law produces a family of combiners parameterized by c. The additive branch (c = 0) yields a cost function like ½(ln x)², while the bilinear branch (c ≠ 0) yields the reciprocal-symmetric J(x) = ½(x + x⁻¹) − 1. This theorem supplies the mathematical criterion that, under the strengthened composition-consistency requirement (non-separate additivity), excludes the additive branch and forces the bilinear one. That choice is required for the subsequent derivation of φ and the RS-native constants.

(3) Reading the Formal Statement

The declaration is:

theorem RCLCombiner_isCoupling_iff (c : ℝ) :
    IsCouplingCombiner (RCLCombiner c) ↔ c ≠ 0

• RCLCombiner c constructs the two-argument function fun u v => 2*u + 2*v + c*u*v.
• IsCouplingCombiner P is the negation of SeparatelyAdditive P, where the latter means P(u,v) = p(u) + q(v) for some unary p,q.
• The proof rewrites via isCouplingCombiner_iff_interactionDefect_nonzero, then splits into both directions using interactionDefect_RCLCombiner and RCLCombiner_nonzero_couples.

(4) Visible Dependencies and Certificates

All supporting declarations appear in the same module:

• SeparatelyAdditive and IsCouplingCombiner (core predicates).
• interactionDefect and its equivalence theorems.
• RCLCombiner and interactionDefect_RCLCombiner.
• The module also exports branchSelectionCert that packages this result together with the contrapositive additive_branch_not_coupling.

No external axioms or sorry appear in the proof.

(5) What the Declaration Does Not Prove

It does not establish the full physical consequences of the bilinear branch, the uniqueness of J up to the residual α freedom, or any link to spacetime emergence or constants. Those steps are handled by separate theorems such as branch_selection and modules outside this slice (e.g., cost functional equation, forcing chain). It likewise does not address empirical calibration or holographic bounds.