The declaration FApply_GApply appears in module IndisputableMonolith.Cost.Ndim.Projector.
(1) In plain English it states that the almost-product operator F, when applied to the result of the golden operator G on any vector v, equals a specific linear combination: half of (F applied to v) plus (√5/2) times v. This is an algebraic interaction identity between the two operators constructed from the same rank-one data (λ, hInv, β).
(2) In Recognition Science the module packages finite-dimensional operator algebra derived from the cost function J. The golden operator G incorporates the factor √5/2 that appears in the golden-ratio self-similarity structure forced by the J-cost functional equation; the identity therefore supplies a concrete algebraic certificate that the almost-product F commutes with G in the manner required by that self-similarity.
(3) The formal statement is read as: for any finite dimension n, scalars lam and inverse-metric kernel hInv, covector β (with the auxiliary scalar mu(lam, hInv, β) nonzero), and any vector v, the equality F(G(v)) = (1/2)·F(v) + (√5/2)·v holds. The proof proceeds by unfolding the definition of GApply and rewriting with the linearity and squaring properties of FApply.
(4) Visible dependencies in the supplied source are the definitions of FApply, GApply, PApply, AApply, mu and sharp, together with the supporting theorems FApply_add, FApply_smul and FApply_square that are invoked in the proof. No external certificates or axioms are required; the proof is a direct equational rewrite inside the module.
(5) The declaration does not prove that the same operators exist in infinite dimensions, that they arise uniquely from the J-cost functional equation, or that they reproduce any specific physical constant or forcing-chain theorem. It likewise supplies no statement about convergence, spectra, or empirical falsifiers.