The declaration GApply_square states that the golden operator satisfies a quadratic identity mirroring the golden-ratio relation.
(1) In plain English: for any vector v, applying the golden operator G twice produces G(v) plus the original v, provided the auxiliary scalar mu is nonzero. This is the operator-level version of φ² = φ + 1.
(2) In Recognition Science the golden operator is built from the cost-induced projector P and the almost-product F = 2P − I inside the finite-dimensional algebra of the Ndim module. The identity therefore encodes the self-similar scaling that the J-cost functional equation forces on rank-one structures.
(3) The formal statement reads: given dimension n, scalar lam, inverse kernel hInv, covector β, a proof hμ that mu(lam, hInv, β) ≠ 0, and vector v, the equality GApply lam hInv β (GApply lam hInv β v) = GApply lam hInv β v + v holds. All parameters are explicit; the proof proceeds by unfolding GApply, invoking the auxiliary identity FApply_GApply, and simplifying with nlinarith on the √5 coefficients.
(4) Visible dependencies inside the supplied source are the preceding theorems FApply_GApply, FApply_square, PApply_idempotent and the definitions of GApply, FApply, PApply, mu and sharp, all in the same module. No external axioms or sorrys appear in the proof.
(5) The declaration proves only this local algebraic identity; it does not prove the existence of the underlying cost functional, the forcing of φ itself, any physical constant, or any link to the recognition-forcing chain.