The declaration FApply_add asserts that the almost-product operator FApply respects vector addition.
(1) In plain English: given scalar lam, inverse kernel hInv, covector β, and vectors v, w in n dimensions, FApply applied to the sum v + w equals the sum of the separate applications FApply(v) + FApply(w). This is the additivity property of the operator F = 2P − I derived from the rank-one projector.
(2) In Recognition Science this matters because FApply arises from the cost-induced projector P in the finite-dimensional operator algebra of the recognition ledger. Additivity ensures the almost-product structure is compatible with the additive ledger operations that underlie J-cost symmetry and recognition events.
(3) The formal statement is read as: for any natural number n, real lam, function hInv : Fin n → Fin n → ℝ, vector β : Vec n, and vectors v, w : Vec n, the equality FApply lam hInv β (v + w) = FApply lam hInv β v + FApply lam hInv β w holds. The proof proceeds by extensionality (ext i), unfolding FApply and invoking PApply_add, followed by ring simplification.
(4) Visible dependencies in the supplied source are the definition of FApply, the theorem PApply_add, and the supporting lemmas PApply_smul and AApply_add. No external certificates beyond these algebraic identities appear in the module.
(5) The declaration does not prove linearity over scalar multiplication (handled separately by FApply_smul), idempotence or involution properties (see FApply_square), infinite-dimensional extensions, or any direct link to physical constants or the forcing chain.