Explanation of FApply_smul

(1) In plain English, the declaration asserts that the almost-product operator F (defined as F = 2P - I from the cost-induced projector P) respects scalar multiplication: scaling the input vector by c and then applying F yields the same result as applying F first and then scaling the output by c. This is a linearity property for the operator induced by the recognition cost structure.

(2) In Recognition Science this matters because the operators A, P, F, G and Metallic are constructed from the J-cost functional and the inverse metric kernel to model finite-dimensional recognition interactions. Linearity of F ensures these operators behave consistently under scaling, which is a prerequisite for embedding them into the larger algebraic framework that derives spatial dimension D = 3 and the golden-ratio operators from the single cost equation.

(3) The formal statement is read as follows: for any natural number n (the dimension), real numbers lam and c, a matrix-like inverse kernel hInv, and vectors β and v, the equality FApply lam hInv β (c • v) = c • FApply lam hInv β v holds. The proof proceeds by extensionality on each coordinate i, simplification using the already-proved PApply_smul, commutativity of multiplication, and the ring tactic.

(4) Visible dependencies in the supplied source are the definition of FApply (which calls PApply) and the theorem PApply_smul. The module imports IndisputableMonolith.Cost.Ndim.Hessian; sibling theorems in the same file include FApply_add, FApply_square and FApply_GApply, all proved by similar unfolding and ring tactics. No external axioms or sorrys appear in this theorem.

(5) This declaration does not prove additivity (handled separately by FApply_add), idempotence of the underlying projector, the quadratic relation A² = μA, or any connection to the golden ratio φ, the forcing chain, or physical constants. It also does not address existence of the kernel hInv or empirical calibration.