Explanation of the theorem FApply_sub
(1) In plain English, the declaration states that the almost-product operator FApply (defined as F = 2P - I from the normalized projector P) respects vector subtraction: applying FApply to the difference of two vectors equals the difference of the individual applications. This is a basic linearity property.
(2) In Recognition Science this matters because the module IndisputableMonolith.Cost.Ndim.Projector builds the finite-dimensional operator algebra for the rank-one tensor picture underlying cost-induced projectors; linearity under subtraction ensures consistent handling of distinctions and vector operations in the recognition-cost framework.
(3) The formal statement is read as: for any dimension n, scalar lam, inverse metric kernel hInv, vector β, and vectors v, w, we have FApply lam hInv β (v - w) = FApply lam hInv β v - FApply lam hInv β w. The proof proceeds by extensionality (ext i) followed by simplification using the definitions of subtraction as addition of negation.
(4) Visible dependencies in the supplied source are the theorems FApply_add and FApply_neg, which are invoked directly in the simp tactic of the proof. No sorry or external certificates appear; the proof is self-contained within the module.
(5) The declaration does not prove idempotence of the projector, the square law FApply_square, or any connection to the forcing chain, constants, or physical emergence results elsewhere in the canon.