Explanation of PApply_FApply in IndisputableMonolith.Cost.Ndim.Projector

(1) What the declaration says in plain English

The theorem asserts that the normalized projector operator PApply is idempotent in a specific way with respect to the almost-product operator FApply: applying PApply after FApply on any vector v simply returns the result of applying PApply to v. In operator terms, P ∘ F = P (under the non-zero mu assumption). This is a local algebraic identity showing that the image of F lies inside the range where P acts as a true projector.

(2) Why it matters in Recognition Science

This property belongs to the finite-dimensional operator algebra that supports the rank-one tensor picture in the cost framework. The module docstring states that A = h^{-1} \tilde g, its quadratic law A² = μ A, and the normalized projector P arise from a covector β and inverse metric kernel hInv. The relation P(F v) = P v helps close the algebra for derived operators (F, G, metallic family) used in higher-level cost-induced structures.

(3) How to read the formal statement

The Lean declaration is:

theorem PApply_FApply {n : ℕ}
    (lam : ℝ) (hInv : Fin n → Fin n → ℝ) (β : Vec n)
    (hμ : mu lam hInv β ≠ 0) (v : Vec n) :
    PApply lam hInv β (FApply lam hInv β v) = PApply lam hInv β v

• {n : ℕ}: finite dimension.
• lam, hInv, β: parameters defining the rank-one operator A and its scalar mu.
• hμ: guard that mu ≠ 0 (ensures P is well-defined).
• v: arbitrary input vector.
• Conclusion: equality of two Vec n values after operator composition. The proof proceeds by extensionality (ext i), unfolding FApply, applying linearity theorems, invoking idempotence of P, and simplifying with ring.

(4) Visible dependencies or certificates in the supplied source

The proof directly references:

• PApply_sub
• PApply_smul
• PApply_idempotent All reside in the same module. The surrounding theorems (FApply_square, PApply_FApply itself) form a closed algebraic certificate for the projector and almost-product operators. No external axioms or sorry appear in this module.

(5) What this declaration does not prove

It does not establish any physical interpretation, connection to the J-cost functional equation, φ-derived constants, or the forcing chain. It supplies only a pure linear-algebra identity inside the N-dimensional projector construction; downstream uses in Recognition Science (e.g., emergence of geometry or recognition bandwidth) are not addressed here.