Explain the Lean theorem `FApply_square` in module `IndisputableMonolith.Cost.Ndim.Projector`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

1. Plain English Meaning

The THEOREM FApply_square states that the operator $F$ is an involution: applying it twice to any vector yields the original vector (i.e., $F^2 = I$). This holds strictly, provided the scaling factor $\mu$ associated with its underlying projector is non-zero.

2. Relevance to Recognition Science

In Recognition Science (RS), fundamental constants like the golden ratio $\varphi$ are derived as THEOREM deliverables forced by structural constraints. The $F$ operator defined here acts as an algebraic "almost-product structure." Because $F^2 = I$, it provides the exact linear algebraic scaffolding needed to construct the "golden operator" $G$ (via GApply), which satisfies $G^2 = G + I$. Thus, this theorem is the functional bridge connecting an arbitrary inverse-metric cost tensor (a MODEL of state deformation) to the recursive, $\varphi$-based algebraic structures that RS uses to derive quantum and gravitational constants.

3. Reading the Formal Statement

• {n : ℕ}: The dimension of the vector space, generalized to any natural number.
• lam : ℝ, hInv : Fin n → Fin n → ℝ, β : Vec n: The geometric inputs defining the operator—a scalar multiplier $\lambda$, an inverse metric kernel $h^{-1}$, and a covector $\beta$.
• hμ : mu lam hInv β ≠ 0: The logical hypothesis requiring that the trace-like scalar mu is non-zero. This prevents division-by-zero when forming the projector.
• v : Vec n: The arbitrary input vector.
• FApply ... (FApply ... v) = v: The conclusion, stating that applying FApply to the result of FApply on v returns exactly v.

4. Visible Dependencies

The proof operates purely by linear algebraic substitution. It unfolds FApply as $F(v) = 2P(v) - v$. When evaluating $F(F(v))$, the proof relies critically on the THEOREM PApply_FApply, which states that the projector $P$ absorbs $F$ ($P(F(v)) = P(v)$). The derivation is conceptually tight: $F(F(v)) = 2P(F(v)) - F(v) = 2P(v) - (2P(v) - v) = v$. The normalization by $\mu$ is encapsulated within PApply, governed by the assumption hμ.

5. What This Declaration Does Not Prove

• Existence of non-zero $\mu$: The theorem assumes $\mu \neq 0$ but does not prove such a configuration is universally forced.
• Orthogonality: It defines a general involution but does not prove that $F$ is an orthogonal reflection. That would require enforcing symmetric positive-definite constraints on $h^{-1}$.
• Dimensionality: It is valid for any dimension $n$. It does not force spatial dimension $D=3$ (which the RS framework proves via a separate topological THEOREM using S¹ cohomology, not linear algebra).