Explanation of FApply_neg in IndisputableMonolith.Cost.Ndim.Projector

(1) Plain English

The theorem asserts that the almost-product operator F (defined as F = 2P - I from the normalized projector P) is homogeneous with respect to negation: F(-w) = -F(w). This is a basic linearity property for the operator algebra built from a covector β and inverse metric kernel hInv.

(2) Why It Matters in Recognition Science

In Recognition Science this operator algebra encodes the finite-dimensional rank-one tensor picture induced by the J-cost functional equation. The F operator serves as the base for the golden operator GApply and the metallic family MetallicApply, which capture self-similar scaling tied to φ. Negation homogeneity ensures the algebra is consistent under sign changes, supporting the reciprocal symmetry J(r) = J(1/r) and the derivation of structural features such as projectors and almost-products without introducing extra parameters.

(3) How to Read the Formal Statement

The declaration reads:

theorem FApply_neg {n : ℕ}
    (lam : ℝ) (hInv : Fin n → Fin n → ℝ) (β w : Vec n) :
    FApply lam hInv β (-w) = -FApply lam hInv β w := by
  simpa using FApply_smul lam hInv β (-1) w

In standard notation: for dimension n, scalar λ (lam), inverse kernel h⁻¹, covector β, and vector w, the equality F(λ, h⁻¹, β, −w) = −F(λ, h⁻¹, β, w) holds. The proof is a one-line instantiation of the scalar-multiplication theorem FApply_smul at scalar −1.

(4) Visible Dependencies or Certificates

• Direct dependency: FApply_smul (used via simpa).
• Upstream: PApply_smul and PApply_add, which establish the same homogeneity for the projector P.
• Module context: The file IndisputableMonolith.Cost.Ndim.Projector imports IndisputableMonolith.Cost.Ndim.Hessian for Vec, dot, and sharp.
• Pattern: Parallel negation theorems exist for AApply_neg, PApply_neg, and FApply_sub, confirming a uniform linearity suite.

(5) What This Declaration Does Not Prove

It establishes only the algebraic negation property for the finite-dimensional FApply operator. It does not prove any link to ledger events, recognition forcing, or the J-cost uniqueness theorem; it does not address infinite-dimensional or continuous limits; it supplies no physical interpretation, empirical falsifier, or connection to the forcing chain in IndisputableMonolith.Foundation.RealityFromDistinction.