1. Plain English Statement
The theorem FApply_sub states that the almost-product operator $F$ is linear with respect to vector subtraction. Applying $F$ to the difference of two vectors $v - w$ yields the identical result as applying $F$ to $v$ and $w$ separately and subtracting the resulting vectors.
2. Role in Recognition Science
In Recognition Science, spatial structure is modeled through operator algebras derived from rank-one tensors. Given a one-form $\beta$ and an inverse metric $h^{-1}$, the framework constructs a base operator $A$, its normalized projector $P$, and an induced almost-product operator $F = 2P - I$. The algebraic behavior of $F$ is structurally necessary because it forms the core of the golden and metallic operator families. FApply_sub guarantees that $F$ rigorously respects subtraction, satisfying the basic linearity requirements needed to manipulate these operators in downstream proofs.
3. Reading the Formal Statement
theorem FApply_sub {n : ℕ}
(lam : ℝ) (hInv : Fin n → Fin n → ℝ) (β : Vec n)
(v w : Vec n) :
FApply lam hInv β (v - w) = FApply lam hInv β v - FApply lam hInv β w
{n : ℕ}: The dimension of the underlying space.lam : ℝ: A scalar parameter $\lambda$.hInv : Fin n → Fin n → ℝ: The inverse metric kernel $h^{-1}$.β : Vec n: The defining covector $\beta$.v w : Vec n: The input vectors being subtracted.- The conclusion asserts the identity $F(v - w) = F(v) - F(w)$.
4. Visible Dependencies and Certificates
The proof is a direct, one-line structural reduction. It unwraps subtraction into the addition of a negated vector (v + (-w)) and relies entirely on two adjacent linearity lemmas:
- FApply_add: establishes $F(v + w) = F(v) + F(w)$.
- FApply_neg: establishes $F(-w) = -F(w)$.
5. What This Declaration Does Not Prove
FApply_sub isolates subtraction linearity. It does not prove:
- That $F$ scales linearly with scalar multiplication. That is handled by FApply_smul.
- That $F$ acts as a true mathematical involution (i.e., $F^2 = I$). That property requires a non-zero trace assumption and is proven in FApply_square.
- Any properties of the golden ($G$) or metallic operators themselves, which only use $F$ as a component block.