1. Plain English Statement
The theorem GApply_square proves that the "golden operator" ($G$) satisfies the quadratic equation $G^2 = G + I$. In vector terms, applying $G$ twice to any arbitrary vector $v$ yields the exact same result as applying $G$ once to $v$ and then adding $v$ itself.
2. Relevance to Recognition Science (RS)
RS derives the structural features of physics from a parameter-free cost foundation, prominently featuring the golden ratio $\varphi$ in the scaling of fundamental constants (such as $\hbar$ and $G$). This theorem shows why the golden ratio emerges geometrically: the cost function defines a covector, which combines with an inverse metric to form a rank-one tensor. This tensor induces a projector and an almost-product operator $F$. The operator $G$ is built from $F$, and because it satisfies $G^2 - G - I = 0$, its eigenvalues are mathematically forced to be the golden ratio $\varphi$ and $-1/\varphi$. This embeds $\varphi$ as a fundamental invariant of the operator algebra, providing the scaffolding for downstream constant derivations.
3. Reading the Formal Statement
{n : ℕ}: The dimension of the vector space.lam : ℝ,hInv : Fin n → Fin n → ℝ,β : Vec n: The components generating the underlying rank-one operator (a scalar $\lambda$, an inverse metric $h^{-1}$, and a covector $\beta$).hμ : mu lam hInv β ≠ 0: The condition that the scalar trace-like quantity $\mu$ is non-zero, which ensures the normalized projector is well-defined.v : Vec n: Any test vector in the space.GApply ... (GApply ... v) = GApply ... v + v: The conclusion stating $G(G(v)) = G(v) + v$.
4. Visible Dependencies
- The proof directly relies on FApply_square, which establishes that the underlying almost-product operator $F$ is an involution ($F^2 = I$).
- It also utilizes FApply_GApply to expand the composition of $F$ acting on the result of $G(v)$.
5. What this Declaration Does Not Prove
- It does not prove that the spatial dimension $D = 3$; the theorem holds for any arbitrary finite dimension $n$.
- It does not derive the specific physical constants (like $\hbar$, $G$, or $c$) from $\varphi$. Those derivations occur in downstream foundational modules (e.g.,
ConstantDerivations). - It establishes purely algebraic geometry and does not invoke the physical Universal Forcing chain itself.