normalizedProjector_goldenOperator_sq
plain-language theorem explainer
If an endomorphism satisfies A² = μA with μ ≠ 0, its μ-normalized form induces a golden operator G with G² = G + I. Algebraists and RS foundation workers cite this when lifting Hessian rank-one projectors to the golden polynomial. The proof is a one-line composition of the normalization-is-projector lemma with the projector-to-golden identity.
Claim. Let $A$ be an endomorphism of a real module $V$ and $\mu \in \mathbb{R}$ with $\mu \neq 0$. If $A^2 = \mu A$, write $P = \mu^{-1} A$ and $G = \varphi P + (1-\varphi)(I-P)$. Then $G^2 = G + I$.
background
This module isolates the algebraic core of the paper route from reciprocal-cost Hessian geometry to golden structure. A projector is an endomorphism $P$ with $P^2 = P$. The golden operator built from $P$ is $G = \varphi P + (1-\varphi)(I-P)$, with $\varphi$ the golden ratio from the RS constants module.
Normalization takes any operator obeying the scalar square law $A^2 = \mu A$ ($\mu \neq 0$) to $P = \mu^{-1} A$. Upstream, normalizedProjector_isProjector proves that this $P$ is a genuine projector. Separately, goldenOperator_sq proves that every projector induces a golden operator: $G^2 = G + I$. The present statement simply chains those two facts.
The ambient space $V$ is the real module used throughout the cost-projector development (shared with the circle-winding and gap-derivation layers). No Hessian manifold structure is assumed here; only endomorphism algebra.
proof idea
One-line term proof. Apply normalizedProjector_isProjector to the hypotheses $A^2 = \mu A$ and $\mu \neq 0$ to obtain that $P = \mu^{-1} A$ is a projector. Feed that projector witness into goldenOperator_sq, which expands $G = \varphi P + (1-\varphi)(I-P)$ and reduces $G^2 - G - I$ to zero by the projector identity $P^2 = P$. No further case splits or extensionality steps appear at this layer.
why it matters
This is the bridge from "cost geometry supplies a scalar-square operator" to "the golden polynomial is forced." Downstream, projector_golden_certificate packages it as the normalized_operator_golden_structure field of the certificate that any cost-induced normalized projector carries the golden-operator equation. The sibling rankOneEnd_goldenOperator_sq specializes the same fact to nondegenerate rank-one endomorphisms $x \mapsto \ell(x),v$, the algebraic shape of the rank-one Hessian projector in the paper.
In the Recognition forcing stack this sits under the projector-to-golden step that feeds metallic/golden structure on the cost side, ultimately supporting the uniqueness path around T5 (J-uniqueness) and the self-similar fixed point $\varphi$ (T6). It does not itself force dimension or the eight-tick octave; it only closes the algebraic golden identity once a projector is in hand.
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