rankOneEnd_normalized_isProjector

This module is the algebraic core of the paper route from reciprocal cost geometry to golden operators. It works with endomorphisms of a real module $V$, not a full Hessian-manifold formalization. A projector is an endomorphism $P$ with $P^2=P$. From any such $P$ one builds the almost-product $F=2P-I$ (so $F^2=I$) and the golden operator $G=\varphi P+(1-\varphi)(I-P)$ (so $G^2=G+I$).

The rank-one endomorphism $A(x)=\ell(x),v$ is the algebraic shape of the rank-one Hessian projector used in that paper. It always squares to a scalar multiple of itself: $A^2=\ell(v)A$. Normalizing an operator that satisfies $A^2=\mu A$ by the nonzero scalar $\mu$ yields $P=\mu^{-1}A$, which is a projector. The nondegeneracy hypothesis here is exactly $\ell(v)\neq 0$, so that scalar is invertible.

One-line term wrapper. Apply the general normalization theorem (if $A^2=\mu A$ and $\mu\neq 0$ then $\mu^{-1}A$ is a projector) to the rank-one square identity $A^2=\ell(v)A$, taking $\mu=\ell(v)$ and feeding the hypothesis $\ell(v)\neq 0$. No further rewriting is required.

This is the concrete projector step for rank-one cost endomorphisms inside the projector-to-golden chain. Downstream, the multi-coordinate cost-Hessian operator normalizes to a projector by specializing this theorem (paper Corollary 3.1): the cost-Hessian form supplies $\ell$ and the direction $\alpha$ supplies $v$. The same fact is packaged into the projector-golden certificate, which records that any cost-induced normalized projector carries the golden-operator equation $G^2=G+I$.

In the Recognition forcing stack this is pure algebra supporting the golden structure forced by reciprocal cost geometry; it does not itself invoke T5 J-uniqueness or the $\varphi$ fixed point, but it is the operator identity those later steps sit on once the Hessian rank-one shape is in place.