rankOneEnd
plain-language theorem explainer
The rank-one endomorphism sending each vector x to ℓ(x)·v is the algebraic primitive behind the Hessian projector route to golden operators. Anyone citing the projector-to-golden step in the cost-geometry forcing stack uses this map. The definition packages x ↦ ℓ(x) v as a real-linear endomorphism by checking additivity and scalar homogeneity from the module axioms.
Claim. For a real module $V$, a linear functional $\ell : V \to \mathbb{R}$, and a vector $v \in V$, define the rank-one endomorphism $A_{\ell,v} \in \mathrm{End}_{\mathbb{R}}(V)$ by $A_{\ell,v}(x) = \ell(x)\, v$.
background
The module CostProjectorGolden isolates the algebraic core of the paper Golden and Metallic Structures on Hessian Manifolds. Once a projector $P$ satisfies $P^2 = P$, the almost-product $F = 2P - I$ obeys $F^2 = I$, and the golden operator $G = \varphi P + (1-\varphi)(I-P)$ satisfies $G^2 = G + I$. The module proves that projector-to-golden step for endomorphisms of a real module; it is deliberately not a full Hessian-manifold formalization.
A rank-one endomorphism is the map $A : x \mapsto \ell(x), v$ built from a linear functional $\ell$ and a fixed vector $v$. In the Hessian-geometry setting this is the algebraic shape of the cost-Hessian operator (paper eq. (6.3)). Downstream, the multi-coordinate cost Hessian is literally this construction with $\ell$ the cost-Hessian form and $v = \alpha$.
The ambient type is any real module (additive commutative group with $\mathbb{R}$-action). Endomorphisms are $\mathrm{End}_{\mathbb{R}}(V)$. No manifold structure is required at this layer.
proof idea
Definition, not a theorem. The underlying function is $x \mapsto \ell(x),\bullet, v$. Linearity is discharged by two short simp proofs: additivity uses linearity of $\ell$ together with add_smul; homogeneity uses linearity of $\ell$ together with smul_smul. No external lemmas beyond the linear-map and module API.
why it matters
This definition seeds the entire projector-to-golden chain in CostProjectorGolden. The companion identity shows $A^2 = \ell(v), A$; normalizing by the scalar $\mu = \ell(v)$ yields a projector when $\mu \neq 0$; the golden operator built from that projector then satisfies $G^2 = G + I$. Those three facts are packaged as rankOneEnd_square, rankOneEnd_normalized_isProjector, and rankOneEnd_goldenOperator_sq.
ProjectorGoldenCertificate records the almost-product and golden-structure properties for arbitrary projectors, so the rank-one case is the concrete generator of the certificate. Downstream, costHessianOperator in JHessianGoldenMulti instantiates this map with the cost-Hessian form, tying the algebraic core to reciprocal-cost geometry. The equation $G^2 = G + I$ is the algebraic signature of the golden ratio $\varphi$ forced at T6 in the forcing chain (T5 J-uniqueness, T6 self-similar fixed point).
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