costHessianOperator_square
plain-language theorem explainer
The multi-coordinate cost-Hessian operator A satisfies A² = μ A, with μ the Hessian form on the comparison direction α (paper Lemma 3.1). Anyone packaging the golden projector on the recognition cost manifold, or citing the multi-coordinate Phase 4 closure, needs this identity. The proof is a one-line application of the general rank-one endomorphism square law.
Claim. For any vectors $\alpha,t$ in the real inner-product space $V$, the cost-Hessian operator $A_{\alpha,t}$ obeys $A_{\alpha,t}^{2}=\mu\,A_{\alpha,t}$, where $\mu=\ell_{\alpha,t}(\alpha)$ and $\ell_{\alpha,t}(X)=\cosh(\alpha\cdot t)\,\langle\alpha,X\rangle$ is the lowered rank-one Hessian form.
background
This module lifts one-dimensional J-Hessian φ-forcing to the genuine multi-coordinate recognition cost manifold (Washburn–Zlatanović, arXiv:2606.02150). The reciprocal cost is $J=\tfrac12(R+R^{-1})-1$ with $R=\prod x_i^{\alpha_i}$, or in log coordinates $J(t)=\cosh(\alpha\cdot t)-1$. Its Hessian is the rank-one tensor $\nabla^2 J=\cosh(\alpha\cdot t),(\alpha\otimes\alpha)$, positive semidefinite of rank one in every dimension.
Against the identity reference metric one obtains the $(1,1)$-operator $A X=\cosh(\alpha\cdot t),\langle\alpha,X\rangle,\alpha$ (paper eq. (6.3)). The associated linear form is $\ell(X)=\cosh(\alpha\cdot t),\langle\alpha,X\rangle$, so $A$ is the rank-one endomorphism $X\mapsto\ell(X),\alpha$. Upstream, rankOneEnd_square records the pure algebra fact that any such endomorphism squares to $\ell(v)$ times itself.
proof idea
One-line wrapper. The operator is defined as the rank-one endomorphism built from the cost-Hessian form $\ell=\mathrm{costHessianForm},\alpha,t$ and the vector $\alpha$. Instantiate the general identity rankOneEnd_square at that form and that vector; the composition identity $A^2=\ell(\alpha),A$ drops out by the definition of rank-one endomorphism multiplication.
why it matters
Direct input to jHessianGoldenMultiCertificate, which discharges the golden-structure φ-forcing hypotheses in arbitrary dimension. The certificate field operator_square is exactly this lemma. Combined with strict positivity of $\mu=\cosh(\alpha\cdot t),|\alpha|^2$ (from $\cosh>0$ and $\alpha\neq 0$) and the normalized projector step, one obtains $P=A/\mu$ and the golden operator $G=\varphi P+(1-\varphi)(I-P)$ with $G^2=G+I$, forcing the positive eigenvalue to be $\varphi$ (paper Corollary 3.1 and §4). This is the multi-coordinate analogue of the unit curvature $J''(1)=1$ used in the one-dimensional path, and it holds over any real inner-product space, closing Phase 4 without new axioms.
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