costHessianOperator
plain-language theorem explainer
The multi-coordinate recognition cost Hessian is realized as the rank-one endomorphism A X = cosh(α·t) ⟨α, X⟩ α on an arbitrary real inner-product space. Anyone citing Phase 4 multi-coordinate φ-forcing or paper eq. (6.3) needs this operator. It is a one-line application of the algebraic rank-one endomorphism constructor to the lowered Hessian form.
Claim. For a comparison direction $\alpha$ and log-coordinate point $t$ in a real inner-product space $V$, define the endomorphism $A_{\alpha,t}:V\to V$ by $A_{\alpha,t}(X)=\cosh(\alpha\cdot t)\,\langle\alpha,X\rangle\,\alpha$ (paper eq. (6.3), identity reference metric).
background
This module lifts one-dimensional J-Hessian φ-forcing to the genuine multi-coordinate recognition cost manifold (Washburn–Zlatanović, arXiv:2606.02150). The n-dimensional reciprocal cost is $J=\tfrac12(R+R^{-1})-1$ with $R=\prod x_i^{\alpha_i}$, which in logarithmic coordinates $t_i=\log x_i$ becomes $J(t)=\cosh(\alpha\cdot t)-1$. 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.
The sibling form $\ell(X)=\cosh(\alpha\cdot t),\langle\alpha,X\rangle$ is that tensor with one index lowered. Upstream, rankOneEnd builds the endomorphism $x\mapsto\ell(x),v$ from a linear form and a vector; that is the algebraic shape of the golden-structure Hessian projector route. Pairing $\ell$ with direction $\alpha$ and the identity reference metric yields the $(1,1)$-tensor $A=h_0^{-1}\nabla^2 J$.
proof idea
One-line definitional wrapper: feed the cost-Hessian linear form and the comparison vector $\alpha$ into the rank-one endomorphism constructor. The resulting map is $X\mapsto\ell(X)\cdot\alpha$ with $\ell=\cosh(\alpha\cdot t),\langle\alpha,\cdot\rangle$, which is exactly $A$ of paper eq. (6.3). No tactic proof; the body is the constructor application.
why it matters
This operator is the raw multi-coordinate Hessian endomorphism that Phase 4 needs. Downstream it feeds the square identity $A^2=\mu A$ (paper Lemma 3.1), the normalized-projector corollary (paper Corollary 3.1), and the multi-coordinate φ-forcing theorem that the induced golden operator satisfies $G^2=G+I$ for arbitrary $\alpha\neq 0$, arbitrary log-point $t$, and arbitrary real inner-product space. The certificate structure packages the strict positivity of the curvature scalar and of $\mu=\cosh(\alpha\cdot t)|\alpha|^2$.
The decisive nondegeneracy source is $\cosh>0$ together with $\alpha\neq 0$: multi-coordinate analogue of the unit curvature $J''(1)=1$ used in the one-dimensional golden Hessian. That closes Phase 4 without a supplied operator or a single-coordinate restriction, and sits on the T5–T6 forcing path (J-uniqueness to φ as self-similar fixed point).
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.