stencilMomentTensor_psd
plain-language theorem explainer
The Freudenthal stencil moment tensor $A_0$ is positive semidefinite: every real $v\in\mathbb{R}^3$ satisfies $v^T A_0 v\ge 0$. Continuum-limit analysts in the QG Test G campaign cite this as the kernel-level PSD witness for the anisotropic seven-class stencil. The proof rewrites the quadratic form as a weighted sum of squares and invokes nonnegativity of the dual weights.
Claim. For every real vector $v\in\mathbb{R}^3$, the quadratic form of the stencil moment tensor $A_0=\sum_d c_d\, d d^T$ is nonnegative: $0\le \sum_{i,j=1}^{3}(A_0)_{ij} v_i v_j$.
background
This module is Stage 1 of panel-locked Test G in the QG full-theory campaign: the action-level continuum limit of the frozen quadratic energy on the canonical periodic Freudenthal triangulation, for every side length $N>2$. The quadratic form of the canonical Regge Hessian is identified exactly with a seven-class nearest-displacement stencil $\sum_x\sum_{d\in D} c_d,(u(x+d)-u(x))^2$, with weights $c_d=\sqrt{\ell_d^2}$ read off the Hessian dual weights ($\ell^2\in{1,1,1,2,2,2,3}$ for axis, face-diagonal, and body-diagonal classes).
The stencil moment tensor is $A_0=\sum_{d\in D} c_d, d d^T$, with exact closed form $A_0=(1+\sqrt{2})I+(\sqrt{2}+\sqrt{3})J$. Upstream, stencilMomentTensor_quadratic_eq supplies the kernel identity $v^T A_0 v=\sum_d c_d\langle d,v\rangle^2$, and stencilWeight_nonneg records $c_d=\sqrt{\ell_d^2}\ge 0$.
proof idea
Term-mode, three steps. Rewrite the double sum via the exact sum-of-squares identity $v^T A_0 v=\sum_d c_d\langle d,v\rangle^2$. Reduce to nonnegativity of a finite sum over the seven displacement classes. Each summand is a product of a stencil weight and a square; the weight is nonnegative by the square-root definition, and the square is nonnegative by positivity, so the product is nonnegative.
why it matters
Stage-1 preflight for the tensor-first anisotropic continuum limit (candidate C8). Downstream, the documentation record StencilPreflightStatus packages this PSD fact with the general-$N$ stencil identity and the a-priori normalization $\rho(N)=1/N$ as certified flags. Without $A_0\succeq 0$, the continuum quadratic form would not be a legitimate elliptic energy, and the Stage-2 energy-limit companion could not pass to a well-posed continuum action. Scope remains partial: the pillar-2 path-sum flag stays red pending a refinement-indexed measure-weighted sum over inequivalent triangulation classes. No Recognition forcing-chain landmark (T0–T8) is touched directly; this is pure discrete-to-continuum gravity infrastructure.
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