stencilMomentTensor
plain-language theorem explainer
The stencil moment tensor A₀ is the 3×3 matrix Σ_d c_d d dᵀ built from the seven canonical Freudenthal displacement classes and their Hessian dual weights. Continuum-limit work on the anisotropic Regge energy cites it as the coefficient of ∫⟨∇f, A₀ ∇f⟩ on the unit cube. The definition is a direct finite sum: weight times outer-product entry over Fin 7.
Claim. Define the stencil moment tensor $A_0:\{0,1,2\}^2\to\mathbb{R}$ by $A_0(i,j)=\sum_{d\in D} c_d\, d_i\, d_j$, where $D$ is the set of seven canonical Freudenthal displacement classes (three axis, three face-diagonal, one body-diagonal), $d_i$ are their real coordinates in $\{0,1\}$, and $c_d=\sqrt{\ell_d^2}$ are the Hessian dual stencil weights ($\ell^2\in\{1,1,1,2,2,2,3\}$).
background
This module is Stage 1 of panel-locked Test G in the QG full-theory campaign (Phase 2b): the action-level continuum limit of the frozen quadratic energy on the canonical periodic Freudenthal triangulation, for every side length $N>2$. The pillar-2 path-sum flag stays red; the scope is the tensor-first anisotropic stencil identity and its moment tensor.
The seven displacement classes are tabulated by dispReal: unit axis vectors $e_1,e_2,e_3$, face diagonals $e_i+e_j$, and the body diagonal $e_1+e_2+e_3$. Sibling stencilWeight supplies the dual weights $c_d=\sqrt{\ell_d^2}$ read off the canonical Regge Hessian incidence structure ($\ell^2=1,1,1,2,2,2,3$). The quadratic form of that Hessian is already identified with the seven-class nearest-displacement stencil $\sum_x\sum_d c_d(u(x+d)-u(x))^2$.
The moment tensor is the second-moment matrix of those weighted displacements. In continuum language it is the constant coefficient of the anisotropic Dirichlet form $\langle\nabla f,A_0\nabla f\rangle$ that the scaled lattice energy is expected to approach.
proof idea
Pure definition, not a proof. The body is the finite sum $\sum_{d:\mathrm{Fin},7} c_d\cdot(\mathrm{dispReal},d,i)\cdot(\mathrm{dispReal},d,j)$: each entry is the corresponding component of the outer product $d,d^\top$, weighted by the stencil coefficient and summed over the seven classes. No lemmas are applied; evaluation and the closed form $A_0=(1+\sqrt{2})I+(\sqrt{2}+\sqrt{3})J$ are separate theorems.
why it matters
A₀ is the algebraic kernel of the Stage-2 continuum limit in FreudenthalEnergyLimit. Downstream, continuumTarget is defined independently as $A_0[0,0]\cdot 2\pi^2$; integral_witness_energy_density checks that the interval integral of $\sum_{i,j}A_0[i,j]\partial_i f,\partial_j f$ equals that constant; rateConstant is $A_0[0,0]\cdot(2\pi)^4/24$; and the witness energy theorems (freudenthalStencilEnergy_witness, freudenthal_witness_energy_rate_integral_form, scaledCanonicalEnergy_witness_closed_form) reduce lattice sums to multiples of $A_0[0,0]$ or the full bilinear form.
In the Recognition framework this is the concrete $D=3$ spatial moment structure forced by the Freudenthal triangulation of the periodic 3-torus (T8 landmark). It turns the panel-locked stencil identity into a numerical continuum target without fitting. The open residual is the red pillar-2 path-sum flag: refinement-indexed measure-weighted sums over inequivalent triangulation classes are out of scope here.
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