stencilNormalization
plain-language theorem explainer
Defines the a-priori normalization factor ρ(N)=1/N for the side-N canonical periodic Freudenthal family on the unit 3-torus. Gravity continuum-limit work cites it as the unique dimensional prefactor converting the raw Regge-Hessian quadratic form into a physical energy density. The body is the literal reciprocal of N; no fitting or limit computation enters.
Claim. For each natural number $N$, the stencil normalization is $\rho(N) := 1/N$. Equivalently, if $h = 1/N$ is the lattice spacing of the side-$N$ triangulation of the unit $3$-torus, then $\rho(N) = h$.
background
This module is Phase 2b of the QG full-theory campaign (panel-locked Test G, stage 1): the action-level continuum limit of the frozen quadratic energy on the canonical Freudenthal triangulation family. Scope is partial; the pillar-2 path-sum flag remains red.
The canonical Regge Hessian on the side-$N$ periodic Freudenthal triangulation yields a quadratic form $Q_N(u)$ that equals a seven-class nearest-displacement stencil $\sum_x \sum_{d} c_d (u(x+d)-u(x))^2$, with weights $c_d = \sqrt{\ell_d^2}$ for edge-length squares $1,1,1,2,2,2,3$. The physical continuum energy density is $h^3 \sum c_d ((\Delta u)/h)^2$. Expanding shows this differs from the raw stencil sum by exactly one power of the mesh size $h=1/N$.
Hence $\rho(N)=1/N$ is fixed by dimensional counting before any continuum limit is taken. The sibling meshSize is the same quantity under the name $h$.
proof idea
Pure definition: the body is the real reciprocal of the natural number $N$. No lemmas, tactics, or computation. The surrounding doc-comment supplies the dimensional justification (one hinge-measure length factor per Hessian summand) that makes $\rho(N)=1/N$ the unique a-priori choice rather than a fitted constant.
why it matters
Stage-1 panel-locked observable of Test G requires the identity $\rho(N)\cdot Q_N(u)=h^3\sum_x\sum_d c_d((u(x+d)-u(x))/h)^2$ for every $N>2$. This definition supplies $\rho(N)$ a priori, so nothing is fitted after the fact.
Downstream, freudenthal_stencil_identity multiplies the canonical Hessian quadratic by this factor; scaledCanonicalEnergy packages the product as the normalized energy sampled in stage 2; scaledCanonicalEnergy_eq_scaled_stencil and the closed-form witness energy in FreudenthalEnergyLimit inherit it. The status record StencilPreflightStatus flags that normalization was declared with dimensional justification, not post-hoc fitting.
In the broader RS gravity program this is the mesh-to-continuum bridge for the anisotropic Freudenthal action (tensor-first candidate C8), sitting under the $D=3$ spatial setting forced at T8.
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