freudenthal_stencil_identity
plain-language theorem explainer
For every lattice side N>2 and every real vertex field u, the a-priori normalized canonical Regge-Hessian quadratic energy equals the seven-class Freudenthal finite-difference stencil with mesh h=1/N. Gravity continuum-limit work cites this as the panel-locked stage-1 identity of Test G. The proof rewrites via the already-proved un-normalized stencil equality, unfolds ρ(N)=1/N and h, then finishes by field arithmetic under the sum.
Claim. For every integer $N>2$ and every real-valued field $u$ on the $N\times N\times N$ periodic Freudenthal vertex set, $$\rho(N)\,Q_N(u)=h^3\sum_x\sum_{d\in D}c_d\Bigl(\frac{u(x+d)-u(x)}{h}\Bigr)^2,$$ where $h=1/N$, $\rho(N)=1/N$, $Q_N$ is the quadratic form of the canonical Regge Hessian on the canonical periodic Freudenthal triangulation, $D$ is the seven displacement classes, and $c_d$ are the Hessian dual stencil weights $\sqrt{\ell_d^2}$ with $\ell_d^2\in\{1,1,1,2,2,2,3\}$.
background
This module is Phase 2b of the QG full-theory campaign: panel-locked Test G stage 1 (candidate C8), the tensor-first anisotropic action continuum limit on the frozen quadratic energy of the canonical Freudenthal family. Stage 1 is exact and general-$N$; stage 2 (companion module) samples continuum fields into the normalized energy.
The geometric substrate is the side-$N$ periodic Freudenthal triangulation with its incidence-consistent structure. The canonical Regge Hessian on that complex yields a quadratic form $Q_N$. An upstream identity already equates $Q_N(u)$ with the un-normalized seven-class stencil energy $\sum_x\sum_d c_d,(u(x+d)-u(x))^2$, where the weights $c_d=\sqrt{\ell_d^2}$ are read from the Hessian dual edge lengths (three axis, three face-diagonal, one body-diagonal classes). Mesh size is $h=1/N$; the normalization factor $\rho(N)=\mathrm{stencilNormalization},N=1/N$ is fixed a priori on dimensional grounds, not fitted after the fact.
The local goal is the panel-locked normalized identity that inserts the difference quotients $(u(x+d)-u(x))/h$ and the volume factor $h^3$, so the continuum limit can be taken without re-tuning constants.
proof idea
One short algebraic reduction, not a new geometric argument.
- Rewrite the left-hand side by the upstream theorem that equates the canonical Regge-Hessian quadratic form with the un-normalized Freudenthal stencil energy (seven displacement classes, weights $c_d$).
- Unfold the definitions of that stencil energy, of $\rho(N)=1/N$, and of mesh size $h=1/N$.
- Record $N\neq 0$ in $\mathbb{R}$ (from $N>2$).
- Pull the outer scalar through the double finite sum, then pointwise under the sum apply field simplification: the factors of $N$ and $1/N$ rearrange into $h^3\cdot((\Delta u)/h)^2$ against the stencil weights.
No incidence or triangulation reasoning is repeated here; that work lives in the upstream stencil equality.
why it matters
This is the panel-locked stage-1 observable of Test G: the exact general-$N$ identity that converts the canonical Regge Hessian into a normalized anisotropic finite-difference energy ready for continuum analysis. Downstream it is recorded as the boolean flag general_N_identity_derived inside StencilPreflightStatus, which packages stage-1 preflight (identity, a-priori $\rho(N)=1/N$, and the exact moment tensor $A_0=(1+\sqrt{2})I+(\sqrt{2}+\sqrt{3})J$).
Together with stage 2 it supports the action-level continuum limit on the frozen quadratic energy of the Freudenthal family. Scope remains partial: the pillar-2 path-sum flag stays red until a refinement-indexed measure-weighted sum over inequivalent triangulation classes is available. Within Recognition gravity, the result anchors the discrete-to-continuum bridge for the anisotropic stencil without post-hoc coefficient fitting, on the same geometric family used for the $D=3$ spatial setting.
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