frozen_identification_stencil
plain-language theorem explainer
Along the conformal edge-length family on the periodic Freudenthal torus (N>2), the true nonlinear Regge action equals half the anisotropic seven-class Freudenthal stencil energy plus the frozen quadratic remainder. Gravity analysts cite this when separating conformal-sector anisotropy from the isotropic TT continuum symbol. The proof is a two-step rewrite through the frozen identification and the canonical Hessian–stencil equality.
Claim. For lattice size $N>2$ and any vertex potential $u$ on the $N\times N\times N$ torus, the true nonlinear Regge action evaluated on the typed conformal edge-length field built from $u$ equals $\tfrac12$ times the Freudenthal seven-class stencil energy of $u$, plus the Regge action remainder of the canonical periodic triangulation, its incidence data, the canonical Regge Hessian, and the potential associated to $u$.
background
This module is Stage 1 of the Regge TT continuum-symbol campaign: it defines the true nonlinear 3D Regge action on the canonical periodic Freudenthal torus as a function of an arbitrary edge squared-length field, $S(\ell)=\sum_e\sqrt{\ell_e},(2\pi-\sum_{t\ni e}\theta)$, with dihedral angles from Cayley–Menger cofactors. At conformal fields the action recovers the existing concrete Regge action.
The conformal family typedConformalEdgeField restricts edge lengths to a scalar potential $u$ on vertices. The frozen model expands the action about the flat point along that family: a quadratic Hessian term plus a remainder. Independently, Test G supplies the anisotropic seven-class Freudenthal stencil energy whose continuum moment tensor is $A_0=(1+\sqrt{2})I+(\sqrt{2}+\sqrt{3})J$.
The companion identification equates the frozen quadratic piece to a canonical Hessian quadratic form; a further lemma identifies that form with the stencil energy when $N>2$. Numerically observed isotropic TT physics is stated to live outside the conformal family, so frozen anisotropy and continuum isotropy can coexist.
proof idea
Term-mode proof by two rewrites. First apply the frozen-model identification: the true Regge action on the typed conformal field equals half the canonical Hessian quadratic form of the potential plus the Regge action remainder. Second rewrite that Hessian quadratic via the equality of the canonical Regge Hessian quadratic form with the Freudenthal stencil energy (valid for $N>2$). No further algebraic work; the two lemmas compose directly.
why it matters
Closes the stencil form of the frozen-model identification inside the Regge TT symbol preflight. It pins the conformal-sector quadratic cost to the anisotropic seven-class stencil of Test G, so any continuum analysis of the frozen model inherits the known moment tensor $A_0=(1+\sqrt{2})I+(\sqrt{2}+\sqrt{3})J$. Downstream, this separates what the conformal restriction can and cannot say about the open target ReggeTTContinuumIsotropyTarget: the C10 numerics report isotropic TT symbol $K(0)=-(1/4)I_{TT}$ matching linearized Einstein–Hilbert, but that physics is outside the conformal family. The theorem therefore explains how frozen anisotropy and continuum TT isotropy are compatible rather than contradictory. No used-by edges yet; it is a leaf identification ready for continuum-symbol arguments.
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