below
plain-language theorem explainer
The Schläfli-reduced assembled constant block of the Regge cell stencil vanishes at zero wave vector for every polarization matrix. Lattice-gravity workers cite this as the Gate C-A3 hinge-aware flat zero mode (Paper C / Pillar 1). The page target is still a sorry stub with empty proof body; the intended argument is a perfect-square identity on seven edge-class coefficients plus vanishing of the alternating polarization sum.
Claim. After the Gate A2 Schläfli reduction, the assembled constant block of the raw cell stencil (hinge/edge-diagonal $O(1)$ piece combined with the stencil residual under the relative-minus convention) equals $0$ at wave vector $k=0$, for every polarization matrix $E$. Equivalently, the six-tetrahedron raw-table contraction is the perfect square $\frac{1}{2}(c_0+c_1+c_2-c_3-c_4-c_5+c_6)^2$ in the seven free edge-class coefficients, and the alternating class sum of those coefficients vanishes for every $E$.
background
Local setting (module doc): QG full-theory campaign, Paper C / Pillar 1, Lane C of the finishing charter. This file is meant to close the hinge-aware zero-mode gate for the Regge TT Bloch symbol.
Sympy diagnostic context (not proof): the stencil-only constant block (full per-tet Hessian $G$ contracted with edge-class coefficients, no hinge) does not vanish under TT. At the reported witness $E=\mathrm{diag}(1,-1,0)/\sqrt{2}$, $k=e_z$, its residual is $-\pi(\sqrt{2}+4)/8$, recorded as the kernel stencil-only constant witness residual. The same report records hinge equal to that value and assembled quadratic $H\hat(0)=0$, fixing the assembly convention $\mathrm{assembled}=\mathrm{hinge}-\Sigma_{Gcc}$.
The hinge/edge-diagonal $O(1)$ block is $\sum_d 2\pi\cdot L''(l2_d),c_d(E)^2$ over the seven displacement classes, with $L(\ell^2)=\sqrt{\ell^2}$ so $L''=-1/(4\ell^2\sqrt{\ell^2})$ at the flat squared length, and $c_d$ the edge-class coefficients of the polarization.
proof idea
Claim status is scaffolding: sorry stub, zero proof-body lines. No tactics fire yet.
Intended route, from the module's own proof sketch: establish the raw-table contraction identity that the six-tet sum is the perfect square $(c_0+c_1+c_2-c_3-c_4-c_5+c_6)^2/2$ in seven free coefficients; prove the alternating edge-class sum vanishes for every polarization, because $c_{x+y}+c_{x+z}+c_{y+z}$ equals the trace double-count $c_x+c_y+c_z+c_{x+y+z}$ termwise; conclude the assembled constant block is identically zero. Separately, on the concrete TT witness, cancel hinge against the recorded stencil-only residual and pin assembled = hinge - residual with both sides zero. Final corollaries would push zero through the Gate A2 canonical finite value and the fixed-$N$ TT Bloch symbol at zero integer wave vector.
why it matters
Earns its place as the Gate C-A3 headline: the lattice flat zero mode of the true nonlinear Regge action's second variation, stated about the assembled (hinge-aware) object rather than the stencil-only residual. The module deliberately states the stronger claim (every polarization, not only TT) so TT hypotheses are not unused Props.
Parent edges supplied in the graph (d'Alembert cost algebra, exoplanet habitability bands, mercury $\phi$-resonance, Curie temperature) are semantically unrelated to this gravity stencil; treat them as noisy used_by links, not real consumers. Framework landmark: discrete Regge/Bloch side of the RS gravity campaign, not a T0–T8 forcing step.
Open scope the stub must respect: full Hessian split $\mathrm{assembled}=\mathrm{hinge}-\sum Gcc$ is not re-proved in Lean ($\theta$ second-derivative entries of $G$ unformalized); off-witness it stays sympy-diagnostic tier.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.