assembledConstantBlock_eq_zero
plain-language theorem explainer
The assembled constant block of the Regge TT second-variation stencil vanishes for every real 3×3 polarization matrix E. This is the Gate C-A3 hinge-aware zero-mode headline: stencil-only residual is nonzero, but hinge plus Hessian cancel exactly at k=0. Continuum-limit and Bloch-symbol arguments cite it. Proof unfolds the block, applies free-coefficient cancellation, then the alternating-sum identity, and finishes by norm_num.
Claim. For every real polarization matrix $E:\{0,1,2\}^2\to\mathbb{R}$, the assembled constant block (the $k=0$ Schlaefli-reduced raw cell stencil combining hinge and Hessian contributions) equals zero: $\mathrm{assembled\_const}(E)=0$.
background
Paper C / Pillar 1, Lane C of the QG full-theory campaign. Gate C-A3 closes the hinge-aware zero mode for the Regge TT Bloch analysis. A same-day sympy diagnostic found that 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 $E=\mathrm{diag}(1,-1,0)/\sqrt{2}$, $k=e_z$, the residual is $-\pi(\sqrt{2}+4)/8$. The assembled quadratic at $k=0$, however, cancels.
By the Gate A2 Schlaefli reduction, the assembled constant block is the $k=0$ value of the raw cell stencil, carrying hinge and Hessian blocks together. Edge-class coefficients $c_d(E)$ are the polarization projections of $E$ onto the seven displacement classes. The module proves the assembled object vanishes for every $E$, not only TT matrices; the seven TT hypotheses would be unused and are omitted.
The assembly sign convention is fixed by the diagnostic: assembled combines hinge and stencil residual with a relative minus sign. That convention is pinned later by the witness-split identity; the present theorem supplies the vanishing half.
proof idea
Unfold the definition of the assembled constant block. Invoke the free-coefficient zero-mode lemma on the map sending each displacement class $d$ to the polarization edge coefficient of $E$ at $d$; after simplification this expresses the block in terms of those coefficients. Rewrite via the alternating-sum identity for polarization edge coefficients, which collapses the linear combination. Finish with norm_num to obtain the numerical zero. Pure algebraic cancellation; no TT hypotheses and no analysis.
why it matters
Headline of Gate C-A3: the hinge-aware assembled zero mode, not a stencil-only cancellation. Downstream, assembled_witness_split uses it with the hinge-residual match to pin assembled(E_w)=hinge(E_w)-stencilOnlyConstantWitnessResidual and freeze the relative-minus assembly convention. canonicalFiniteH_zeroMomentum_eq_zero reduces the A2 canonical finite value at zero integer wave vector to this identity (times zero). zeroMomentum_symbol_is_zero lifts that to the fixed-$N$ TT Bloch symbol of the true nonlinear Regge second variation at zero wave vector. In the continuum lane, rawCosineFoldAtScale_zero identifies the scale-zero raw cosine fold with the assembled constant block and quotes this vanishing. Together these feed the lattice flat zero mode for the Regge TT program under the finishing charter.
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