seedOrbitAssembly
plain-language theorem explainer
Cell-local seed-orbit contribution to the flat Hessian of the 4D Regge action, with combinatorial incidence hard-wired as a cutoff. Anyone assembling the Freudenthal-cell second variation cites this MODEL skeleton that forces off-support edge classes to vanish. It specializes the bilinear orbit form: area weights are scaled by seed-hinge incidence, and deficit kernels are zeroed when either index is off-support.
Claim. Given area weights $A_e$, deficit kernels $K_{ef}$, and edge variations $c_e$ on the fifteen edge classes, the seed-orbit assembly equals $\sum_{e,f} \iota(e)\,A_e\,K^{\mathrm{cut}}_{ef}\,c_e c_f$, where $\iota(e)\in\{0,1,2\}$ is the combinatorial incidence of class $e$ in the two seed-containing simplices, and $K^{\mathrm{cut}}_{ef}=0$ whenever $\iota(e)=0$ or $\iota(f)=0$, else $K_{ef}$.
background
The module is the next kernel-checked increment after the 15-class Regge edge stencil in the QG full-theory campaign. It works inside one Freudenthal/Kuhn 4-cube cell: 24 monotone 4-simplices from axis permutations, with a distinguished seed hinge (triangle on vertices $0$, $e_0$, $e_0+e_1$). Exactly two of the 24 simplices contain that hinge.
Combinatorial incidence counts, for each of the 15 edge classes, how many of those two seed-simplices carry the class as a local edge. Three classes are combinatorial decoys (incidence zero); the three hinge-boundary classes each have multiplicity two. The support is nonempty and invariant under the axis swap $2\leftrightarrow 3$ that fixes the seed hinge.
Upstream, the flat-Hessian orbit form is the bilinear contraction $\sum_{e,f} A_e K_{ef} c_e c_f$, the 4D skeleton of the 3D Schläfli-reduced second variation. Both $A$ and $K$ remain OPEN parameters: true dihedral and Cayley–Menger kernels are not yet plugged in.
proof idea
Definitional wrapper, not a proved statement. It specializes the flat-Hessian orbit form in three places: (i) each area weight is multiplied by the natural-number seed-hinge incidence of that class, cast to $\mathbb{R}$; (ii) the deficit kernel is replaced by a cutoff that returns $0$ if either index has vanishing incidence, else the supplied $K$; (iii) the variation vector $c$ is passed through unchanged. The resulting double sum therefore receives no contribution from off-support classes.
why it matters
This MODEL definition is the assembly skeleton named in the module header: it contracts OPEN per-hinge area weights against OPEN deficit kernels, forced to vanish off the incidence support. Downstream, the decoy-area theorem shows a decoy-only bump in the area weight is annihilated by the incidence cutoff, and the support-projection theorem shows the assembly depends on $c$ only through supported classes.
It advances deliverable B (combinatorial support gates that are kernel-checked) without closing the OPEN numeric kernels. It does not complete the flat Hessian of the 4D Regge action, does not prove $S_{\mathrm{RS}}$ converges to Einstein–Hilbert in 4D, and does not flip gap_action_recovery. The incidence cutoff is combinatorial scaffolding for later true second-variation kernels.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.