linkEdges
plain-language theorem explainer
Link edges of the hinge triangle in the three-pent complex: the 2-vertex sets disjoint from the hinge whose union with the hinge is one of the three 4-simplices. Cited by anyone checking the interior-hinge gate for glued-pent Regge action. Pure filter definition on 2-subsets of six vertices, same shape as the two-pent module with the complex swapped.
Claim. Let $H=\{0,1,2\}$ be the hinge triangle and $\mathcal{C}$ the three-pent complex on vertex set $\{0,\ldots,5\}$. The link-edge set is $\{E\subseteq\{0,\ldots,5\}:|E|=2,\,E\cap H=\emptyset,\,H\cup E\in\mathcal{C}\}$.
background
In the RS gravity stack, glued-pent expressions may be called Regge action only when the hinge link is a cycle (genuine interior hinge), not a path (boundary). The committed two-pent module already shows two 4-simplices give a path link; a counting lemma proves three pents are necessary for an interior hinge. This module is the positive half of that gate: the minimal complex whose hinge link is a cycle.
The hinge is the triangle $H={0,1,2}$. Three 4-simplices sit on $\mathrm{Fin},6$: $A={0,1,2,3,4}$, $B={0,1,2,4,5}$, $C={0,1,2,3,5}$. Adjacent pairs share exactly one tetrahedron containing $H$; the triple intersection is exactly $H$. Each pent contributes one residual pair outside $H$, and those pairs become the candidate link edges.
Upstream, the hinge and the two-pent filter shape are imported from GluedPentsHingeWitness; the three-pent complex is the local carrier.
proof idea
Definitional construction, not a proved statement. Take the powerset of 2-element subsets of $\mathrm{Fin},6$ and retain those $E$ with $E\cap H=\emptyset$ and $H\cup E$ a member of the three-pent complex. Identical filter shape to the two-pent link-edge definition, with the complex argument replaced by the three-pent carrier.
why it matters
Carrier set for the positive interior-hinge witness (panel P1-remainder, live bet C12). Downstream results identify this set with the triangle cycle ${{3,4},{4,5},{3,5}}$, prove every link vertex has degree 2, and discharge the cyclic-link predicate, so $H$ is a genuine interior hinge. Paired with the two-pent path witness and the necessity counting lemma, it closes minimality: three pents are necessary and attained. Feeds the module's cycle theorem and residual-edge identification, and is the three-pent counterpart of the two-pent link-edge set used by the non-cycle theorem.
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