absoluteTriple
plain-language theorem explainer
Names the absolute vertex-mask triple of a triangle slot in a Kuhn 4-simplex: the three vertex bitmasks as a sorted ℕ-triple. Orbit-classification arguments cite it when comparing absolute S₄ action on triangles rather than difference-mask type alone. The body is a one-line alias of the ordered vertex-mask extractor.
Claim. For each Kuhn simplex index $s \in \{0,\ldots,23\}$ and each oriented triangle slot $t \in \{0,\ldots,9\}$ inside that simplex, the absolute triple is the ordered triple of vertex bitmasks $(m_0,m_1,m_2) \in \mathbb{N}^3$ of the three vertices of that triangle.
background
This module classifies triangle hinges in one unit 4-cube Kuhn triangulation up to lattice translation and triangulation-preserving symmetry. It is a combinatorial prerequisite for assembling the flat Hessian of the 4D Regge action from per-orbit star kernels; it does not evaluate those kernels or complete the Hessian.
A Kuhn cell has 24 simplices. Each simplex carries a vertexMask API: vertices are bitmasks in ${0,\ldots,15}$. A triangle slot is one of the $C(5,3)=10$ index triples in a 5-vertex simplex. The upstream extractor returns the three masks in index order: $(v(s,i),v(s,j),v(s,k))$ for the slot's index triple.
Difference-mask type uses the successive XORs $(a,b)=(m_1\oplus m_0,m_2\oplus m_1)$ and their popcounts. Absolute triples keep the actual masks, so coordinate permutation can be tested directly on the triple rather than only on the type pair $(|a|,|b|)$.
proof idea
One-line definitional wrapper: the absolute triple is exactly the ordered vertex-mask triple of slot $t$ in simplex $s$. No extra sorting or normalization is performed at this layer; the name marks the object as the absolute (not difference-type) representative used by later orbit arguments.
why it matters
Feeds the honest refinement that coordinate permutation is not transitive on absolute $(1,1)$ triangles inside a cell. The parent theorem exhibits the seed hinge with absolute triple $(0,1,3)$ and the interior chain $(1,3,7)$, both of difference type $(1,1)$, and proves no $S_4$ bit-position permutation sends one triple to the other.
That distinction refines deliverable A of the module: lattice orbits under difference type give six (or four with complement) orbits, but absolute $S_4$ action on vertex masks splits the $(1,1)$ class further. The split matters when assembling per-orbit star kernels for the flat Hessian; conflating absolute orbits with difference-type orbits would mis-count independent kernel contributions.
Scope remains combinatorics of one Kuhn 4-cube. It does not touch $S_{\mathrm{RS}}\to$ Einstein–Hilbert recovery or gap-action flags.
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