hinge4DOrbitClassificationStatus
plain-language theorem explainer
Status ledger for the 4D Regge triangle-hinge orbit classification on one Kuhn cell. It records three combinatorial theorems (six S4 orbits by popcount with counts 72/48/48/24/24/24; four orbits under S4 plus complement; within-cell absolute triangles need not be one S4 orbit), two open items (non-seed star kernels; full flat Hessian), and an explicit non-claim on continuum EH recovery. Downstream flag checks pin length and the open-kernel string. Pure list definition, no proof content.
Claim. A fixed six-string status list for the module: (i) six $S_4$ lattice orbits of Kuhn-cell triangle hinges classified by difference-mask popcount type, with per-type cell counts $72/48/48/24/24/24$; (ii) the larger group $S_4\rtimes\{\mathrm{id},\mathrm{complement}\}$ merges types $(1,2)\sim(2,1)$ and $(1,3)\sim(3,1)$ to four orbits; (iii) absolute within-cell triangles of fixed type need not form a single $S_4$ orbit; plus two open items (per-orbit star kernels for the five non-seed $S_4$ orbits; flat Hessian assembly over all hinge orbits) and one non-claim that continuum Einstein-Hilbert recovery is not asserted here.
background
The ambient module classifies triangle hinges in the Freudenthal/Kuhn triangulation of the unit 4-cube. Scope is combinatorics only: difference masks of monotone vertex-mask chains inside the 24 Kuhn simplices, up to lattice translation and triangulation-preserving symmetry. Every index-triple triangle is a chain $m_0\subset m_1\subset m_2$ with disjoint nonzero difference masks $(a,b)=(m_1\oplus m_0,m_2\oplus m_1)$; its type is the popcount pair $(|a|,|b|)\in{(1,1),(1,2),(2,1),(1,3),(3,1),(2,2)}$.
There are exactly $24\cdot C(5,3)=240$ oriented triangle slots. The $S_4$ action on bit positions preserves type and is transitive on realizable pairs of each type (six orbits); the seed hinge ${0,e_0,e_0+e_1}$ has type $(1,1)$. Bitwise complement $m\mapsto m\oplus 15$ swaps $(i,j)$ with $(j,i)$, so under $S_4\rtimes{\mathrm{id},\mathrm{complement}}$ one obtains four lattice orbits.
This status board sits in the QG full-theory campaign as the combinatorial prerequisite for assembling the flat Hessian from per-orbit star kernels. It imports the 24-simplex/vertexMask API and the 15-class mask utilities; it does not redefine them. Packaged continuum targets such as $S_{\mathrm{RS}}\to\mathrm{EH}_{4d}$ remain open elsewhere and are explicitly non-claimed here.
proof idea
No proof: the declaration is a List String literal. Six hard-coded status lines encode the module's tier tags (three THEOREM lines, two OPEN lines, one NONCLAIM). Downstream, a one-line decide theorem checks that the list has length 6 and that the open star-kernel string is a member. Nothing is derived algebraically at this site.
why it matters
This is the human- and machine-readable ledger for deliverable A of the Regge 4D hinge campaign: what is proved about lattice orbits, what remains open before flat-Hessian assembly, and what is deliberately not claimed. The sole direct consumer is hinge4DOrbitClassificationStatus_flags, which pins length and membership of the open non-seed star-kernel item so the status board cannot silently drift.
In the broader Recognition gravity stack, per-orbit star kernels and the flat Hessian of the 4D Regge action are the combinatorial bridge toward weak-field continuum recovery. The module doc and the NONCLAIM line both refuse to assert $S_{\mathrm{RS}}$ converges to Einstein-Hilbert in 4D or to flip gap_action_recovery. The board therefore keeps the forcing-chain gravity side honest: six (resp. four) orbits are settled combinatorics; kernel evaluation beyond the committed seed orbit and full Hessian assembly stay open.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.