hinge4DStarKernelStatus
plain-language theorem explainer
Status board for the 4D Regge full-star hinge kernel on the seed triangle. It records three closed deliverables (star enumeration, flatness gate, full-star class kernel), two intentionally open scopes (other hinge orbits, flat Hessian assembly), and two unproved campaign goals (EH 4D convergence, gap-action recovery). Downstream flag theorems cite it to reify the campaign state. The body is a pure structure instance with boolean literals.
Claim. The campaign status for the 4D Regge star deficit kernel on the seed hinge $\{0,e_0,e_0+e_1\}$ is the record: star enumeration closed; flatness gate closed; full-star class kernel closed; other hinge orbits open; flat Hessian assembly open; $S_{\mathrm{RS}}\to\mathrm{EH}_{4d}$ convergence false; gap-action recovery false.
background
This module is the next kernel-checked increment in the QG full-theory campaign after the dihedral cosine kernel. Scope is the seed triangle hinge ${0,e_0,e_0+e_1}$ and its full periodic Freudenthal star in the integer lattice: four containing unit cubes and six incident 4-simplices. The Freudenthal incidence layer, 15-class edge stencil, and two-simplex dihedral cosine calculus are imported, not redefined.
Deliverables already proved in the module include exact star enumeration (six cube-translate/Kuhn-simplex pairs), the flat cosine multiset (four simplices at $1/\sqrt{2}$, two at $0$), the flatness gate (star angle sum exactly $2\pi$), and the full-star deficit class kernel on classes $(2,3,6,7,10,11,14)$ with values $(-1,-1,+1,-1,+1,+1,-1)$.
Hinge4DStarKernelStatus is the boolean scoreboard structure: three closed flags, two open-scope flags, and two campaign-goal flags (convergesEH4d, gapActionRecovery).
proof idea
No proof work. The definition is a structure instance that hard-codes the seven booleans to match the module's tier tags and deliverable list: the three closed kernel items are true, the two open-scope items are true (meaning still open), and the two unproved campaign goals are false.
why it matters
Gives a single named status object that the sibling theorem hinge4DStarKernelStatus_flags reifies as a conjunction of equalities. Downstream readers and CI can query campaign progress without rereading the module doc. The flags encode the binding tier tags: other hinge orbits remain open, flat Hessian assembly over all hinges is not done, and neither $S_{\mathrm{RS}}\to\mathrm{EH}_{4d}$ nor gap-action recovery is claimed. In the Recognition gravity stack this is bookkeeping for the Regge 4D star kernel, not a physics identity; it keeps the proved seed-star class kernel cleanly separated from the still-open lattice-wide and continuum limits.
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