Hinge4DStarKernel13Status
plain-language theorem explainer
Status record for the type-(1,3) Regge 4D periodic-lattice star deficit kernel campaign. It packages seven Boolean gates: three closed (star enumeration, flatness, full-star class kernel) and four still open (type-(3,1) transport, flat Hessian assembly, EH4d convergence, gap-action recovery). Gravity analysts cite it as the single checklist for what this module has kernel-checked. The declaration is a plain structure definition with no proof content.
Claim. A status bundle of seven Booleans for the type-$(1,3)$ 4D Regge hinge star: whether star enumeration is closed, whether the flatness gate is closed, whether the full-star deficit class kernel is closed, whether type-$(3,1)$ transport remains open, whether flat Hessian assembly remains open, whether convergence of the RS action to 4D Einstein–Hilbert is claimed, and whether gap-action recovery is claimed.
background
The module sits in the QG full-theory campaign for Regge calculus on the 4D Freudenthal lattice. It treats the type-$(1,3)$ triangle hinge with absolute masks ${0,1,15}$ (difference masks $(1,14)$, local flat squared lengths $(1,3,4)$) and its full periodic star of Kuhn simplices. Prior increments handled the type-$(1,1)$ seed orbit and the orbit classification layer; this file imports the 15-class edge stencil, dihedral and flat kernels, and never redefines their API.
Deliverables already kernel-checked in the module include: exactly six origin-cube simplices contain the hinge; only the origin among ${-1,0,1}^4$ cube translates does; all six flat cosines equal $1/2$; the star angle sum is exactly $2\pi$; and the full-star deficit class kernel on classes $(1,3,5,7,9,11,13)$ takes values $(-\sqrt{3},-\sqrt{3},+\sqrt{3},-\sqrt{3},+\sqrt{3},+\sqrt{3},-\sqrt{3})$.
The complementary type-$(3,1)$ hinge is related by mask complement in the classification layer, but transport of this kernel across that duality is explicitly left open, as are flat Hessian assembly over all hinges, $S_{\mathrm{RS}}\to\mathrm{EH}_{4d}$, and gap-action recovery.
proof idea
No proof: this is a structure definition. Seven named Boolean fields record campaign gates. The sole downstream inhabitant hinge4DStarKernel13Status assigns the concrete flags (three closed, four open) by structure literal; nothing is proved at this declaration.
why it matters
Gives a single, machine-readable scoreboard for the type-$(1,3)$ star-kernel increment in the Regge gravity analysis stack. Downstream, hinge4DStarKernel13Status is the concrete witness that star enumeration, the flatness gate, and the full-star class kernel are closed, while type-$(3,1)$ transport, flat Hessian assembly, EH4d convergence, and gap-action recovery stay open.
That matches the module's binding tier tags: the file advances kernel-checked geometry (six-simplex star, $6\cdot\arccos(1/2)=2\pi$, class kernel on the seven odd classes) without claiming the larger Recognition-Science gravity closures. In the broader RS forcing picture this is scaffolding toward discrete curvature matching continuum EH in $D=3+1$, not a step of the T0–T8 chain itself. Referees can read the Booleans as the honest boundary between proved kernel facts and still-open assembly theorems.
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