hinge4DStarKernel12Status

This module is the next kernel-checked increment in the QG full-theory Regge campaign after the type-$(1,1)$ seed orbit. Scope is the type-$(1,2)$ triangle hinge ${0, e_0, e_0+e_1+e_2}$ (masks $0,1,7$; difference masks $(1,6)$) and its full periodic Freudenthal star in the integer lattice: two containing unit cubes, four incident $4$-simplices. It imports the Freudenthal incidence layer, the $15$-class stencil, and the committed Gram-projection / cleared-denominator cosine calculus without redefining their API.

The status structure is a boolean dashboard with eight fields: three closed-work flags (star enumeration, flatness gate, full-star class kernel), three open-work flags (type-$(2,1)$ complement orbit, other hinge orbits, flat Hessian assembly), and two non-claims (convergesEH4d, gapActionRecovery). Module tier tags bind THEOREM content to kernel-checked results with no sorry, no new axioms, and expected footprint [propext, Classical.choice, Quot.sound].

Deliverables already proved in-module include: exactly four (cube translate, Kuhn simplex) pairs contain the representative hinge; all four simplices have flat cosine $0$; star angle sum equals $2\pi$ ($4\cdot\arccos 0$); full-star deficit class kernel on all $15$ stencil classes with values $\pm\sqrt{2}/2$; plus nonvacuity, swap-$1\leftrightarrow 2$ symmetry, uniform-scaling decoy, and homothety stationarity gates.

One-line structure literal (definition, not a theorem). Instantiates Hinge4DStarKernel12Status by assigning the eight boolean fields to the values that match the module's binding tier tags: three closed flags true, three open flags true, and the two non-claim fields false. No lemmas, tactics, or algebraic work.

Gives a single machine-readable snapshot of what this type-$(1,2)$ star-kernel module has closed versus left open. Downstream, hinge4DStarKernel12Status_flags re-exports the closed/open conjunction so callers can assert the dashboard without unpacking fields by hand.

In the Recognition gravity stack this sits inside the Regge discrete-action path toward continuum Einstein-Hilbert recovery. The module doc is explicit that the present increment does not complete flat Hessian assembly over all hinges, does not prove $S_{RS}$ converges to EH in $4$d, does not flip gap-action recovery, and does not reverse-engineer weights from Einstein-Hilbert. The type-$(2,1)$ complement orbit is deliberately not transported here. The record therefore fences the proved kernel (enumeration, flatness, $15$-class deficit values $\pm\sqrt{2}/2$) from the still-open campaign steps that would feed a full $4$d continuum limit.