Hinge4DFlatKernelStatus
plain-language theorem explainer
Status record for the 4D Regge hinge flat-kernel campaign: five boolean gates tracking Freudenthal 24-simplex enumeration, seed-hinge incidence closure, openness of true deficit kernels, EH4d convergence, and gap-action recovery. Gravity auditors cite it to read the honest closed/open split at a glance. It is a plain structure definition; the concrete assignment lives in the companion value.
Claim. A five-flag status record for the 4D Regge flat-Hessian skeleton: (i) the 24 Freudenthal/Kuhn monotone 4-simplices of the unit 4-cube are enumerated; (ii) seed-hinge incidence multiplicities are closed; (iii) the true per-hinge deficit kernels remain open; (iv) convergence of the RS action to 4D Einstein–Hilbert is not claimed; (v) gap-action recovery is not claimed.
background
This module is the next kernel-checked increment after the 15-class 4D Regge edge stencil. It works inside a Freudenthal/Kuhn triangulation of the unit 4-cube: 24 monotone 4-simplices (axis permutations), each with five nested vertices and ten local edges drawn from the 15 stencil classes.
The seed hinge is the triangle on vertices $0$, $e_0$, $e_0+e_1$ (masks $0,1,3$). Exactly two of the 24 simplices contain it. Incidence multiplicities count, for each edge class, how often it appears as a local edge in those seed simplices; three classes are combinatorial decoys (multiplicity zero), and the three hinge-boundary classes each have multiplicity two.
Per the module contract, the combinatorial layer is THEOREM-grade, while the flat-Hessian assembly formula is MODEL-only: it contracts open per-hinge area weights against open per-hinge deficit (dihedral/Cayley–Menger) kernels and forces vanishing off the incidence support. This structure packages that closed/open split as named booleans.
proof idea
No proof: this is a structure declaration with five Bool fields and no inhabitants or obligations in the signature. The companion definition assigns the honest values (true for the two combinatorial gates and the open-kernel flag; false for EH4d convergence and gap-action recovery).
why it matters
Gives a single typed dashboard for deliverable B of the QG full-theory campaign on 4D Regge hinges. Downstream, hinge4DFlatKernelStatus fills the record so readers see at once: Freudenthal 24-enumeration and seed-hinge incidence are closed; true deficit kernels stay OPEN; neither $S_{\mathrm{RS}}\to\mathrm{EH}_{4d}$ nor gap-action recovery is flipped.
That matches the module’s binding tier tags and the explicit non-claims (no completed flat Hessian, no reverse-engineered Einstein–Hilbert weights). In the broader Recognition gravity stack it keeps the combinatorial support honest while the numeric class weights from dihedral/Cayley–Menger calculus remain future work.
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