RecognitionMeshExactJBridge4DStatus
plain-language theorem explainer
Status record for the 4D Recognition-mesh exact-J bridge to the Option-C midpoint Bloch continuum face. It packages six Boolean flags: mesh carrier present, amplitude Hessian closed, iterated EH limit closed, exact-J equals true-Regge Hessian closed, gap-action recovery, and Schläfli elevation still open. Downstream audit code cites the concrete instance; the declaration itself is a plain structure with no proof content.
Claim. A status bundle of six Booleans for the 4D Recognition-mesh exact-$J$ bridge: whether the mesh carrier is defined; whether the amplitude Hessian, the iterated Einstein--Hilbert continuum limit, and the identification of exact-$J$ with the true Regge Hessian remain open; whether gap-action recovery holds; and whether Schläfli elevation of the model action remains open.
background
This module sits in the quantum-gravity full-theory campaign at the Recognition gate of 4D continuum closure. It builds a canonical Recognition mesh on the periodic Freudenthal 4-torus and attaches a value-level action whose amplitude Hessian is the geometric Option-C midpoint Bloch symbol on that torus family.
Binding honesty is explicit: the model identification takes the exact-$J$ mesh action to be the exact midpoint Bloch symbol on edge classes at amplitude $\varepsilon$. Elevating that Hessian to the literal nonlinear Regge action via Schläfli is not claimed. Arbitrary test-variation pullbacks are excluded by preflight. The preferred limit shape is amplitude Hessian at fixed mesh, then $N\to\infty$.
Sibling constructions supply the mesh carrier, the true-Regge quadratic Hessian, and the exact-$J$ action on the mesh. Upstream continuum and Bloch-symbol assemblies (flat Hessian, torus bridge, midpoint $m^2$ TT identity) feed the closed faces that this status records.
proof idea
No proof: this is a structure definition packing six Bool fields. Field doc-comments record the intended closed/open reading (amplitude Hessian closed as equal to mesh true-Regge; iterated EH Tendsto closed at the scale-explicit Option-C face; exact-$J$ equals true Regge by model identification; Schläfli elevation not claimed). The concrete witness is the downstream value that sets the closed flags to false (meaning not open) and the carrier flag to true.
why it matters
The structure is the audit surface for the Recognition-mesh exact-$J$ $\to$ Option-C midpoint Bloch bridge at value level. Its sole consumer is the module-level status instance, which reports mesh carrier defined and the three main analytic faces closed (amplitude Hessian, iterated EH, exact-$J$ equals true Regge), while leaving gap-action recovery and Schläfli elevation as honesty markers.
In the broader RS gravity program this gate separates what is theorem-level (Hessian existence and equality by construction; $N\to\infty$ Tendsto via discrete torus bridge plus midpoint $m^2$ TT/gauge faces) from what remains open (Schläfli elevation of the model action; inhabiting full $S_{\mathrm{RS}}\to\mathrm{EH}$ convergence in 4D). It does not flip gap-action recovery. Relative to the forcing chain, this is continuum/Regge analysis downstream of the eight-tick and $D=3$ landmarks, not a foundation step.
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