meshGeometricDeficit_regge_convention
plain-language theorem explainer
The mesh geometric deficit at deformation parameter h equals the classical Regge hinge deficit 2π minus four times the dihedral angle of the deformed four-tet star. Discrete-gravity and recognition-mesh workers cite this to pin the residual to squared-edge Regge data with no log-ratio scaffolding. The proof is pure definitional reflexivity: the mesh deficit is the banked star deficit written in that convention.
Claim. For every real deformation parameter $h$, the mesh geometric deficit equals $2\pi - 4\,\theta(h)$, where $\theta(h)$ is the dihedral angle (from squared-edge data) of one tetrahedron in the four-tet star at the hinge edge, with rim squared length $p(h)=\frac{3}{2}(1-h)$.
background
In Regge calculus the curvature at a hinge is the angular deficit $2\pi$ minus the sum of dihedral angles meeting there. For the four-tet star used as the recognition Freudenthal mesh seed, four congruent tetrahedra meet at the hinge, so the deficit is $2\pi-4\theta$.
The deformation family is parameterized by a real $h$. Rim squared length is $p(h)=\frac{3}{2}(1-h)$, with flat value $p(0)=\frac{3}{2}$ (where the hinge dihedral cosine vanishes). Squared-edge data of one star tetrahedron fix hinge and spoke lengths squared to 1 and the rim to $p$. The hinge dihedral angle is $\arccos$ of the Cayley-Menger cosine on that 6-tuple of squared edges.
This module attacks Wave B residual R1: identify the mesh geometric deficit with the signed Regge-convention star deficit, free of $x$-ratio or log scaffolding. The mesh deficit is defined to be exactly that star-deficit function on the deformation line.
proof idea
One-line reflexivity. The mesh geometric deficit is defined as the banked star deficit, and that star deficit is definitionally $2\pi$ minus four times the hinge dihedral angle of the squared-edge star at rim $p(h)$. Both sides of the equality unfold to the same term, so rfl closes the goal with no lemmas or rewriting.
why it matters
Pins the Regge-convention reading of the mesh geometric deficit inside Wave B residual R1 of the QG full-completion session: the residual is the signed hinge deficit on the recognition Freudenthal star, written purely in squared-edge geometry. Sibling facts in the same module (equality to star deficit, arcsin form, oddness, flat vanishing, sign certificates, and the typed-residual identification) rest on this convention pin.
It does not yet lift star deficit onto a triangulation field of the recognition Freudenthal mesh, nor flip gap1_bridge_derived, nor inhabit constitutive-coupling interfaces. The module records the open remainder as the encoded Freudenthal lift onto a concrete Regge deficit-angle map; that join is not yet expressible because the mesh carrier has no triangulation field and the four-tet package stops at the abstract-star convention.
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