hingeMeasureUnderConformal
plain-language theorem explainer
Under a vertex-conformal ansatz on a finite 3D triangulation, the Regge hinge measure on a global edge is the bare edge length times exp of the mean vertex potential at its two endpoints. Anyone writing the concrete Regge action or its conformal first variation multiplies this factor by the deficit angle. The body is a one-line product of sqrt of the incidence-consistent squared edge length with that exponential weight.
Claim. Let $K$ be an incidence-consistent 3D triangulation, $\xi$ a real potential on its vertices, and $e$ a global edge with endpoints $u,v$. The conformal hinge measure of $e$ is $\sqrt{L_e^2}\,\exp\bigl((\xi(u)+\xi(v))/2\bigr)$, where $L_e^2$ is the squared length of $e$ in the background metric of $K$.
background
This module builds a concrete 3D Regge action under a vertex-conformal ansatz and feeds its second variation into the existing Hessian interface. In classical Regge calculus the action is a sum over hinges of (hinge measure) times (deficit angle). In three dimensions the hinges are edges, so the hinge measure is simply the edge length.
The vertex-conformal ansatz deforms the metric by assigning a real potential $\xi$ to each vertex and rescaling lengths by the geometric mean of the endpoint factors. Concretely, if $u,v$ are the endpoints of edge $e$, the conformal length is the bare length times $\exp((\xi(u)+\xi(v))/2)$. Incidence consistency of $K$ supplies a well-defined global squared edge length for each edge index, so the square root is unambiguous.
Spatial dimension $D=3$ is the forced value from the Recognition forcing chain (T8), which is why the triangulation type and the hinge geometry are three-dimensional throughout this module.
proof idea
Pure definition, not a proved theorem. Read the two endpoints of the edge index from the triangulation's edge-vertex map, take the square root of the incidence-consistent global squared edge length, and multiply by the exponential of the arithmetic mean of the vertex potentials at those endpoints. No lemmas are applied.
why it matters
This factor is the length weight inside the concrete Regge action: the action is the sum over edges of (conformal hinge measure) times (deficit angle). Downstream first-variation work multiplies the same measure (often at zero potential) into Schläfli cancellation and incidence-bookkeeping propositions, and into the directional-derivative package for deficit angles. Those packages are what turn local tetrahedral dihedral derivatives into a global first-variation formula under the conformal ansatz.
In the Recognition geometry stack this is the analytic hinge weight needed before the Hessian target can be filled: once the action and its second variation exist in closed form, the module can discharge the existing Regge-Hessian data interface for finite 3D triangulations. The construction sits in the $D=3$ regime forced by T8; it does not itself derive dimension.
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