MixedHingeDeficitFromDeficitPackageTarget
plain-language theorem explainer
Names the deficit-package form of the surviving mixed hinge/deficit identity: the edge sum of hinge-measure directional derivatives times package deficit derivatives equals the canonical Regge Hessian quadratic. Cited by anyone discharging the nonlinear second-variation endpoint via an explicit first-variation package. Pure Prop definition; no proof content.
Claim. For an incidence-consistent 3D triangulation $K$ and a deficit-angle directional derivative package $D$ on $K$, every vertex potential $\xi$ satisfies $\sum_e(\partial_{\mathrm{hinge}}\mu)(\xi,e)\,D.(\partial\delta)(\xi,e)=Q_{H_{\mathrm{can}}(K)}(\xi)$, where the left-hand side uses the package's first-variation deficit derivative and the right-hand side is the quadratic form of the canonical Regge Hessian.
background
The module isolates the remaining hard step for the full nonlinear Regge action: the second directional derivative of the action at the flat potential must equal the canonical incidence Hessian. After second-order Schläfli cancellation, only a mixed hinge/deficit term survives.
The sibling target states that mixed term with opaque line derivatives at zero on both hinge measure and deficit angle. This definition rewrites the same identity against an explicit DeficitAngleDirectionalDerivativePackage, so the deficit factor is the package field rather than a bare line derivative at the origin. The right-hand side remains the quadratic form of the canonical Regge Hessian on the triangulation.
Related forms replace that Hessian quadratic by the canonical graph Dirichlet energy or by an edge-stencil expression; those are separate targets bridged in the same file.
proof idea
Definition of a proposition, not a proved theorem. The body is a single universal quantifier over vertex potentials asserting equality of the mixed edge sum with the canonical Hessian quadratic. No tactics, no lemmas, no algebraic reduction: it only names the identity that bridge lemmas either assume or discharge.
why it matters
Sits in the middle of the nonlinear Hessian interface. Downstream, mixedHingeDeficitCanonicalHessian_of_deficitPackage lifts this package form back to the opaque-deriv target; mixedHingeDeficitFromDeficitPackage_of_dirichlet and mixedHingeDeficitFromDeficitPackage_of_edgeStencil supply this target from Dirichlet-energy and edge-stencil forms. Closing any path advances the exact second chain-rule endpoint from which ReggeActionSecondVariationInput follows. In the RS geometry stack this residual identity lives on incidence-consistent 3D triangulations, matching the forced spatial dimension $D=3$.
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