reggeActionFirstVariationInput_of_incidenceBookkeeping

This module targets vanishing of the first variation of the full nonlinear Regge action at the flat conformal potential. The geometric engine is Schläfli cancellation plus zero deficit; the file records the exact analytic statement and the named inputs needed until the derivative calculation is fully expanded from closed-form local Schläfli identities.

ConformalSchlaefliIncidenceBookkeeping is the global incidence identity that turns the edge-indexed deficit variation into a sum of local tetrahedral Schläfli sums: for every vertex potential $\eta$, the sum over edges of hinge measure (at zero potential) times the deficit directional derivative equals minus the double sum over tetrahedra and their six faces of $\sqrt{\mathrm{sqEdge}}\cdot A.\mathrm{angleDeriv}$. The structure ReggeActionFirstVariationInput packages the intended conclusion ReggeActionCriticalAtZero.

Upstream, local angle derivatives at flat configurations are assembled into squared-edge then length chain-rule packages, then into a local dihedral directional-derivative package. The theorem conformalSchlaefliCancellation_of_lengthChain_of_bookkeeping converts bookkeeping plus that package into the cancellation hypothesis required by the local-angles constructor.

One-line assembly definition. It calls reggeActionFirstVariationInput_of_localAngles on $K$, incidence, and flatness, supplying two arguments built from the same nested chain:

1. the local dihedral directional-derivative package obtained by localDihedralDirectionalDerivativePackage_of_lengthChain of localAngleLengthChainRulePackage_of_sqEdge of localAngleSqEdgeChainRulePackage_of_flat;
2. the cancellation proof conformalSchlaefliCancellation_of_lengthChain_of_bookkeeping applied to the length-chain package and the given bookkeeping hypothesis hBook.

No new analysis is performed; the def only wires bookkeeping into the already-named local-angles input constructor.

Sits on the critical path to the module target: vanishing first variation of the nonlinear Regge action at the flat conformal potential. Downstream, reggeActionFirstVariationInput_of_edgeSlotBookkeeping reduces edge-slot bookkeeping to this incidence form, so every coarser bookkeeping route funnels through here before the criticality structure is obtained.

In the Recognition framework this is the discrete-geometry half of the $D=3$ story (forcing chain T8): Regge calculus on a 3D triangulation, with Schläfli identities supplying the cancellation that makes the flat conformal background critical. The module doc states the intended lower-level proof shape explicitly: differentiate hinge-length and local dihedral factors, use zero deficit on the hinge term, then global Schläfli on the dihedral term. This definition closes the incidence-bookkeeping gate on that path.