reggeAction_taylor_decomposition
plain-language theorem explainer
Exact algebraic split of the nonlinear Regge action on an incidence-consistent 3D triangulation into its flat (zero-potential) value, half a candidate Hessian quadratic form, and a nonlinear remainder. Cited by the conformal Hessian pipeline and the frozen TT-symbol preflight. Proof is definitional: unfold the remainder and cancel by ring.
Claim. For any incidence-consistent finite 3D triangulation $K$, any candidate vertex bilinear form $H$, and any vertex potential $\xi$, the conformal Regge action equals its value at the zero potential plus $\tfrac12$ times the quadratic form of $H$ on $\xi$, plus the nonlinear remainder of the action relative to that candidate quadratic.
background
The module isolates the final analytic Hessian step for a finite 3D Regge triangulation under the conformal ansatz: edge lengths are deformed by a vertex potential, dihedral angles and hinge measures respond, and the Regge action is the sum of deficit-angle times hinge-measure contributions.
The zero potential is the flat reference configuration. A candidate Hessian is any real bilinear form $H$ on the finite vertex set; its quadratic form is the natural second-order piece one hopes matches the true second variation. The nonlinear remainder is defined precisely so that action = flat value + half that quadratic + remainder, for every candidate $H$.
Upstream cost algebra supplies the shifted cost $H(x)=J(x)+1$ and the Recognition Composition Law, but this lemma itself is pure bookkeeping on the concrete action package, not a new identity for $J$.
proof idea
One-line term proof. Unfold the definition of the nonlinear remainder (which is literally action minus flat value minus half the candidate quadratic), then ring cancels the three terms. No geometric lemma is invoked; the equality is the defining rearrangement of the remainder.
why it matters
This is the algebraic scaffold every later nonlinear Hessian argument sits on. Downstream, nonlinearRegge_exact_canonical_split specializes the candidate quadratic to the canonical $J$/Dirichlet term; actionAlongLine_canonical_split restricts the same split to one-parameter conformal lines so that only the second variation of the remainder remains analytic work. Gravity's frozen TT-symbol preflight names the same identity with the frozen graph-Laplacian Hessian and explicitly warns that the equation is tautological until the remainder is expanded.
In the Recognition geometry chain this is the concrete 3D Regge step that turns the conformal action package into the existing Hessian-data interface, preparatory to matching continuum curvature and the $D=3$ forcing landmark. It does not yet prove the remainder is cubic or that the candidate equals the true second variation; those are the open analytic closures it exposes.
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