Geometry Regge Remainder Closure Audit Nonlinear Regge Local Hessian Taylor Inpu

A closed local input theorem

In numerical relativity, the Regge action is a discrete approximation to the Einstein-Hilbert action built from edge lengths of a triangulation. Its local behavior near a flat configuration is governed by a Taylor expansion: a quadratic core from the Hessian plus a cubic remainder. The theorem nonlinearReggeLocalHessianTaylorInputs_closed in the framework's machine-checked library of formal theorems certifies that this cubic remainder is analytically closed, meaning its bounding target is derivable from the flat configuration and standard first- and second-variation jet inputs.

The theorem takes a triangulation K with incidence consistency, a flat configuration, a nonlinear directional Hessian theorem, and a first-variation remainder input, and produces the local Hessian/Taylor input certificate. Its proof is a direct handoff from a broader closure audit: the analytic remainder branch has its own buildable closure certificate, so downstream users do not need to import the full progress audit or split the line-Taylor cases themselves. This is a formal statement in the library, not a numerical computation.

What the theorem does not claim is significant. It does not assert that the Regge action equals the Einstein-Hilbert action, nor that the Hessian theorem or flat configuration hold for any particular triangulation; those are inputs, not conclusions. It does not claim the cubic remainder is zero, only that it is bounded. It does not establish any global correspondence between Regge calculus and continuum gravity, only a local quadratic-core correspondence with a controlled cubic remainder. The theorem is a closure certificate for the analytic remainder branch, not a physical derivation.

In plain terms, the declaration is a bookkeeping guarantee: given the standard local inputs, the remainder analysis is complete and self-contained. It lets downstream work on nonlinear Regge correspondence proceed without re-auditing the remainder branch. The payoff is modularity: one file certifies the analytic remainder, and other layers can rely on it.

/-- Direct handoff theorem from the closed analytic-remainder certificate to the
local Hessian/Taylor input package used by the nonlinear correspondence layer.
The only inputs left are the non-remainder data: flatness, the nonlinear
Hessian theorem, and first-variation vanishing for the canonical remainder. -/
theorem nonlinearReggeLocalHessianTaylorInputs_closed
    (K : ReggeTriangulation3D.Triangulation3D) (hK : IncidenceConsistent K)
    (hFlat : FlatConfiguration K hK)
    (hHessian : NonlinearReggeDirectionalHessianTheorem K hK)
    (hFirst : ReggeActionFirstVariation.ReggeActionRemainderFirstVariationInput K hK
      (canonicalReggeHessian K hK)) :
    ReggeActionCubicTaylorBound.NonlinearReggeLocalHessianTaylorInputs K hK hFlat := by
  let hSecond : ReggeActionRemainderSecondVariationInput K hK :=
    reggeActionRemainderSecondVariationInput_of_flat_nonlinearHessian K hK hFlat
      hHessian
  exact ReggeActionCubicTaylorBound.nonlinearReggeLocalHessianTaylorInputs_of_hessian_and_taylor
    K hK hFlat hHessian
    (nonlinearReggeCubicTaylorTheorem_closed K hK hFlat hFirst hSecond)

/-- Direct handoff theorem from the closed analytic-remainder certificate to the
local Hessian/Taylor input package used by the nonlinear correspondence layer.
The only inputs left are the non-remainder data: flatness, the nonlinear
Hessian theorem, and first-variation vanishing for the canonical remainder. -/
theorem nonlinearReggeLocalHessianTaylorInputs_closed
    (K : ReggeTriangulation3D.Triangulation3D) (hK : IncidenceConsistent K)
    (hFlat : FlatConfiguration K hK)
    (hHessian : NonlinearReggeDirectionalHessianTheorem K hK)
    (hFirst : ReggeActionFirstVariation.ReggeActionRemainderFirstVariationInput K hK
      (canonicalReggeHessian K hK)) :
    ReggeActionCubicTaylorBound.NonlinearReggeLocalHessianTaylorInputs K hK hFlat := by
  let hSecond : ReggeActionRemainderSecondVariationInput K hK :=
    reggeActionRemainderSecondVariationInput_of_flat_nonlinearHessian K hK hFlat
      hHessian
  exact ReggeActionCubicTaylorBound.nonlinearReggeLocalHessianTaylorInputs_of_hessian_and_taylor
    K hK hFlat hHessian
    (nonlinearReggeCubicTaylorTheorem_closed K hK hFlat hFirst hSecond)

What this page does not claim

The theorem does not assert equality between the Regge action and the Einstein-Hilbert action. It does not claim the cubic remainder vanishes, only that it is bounded. It does not establish a global correspondence between Regge calculus and continuum gravity.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Geometry/ReggeRemainderClosureAudit.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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