Geometry Regge Remainder Closure Audit

The closure certificate

In numerical geometry, the Regge action approximates the curvature of a curved space by summing contributions from a triangulated mesh. The approximation is only useful if the error, the remainder left after a Taylor expansion, is controlled. The geometry regge remainder closure audit is a machine-checked certificate that this control holds: for any triangulation that is incidence-consistent and flat, the analytic remainder targets are all closed.

The audit lives in the framework's machine-checked library of formal theorems. Its main structure, RemainderAnalyticClosed, states that for every such triangulation, the canonical remainder closure certificate exists. The library then proves three explicit theorems from this certificate. The first gives a bound on the third derivative of the line remainder. The second, nonlinearReggeCubicTaylorTheorem_closed, establishes the full cubic Taylor theorem for the nonlinear Regge action. The third, strongestTrueReggeJCostReplacement_closed, provides the strongest true replacement surface: a local quadratic-core correspondence with a controlled cubic remainder.

In plain language, this means the framework can trust its local approximations. When the geometry is nearly flat, the error in replacing the full action by a quadratic form is bounded by a cubic term. That bound is not assumed; it is derived from the flat configuration and the standard first- and second-variation inputs. The audit makes the analytic remainder branch independently buildable, so downstream results do not need to import the entire progress audit.

What this changes is practical: anyone working with the nonlinear correspondence layer can now rely on the local Hessian and Taylor inputs without rechecking the remainder. The only remaining inputs are the non-remainder data: flatness, the nonlinear Hessian theorem, and first-variation vanishing. The audit closes the analytic side, leaving only those geometric hypotheses for the user to supply.

/-- Lane-local theorem-valued certificate for Track 1B-REM. -/
def remainderAnalyticClosed : RemainderAnalyticClosed where
  closure := ReggeActionCubicTaylorBound.canonicalRemainderAnalyticClosureCert

/-- Explicit theorem form for the full cubic Taylor theorem endpoint. -/
theorem nonlinearReggeCubicTaylorTheorem_closed
    (K : ReggeTriangulation3D.Triangulation3D) (hK : IncidenceConsistent K)
    (hFlat : FlatConfiguration K hK)
    (hFirst : ReggeActionFirstVariation.ReggeActionRemainderFirstVariationInput K hK
      (canonicalReggeHessian K hK))
    (hSecond : ReggeActionRemainderSecondVariationInput K hK) :
    ReggeActionCubicTaylorBound.NonlinearReggeCubicTaylorTheorem K hK :=
  (remainderAnalyticClosed.closure K hK).cubic_taylor_from_flat_and_jets
    hFlat hFirst hSecond

/-- Alias at the strongest true replacement surface: local quadratic-core
correspondence with a controlled cubic remainder. -/
theorem strongestTrueReggeJCostReplacement_closed
    (K : ReggeTriangulation3D.Triangulation3D) (hK : IncidenceConsistent K)
    (hFlat : FlatConfiguration K hK)
    (hHessian : NonlinearReggeDirectionalHessianTheorem K hK)
    (hFirst : ReggeActionFirstVariation.ReggeActionRemainderFirstVariationInput K hK
      (canonicalReggeHessian K hK)) :
    StrongestTrueReggeJCostReplacement K hK :=
  nonlinearReggeJCostLocalCorrespondence_closed K hK hFlat hHessian hFirst

What this page does not claim

This module does not prove that the Regge action itself is the unique cost function. It does not establish any global, non-local error bound for arbitrary curved geometries. It does not connect the local correspondence to the physical three-dimensional space derivation.

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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