Geometry Regge Remainder Closure Audit Remainder Analytic Closed

A closed remainder certificate

In numerical analysis and discrete geometry, a remainder is the error left over when you cut a Taylor expansion short. Regge calculus, named after Tullio Regge in 1961, approximates curved spacetime by a triangulated network, and the action that measures its energy has such remainder terms. The declaration RemainderAnalyticClosed is a machine-checked certificate that, for any triangulation satisfying an incidence consistency condition, the analytic remainder targets are closed, meaning they are derived from the flat configuration plus standard first- and second-variation inputs.

The certificate is a structure that packages a closure proof: for every triangulation K with incidence consistency hK, it provides a CanonicalRemainderAnalyticClosureCert. A companion definition, remainderAnalyticClosed, instantiates this structure with the canonical closure certificate. The file also exposes explicit theorem forms for downstream agents, such as a bound on the third derivative of the remainder line, and a full nonlinear cubic Taylor theorem endpoint.

The most general theorem, nonlinearReggeCubicTaylorTheorem_closed, states that from a flat configuration, a first-variation input, and a second-variation input, one obtains the nonlinear Regge cubic Taylor theorem. A further theorem, nonlinearReggeJCostLocalCorrespondence_closed, hands off the closed certificate to a local correspondence result: it shows that the framework's cost function J, locally, corresponds to a quadratic core with a controlled cubic remainder. The alias strongestTrueReggeJCostReplacement_closed names this as the strongest true replacement surface.

What this declaration does not claim is equally important. It does not prove the nonlinear Hessian theorem itself, nor does it establish first-variation vanishing; those are supplied as inputs. It does not assert that the framework's cost function is globally identical to the Regge action, only that a local correspondence holds with a bounded remainder. The certificate is a lane-local audit artifact, not a standalone physical derivation.

/-- The 1B-REM proof surface needed downstream: every local analytic remainder
target is closed from `FlatConfiguration` plus the standard first- and
second-variation remainder jet inputs. -/
structure RemainderAnalyticClosed where
  closure :
    ∀ (K : ReggeTriangulation3D.Triangulation3D) (hK : IncidenceConsistent K),
      ReggeActionCubicTaylorBound.CanonicalRemainderAnalyticClosureCert K hK

/-- 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

/-- Direct local-correspondence endpoint from the closed 1B-REM certificate.
This keeps downstream users out of the line-Taylor split; they supply only the
non-remainder Hessian theorem and canonical remainder first-variation input. -/
theorem nonlinearReggeJCostLocalCorrespondence_closed
    (K : ReggeTriangulation3D.Triangulation3D) (hK : IncidenceConsistent K)
    (hFlat : FlatConfiguration K hK)
    (hHessian : NonlinearReggeDirectionalHessianTheorem K hK)
    (hFirst : ReggeActionFirstVariation.ReggeActionRemainderFirstVariationInput K hK
      (canonicalReggeHessian K hK)) :
    NonlinearReggeJCostLocalCorrespondence K hK :=
  nonlinearRegge_localCorrespondence_of_localHessianTaylorInputs K hK hFlat
    (nonlinearReggeLocalHessianTaylorInputs_closed K hK hFlat hHessian hFirst)

What this page does not claim

The certificate does not prove the nonlinear Hessian theorem or first-variation vanishing; these are supplied as inputs. The certificate does not assert global identity between the cost function and the Regge action, only local correspondence with a bounded remainder.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: