Geometry Regge Remainder Closure Audit Strongest True Regge Jcost Replacement Cl

The local correspondence

Regge calculus is a discrete approach to general relativity in which spacetime is approximated by a triangulated manifold, and the gravitational action is expressed in terms of edge lengths. A central question is how this discrete action relates to the smooth continuum theory. The declaration strongestTrueReggeJCostReplacement_closed addresses this by establishing a precise, local correspondence between the Regge action and the framework's cost function J.

The theorem states: for any triangulation K of a three-dimensional manifold that is incidence-consistent, if the configuration is flat (meaning the metric is locally Euclidean), and if the nonlinear Hessian theorem and the first-variation remainder input hold, then the NonlinearReggeJCostLocalCorrespondence is satisfied. In plain terms, this means that near a flat configuration, the Regge action's quadratic core (its second-order Taylor expansion) matches the cost function J, and the error is controlled by a cubic remainder term. The proof is a direct chain: it first establishes a bound on the third derivative of the remainder, then uses this to prove a cubic Taylor theorem for the nonlinear Regge action, and finally assembles these into the local correspondence.

The declaration is a theorem in the framework's machine-checked library of formal theorems, meaning its proof has been verified by a computer. Its role is to provide a clean, self-contained certificate for the analytic remainder branch of a larger audit. It does not require importing the broad progress audit, making it a focused building block for downstream work. The inputs it requires are all non-remainder data: flatness, the nonlinear Hessian theorem, and first-variation vanishing for the canonical remainder.

What the declaration does not claim is equally important. It does not assert that the Regge action equals J globally, only that their quadratic cores correspond locally near flat configurations. It does not prove the nonlinear Hessian theorem itself; that is taken as an input. And it does not claim that the correspondence holds for non-flat configurations or that the cubic remainder vanishes; it only bounds it. The theorem is a precise, conditional statement about a local relationship, not a global equivalence.

/-- 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

/-- Explicit theorem form for agents that only need the third-derivative bound
and should not import the broad progress audit. -/
theorem canonicalRemainderLineThirdDerivBound_closed
    (K : ReggeTriangulation3D.Triangulation3D) (hK : IncidenceConsistent K)
    (hFlat : FlatConfiguration K hK) :
    ReggeActionCubicTaylorBound.CanonicalRemainderLineThirdDerivBoundTarget K hK :=
  (remainderAnalyticClosed.closure K hK).line_third_deriv_bound_from_flat hFlat

/-- 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

The theorem does not assert a global equality between the Regge action and J, only a local correspondence of quadratic cores. The theorem does not prove the nonlinear Hessian theorem; it takes it as an input. The theorem does not claim the correspondence holds for non-flat configurations or that the cubic remainder vanishes.

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

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