Geometry Regge Remainder Closure Audit Nonlinear Regge Cubic Taylor Theorem Clos

The cubic remainder bound

In numerical analysis, a Taylor expansion approximates a function near a point using its derivatives, and the remainder is the difference between the true function and the polynomial approximation. For the discrete gravity action used in Regge calculus, a triangulated version of general relativity, the framework's library of formal theorems proves a cubic bound on that remainder. The theorem, nonlinearReggeCubicTaylorTheorem_closed, states that for any incidence-consistent triangulation of three-dimensional space that is flat, plus standard first- and second-variation inputs, the cubic Taylor remainder is analytically bounded. The inputs are the flatness condition, the first-variation input for the canonical Hessian, and the second-variation input; the conclusion is the full cubic Taylor theorem for the nonlinear Regge action.

The proof is constructive: it builds the certificate from a single closed analytic-remainder certificate, remainderAnalyticClosed, which packages the closure for every local analytic remainder target. The library shows that all remainder targets are closed from the flat configuration plus the standard jet inputs. This means the analytic part of the remainder estimate is fully derived, not assumed. The theorem is a formal statement in the machine-checked library, which means every step is verified by the kernel.

What the theorem does not claim is equally important. It does not claim that the nonlinear Regge action equals its quadratic part; it bounds the cubic remainder, so the correspondence is local and quadratic-core with a controlled error. It does not prove that any actual triangulation in the physical world is flat; flatness is an input hypothesis, not a derived conclusion. It does not claim that the first- and second-variation inputs are automatically satisfied; they are explicit assumptions. The theorem is a conditional statement: given the hypotheses, the cubic remainder bound follows. It is a closure certificate for the analytic remainder branch, not a claim about the physical content of Regge calculus itself.

The practical consequence is that downstream users can rely on the cubic remainder bound without re-proving the analytic estimates. The theorem hands off the closed analytic-remainder certificate directly to the local Hessian and Taylor input package, so that only the non-remainder data, flatness, the nonlinear Hessian theorem, and first-variation vanishing, remain as inputs. This makes the analytic core of the remainder estimate a settled fact within the framework, and the remaining work is purely about the geometric and variational hypotheses, not about the analytic bounds.

/-- 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

/-- 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

What this page does not claim

The theorem does not prove that the nonlinear Regge action equals its quadratic part; it only bounds the cubic remainder. The theorem does not show that any actual triangulation is flat; flatness is an input hypothesis. The theorem does not establish that the first- and second-variation inputs are automatically satisfied for any triangulation.

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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