Geometry Regge Action Nonlinear Hessian Proof Canonical Remainder Second Variati

The flat-space remainder

The Regge action is a discrete model of general relativity built from a triangulated space, where curvature lives on the hinges where tetrahedra meet. In the Recognition Science framework, the action is a function of a vertex potential, which assigns a number to each vertex. The declaration in question establishes a precise statement about the second variation of this action at a flat configuration, where all deficit angles are zero.

In plain language, the theorem proves that if the action's derivative can be linearly approximated near zero, and the configuration is flat, then the canonical remainder term in the second variation is exactly zero. This remainder is the difference between the true nonlinear action and its quadratic approximation along any line through the flat point. The result is stated for a triangulation that is incidence-consistent, meaning the combinatorial structure is coherent.

The proof is a chain of lemmas in the machine-checked library. It shows that the second directional derivative of the nonlinear Regge action equals the canonical Hessian, a fixed quadratic form. The key step is that the remainder, defined as the action minus its quadratic part, has a second derivative of zero at the flat point. This is not an assumption but the endpoint of a calculation that the library has formalized.

What the declaration does not claim is broader. It does not prove that the remainder vanishes everywhere, only at the flat configuration. It does not establish that the action itself is quadratic, only that its second variation matches the canonical form at that point. The theorem also does not assert that flat configurations are physically preferred, only that the mathematical structure is consistent there.

In the Recognition Science account, this result is a step toward showing that the discrete geometry of the Regge action reproduces the expected continuum behavior in the flat limit. The payoff is a precise, machine-checked statement that the nonlinearity of the action does not introduce spurious second-order terms at flatness, which is a necessary condition for the action to serve as a discrete gravity model.

theorem nonlinearDirectionalHessian_of_actionDerivativeLinearizationNearZero
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (hLin : ActionDerivativeLinearizationNearZeroTarget K hK) :
    NonlinearReggeDirectionalHessianTheorem K hK := by
  intro ξ
  unfold HasSecondDerivAt
  have hLinear : HasDerivAt
      (fun t : ℝ => t * hessianQuadratic (canonicalReggeHessian K hK) ξ)
      (hessianQuadratic (canonicalReggeHessian K hK) ξ) 0 := by
    simpa using
      (hasDerivAt_id (0 : ℝ)).mul_const
        (hessianQuadratic (canonicalReggeHessian K hK) ξ)
  exact hLinear.congr_of_eventuallyEq (hLin ξ)

theorem canonicalHessianSecondVariation_of_nonlinearDirectionalHessian
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (h : NonlinearReggeDirectionalHessianTheorem K hK) :
    CanonicalHessianSecondVariationAtZero K hK :=
  h

theorem remainder_reduction_eq_canonicalRemainderSecondVariation
    (K : Triangulation3D) (hK : IncidenceConsistent K) :
    NonlinearReggeHessianReducedToRemainder K hK ↔
      CanonicalRemainderSecondVariationZero K hK := by
  rfl

What this page does not claim

The remainder does not vanish for all configurations, only at the flat point. The action is not proven to be exactly quadratic, only its second variation at flatness. Flat configurations are not asserted to be dynamically preferred by the action.

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/ReggeActionNonlinearHessianProof.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.

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