Geometry Regge Action Nonlinear Hessian Proof Action Derivative Tangency To Quad

The flat-space tangency

In the geometry of curved spaces, the Regge action is a way to approximate the physics of a continuous space by breaking it into flat, four-sided building blocks called tetrahedra. The action is a number assigned to a triangulated space, built from the lengths of its edges and the angles between its faces. A central question is how this number changes when the shape of the space is slightly deformed. The Hessian is the matrix of second derivatives that describes this change, generalizing the curvature of a simple curve to a multi-dimensional space.

The declaration actionDerivativeTangencyToQuadratic_of_flat_productRuleDerivativeCanonicalHessian establishes a precise tangency condition. It proves that, for a flat configuration, the derivative of the Regge action along any line in the space of possible deformations is tangent to the derivative of a specific, simpler quadratic form. This quadratic form is built from the canonical Hessian, a standard object in the framework's geometry. The theorem states that the difference between the action's derivative and this quadratic form's derivative vanishes to first order at the flat point. This is a local statement, meaning it holds in an infinitesimal neighborhood of the flat configuration.

This result is not an isolated fact but a deliberate step in a larger proof. The documentation describes it as the "exact endpoint of the second chain-rule calculation." Once this tangency is established, it directly feeds into the NonlinearReggeDirectionalHessianTheorem, which asserts that the second directional derivative of the full nonlinear action equals the canonical Hessian's quadratic form. This is the key bridge: it shows that the complicated nonlinear action, near flat space, behaves exactly like its simpler quadratic approximation. The proof is machine-checked, meaning a computer program has verified every logical step, leaving no gaps for human error.

What this theorem does not claim is equally important. It does not prove the full nonlinear Regge Hessian theorem on its own; it only establishes the tangency condition that is one of its inputs. The theorem also assumes a flat configuration and does not address the behavior of the action in curved regions. It does not claim that the Regge action itself is a quadratic function everywhere; it only shows that its second derivative at the flat point matches that of the quadratic form. The tangency is a local property, not a global one.

theorem nonlinearDirectionalHessian_of_actionDerivativeFirstOrderTangency
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (hTan : ActionDerivativeFirstOrderTangencyTarget 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)
  have hsum := (hTan ξ).add hLinear
  convert hsum using 1
  · ext t
    let q := hessianQuadratic (canonicalReggeHessian K hK) ξ
    change deriv (actionAlongLine K hK ξ) t =
      deriv (actionAlongLine K hK ξ) t - t * q + t * q
    ring
  · ring

/-- Exact second-directional-variation statement for the full nonlinear Regge
action. -/
def NonlinearReggeDirectionalHessianTheorem
    (K : Triangulation3D) (hK : IncidenceConsistent K) : Prop :=
  ∀ ξ : VertexPotential K,
    HasSecondDerivAt (actionAlongLine K hK ξ)
      (hessianQuadratic (canonicalReggeHessian K hK) ξ) 0

What this page does not claim

The theorem does not prove the full nonlinear Regge Hessian theorem on its own. The theorem does not claim the Regge action is quadratic everywhere, only that its second derivative matches at the flat point. The theorem does not address the behavior of the action in curved regions.

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.

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: