Geometry Regge Action Cubic Taylor Bound Canonical Remainder Line Quadratic Tayl

The cubic remainder bound

In the numerical study of curved space, the Regge action is a standard way to approximate the Einstein-Hilbert action of general relativity on a triangulated manifold, a space built from flat tetrahedra. The action assigns a number to each triangulation, and one often wants to expand that number in powers of a small change to the vertex positions. The quadratic term in that expansion is the Hessian, a matrix of second derivatives that controls stability and small oscillations. The question is whether the part left over after subtracting the constant, linear, and quadratic terms is small enough to ignore.

The declaration in question, canonicalRemainderLineQuadraticTaylorZero_of_flat_first_and_directionalHessian, is a theorem in the framework's machine-checked library of formal theorems. It states that when the geometry starts flat, meaning the initial triangulation has zero curvature, and a certain directional Hessian condition holds, then the Taylor expansion of the remainder along any line through the flat configuration has zero constant, linear, and quadratic terms. In plain language, the error term in the quadratic approximation begins at third order in the perturbation size. The theorem is a formal statement in the language of the library, proved for the specific remainder function that subtracts the canonical Hessian from the full Regge action.

The practical content is a cubic bound. The theorem lineCubicEstimate_of_lineTaylorData then converts that zero quadratic Taylor data into an explicit estimate: for small perturbations, the norm of the remainder is bounded by a constant times the cube of the perturbation norm. This is the kind of estimate that justifies stopping a series at the quadratic term in numerical relativity or in a stability analysis. The bound is local, holding only for perturbations smaller than some radius, and the constant is not computed explicitly by the theorem, only shown to exist.

In Recognition Science, this theorem is one step in a longer chain that aims to show the Regge action, and hence discrete gravity, has a well-behaved local expansion. The framework models physical structure from a ledger of recognition events, and this particular result is a technical lemma about the smoothness of a geometric functional. It does not by itself prove any global property of the action, nor does it compute the value of the cubic bound. It establishes that the remainder is locally cubic, not that the cubic term is small in any absolute sense.

The theorem is a local statement about a single flat configuration. It says nothing about what happens far from that configuration, and it does not identify the constant in the bound. It also does not claim that the Hessian itself is nondegenerate or that the flat configuration is a minimum of the action. Those are separate questions. What the theorem does give is a precise, machine-checked guarantee that the error in the quadratic approximation is controlled at third order, which is the condition needed for the second-order expansion to be a reliable starting point.

def CanonicalRemainderLineQuadraticTaylorZeroTarget
    (K : Triangulation3D) (hK : IncidenceConsistent K) : Prop :=
  ∃ r : ℝ, 0 < r ∧
    ∀ ξ : VertexPotential K, ‖ξ‖ < r →
      taylorWithinEval
        (fun t : ℝ =>
          reggeActionRemainder K hK (canonicalReggeHessian K hK)
            (linePotential K ξ t))
        2 (Set.Icc (0 : ℝ) 1) 0 1 = 0

theorem lineCubicEstimate_of_lineTaylorData
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (hData : CanonicalRemainderLineTaylorDataTarget K hK) :
    CanonicalRemainderLineCubicEstimateTarget K hK := by
  rcases hData with ⟨r, M, hr, hM, hdata⟩
  refine ⟨r, M / 6, hr, div_nonneg hM (by norm_num), ?_⟩
  intro ξ hξ
  rcases hdata ξ hξ with ⟨hCont, hTaylorZero, hBound⟩
  have h :=
    abs_value_le_cubic_of_taylor_data
      (fun t : ℝ =>
        reggeActionRemainder K hK (canonicalReggeHessian K hK)
          (linePotential K ξ t))
      M (‖ξ‖ ^ (3 : ℕ)) hCont hTaylorZero hBound
  simpa [Real.norm_eq_abs] using h

/-- One-dimensional line form of the remaining cubic Taylor estimate.

For every conformal direction `ξ`, restrict the canonical remainder to the line
`t ↦ tξ`.  A standard one-variable Taylor theorem with zero value, first
variation, and second variation at `t = 0` should prove this estimate from a
uniform bound on third derivatives along the segment. -/
def CanonicalRemainderLineCubicEstimateTarget
    (K : Triangulation3D) (hK : IncidenceConsistent K) : Prop :=
  ∃ (r C : ℝ), 0 < r ∧ 0 ≤ C ∧
    ∀ ξ : VertexPotential K, ‖ξ‖ < r →
      ‖reggeActionRemainder K hK (canonicalReggeHessian K hK)
          (linePotential K ξ 1)‖ ≤
        C * ‖ξ‖ ^ (3 : ℕ)

What this page does not claim

The theorem does not compute the constant in the cubic bound. The theorem does not prove the Hessian is nondegenerate or that the flat configuration is a minimum. The theorem does not apply to configurations far from the flat one.

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/ReggeActionCubicTaylorBound.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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