Geometry Regge Action Cubic Taylor Bound Canonical Remainder Line Has Deriv At Z

The cubic remainder bound

In numerical analysis, a Taylor expansion approximates a smooth function near a point by a polynomial, and the remainder is the difference between the function and that polynomial. The classical Taylor theorem says that if a function is three times differentiable, the remainder at a point is bounded by a constant times the cube of the distance from that point. This is the cubic Taylor bound, and it is what makes second-order approximations reliable: the error does not merely shrink, it shrinks at a known, fast rate.

The framework's machine-checked library of formal theorems proves the same cubic bound for the remainder of a discrete gravity action, known as the Regge action. In this setting, a triangulation of three-dimensional space has a potential at each vertex, and the action is a function of these vertex potentials. The theorem canonicalRemainderLine_hasDerivAt_zero_of_remainderFirstVar establishes that, for a flat configuration, the remainder along any line through the origin is three times continuously differentiable at zero. This means the remainder has a well-behaved third derivative, which is the technical condition needed to apply the cubic Taylor estimate.

In Recognition Science, this is the final analytic step that turns a local bound on the third derivative into a global statement about the remainder. The theorem NonlinearReggeCubicTaylorTheorem (a definition that packages the bound) asserts that for any small perturbation, the norm of the remainder is at most a constant times the cube of the perturbation's norm. The proof combines three ingredients: the remainder vanishes at zero, its quadratic Taylor term is zero, and its third derivative is bounded locally. Together these yield the cubic estimate, which is the precise sense in which the canonical Hessian captures the action's second-order behavior.

What this theorem does not claim is that the remainder itself is zero, or that the cubic bound holds for large perturbations. It also does not assert that the Regge action is exactly quadratic; it only controls the error term. The bound is local, valid only within a small radius around the flat configuration. This is a statement about the remainder's smoothness and growth rate, not about the global geometry of the action functional.

/-- ContDiffAt of the line-restricted canonical remainder at `t = 0`. -/
theorem canonicalRemainderLine_contDiffAt_zero_of_flatConfiguration
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (hFlat : FlatConfiguration K hK) (ξ : VertexPotential K) :
    ContDiffAt ℝ (⊤ : ℕ∞)
      (fun t : ℝ =>
        reggeActionRemainder K hK (canonicalReggeHessian K hK)
          (linePotential K ξ t)) 0 := by
  have hR : ContDiffAt ℝ (⊤ : ℕ∞)
      (fun ξ' : VertexPotential K =>
        reggeActionRemainder K hK (canonicalReggeHessian K hK) ξ')
      (zeroPotential K) :=
    canonicalRemainder_contDiffAt_zero_of_flatConfiguration K hK hFlat
  have hLine : ContDiffAt ℝ (⊤ : ℕ∞)
      (fun t : ℝ => linePotential K ξ t) 0 := by
    rw [contDiffAt_pi]
    intro i
    show ContDiffAt ℝ (⊤ : ℕ∞) (fun t : ℝ => t * ξ i) 0
    fun_prop
  -- Rewrite the target point using `linePotential K ξ 0 = zeroPotential K`.
  have hLine0 : linePotential K ξ 0 = zeroPotential K := linePotential_zero K ξ
  have hR' : ContDiffAt ℝ (⊤ : ℕ∞)
      (fun ξ' : VertexPotential K =>
        reggeActionRemainder K hK (canonicalReggeHessian K hK) ξ')
      (linePotential K ξ 0) := by
    rw [hLine0]; exact hR
  exact hR'.comp 0 hLine

/-- Exact Taylor theorem needed for the nonlinear Regge remainder.  This is
the finite-dimensional third-order Taylor estimate specialized to the canonical
Regge remainder. -/
def NonlinearReggeCubicTaylorTheorem
    (K : Triangulation3D) (hK : IncidenceConsistent K) : Prop :=
  LocalCubicRemainderBound K hK

What this page does not claim

The remainder is zero, only that it is bounded by a cubic term. The cubic bound holds for all perturbations, only for those within a small radius. The Regge action is exactly quadratic, only that its error is controlled.

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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