Geometry Regge Action Cubic Taylor Bound Canonical Remainder Line Cont Diff At Z

The smooth remainder

In Regge calculus, a curved spacetime is approximated by a triangulation, a network of flat tetrahedral cells. The Regge action assigns a number to such a triangulation, and the difference between the true action and its quadratic approximation is called the remainder. The declaration canonicalRemainderLine_contDiffAt_zero_of_flatConfiguration is a theorem in the framework's machine-checked library of formal theorems. It states that for any triangulation that is flat, meaning all its cells lie in a single plane, this remainder, when restricted to a straight line of vertex potentials passing through the zero potential, is infinitely differentiable at the zero point.

To unpack the statement: a vertex potential is a small displacement assigned to each vertex of the triangulation. The line potential is the straight-line path from the zero potential to a given displacement, parameterized by a real number t from 0 to 1. The theorem considers the remainder as a function of t along this line. It proves that this function is smooth (has derivatives of all orders) at t = 0, provided the starting triangulation is flat. This is not a statement about the whole space of potentials, only about this one-dimensional slice through it.

The theorem is a component in a larger proof chain. The goal is a cubic Taylor bound: showing that the remainder is bounded by a constant times the cube of the displacement's norm, for small displacements. This bound is the final analytic step needed after the Hessian (the matrix of second derivatives) of the action has been identified. The smoothness result, along with a separate bound on the third derivative, provides the regularity needed to apply the Taylor theorem with a remainder. The theorem's proof relies on the chain rule and the fact that the line potential is a continuous linear map.

What the theorem does not claim is equally important. It does not establish smoothness for the remainder as a function on the full space of vertex potentials, only along the specific line. It does not prove the cubic bound itself; that is a separate target. It also does not say anything about the physical content of the Regge action or its remainder, only about the mathematical properties of the remainder function under the stated flatness condition. The flatness hypothesis is essential: the theorem's conclusion does not follow, and is not claimed, for a general curved triangulation.

Within the Recognition Science framework, this result is a piece of the infrastructure for showing that the Regge action, a classical discretization of gravity, has the expected analytic behavior. It is a technical lemma, not a physical law. Its role is to enable the cubic error estimate, which in turn supports the framework's treatment of the action's second variation. The theorem is a formal statement, checked by the kernel, and its meaning is confined to the mathematics of the remainder function on a line.

/-- 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

/-- 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

/-- 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

What this page does not claim

The theorem does not prove the cubic Taylor bound itself, which is a separate target. The theorem does not make any statement about the physical content of the Regge action or its remainder. The theorem does not apply to non-flat triangulations, where the conclusion is not claimed to hold.

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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