Gravity Analysis Regge Ttlocal Symbol Existence Edge Angle Contribution Plane Wa

A smooth angle contribution

In the Regge calculus approach to quantum gravity, spacetime is built from flat tetrahedral blocks, and the gravitational field is encoded in the lengths of their edges. A plane wave is a simple disturbance that changes these lengths in a regular, oscillating pattern. The question is whether the angles between tetrahedron faces, which control the curvature, respond smoothly as the wave's amplitude is turned on from zero.

The theorem edgeAngleContribution_planeWave_contDiffAt answers this in the affirmative. It states that for a plane wave of any amplitude, the contribution of a single tetrahedron to the angle sum around a given edge is a smooth function of that amplitude, at the point where the amplitude is zero. Smoothness here means the contribution has derivatives of every finite order at that point. The proof works because at zero amplitude the tetrahedra form a perfectly regular, non-degenerate configuration, where all angles are well-defined and no edge has zero length.

This is a local regularity result. It does not compute the value of the angle contribution, nor does it say anything about the angle itself. It only guarantees that the contribution varies in a well-behaved way as the wave begins to act. This smoothness is a necessary precondition for defining a meaningful physical quantity, the Bloch symbol, which characterizes how the wave's energy and momentum are distributed.

Within the Recognition Science framework, this theorem is a step in a larger program to derive the properties of gravitational waves from a discrete, computational starting point. The framework's machine-checked library of formal theorems proves this result, and the proof relies only on the standard axioms of the underlying logic. The theorem is a statement about the existence of a smooth limit, not about the value of that limit.

The practical consequence is that the program can proceed to define the Bloch symbol for a plane wave at a fixed lattice spacing. The next, and still open, step is to evaluate that symbol and show it approaches a specific continuum value as the lattice spacing shrinks. This theorem clears the way for that calculation, but does not perform it.

/-- The per-tetrahedron angle contribution to one edge's angle sum is
`ContDiffAt` at `0` (finite case split on the slot lookup; `none` branch is
the constant `0`). -/
theorem edgeAngleContribution_planeWave_contDiffAt (E : Fin 3 → Fin 3 → ℝ)
    (k : Fin 3 → ℝ) (e : PeriodicEdge N N N) (cellTet : PeriodicTet N N N)
    (n : ℕ∞) :
    ContDiffAt ℝ n
      (fun t : ℝ =>
        edgeAngleContributionOfField N (planeWaveEdgeField N E k t) e cellTet)
      0 := by
  unfold edgeAngleContributionOfField
  cases h : canonicalEdgeSlot? e cellTet.1 cellTet.2 with
  | none => simpa [h] using contDiffAt_const (c := (0 : ℝ))
  | some f =>
      simpa [h] using
        tetDihedralAngle_planeWave_contDiffAt N E k cellTet f n

/-- At `t = 0` the local tuple is exactly the flat Freudenthal tuple. -/
theorem planeWaveTetSqEdges_zero (E : Fin 3 → Fin 3 → ℝ) (k : Fin 3 → ℝ)
    (cellTet : PeriodicTet N N N) :
    tetSqEdgesOfField N (planeWaveEdgeField N E k 0) cellTet =
      freudenthalTetSqEdges := by
  funext f
  rw [planeWaveTetSqEdges_apply, zero_mul, add_zero]

/-- The per-tetrahedron angle contribution to one edge's angle sum is
`ContDiffAt` at `0` (finite case split on the slot lookup; `none` branch is
the constant `0`). -/
theorem edgeAngleContribution_planeWave_contDiffAt (E : Fin 3 → Fin 3 → ℝ)
    (k : Fin 3 → ℝ) (e : PeriodicEdge N N N) (cellTet : PeriodicTet N N N)
    (n : ℕ∞) :
    ContDiffAt ℝ n
      (fun t : ℝ =>
        edgeAngleContributionOfField N (planeWaveEdgeField N E k t) e cellTet)
      0 := by
  unfold edgeAngleContributionOfField
  cases h : canonicalEdgeSlot? e cellTet.1 cellTet.2 with
  | none => simpa [h] using contDiffAt_const (c := (0 : ℝ))
  | some f =>
      simpa [h] using
        tetDihedralAngle_planeWave_contDiffAt N E k cellTet f n

/-- The per-tetrahedron angle contribution to one edge's angle sum is
`ContDiffAt` at `0` (finite case split on the slot lookup; `none` branch is
the constant `0`). -/
theorem edgeAngleContribution_planeWave_contDiffAt (E : Fin 3 → Fin 3 → ℝ)
    (k : Fin 3 → ℝ) (e : PeriodicEdge N N N) (cellTet : PeriodicTet N N N)
    (n : ℕ∞) :
    ContDiffAt ℝ n
      (fun t : ℝ =>
        edgeAngleContributionOfField N (planeWaveEdgeField N E k t) e cellTet)
      0 := by
  unfold edgeAngleContributionOfField
  cases h : canonicalEdgeSlot? e cellTet.1 cellTet.2 with
  | none => simpa [h] using contDiffAt_const (c := (0 : ℝ))
  | some f =>
      simpa [h] using
        tetDihedralAngle_planeWave_contDiffAt N E k cellTet f n

What this page does not claim

The theorem does not provide the value of the angle contribution or the Bloch symbol. The theorem does not establish any result about the continuum limit as the lattice spacing goes to zero. The theorem does not apply to wave profiles that are not plane waves.

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

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