Gravity Analysis Regge Edge Ttattachment4 D Plane Wave Axis Edge Pert Decomposit

The plane-wave split

In the Recognition Science framework's machine-checked library of formal theorems, a plane wave is a simple periodic disturbance, like a ripple moving across a lattice. The theorem planeWaveAxisEdgePert_decomposition concerns a four-dimensional torus, a space that wraps around on itself in four directions. It takes a symmetric matrix H, which describes a small perturbation, and a wave covector m, which sets the direction and frequency of the wave. The theorem states that the perturbation's effect on an edge aligned with one of the torus's axes equals the sum of three separate effects: one from the transverse-traceless part, one from the gauge part, and one from the residual trace part, each projected appropriately. This is a linear decomposition, meaning the total effect is simply the sum of the parts, and it holds exactly on the lattice, not just in a continuous limit.

The theorem is a formal statement, checked by a computer, with no gaps in its proof. It builds on an earlier result, edgeLoad_decomposition, which establishes the same splitting for the edge load itself, before the plane-wave factor is applied. The key insight is that the edge load, a quadratic form measuring the squared length of a displacement, is linear in the matrix H. Because of this linearity, the decomposition of H into its three parts carries over directly to the decomposition of the plane-wave perturbation. The theorem also includes a specific identity for the gauge part: on an axis edge, its edge load equals 2 times the product of the wave component and the gauge vector component along that axis. This is an exact finite-difference identity on the lattice, not a continuum statement.

In Recognition Science, this theorem is a step in a larger campaign to connect the framework's discrete ledger of recognition events to the continuous geometry of general relativity. The framework models spacetime as a lattice of discrete events, and this theorem is part of showing how small perturbations on that lattice behave. However, the theorem does not claim to prove the full recovery of Einstein's equations. It does not establish that the lattice action converges to the Einstein-Hilbert action in four dimensions, nor does it prove the broader statement S_RS_converges_EH_4d. The theorem is a precise, limited result about the algebraic structure of plane-wave perturbations on a single edge, not a complete theory of quantum gravity.

What this means for a reader is that the framework has a rigorous, machine-checked handle on one piece of the puzzle: how a plane-wave perturbation splits into gauge and physical parts on a lattice edge. This is a necessary technical step, but it is far from the whole story. The theorem's value lies in its exactness and its clear scope. It shows that the framework can produce precise statements about the algebraic structure of perturbations, which is a foundation for future work, but it does not yet bridge the gap to the continuous theory of gravity.

theorem planeWaveAxisEdgePert_decomposition (m : Fin 4 → ℝ) (H : Mat4)
    (hH : IsSymmetric H) (hm : momentumSq m ≠ 0) (x : Fin 4 → ℝ)
    (a : Fin 4) :
    planeWaveAxisEdgePert H m x a =
      planeWaveAxisEdgePert (ttProject m H) m x a +
        planeWaveAxisEdgePert (gaugePart m (gaugeVector m H)) m x a +
        planeWaveAxisEdgePert (residualTrace m H • transverseProjector m)
          m x a := by
  unfold planeWaveAxisEdgePert
  rw [edgeLoad_decomposition m H hH hm]
  ring

theorem edgeLoad_decomposition (m : Fin 4 → ℝ) (H : Mat4)
    (hH : IsSymmetric H) (hm : momentumSq m ≠ 0) (d : Fin 4 → ℝ) :
    edgeLoad H d =
      edgeLoad (ttProject m H) d +
        edgeLoad (gaugePart m (gaugeVector m H)) d +
        edgeLoad (residualTrace m H • transverseProjector m) d := by
  have h := (exists_edgeTTDecomposition m H hH hm).1
  -- Rewrite only the left-hand `H`, not the occurrences inside `ttProject m H`.
  conv_lhs => rw [h]
  rw [edgeLoad_add, edgeLoad_add]

/-- Plane-wave edge perturbation of a gauge matrix on an axis edge. -/
theorem planeWaveAxisEdgePert_gaugePart (m v x : Fin 4 → ℝ) (a : Fin 4) :
    planeWaveAxisEdgePert (gaugePart m v) m x a =
      2 * m a * v a * Real.cos (axisMidpointPhase m x a) := by
  unfold planeWaveAxisEdgePert
  rw [edgeLoad_gaugePart_axis]

theorem planeWaveAxisEdgePert_decomposition (m : Fin 4 → ℝ) (H : Mat4)
    (hH : IsSymmetric H) (hm : momentumSq m ≠ 0) (x : Fin 4 → ℝ)
    (a : Fin 4) :
    planeWaveAxisEdgePert H m x a =
      planeWaveAxisEdgePert (ttProject m H) m x a +
        planeWaveAxisEdgePert (gaugePart m (gaugeVector m H)) m x a +
        planeWaveAxisEdgePert (residualTrace m H • transverseProjector m)
          m x a := by
  unfold planeWaveAxisEdgePert
  rw [edgeLoad_decomposition m H hH hm]
  ring

What this page does not claim

The theorem does not establish the continuum limit of the lattice action to the Einstein-Hilbert action. The theorem does not prove the full statement S_RS_converges_EH_4d. The theorem does not apply to edges that are not aligned with the torus's axes.

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/ReggeEdgeTTAttachment4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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