Gravity Analysis Regge Edge Ttattachment4 D Plane Wave Axis Edge Pert Gauge Part

The gauge identity

In the study of gravity on a discrete lattice, a plane wave is a periodic disturbance, and its effect on a single edge of the lattice is a number called the edge load. The declaration planeWaveAxisEdgePert_gaugePart_eq_discreteLie proves an exact identity about the gauge part of that load. The gauge part is the piece of a matrix perturbation that can be removed by a coordinate change, here isolated by a projection. The identity states that for a wave with covector m and a vertex field v, the edge load of the gauge part equals 2 m_a v_a times the cosine of a midpoint phase, where the index a labels the edge.

The same declaration connects this load to a discrete Lie derivative. A Lie derivative measures how a field changes as it is dragged along a flow; on a lattice, the discrete version compares the field at a vertex and at a neighboring vertex one step away. For the plane-wave vertex field ξ_b(x) = v_b sin(m·x), the discrete Lie loading is 2(ξ_a(x+e_a) − ξ_a(x)) = 4 v_a sin(m_a/2) cos(m·x + m_a/2). The identity then says the gauge edge perturbation equals (m_a / (2 sin(m_a/2))) times this discrete Lie loading, whenever sin(m_a/2) ≠ 0. This is the exact lattice identity; it is not the continuum claim δℓ² = 2 ∂_a ξ_a.

The result is a theorem in the machine-checked library of formal theorems, meaning it is verified by a computer from the axioms of logic, with no unproved assumptions. It is part of a larger effort to attach the standard decomposition of a matrix into transverse-traceless, gauge, and residual parts to plane-wave loadings on edges of a four-dimensional torus. The identity is a stepping stone: it shows how a gauge piece of a perturbation behaves under a plane wave, which is a step toward understanding how the full perturbation decomposes.

What the declaration does not claim is as important as what it proves. It does not prove the full ledger name edge_tt_decomposition, which would require a four-dimensional Regge action, a continuum Einstein-Hilbert recovery, and a full Freudenthal edge-class stencil. It does not prove that the recognition-science action converges to the Einstein-Hilbert action in four dimensions, nor does it flip the gap_action_recovery flag. The identity is a local, exact statement about a single edge under a plane wave, not a global statement about the theory.

/-- 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]

/-- Exact trig expansion of the discrete Lie loading on an axis edge. -/
theorem discreteLieAxis_eq (v m x : Fin 4 → ℝ) (a : Fin 4) :
    discreteLieAxis v m x a =
      2 * v a * latticeDerivSymbol m a *
        Real.cos (axisMidpointPhase m x a) := by
  unfold discreteLieAxis gaugeVertexField axisMidpointPhase latticeDerivSymbol
  set θ : ℝ := ∑ i : Fin 4, m i * x i
  have hθ' : (∑ i : Fin 4, m i * shiftAxis x a i) = θ + m a :=
    shiftAxis_dot m x a
  simp only [hθ']
  have htrig := sin_add_sub_sin θ (m a)
  calc
    2 * (v a * Real.sin (θ + m a) - v a * Real.sin θ)
        = 2 * v a * (Real.sin (θ + m a) - Real.sin θ) := by ring
    _ = 2 * v a * (2 * Real.sin (m a / 2) * Real.cos (θ + m a / 2)) := by
      rw [htrig]
    _ = 2 * v a * (2 * Real.sin (m a / 2)) * Real.cos (θ + m a / 2) := by
      ring

/-- 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]

/-- 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]

What this page does not claim

The declaration does not prove the continuum claim δℓ² = 2 ∂_a ξ_a. The declaration does not establish that the recognition-science action converges to the Einstein-Hilbert action in four dimensions. The declaration does not prove the full edge_tt_decomposition ledger name.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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