Gravity Analysis Edge Ttdecomposition Closer4 D

The edge decomposition closer

In general relativity, gravitational waves are transverse and traceless: they wiggle perpendicular to their direction of travel and leave volumes unchanged. A standard tool for studying such waves is the transverse-traceless (TT) decomposition, which splits any symmetric tensor into a physical TT part and a remainder that can be removed by a coordinate choice. The module named gravity analysis edge ttdecomposition closer4 d proves, in a machine-checked library of formal theorems, that this decomposition exists at the boundary of a four-dimensional spacetime region, for every wave whose momentum is nonzero.

The proof works in four steps. First, it shows that any wave with nonzero momentum admits an algebraic TT decomposition, meaning the tensor can be written as a physical part plus a pure-gauge part. Second and third, it constructs two explicit polarization states, the plus and cross modes familiar from gravitational wave detectors, and shows each is a valid TT polarization. Fourth, it builds a decoy: a gauge field that is not transverse, so it cannot be mistaken for a physical wave. The theorem then assembles these pieces into a single statement: the edge TT decomposition always holds.

In Recognition Science, this result lives in the ledger layer. The framework treats physical law as a discrete record of recognition events, and this theorem is one entry in that record: it establishes a structural fact about gravitational waves at an edge, not a numerical prediction. The full recovery of the continuum gravitational action from many orbits remains an open target, gated behind the S_RS_converges_EH_4d condition, which is not proved here.

What the module changes is practical: any future work that needs to separate physical wave content from coordinate artifacts at a boundary can rely on this decomposition as a proved fact, not an assumption. It is a small but solid brick in the framework's account of how gravity emerges from recognition.

/-- **THEOREM (named ledger closer, algebraic+attachment layer).** -/
theorem edge_tt_decomposition :
    Regge4DContinuumPreflight.edge_tt_decomposition := by
  refine ⟨?_, ?_, ?_, ?_⟩
  · intro m H hH hm
    have hm' : momentumSq m ≠ 0 := by
      simpa [waveNormSq_eq_momentumSq] using hm
    exact exists_edgeTTDecomposition m H hH hm'
  · exact axisTTPlusNormalized_isTTPolarization
  · exact axisTTCrossNormalized_isTTPolarization
  · exact decoyGauge_not_transverse

/-- **THEOREM (named ledger closer, algebraic+attachment layer).** -/
theorem edge_tt_decomposition :
    Regge4DContinuumPreflight.edge_tt_decomposition := by
  refine ⟨?_, ?_, ?_, ?_⟩
  · intro m H hH hm
    have hm' : momentumSq m ≠ 0 := by
      simpa [waveNormSq_eq_momentumSq] using hm
    exact exists_edgeTTDecomposition m H hH hm'
  · exact axisTTPlusNormalized_isTTPolarization
  · exact axisTTCrossNormalized_isTTPolarization
  · exact decoyGauge_not_transverse

theorem decoyGauge_not_transverse :
    ¬ IsTransverse axisWave decoyGauge := by
  rw [decoyGauge_eq_decoyLongitudinal]
  exact decoyLongitudinal_not_transverse

What this page does not claim

This module does not prove the full continuum gravitational action recovery from many orbits. This theorem does not make any numerical prediction about specific gravitational wave sources. The decomposition applies to waves with nonzero momentum only; the zero-momentum case is not addressed here.

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/EdgeTTDecompositionCloser4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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