Gravity Analysis Edge Ttdecomposition Closer4 D Decoy Gauge Eq Decoy Longitudina

The decoy equivalence

In the framework's machine-checked library of formal theorems, the declaration decoyGauge_eq_decoyLongitudinal proves that two auxiliary objects, the decoy gauge and the decoy longitudinal wave, are equal. Both are constructed from the same underlying axis wave vector, and the proof unfolds their definitions to verify equality component by component. The theorem is a formal identity: it states that these two named constructions coincide, nothing more and nothing less.

The equality matters because it feeds a larger result. A second theorem, decoyGauge_not_transverse, uses this equality to show that the decoy gauge wave is not transverse, meaning it does not lie purely in the spatial plane perpendicular to its propagation direction. This non-transversality is one of four ingredients in the named closer edge_tt_decomposition, which assembles a decomposition of edge waves into transverse-traceless, plus, cross, and decoy parts. The decoy part is deliberately non-transverse; it plays the role of a pure-gauge artifact in the decomposition.

The proof is fully checked by the framework's kernel, with no unverified assumptions. It establishes an algebraic identity and its immediate consequence about transversality. It does not, however, claim anything about the physical significance of the decoy wave, nor does it assert that this decomposition recovers the full continuum theory of gravity. The docstring is explicit: the named closer inhabits only the algebraic and attachment layer, while the full multi-orbit true-weight continuum recovery remains behind a separate gate called S_RS_converges_EH_4d.

In plain terms, the declaration is a precise but narrow step. It tells a reader that two formal constructions are the same, and that this sameness makes one of them provably non-transverse. It does not by itself establish any physical prediction, any numerical value, or any claim about the real world. It is a piece of the formal scaffolding on which larger claims, if they are ever completed, would rest.

theorem decoyGauge_eq_decoyLongitudinal : decoyGauge = decoyLongitudinal := by
  unfold decoyGauge decoyLongitudinal axisGaugeVector
  funext i j
  fin_cases i <;> fin_cases j <;> simp [gaugePart, axisWave]

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

What this page does not claim

The declaration does not establish any physical prediction or numerical value. The declaration does not prove that the full continuum gravity theory is recovered. The declaration does not assign physical significance to the decoy wave.

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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