Gravity Analysis Edge Ttdecomposition Closer4 D Edge Tt Decomposition

The edge decomposition

In general relativity, gravitational waves are transverse: they wiggle perpendicular to their direction of travel, with no longitudinal squeezing. The framework's declaration edge_tt_decomposition establishes the same split for the discrete ledger model, where spacetime is built from elementary edges rather than a smooth continuum. It proves that any edge wave with nonzero momentum decomposes into exactly the two transverse polarizations, plus a third mode that is provably not transverse. That third mode is a pure gauge artifact, a coordinate choice that carries no physical wave content.

The theorem is a named closer, meaning it assembles previously proved pieces into one statement. It combines an algebraic decomposition of the wave into plus and cross polarizations, normalizes those polarizations, attaches them to plane-wave edges, and then adds the decoy longitudinal mode. The key result is that this decoy mode is not transverse, so it cannot masquerade as a physical gravitational wave. The proof is machine-checked in the framework's library of formal theorems, with no axioms beyond the standard logical ones.

In Recognition Science, this is the algebraic and attachment layer for a larger goal. The declaration does not claim that the full continuum recovery of Einstein's equations from the discrete ledger is complete. That remains an open gate, named S_RS_converges_EH_4d, which would establish the true-weight continuum limit. The edge decomposition is a necessary step, not the final verdict: it shows the discrete waves have the right transverse structure, but it does not yet show that summing over all orbits of the ledger reproduces general relativity in the smooth limit.

/-- **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

The theorem does not prove that the discrete ledger reproduces full general relativity in the continuum limit. The theorem does not establish uniqueness of the decomposition across all possible edge attachments. The theorem does not claim that the decoy mode has any physical gravitational effect.

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)

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: