Gravity Analysis Edge Ttdecomposition4 D Decoy Longitudinal Not Transverse

The decoy and its failure

In the study of gravitational waves, a common mathematical task is to decompose a symmetric 4×4 matrix into parts that are transverse and traceless. Transverse means the wave's effect is perpendicular to its direction of travel, and traceless means it does not change volume. The declaration decoyLongitudinal_not_transverse establishes a precise negative fact: a particular matrix, called the decoy longitudinal, is not transverse with respect to a chosen wave direction. This is a theorem in the machine-checked library of formal theorems, meaning it is a fully verified statement within the framework's algebraic layer.

To understand the decoy, consider the wave direction defined by the vector (1, 0, 0, 0) in four-dimensional Euclidean space. The decoy longitudinal matrix is built from this direction using a standard gauge construction, which produces a symmetric matrix. The theorem proves that this matrix fails the transversality condition: when the matrix acts on the wave direction, the result is not zero. In plain terms, the decoy looks like it could be a legitimate wave component, but it is not, because it has a longitudinal piece along the direction of travel.

The importance of this theorem is that it validates the decomposition algorithm. The framework's library also proves that when the transverse-traceless projection is applied to this decoy, the result is transverse and traceless. This means the algorithm correctly identifies and removes the longitudinal part. The decoy is a test case: if the projection failed on it, the algorithm would be unreliable. The theorem shows the algorithm works on this challenging input.

This result is purely algebraic. It concerns real 4×4 matrices and a fixed Euclidean wave vector. It does not claim anything about physical gravitational waves in curved spacetime, nor does it prove that the framework's full theory converges to Einstein's equations. The theorem is a building block in a larger campaign, not a statement about the physical universe.

theorem decoyLongitudinal_not_transverse :
    ¬ IsTransverse axisWave decoyLongitudinal := by
  intro h
  have h0 := h 0
  simp [decoyLongitudinal, gaugePart, axisWave, Fin.sum_univ_four] at h0

theorem decoyLongitudinal_symmetric : IsSymmetric decoyLongitudinal :=
  gaugePart_symmetric _ _

theorem decoy_ttProject_isTT :
    IsTT axisWave (ttProject axisWave decoyLongitudinal) :=
  ttProject_isTT axisWave decoyLongitudinal decoyLongitudinal_symmetric
    (by simp [axisWave_momentumSq])

What this page does not claim

This theorem does not apply to physical gravitational waves in curved spacetime. This theorem does not prove the framework's theory converges to Einstein's equations. This theorem does not claim anything about the Lorentzian or null case.

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/EdgeTTDecomposition4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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