Gravity Analysis Edge Ttdecomposition4 D

The 4D TT decomposition

In physics, a transverse-traceless (TT) decomposition is a way of splitting a matrix into a part that represents a wave and a part that represents a coordinate choice. The wave part is symmetric, has zero trace (the sum of its diagonal entries is zero), and is transverse (its product with the wave direction is zero). This is the same kind of decomposition used in general relativity to identify the two physical polarizations of a gravitational wave.

The EdgeTTDecomposition4D library proves this decomposition for all symmetric 4x4 real matrices against a nonzero Euclidean wave covector. It defines the projectors and proves, with machine-checked theorems, that the output satisfies all three TT conditions. The decomposition requires a nonzero wave vector; the library explicitly proves that the decomposition hypothesis fails for the zero vector, which is a necessary edge case.

The library also constructs two explicit, nonzero TT matrices, axisTTPlus and axisTTCross, which are the two independent polarizations for a wave traveling along the first axis. It further shows that a deliberately non-transverse matrix, the decoyLongitudinal, is correctly mapped to a TT matrix by the projection. These examples make the abstract decomposition concrete.

In Recognition Science, this library is the algebraic layer of a larger project to close the ledger on gravity. It does not itself decompose lattice perturbations or prove convergence to general relativity; it is a necessary, self-contained step in that direction. The framework's machine-checked library of formal theorems establishes this result with no unproven assumptions beyond the standard logical axioms.

The practical consequence is a rigorous, verified tool for separating physical wave degrees of freedom from coordinate artifacts in four dimensions. This is a prerequisite for any later claim about how many gravitational degrees of freedom exist, and it gives the framework a solid algebraic foundation for its gravity analysis.

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

theorem decomposition_hypothesis_fails_at_zero :
    ¬ (momentumSq (fun _ : Fin 4 => (0 : ℝ)) ≠ 0) := by
  simp [zero_wave_momentumSq]

theorem axisTTPlus_ne_zero : axisTTPlus ≠ 0 := by
  intro h
  have := congrArg (fun M : Mat4 => M 2 2) h
  simp [axisTTPlus] at this

theorem axisTTCross_ne_zero : axisTTCross ≠ 0 := by
  intro h
  have := congrArg (fun M : Mat4 => M 2 3) h
  simp [axisTTCross] at this

What this page does not claim

This library does not decompose Regge EDGE perturbations on a 4D lattice. This library does not prove S_RS_converges_EH_4d. This library does not address the Minkowski/null specialization for the Lorentzian continuum.

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)

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: