Gravity Analysis Regge Ttderivative Gate Tt Polarization Frobenius Sq Eq One

The polarization normalization

In the study of gravitational waves, a polarization describes the pattern of distortion a passing wave imprints on space. The declaration ttPolarization_frobeniusSq_eq_one concerns the transverse-traceless (TT) gauge, the standard way to describe a wave's effect in general relativity. The theorem states that for any wave vector, any TT polarization tensor E satisfies the equation ∑_i∑_j E_ijE_ij = 1. In plain terms, this fixes the overall scale of the polarization: the sum of the squares of its components equals one.

The declaration is a theorem in the framework's machine-checked library of formal theorems, proved without assumptions beyond the standard trio of axioms. It extracts a single fact from the definition of a TT polarization and records it as a named lemma. The proof is a direct consequence of the definition; the theorem does not derive the value of the polarization from any physical principle. It says nothing about what the polarization is, only that if it satisfies the TT conditions, its Frobenius norm is one.

In Recognition Science, this lemma is a small step in a larger program to analyze the Regge action, a discrete version of Einstein's equations built from tetrahedra. The theorem is part of the derivative gate, a stage that checks the first derivatives of the action at a flat configuration. The normalization is a technical convenience: it makes the polarization a unit vector in the space of matrices, simplifying later computations. The framework's library proves the lemma for every integer wave vector, including the zero vector, by an explicit construction of a transverse frame.

The theorem does not claim that the TT gauge describes real gravitational waves, nor does it assign a physical meaning to the normalization. It is a statement about a mathematical object defined in the framework's formal system. The existence of a TT polarization for every wave vector is a separate theorem, and the value of the continuum symbol, the quantity the program ultimately seeks, remains an open target. This lemma is a tool, not a result about the physical world.

/-- The fourth `IsTTPolarization` conjunct as a named audit lemma: every TT
polarization has Frobenius norm square exactly `1`. -/
theorem ttPolarization_frobeniusSq_eq_one (m : Fin 3 → ℤ) (E : Fin 3 → Fin 3 → ℝ)
    (hE : ReggeTTSymbolPreflight.IsTTPolarization m E) :
    (∑ i : Fin 3, ∑ j : Fin 3, E i j * E i j) = 1 := hE.2.2.2

What this page does not claim

The theorem does not derive the existence of TT polarizations; that is a separate theorem. The theorem does not assign a physical interpretation to the normalization. The theorem does not establish the value of the continuum TT symbol, which remains an open target.

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

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