Gravity Analysis Regge Ttsymbol Specification Audit Is Ttpolarization Frobenius

The normalization pin

In the study of gravitational waves, a polarization describes how the wave's stretching and squeezing is oriented in space. For a wave traveling in three dimensions, this orientation is encoded in a 3 by 3 matrix. The matrix must satisfy several conditions: it must be symmetric, have zero trace (no overall expansion or contraction), and be transverse (perpendicular to the wave's direction of travel). These conditions define a transverse-traceless polarization, often abbreviated as TT.

There is a problem with these conditions alone: they do not fix the matrix's size. If you take a valid polarization matrix and multiply every entry by 2, you get another matrix that is also symmetric, traceless, and transverse. This matters because a proposed physical constant, the value -1/4 for a certain wave property, would be meaningless if the size of the polarization could vary. The same wave would report different values depending on how it was scaled.

The declaration isTTPolarization_frobenius_pinned resolves this. It is a theorem in the framework's machine-checked library of formal theorems. The theorem states that any polarization matrix satisfying the TT conditions also has a Frobenius norm equal to 1. The Frobenius norm is a measure of size: the square root of the sum of the squares of all nine entries. Pinning this norm to 1 means the matrix has a fixed, standard size.

This pin is not arbitrary. The library also proves that scaling a polarization matrix by a factor c changes its Frobenius norm by c squared. If the norm is fixed at 1, then the only allowed scalings are c equals 1 or c equals -1. Both leave the norm unchanged. This makes the -1/4 target well-posed: it can be stated as a fixed number because the polarization it refers to has a fixed size.

What this theorem does not do is prove that the value -1/4 is correct. The pin only ensures that the statement is meaningful, not that it is true. The actual value of the continuum TT Bloch symbol remains an open target. The theorem is a necessary condition for the claim to make sense, not evidence for its truth.

/-- (e), pin re-export: `IsTTPolarization` pins the Frobenius square-sum
to `1` (fourth conjunct of the definition). -/
theorem isTTPolarization_frobenius_pinned (m : Fin 3 → ℤ)
    (E : Fin 3 → Fin 3 → ℝ) (h : IsTTPolarization m E) :
    frobeniusSq E = 1 :=
  h.2.2.2

/-- (e), pin re-export: `IsTTPolarization` pins the Frobenius square-sum
to `1` (fourth conjunct of the definition). -/
theorem isTTPolarization_frobenius_pinned (m : Fin 3 → ℤ)
    (E : Fin 3 → Fin 3 → ℝ) (h : IsTTPolarization m E) :
    frobeniusSq E = 1 :=
  h.2.2.2

/-- The Frobenius square-sum scales quadratically under `E ↦ c·E`. -/
theorem frobeniusSq_smul (c : ℝ) (E : Fin 3 → Fin 3 → ℝ) :
    frobeniusSq (c • E) = c ^ 2 * frobeniusSq E := by
  unfold frobeniusSq
  rw [Finset.mul_sum]
  refine Finset.sum_congr rfl fun i _ => ?_
  rw [Finset.mul_sum]
  refine Finset.sum_congr rfl fun j _ => ?_
  show (c * E i j) * (c * E i j) = c ^ 2 * (E i j * E i j)
  ring

What this page does not claim

The theorem does not prove that the -1/4 value is correct. The theorem does not provide numerical evidence for the -1/4 value. The theorem does not apply to polarizations that are not transverse-traceless.

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

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