Gravity Analysis Regge4 Dcontinuum Preflight Axis Ttplus Normalized Is Ttpolariz

The normalized polarization

In the study of gravitational waves, the two independent wave shapes are called the plus and cross polarizations. The plus polarization is a specific pattern of stretching and squeezing. The declaration axisTTPlusNormalized is a definitional choice that fixes a normalized version of this plus polarization in four dimensions. It scales the basic plus polarization matrix by the inverse of the square root of two, so that its Frobenius norm, a measure of the matrix's size, equals exactly one. This normalization is a convention, not a physical discovery.

The declaration also establishes a property: the normalized plus polarization is transverse and traceless. In plain terms, the wave's effect is perpendicular to its direction of travel, and it does not change the volume of space. The library defines this property as IsTTPolarization4D, which requires both the transverse and traceless conditions and the unit norm. The declaration is part of a larger preflight setup for a four-dimensional continuum gravity analysis, where the goal is to check whether a discrete lattice model reproduces the Einstein-Hilbert action of general relativity. The normalized polarization is a test object, not a claim about the physical universe.

What the declaration does not claim is equally important. It does not prove that the continuum limit is recovered. The preflight module explicitly states that nothing here proves continuum recovery. It does not claim that the lattice model matches the Einstein-Hilbert action; that remains an open target. It does not claim that the plus polarization is the only valid wave shape, as the cross polarization is also defined and normalized. It does not claim that the normalization has physical meaning beyond being a mathematical convenience for the analysis. The declaration is a piece of bookkeeping, a definitional step in a larger formal investigation.

/-- Axis plus polarization normalized to Frobenius 1. -/
def axisTTPlusNormalized : Mat4 :=
  (Real.sqrt 2)⁻¹ • axisTTPlus

/-- Axis plus polarization normalized to Frobenius 1. -/
def axisTTPlusNormalized : Mat4 :=
  (Real.sqrt 2)⁻¹ • axisTTPlus

/-- Continuum TT polarization: algebraic TT plus Frobenius normalization.
Without the pin, a fixed continuum coefficient is ill-posed. -/
def IsTTPolarization4D (m : Wave4) (E : Mat4) : Prop :=
  IsTT m E ∧ frobeniusNormSq E = 1

/-- Packaged OPEN name for the ledger edge closer: algebraic TT split
attached to all 15 Regge edge classes of the true-weight Hessian, with
two independent TT polarizations and gauge/trace annihilation. -/
def edge_tt_decomposition : Prop :=
  (∀ (m : Wave4) (H : Mat4),
      IsSymmetric H → waveNormSq m ≠ 0 →
        H = ttProject m H + gaugePart m (gaugeVector m H) +
            residualTrace m H • transverseProjector m ∧
              IsTT m (ttProject m H)) ∧
    IsTTPolarization4D axisWave axisTTPlusNormalized ∧
      IsTTPolarization4D axisWave axisTTCrossNormalized ∧
        (¬ IsTransverse axisWave decoyGauge)

What this page does not claim

This declaration does not prove that the continuum limit of the lattice model is recovered. This declaration does not claim that the plus polarization is the only possible wave shape. This declaration does not assign physical meaning to the normalization beyond mathematical convenience.

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/Regge4DContinuumPreflight.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

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