Gravity Analysis Regge Ttsymbol Preflight Plane Wave Edge Field Neg Polarization

The sign symmetry

A plane wave is a periodic disturbance that moves through a medium, and its polarization describes the orientation of that disturbance. In the discrete gravity model studied here, the medium is a lattice of tetrahedra, and the disturbance is a small change in the squared length of each edge. The declaration planeWaveEdgeField_neg_polarization is a formal statement, checked by a machine, that this model has a simple symmetry: if you flip the sign of the polarization matrix, the resulting plane wave has the same energy profile, up to a sign in the amplitude that cancels in the quadratic form.

Concretely, the model defines a family of edge-length fields that oscillate like a cosine wave, with an amplitude controlled by a parameter t and a polarization matrix E. The key quantity is the second difference of the action, a measure of how the energy responds to the wave. The theorem proves that this second difference is even in the amplitude: replacing t with -t leaves it unchanged. Since flipping the sign of E is equivalent to flipping the sign of t, the same evenness applies to the polarization. This is a necessary condition for the model to have a well-defined quadratic approximation, where the energy depends only on the square of the amplitude, not on its sign.

The theorem is a small but essential piece of a larger investigation. The framework's library also proves that the plane wave passes through the flat configuration at zero amplitude, and that the second difference is even in the amplitude. These are the algebraic facts that any sensible quadratic model must satisfy. What the declaration does not claim is that the limit of the second difference exists as the amplitude goes to zero, nor that it equals any particular value. That limit, the so-called Bloch symbol, is the subject of a separate, explicitly open target. The numerical evidence suggests the value is -1/4, but that is not proved here.

In plain terms, this declaration establishes a symmetry that any quadratic approximation must respect, and it does so with a machine-checked proof. It does not establish the value of the quadratic coefficient itself. That distinction, between a symmetry that is proved and a limit that is only conjectured, is the honest boundary of this result.

/-- The second-difference quadratic form is even in the amplitude. -/
theorem ttSecondDifference_even (E : Fin 3 → Fin 3 → ℝ) (k : Fin 3 → ℝ)
    (t : ℝ) :
    ttSecondDifference N E k (-t) = ttSecondDifference N E k t := by
  unfold ttSecondDifference
  rw [neg_neg, neg_sq]
  ring

/-- The action profile vanishes at amplitude `0` (flat point of the
family). -/
theorem planeWaveActionProfile_zero (E : Fin 3 → Fin 3 → ℝ)
    (k : Fin 3 → ℝ) :
    planeWaveActionProfile N E k 0 = 0 := by
  unfold planeWaveActionProfile
  rw [planeWaveEdgeField_zero_amplitude, trueReggeAction_flatEdgeField]

What this page does not claim

The existence or value of the Bloch symbol limit is not proved here. The numerical value -1/4 is not established as a theorem. The declaration does not claim that the true action's quadratic approximation is isotropic.

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/ReggeTTSymbolPreflight.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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