Gravity Analysis Regge Ttsymbol Preflight Tt Second Difference Neg Polarization

The sign symmetry

The declaration ttSecondDifference_neg_polarization is a formal theorem about a specific mathematical object: the second difference of a gravity action when probed by a plane wave. In the framework's library, a machine-checked collection of formal theorems, it proves that changing the polarization matrix E to its negative, written E ↦ −E, leaves the second difference unchanged. This is a sign symmetry: the quantity is even in the polarization amplitude. The proof is a short algebraic identity, and the theorem carries the standard axiom footprint of the library's kernel, with no additional assumptions.

The object itself is a preflight check for a larger program. The framework models gravity through a discrete Regge action, a sum over tetrahedra of edge lengths times angle deficits. To study how this action responds to a gravitational wave, one perturbs the flat edge lengths by a plane wave with a given polarization and momentum. The second difference, a discrete analogue of a second derivative, measures the quadratic response of the action to that perturbation. The theorem ensures that this response does not care whether the wave's polarization is flipped in sign, a property any sensible energy-like quantity must have.

What the theorem does not claim is just as important. It does not establish the value of that second difference, nor does it show that the limit as the lattice spacing goes to zero exists. Those are separate, named open targets in the library. The theorem also does not prove that the full nonlinear Regge action is isotropic in the continuum limit, a property that a numerical probe has suggested but which remains formally unproved. The sign symmetry is a well-formedness check, not a physical prediction.

In the broader campaign, this lemma is one of several small, verified pieces that together make a larger conjecture credible. The conjecture, that the true nonlinear Regge action yields a specific isotropic coefficient in the continuum limit, remains open. This theorem is a necessary but far from sufficient step toward it.

/-- 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

What this page does not claim

The theorem does not establish the value of the second difference, only its evenness in the polarization sign. The theorem does not prove the existence or value of the continuum TT Bloch symbol limit. The theorem does not prove that the true nonlinear Regge action is isotropic in the continuum limit.

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)

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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: