Gravity Analysis Regge Bloch All Orbit Symbol4 D Factorized Bloch Fold All Gauge

The zero-momentum gauge identity

In lattice gravity, one often studies how a discrete version of the Einstein action behaves when matter or geometry is probed by waves of different wavelengths. The declaration factorizedBlochFoldAll_gauge_zeroMomentum is a machine-checked theorem about one such probe. It says that for a particular test configuration, called decoyGauge, the entire summed quantity factorizedBlochFoldAll equals zero when the wave momentum is exactly zero.

The quantity being summed is a Bloch symbol, a standard tool in solid-state and lattice physics that packages how a wave of a given momentum interacts with a periodic structure. Here the structure is a four-dimensional hypercubic lattice, and the sum runs over all possible hinge types, the elementary geometric joints of the lattice. The theorem is a special case of a broader result: at zero momentum, the full sum reduces to a simpler quadratic form, and for this particular test configuration that quadratic form is identically zero.

The practical significance is a sanity check. The decoyGauge configuration is deliberately constructed to be a gauge-like direction, one that should not contribute physical content. The theorem confirms that at zero momentum, it indeed contributes nothing. This is a consistency requirement: if this test configuration had produced a nonzero value, the whole construction would be suspect. The theorem is proved in the framework's machine-checked library of formal theorems, with the expected axiom footprint of [propext, Classical.choice, Quot.sound].

What the theorem does not claim is equally important. It does not say that the decoyGauge configuration vanishes at nonzero momentum; that is a separate, open question. It does not establish any result about the continuum limit, the smooth spacetime that lattice theories are meant to approximate. And it does not prove that the full Bloch symbol recovers the Einstein-Hilbert action in that limit. The theorem is a narrow, precise algebraic identity, not a statement about the physical content of the theory.

theorem factorizedBlochFoldAll_gauge_zeroMomentum :
    factorizedBlochFoldAll decoyGauge (fun _ => (0 : ℝ)) = 0 := by
  rw [factorizedBlochFoldAll_zeroMomentum, trueWeightZeroMomQuadratic_decoyGauge]

/-- All-orbit fold at zero momentum is the committed true-weight Hessian. -/
theorem factorizedBlochFoldAll_zeroMomentum (H : Mat4) :
    factorizedBlochFoldAll H (fun _ => (0 : ℝ)) =
      trueWeightZeroMomQuadratic H := by
  unfold factorizedBlochFoldAll trueWeightZeroMomQuadratic
  exact Finset.sum_congr rfl fun ty _ =>
    factorizedBlochFoldOrbit_zeroMomentum ty H

theorem factorizedBlochFoldAll_gauge_zeroMomentum :
    factorizedBlochFoldAll decoyGauge (fun _ => (0 : ℝ)) = 0 := by
  rw [factorizedBlochFoldAll_zeroMomentum, trueWeightZeroMomQuadratic_decoyGauge]

What this page does not claim

The theorem does not claim that decoyGauge vanishes at nonzero momentum. It does not establish any result about the continuum limit or recovery of the Einstein-Hilbert action. It does not prove that the full Bloch symbol is physically meaningful.

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/ReggeBlochAllOrbitSymbol4D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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