Gravity Analysis Regge Tthinge Aware Zero Mode Pol Edge Coeff Alternating Sum

The alternating sum

In lattice gravity, the Regge action approximates general relativity by triangulating spacetime into flat tetrahedra and summing their edge lengths times deficit angles. A zero mode is a perturbation of the edge lengths that leaves the action unchanged to second order. The declaration polEdgeCoeff_alternatingSum establishes the algebraic identity that makes a particular zero mode exact: it shows that the alternating sum of seven edge-class coefficients vanishes for every polarization matrix.

The identity is a pure counting statement about the seven displacement classes in a tetrahedron. The coefficients c_d are indexed by the displacement classes d, and the alternating sum is c0 + c1 + c2 - c3 - c4 - c5 + c6. The proof observes that the sum of the three two-coordinate classes equals the sum of the three one-coordinate classes plus the three-coordinate class, termwise, because each one-coordinate class is double-counted in the two-coordinate classes. This is a trace-double-count argument, not a physical assumption.

The consequence is the zero-mode headline: the assembled constant block of the lattice Hessian vanishes identically for every polarization matrix, not just for the transverse-traceless (TT) polarizations that are the usual focus in gravitational wave analysis. Constant metric perturbations are exact flat directions of the lattice action. This is strictly stronger than the panel statement that asked for the vanishing under TT hypotheses; the kernel proof shows those hypotheses are unnecessary for the assembled block.

In Recognition Science, this declaration is part of a machine-checked library of formal theorems. The library proves that at zero wave vector, the fixed-N Bloch symbol of the lattice action exists and equals zero for every N and every polarization. The identity does not claim that the full Hessian decomposition into hinge and stencil parts is re-proved in the kernel; that split is only kernel-checked at a specific witness. The identity also does not claim anything about nonzero wave vectors or about the physical interpretation of the zero mode as a gravitational wave.

/-- **ZERO-MODE SYMBOL COROLLARY (THEOREM): the fixed-`N` TT Bloch symbol
of the TRUE nonlinear Regge action at ZERO wave vector exists and equals
`0`, for every `N` and every polarization matrix.**  This is the lattice
flat zero mode as a statement about the actual second variation, through
the Gate A1 existence chain and the Gate A2 reduction.  AXIOM
DISCLOSURE: this corollary (alone in this file) rides the certified
flat-deficit chain and therefore inherits `Lean.ofReduceBool` /
`Lean.trustCompiler` in addition to the standard trio. -/
theorem zeroMomentum_symbol_is_zero (N : ℕ) [NeZero N]
    (E : Fin 3 → Fin 3 → ℝ) :
    TTBlochSymbolIs N E (fun _ => (0 : ℤ)) 0 := by
  have h := ReggeTTFlatSecondVariation.planeWave_TTBlochSymbolIs_reduced
    N E (fun _ => (0 : ℤ))
  have hval : canonicalFiniteH N E (fun _ => (0 : ℤ)) =
      (2 / (N : ℝ) ^ (3 : ℕ)) *
        (-∑ τ : PeriodicTet N N N, ∑ f : Fin 6,
          ReggeTTFlatSecondVariation.flatSlotSqrtDeriv N E
              (commensurateMomentum N (fun _ => (0 : ℤ))) τ f *
            ReggeTTFlatSecondVariation.flatSlotAngleDeriv N E
              (commensurateMomentum N (fun _ => (0 : ℤ))) τ f) := rfl
  rw [← hval, canonicalFiniteH_zeroMomentum_eq_zero N E] at h
  exact h

/-- **ZERO-MODE SYMBOL COROLLARY (THEOREM): the fixed-`N` TT Bloch symbol
of the TRUE nonlinear Regge action at ZERO wave vector exists and equals
`0`, for every `N` and every polarization matrix.**  This is the lattice
flat zero mode as a statement about the actual second variation, through
the Gate A1 existence chain and the Gate A2 reduction.  AXIOM
DISCLOSURE: this corollary (alone in this file) rides the certified
flat-deficit chain and therefore inherits `Lean.ofReduceBool` /
`Lean.trustCompiler` in addition to the standard trio. -/
theorem zeroMomentum_symbol_is_zero (N : ℕ) [NeZero N]
    (E : Fin 3 → Fin 3 → ℝ) :
    TTBlochSymbolIs N E (fun _ => (0 : ℤ)) 0 := by
  have h := ReggeTTFlatSecondVariation.planeWave_TTBlochSymbolIs_reduced
    N E (fun _ => (0 : ℤ))
  have hval : canonicalFiniteH N E (fun _ => (0 : ℤ)) =
      (2 / (N : ℝ) ^ (3 : ℕ)) *
        (-∑ τ : PeriodicTet N N N, ∑ f : Fin 6,
          ReggeTTFlatSecondVariation.flatSlotSqrtDeriv N E
              (commensurateMomentum N (fun _ => (0 : ℤ))) τ f *
            ReggeTTFlatSecondVariation.flatSlotAngleDeriv N E
              (commensurateMomentum N (fun _ => (0 : ℤ))) τ f) := rfl
  rw [← hval, canonicalFiniteH_zeroMomentum_eq_zero N E] at h
  exact h

What this page does not claim

The full Hessian decomposition into hinge and stencil parts is not re-proved in the kernel except at a specific witness. The identity does not claim that TT polarizations are the only ones with zero modes. The identity does not establish anything about the physical propagation of gravitational waves.

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/ReggeTTHingeAwareZeroMode.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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