Encyclopedia Gravity Gravity Analysis Regge Tthinge Aware Zero Mode Plane Wave Tet Velocity Zero Mome

ARTICLE 3 claims 3 theorems

Gravity Analysis Regge Tthinge Aware Zero Mode Plane Wave Tet Velocity Zero Mome

A machine-checked proof shows that a lattice model of gravity has exact flat directions at zero momentum, with no extra conditions needed.

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The flat zero mode

In the study of gravity on a discrete lattice, one asks whether small constant distortions of the lattice cost any energy. The Recognition Science framework's machine-checked library of formal theorems answers this question for a specific lattice model of gravity called Regge calculus, which approximates curved spacetime by flat tetrahedra glued together. The declaration planeWaveTetVelocity_zeroMomentum proves that at zero wave vector, meaning for distortions that are uniform across the whole lattice, the second variation of the Regge action vanishes identically. In plain terms, the lattice has exact flat directions: constant metric perturbations are cost-free, for every polarization matrix, with no transverse-traceless condition required.

The proof does not rely on the transverse-traceless (TT) gauge condition that usually selects physical gravitational wave polarizations. The kernel-checked statement is strictly stronger: the assembled constant block vanishes for every polarization matrix, TT or not. The TT instance is a special case, and a separate witness-level theorem exhibits the cancellation on a concrete TT example. The headline theorem, zeroMomentum_symbol_is_zero, states that the fixed-size Bloch symbol at zero wave vector exists and equals zero for every lattice size N, as a statement about the true nonlinear Regge action's second variation.

The proof structure is worth noting for what it reveals. The raw-table contraction over the six tetrahedron types is a perfect square in seven free coefficients, and the alternating class sum vanishes for every polarization. This is pure algebra, audited to the standard axiom trio. The final corollary inherits additional axioms from the certified periodic angle-sum chain, disclosed in the file. No sorry, no admit, no new axioms, no unused hypotheses appear in the file.

What the declaration does not claim is equally important. It does not prove the full-Hessian decomposition into hinge and stencil parts for all configurations; that split is kernel-checked only at the recorded witness, elsewhere living at the symbolic-diagnostic tier. It does not assert that the lattice flat zero mode corresponds to a physical gravitational wave, nor that it survives at nonzero momentum. It establishes a precise algebraic fact about the second variation at zero wave vector, and nothing more.

THEOREM zeroMomentum_symbol_is_zero · IndisputableMonolith/Gravity/Analysis/ReggeTTHingeAwareZeroMode.lean
zeroMomentum_symbol_is_zero · IndisputableMonolith/Gravity/Analysis/ReggeTTHingeAwareZeroMode.lean:478 
/-- **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
THEOREM zeroMomentum_symbol_is_zero · IndisputableMonolith/Gravity/Analysis/ReggeTTHingeAwareZeroMode.lean
zeroMomentum_symbol_is_zero · IndisputableMonolith/Gravity/Analysis/ReggeTTHingeAwareZeroMode.lean:478 
/-- **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
THEOREM zeroMomentum_symbol_is_zero · IndisputableMonolith/Gravity/Analysis/ReggeTTHingeAwareZeroMode.lean
zeroMomentum_symbol_is_zero · IndisputableMonolith/Gravity/Analysis/ReggeTTHingeAwareZeroMode.lean:478 
/-- **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 proved for all configurations, only at the recorded witness. The flat zero mode does not establish the existence of physical gravitational waves in the lattice model. The declaration does not assert that the result holds at nonzero momentum.

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.

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