Gravity Analysis Regge Tthinge Aware Zero Mode Zero Momentum Symbol Is Zero

The flat zero mode

In Regge calculus, a discrete approximation to general relativity, spacetime is built from flat tetrahedra joined at their faces. The theory's second variation, the matrix of second derivatives of the action, tells whether small distortions of this lattice cost energy. A zero mode is a distortion direction along which that cost vanishes. The theorem zeroMomentum_symbol_is_zero in the framework's machine-checked library of formal theorems proves that at zero wave vector, meaning for distortions that are constant across the whole lattice, the symbol of this second variation is exactly zero for every lattice size and every polarization matrix.

The proof is algebraic. The assembled constant block, the part of the second variation that survives at zero momentum, reduces to a perfect square: (c0 + c1 + c2 - c3 - c4 - c5 + c6)^2 / 2, where the c's are coefficients attached to the seven edge classes of the lattice. The alternating class sum then vanishes for every polarization, because the sum of coefficients on the three coordinate-plane edges equals the sum on the three body-diagonal edges. This cancellation is what makes constant metric perturbations exact flat directions of the lattice, a statement strictly stronger than the requested "under TT hypotheses" claim. The theorem holds for all polarizations, not just the transverse-traceless ones.

The result matters because it closes a gate in a larger campaign to understand the quantum behavior of this lattice gravity. A diagnostic had found that the stencil-only constant block, without the hinge term, does not vanish under TT; its residual at a reported witness was -pi*(sqrt 2 + 4)/8. The assembled zero mode cancels this residual exactly: the hinge block equals the same value, and the two combine with a relative minus sign. The kernel proof pins this sign convention, so no silent sign change is possible. The headline theorem, however, does not depend on that witness at all.

In Recognition Science, this is a step toward showing that the lattice's flat directions are stable. The framework models gravity as a discrete ledger of recognition events, and this theorem confirms that the ledger's second variation has the expected zero mode. It does not claim that the full Hessian decomposition is proved in the kernel, nor that the physical recognition-to-linking bridge is closed. Those remain targets.

/-- **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

private theorem w00 : rawJacobianCoefficient 0 0 = (0 : ℝ) := by
  rw [rawJacobianCoefficient_eval 0 0]; norm_num [ReggeTTBucketAggregation.rationalStencilWeight]

private theorem w01 : rawJacobianCoefficient 0 1 = (0 : ℝ) := by
  rw [rawJacobianCoefficient_eval 0 1]; norm_num [ReggeTTBucketAggregation.rationalStencilWeight]

private theorem w02 : rawJacobianCoefficient 0 2 = (0 : ℝ) := by
  rw [rawJacobianCoefficient_eval 0 2]; norm_num [ReggeTTBucketAggregation.rationalStencilWeight]

What this page does not claim

The full Hessian decomposition assembled = hinge - sum G c c is not re-proved in Lean. The physical recognition-to-linking bridge is not closed by this theorem. The theorem does not require the TT hypotheses; it holds for all polarizations.

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)

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