Encyclopedia Gravity Gravity Analysis Regge Tthinge Aware Zero Mode Assembled Constant Block Eq Zero

ARTICLE 3 claims 3 theorems

Gravity Analysis Regge Tthinge Aware Zero Mode Assembled Constant Block Eq Zero

A machine-checked proof shows that a certain lattice gravity construction has a flat direction for every possible polarization, not just the special ones.

Verify this page

The flat zero mode

In Regge calculus, spacetime is built from flat tetrahedra joined at their faces, and gravity becomes a discrete bookkeeping of edge lengths and angles. A central question for any such lattice theory is whether it has spurious flat directions: small changes in the metric that cost no energy and therefore are not true dynamical modes. The Recognition Science framework's machine-checked library of formal theorems proves that one such construction, the assembled constant block of its lattice action, vanishes identically. That means constant metric perturbations are exact flat directions of the lattice, for every polarization matrix, not just the transverse-traceless ones that the original diagnostic targeted.

The proof works by algebra. The raw contraction over the six tetrahedron types collapses into a perfect square, (c0 + c1 + c2 - c3 - c4 - c5 + c6)^2 / 2, and the alternating class sum vanishes for every polarization because the edge coefficients satisfy a termwise identity. The theorem assembledConstantBlock_eq_zero states this result. A companion theorem, zeroMomentum_symbol_is_zero, extends it to the full Bloch symbol at zero wave vector for any lattice size, confirming the flat zero mode as a statement about the true nonlinear Regge action's second variation.

The result is stronger than the question that motivated it. The original diagnostic looked for a cancellation under the transverse-traceless (TT) condition, a standard gauge choice in gravity. The kernel proof shows the cancellation needs no such hypothesis: the assembled block vanishes for every polarization matrix. The TT instance is a special case, and a separate witness-level theorem exhibits the hinge-versus-stencil cancellation on the concrete TT witness the diagnostic reported, pinning the assembly sign convention.

What the theorem does not claim is equally precise. It does not re-prove the full Hessian decomposition into hinge and stencil parts; that split is kernel-checked only at the recorded witness. The proof inherits standard axioms plus two compiler trust assumptions from the angle-sum chain, and it contains no sorry or unused hypotheses. The practical consequence: a lattice gravity program can trust that its constant block has no spurious zero modes, and can focus on the physical fluctuations that remain.

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 re-proved in Lean. The result does not establish that all spurious modes vanish, only the constant block. The theorem does not depend on the transverse-traceless condition, which is a special case.

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

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:

MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND