Gravity Analysis Regge4 Dexact Action Symbol Phased Deficit Dot Resolved T11 Zer

The zero-momentum simplification

Regge calculus is a discrete approach to general relativity where spacetime is approximated by a grid of flat chunks, and gravity is encoded in the angles where these chunks meet. In this framework, a key object is the second variation of the action, which determines how perturbations of the geometry behave. The declaration phasedDeficitDotResolvedT11_zeroMomentum is a theorem about one specific piece of this variation, the cross-term that mixes area changes with angle deficits.

The theorem states that when the perturbation has zero momentum, meaning it is constant across the grid, the complicated phase bookkeeping simplifies dramatically. Instead of summing over all the individual hinge contributions, the expression reduces to a sum over just six independent directions, each weighted by a simple geometric factor. This is a structural simplification, not a numerical approximation: it is an exact identity proved in the machine-checked library of formal theorems.

The simplification is important because it isolates the essential degrees of freedom. In the full expression, the phase factors encode how the perturbation varies from one grid point to another. At zero momentum, these variations vanish, and the remaining sum reveals the underlying geometric structure of the grid itself. This is a first step toward understanding more complex, momentum-dependent behavior.

The theorem does not claim that the full action is recovered, nor that this simplification holds for all perturbation types. It applies specifically to the t11 orbit of hinges, one of several distinct types. Other orbits, such as t12, have their own separate theorems. The result is also limited to the zero-momentum case; the behavior at non-zero momentum remains an open target for future work.

/-- At zero wave covector, cosine phases drop and each star-member
resolved deficit equals the ordinary classDot of the pushed assemble. -/
theorem phasedDeficitDotResolvedT11_zeroMomentum (H : Mat4) (x : Wave4)
    (p : Fin 24) :
    phasedDeficitDotResolvedT11 H (fun _ => (0 : ℝ)) x p =
      ∑ μ : Fin 6,
        classDot
          (pushforwardClass
            (ReggeHinge4DStarKernel.assembleStarMember μ) p) H := by
  unfold phasedDeficitDotResolvedT11
  refine Finset.sum_congr rfl fun μ _ => ?_
  simpa using
    phasedClassDot_zeroMomentum
      (pushforwardClass (ReggeHinge4DStarKernel.assembleStarMember μ) p) H
      (fun i =>
        x i + transportOffset p (cubeOffsetT11 (starMemberCubeT11 μ)) i)

/-- At zero wave covector, cosine phases drop and each star-member
resolved deficit equals the ordinary classDot of the pushed assemble. -/
theorem phasedDeficitDotResolvedT11_zeroMomentum (H : Mat4) (x : Wave4)
    (p : Fin 24) :
    phasedDeficitDotResolvedT11 H (fun _ => (0 : ℝ)) x p =
      ∑ μ : Fin 6,
        classDot
          (pushforwardClass
            (ReggeHinge4DStarKernel.assembleStarMember μ) p) H := by
  unfold phasedDeficitDotResolvedT11
  refine Finset.sum_congr rfl fun μ _ => ?_
  simpa using
    phasedClassDot_zeroMomentum
      (pushforwardClass (ReggeHinge4DStarKernel.assembleStarMember μ) p) H
      (fun i =>
        x i + transportOffset p (cubeOffsetT11 (starMemberCubeT11 μ)) i)

What this page does not claim

The theorem does not recover the full Regge action from the cross-term alone. The zero-momentum simplification does not apply to the t12 or other hinge orbits without separate theorems. The behavior of the cross-term at non-zero momentum is not established by this declaration.

Verify this page

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

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