Gravity Discrete Vacuum Einstein Hinge Derivative Matches Incidence Simplified

The hinge derivative identity

In a discrete model of gravity, space is built from flat tetrahedra joined at their faces. The Regge action, a standard tool from 1961, measures curvature by the deficit angle at each hinge: the angle left over when the tetrahedra around an edge are laid flat. The vacuum Einstein equation in this setting says this deficit is zero at every hinge. The question is how the action changes when you nudge the vertex positions, and whether that change can be read off from a simple combinatorial rule.

The theorem hingeDerivative_matches_incidence_simplified answers that question for a specific normalization. It states: if the geometric derivative of the hinge measure equals a certain incidence coefficient times the square root of the edge length, then the derivative is exactly that incidence term. The incidence coefficient is plain: for a given edge, each of its two endpoint vertices contributes one half to the derivative, and all other vertices contribute zero. The theorem is a formal cleanup, removing a zero term and simplifying the equality to its essential form.

The statement is established in the framework's machine-checked library of formal theorems. It is not a physical law by itself. It is a compatibility condition, isolating a normalization choice: the geometric derivative carries a length factor, while the incidence formula is dimensionless. The theorem says these two agree once the length factor is handled. It does not show that the geometric derivative always equals the incidence term; that equality is an input, named HingeDerivativeMatchesIncidence.

The payoff is that the vacuum equation becomes a statement about incidence alone. If the compatibility holds, then the derivative of the action at the flat configuration is a sum over edges of the deficit times the incidence coefficient. The reverse direction, recovering zero deficit from vanishing derivatives, needs an additional rank condition on the triangulation, called IncidenceDeficitSeparating. That condition is real, not automatic. The theorem here is the forward, simplified direction.

theorem hingeDerivative_matches_incidence_simplified
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (hMatch : HingeDerivativeMatchesIncidence K hK) :
    ∀ η : VertexPotential K, ∀ e : Fin K.nE,
      Geometry.ReggeActionFirstVariation.hingeMeasureDirectionalDeriv K hK η e =
        Real.sqrt (hK.globalSqEdge e) * directionalLengthCoefficient K η e := by
  intro η e
  simpa using hMatch η e

/-- Vertex-edge incidence coefficient for the conformal derivative of a global
edge length: each endpoint contributes `1/2`. -/
def vertexEdgeIncidenceDerivative
    (K : Triangulation3D) (e : Fin K.nE) (i : Fin K.nV) : ℝ :=
  if (K.edgeVerts e).1 = i ∨ (K.edgeVerts e).2 = i then (1 / 2 : ℝ) else 0

/-- Compatibility between the geometric hinge derivative and the endpoint
incidence coefficient used in the vacuum equation.  This isolates the
normalization issue: the geometric derivative includes the flat edge length,
whereas the incidence formula is dimensionless. -/
def HingeDerivativeMatchesIncidence
    (K : Triangulation3D) (hK : IncidenceConsistent K) : Prop :=
  ∀ η : VertexPotential K, ∀ e : Fin K.nE,
    Geometry.ReggeActionFirstVariation.hingeMeasureDirectionalDeriv K hK η e =
      deficitAngle K (zeroPotential K) e * 0 +
        Real.sqrt (hK.globalSqEdge e) * directionalLengthCoefficient K η e

theorem zero_deficit_of_critical_of_variationFormula_of_separating
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (hFormula : ReggeFirstVariationFormula K hK)
    (hSep : IncidenceDeficitSeparating K)
    (hCrit : CriticalAtFlat K hK) :
    ZeroDeficitAtFlat K := by
  unfold CriticalAtFlat ReggeActionCriticalAtZero at hCrit
  unfold ZeroDeficitAtFlat
  have hdelta :
      (fun e : Fin K.nE => deficitAngle K (zeroPotential K) e) = 0 := by
    apply hSep
    intro η
    have happly := congrArg (fun L : VertexPotential K →L[ℝ] ℝ => L η) hCrit
    have hzero :
        (fderiv ℝ (reggeAction K hK) (zeroPotential K)) η = 0 := by
      simpa using happly
    have hformula := hFormula.variation_formula η
    rw [hformula] at hzero
    simpa using hzero
  intro e
  exact congrFun hdelta e

What this page does not claim

The theorem does not prove the Regge vacuum equation holds for any physical spacetime. The theorem does not show the geometric derivative always matches the incidence formula; that is a separate input. The theorem does not establish the reverse implication from critical action to zero deficit without the IncidenceDeficitSeparating condition.

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

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