Gravity Restricted Incidence Recovery Zero Deficit Of Critical Of Restricted Var

Restricted recovery

In the framework's account of discrete gravity, a three-dimensional triangulation carries edge lengths, and each edge has a deficit angle: the amount by which the surrounding tetrahedra fail to close flat around that edge. The unrestricted recovery predicate asks vertex probes to recover an arbitrary edge-deficit vector. That is too strong for bulk 3D lattices, where there are generally more edge variables than vertex probes.

The theorem zero_deficit_of_critical_of_restrictedVariationFormula states the mathematically valid version: recovery and separation on an explicitly declared geometric deficit subspace. The declaration proves that if a triangulation is incidence-consistent, if the actual deficit vector lies in a declared subspace, if the Regge first variation formula holds, if that subspace is separating, and if the configuration is critical at flat, then the deficit at flat is zero. The key restriction is that the deficit vector must belong to the declared subspace; the theorem does not apply to arbitrary deficit vectors.

In Recognition Science, the framework models the vacuum Einstein equations as a discrete ledger. The declaration is a reverse vacuum implication restricted to a declared geometric deficit subspace. The actual deficit vector must be shown to belong to the subspace. The theorem proves that under these conditions, a critical point of the Regge action at a flat configuration has zero deficit, which is the discrete analog of the vacuum Einstein equations holding.

The theorem does not claim that the deficit vector always lies in the declared subspace. That is a separate condition, stated as a hypothesis. The theorem also does not prove that a flat configuration exists; it assumes one. It does not claim that the unrestricted recovery problem is solvable, only the restricted version on the declared subspace. The theorem is a conditional statement: if the hypotheses hold, then the conclusion follows.

What the theorem changes is the scope of what can be recovered. In a bulk 3D lattice, the unrestricted problem is too strong. The restricted version, on a declared geometric deficit subspace, is the mathematically valid one. The theorem shows that on that subspace, recovery and separation are equivalent, and that a critical point at flat has zero deficit. This is a precise, machine-checked statement about when the discrete vacuum equations hold.

/-- Reverse vacuum implication restricted to a declared geometric deficit
subspace.  The actual deficit vector must be shown to belong to the subspace. -/
theorem zero_deficit_of_critical_of_restrictedVariationFormula
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (S : DeficitSubspace K)
    (hDeficitInS : S (fun e : Fin K.nE => deficitAngle K (zeroPotential K) e))
    (hFormula : ReggeFirstVariationFormula K hK)
    (hSep : RestrictedIncidenceDeficitSeparating K S)
    (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
    · exact hDeficitInS
    · 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

/-- Reverse vacuum implication restricted to a declared geometric deficit
subspace.  The actual deficit vector must be shown to belong to the subspace. -/
theorem zero_deficit_of_critical_of_restrictedVariationFormula
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (S : DeficitSubspace K)
    (hDeficitInS : S (fun e : Fin K.nE => deficitAngle K (zeroPotential K) e))
    (hFormula : ReggeFirstVariationFormula K hK)
    (hSep : RestrictedIncidenceDeficitSeparating K S)
    (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
    · exact hDeficitInS
    · 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

/-- Reverse vacuum implication restricted to a declared geometric deficit
subspace.  The actual deficit vector must be shown to belong to the subspace. -/
theorem zero_deficit_of_critical_of_restrictedVariationFormula
    (K : Triangulation3D) (hK : IncidenceConsistent K)
    (S : DeficitSubspace K)
    (hDeficitInS : S (fun e : Fin K.nE => deficitAngle K (zeroPotential K) e))
    (hFormula : ReggeFirstVariationFormula K hK)
    (hSep : RestrictedIncidenceDeficitSeparating K S)
    (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
    · exact hDeficitInS
    · 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 that the deficit vector always lies in the declared subspace; that is a hypothesis. The theorem does not prove that a flat configuration exists; it assumes one. The theorem does not claim that the unrestricted recovery problem is solvable, only the restricted version on the declared subspace.

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

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