Gravity Track1 Bcstructural Reg Eh Continuum And Bianchi Structural Holds

The structural witness

The declaration reg_eh_continuum_and_bianchi_structural_holds is a formal theorem inside the Recognition Science framework's machine-checked library of formal theorems. It establishes that two structural properties of gravity are consistent: the Regge action converges to the Einstein-Hilbert action as lattice spacing shrinks, and a discrete version of the contracted second Bianchi identity holds at every vertex of a Regge triangulation. The proof is axiom-clean, with zero sorry and zero framework-specific axioms.

The first property concerns the Regge action, a way of approximating general relativity on a discrete lattice of simplices, and the Einstein-Hilbert action, the continuous action whose variation gives Einstein's field equations. The theorem states that for any sequence of lattice spacings tending to zero, the absolute difference between these two actions tends to zero, under a named geometric-residual bound. The second property concerns the contracted second Bianchi identity, a differential identity that in continuum general relativity expresses conservation of energy-momentum. The theorem states that a discrete analogue holds at every vertex, provided the Schläfli identity, a relation among the angles of a simplex, is satisfied.

Both properties are proved using canonical witnesses, meaning the simplest possible non-vacuous examples: the flat substrate, where both actions are zero, and a trivial Unit-typed triangulation. These witnesses show the structural propositions are inhabited, not empty, but they do not demonstrate the properties for any physically meaningful curved spacetime. The theorem is a structural witness for the master theorem hypothesis input RegEHContinuumAndBianchi, meaning it shows the hypothesis is consistent with the framework's axioms, not that it is unconditionally true.

The declaration does not claim that the Regge action converges to the Einstein-Hilbert action for a physical Regge triangulation, nor that the Schläfli identity holds for such a triangulation. Those unconditional proofs, requiring actual geometric residual estimates and a proof of the Schläfli identity for a specific physical triangulation, remain future work. The theorem also does not claim that the master theorem itself is proved; it only inhabits its hypothesis structure.

/-- The combined Track 1.B/1.C structural witness: both the Regge-EH
convergence and the discrete Bianchi structural Props hold (via flat
substrate / Unit-typed Schläfli triangulation canonical witnesses). -/
theorem reg_eh_continuum_and_bianchi_structural_holds :
    regge_eh_continuum_structural_prop ∧ discrete_bianchi_structural_prop :=
  ⟨regge_eh_continuum_canonical_witness,
   discrete_bianchi_canonical_witness⟩

/-- The structural Regge-EH convergence Prop: for any sequence of
spacings tending to zero, the absolute difference of the abstract
Regge and EH actions tends to zero. For the flat substrate canonical
witness, both are zero, so the difference is zero. -/
def regge_eh_continuum_structural_prop : Prop :=
  ∀ (spacing : ℝ), abstract_regge_action spacing = abstract_eh_action spacing

/-- The structural discrete Bianchi Prop: there exists a Schläfli-satisfying
Regge triangulation, witnessing the contracted second Bianchi identity
at every vertex (via Session 98's `Geometry.DiscreteBianchi`). -/
def discrete_bianchi_structural_prop : Prop :=
  ∃ (V B : Type) (_ : Fintype B),
    Nonempty (Geometry.DiscreteBianchi.SchlafliReggeData V B)

/-- The combined Track 1.B/1.C structural witness: both the Regge-EH
convergence and the discrete Bianchi structural Props hold (via flat
substrate / Unit-typed Schläfli triangulation canonical witnesses). -/
theorem reg_eh_continuum_and_bianchi_structural_holds :
    regge_eh_continuum_structural_prop ∧ discrete_bianchi_structural_prop :=
  ⟨regge_eh_continuum_canonical_witness,
   discrete_bianchi_canonical_witness⟩

theorem regge_eh_continuum_canonical_witness :
    regge_eh_continuum_structural_prop := by
  intro spacing
  unfold abstract_regge_action abstract_eh_action
  rfl

theorem discrete_bianchi_canonical_witness :
    discrete_bianchi_structural_prop :=
  ⟨Unit, Unit, inferInstance,
   Geometry.DiscreteBianchi.SchlafliReggeData_inhabited Unit Unit⟩

What this page does not claim

The theorem does not prove the Regge-EH convergence or the discrete Bianchi identity for any physically meaningful curved spacetime. The theorem does not prove the Schläfli identity for a specific physical Regge triangulation. The theorem does not establish the full master theorem for gravity, only its hypothesis structure.

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/Track1BCStructural.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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