Gravity Track1 Bcstructural Track1 Bc One Statement

The structural witness

In general relativity, the Einstein-Hilbert action is the master formula whose variation produces the field equations. In numerical and quantum approaches, physicists often discretize spacetime into a lattice, and the Regge action is the standard discrete stand-in. A central question is whether the discrete theory converges to the continuous one as the lattice spacing shrinks to zero. A second, independent question is whether the discrete geometry respects the contracted second Bianchi identity, a key consistency condition that in the continuum encodes local conservation of energy and momentum.

The declaration track1BC_one_statement in the framework's machine-checked library of formal theorems establishes a structural witness for both questions. It proves, as a formal theorem, that there exist abstract functions for the Regge and Einstein-Hilbert actions such that their difference tends to zero as spacing goes to zero, and that there exists a discrete triangulation satisfying the Schläfli identity, which implies the discrete Bianchi identity. The proof is non-vacuous: it supplies concrete witnesses, namely a flat substrate where both actions are zero and a trivial triangulation on a single point.

What this does not claim is the unconditional result. The theorem does not prove that any physical Regge triangulation converges to Einstein-Hilbert, nor that the Schläfli identity holds for a realistic curved spacetime. Those remain future work, requiring actual analytic estimates and geometric proofs. The declaration is a consistency check, showing the master theorem's hypothesis is inhabited, not a derivation of gravity from discrete principles.

/-- **TRACK 1.B/1.C STRUCTURAL ONE-STATEMENT**. The combined Track 1.B
(Regge-EH continuum convergence) and Track 1.C (contracted discrete
Bianchi via Schläfli identity) structural Props hold via canonical
witnesses (flat substrate for Regge-EH; Schläfli-satisfying triangulation
for Bianchi). The master theorem hypothesis input
`RegEHContinuumAndBianchi` is inhabited by `regEHContinuumAndBianchiWitness`.
The fully **unconditional** Track 1.B/1.C closure (the geometric
residual estimate + the Schläfli identity for a physical Regge
triangulation) remains future multi-session geometric work. -/
theorem track1BC_one_statement :
    (regge_eh_continuum_structural_prop) ∧
    (discrete_bianchi_structural_prop) ∧
    (Nonempty Gravity.MasterTheorem.RegEHContinuumAndBianchi) :=
  ⟨regge_eh_continuum_canonical_witness,
   discrete_bianchi_canonical_witness,
   ⟨regEHContinuumAndBianchiWitness⟩⟩

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 declaration does not prove unconditional convergence of the Regge action to the Einstein-Hilbert action for any physical lattice. It does not establish the Schläfli identity for a realistic curved triangulation. It does not derive gravity from discrete principles; it only shows a formal consistency.

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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