Gravity Track1 Bcstructural With

What the structural witness shows

General relativity describes gravity as the curvature of spacetime, and its core equation, the Einstein-Hilbert action, is a continuous mathematical object. In numerical simulations, physicists often replace continuous spacetime with a discrete lattice of points, and the Regge action is the standard discrete approximation. The question is whether the discrete picture genuinely reproduces the continuous one as the lattice spacing shrinks to zero, and whether the discrete version preserves the deep geometric identities of the continuum theory.

The Recognition Science framework's machine-checked library of formal theorems addresses this question with a structural witness, a formal object that demonstrates the logical shape of the result under explicit hypotheses. The declaration ledger, a discrete record of events, establishes two structural propositions. First, for any sequence of lattice spacings tending to zero, the abstract Regge action equals the abstract Einstein-Hilbert action, provided a named geometric residual bound holds. Second, there exists a Regge triangulation satisfying the Schläfli identity, a geometric relation between the angles and volumes of a simplex, which witnesses the contracted second Bianchi identity at every vertex, a discrete analogue of a fundamental conservation law in differential geometry.

The proof is axiom-clean: it contains zero sorry placeholders and zero framework-specific axioms, relying only on the standard logical postulates. The canonical witnesses are the flat substrate, where both actions are zero and convergence is trivial, and the Unit-type triangulation, which satisfies the Schläfli identity by construction. These witnesses inhabit the master theorem hypothesis structure, meaning the framework has formally connected the discrete and continuum descriptions in a way that is logically valid but not yet physically substantive.

What the declaration does not claim is equally important. It does not prove the unconditional convergence of the Regge action to the Einstein-Hilbert action for a physical triangulation; that requires an actual geometric residual estimate, which remains future work. It does not prove the Schläfli identity for a specific physical Regge triangulation; the canonical witness uses a trivial Unit type, not a realistic spacetime lattice. The structural witness is a scaffold, not the building: it shows the logical shape of the argument, but the analytic and geometric content that would make it a physical theorem is still open.

/-- 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)

/-- **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⟩⟩

What this page does not claim

The unconditional convergence of the Regge action to the Einstein-Hilbert action for a physical triangulation is not proved. The Schläfli identity for a specific physical Regge triangulation is not proved. The structural witness does not establish that the discrete theory is dynamically equivalent to general relativity.

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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