Gravity Track1 Bcstructural

A structural bridge to Einstein

General relativity describes gravity as the curvature of spacetime, captured by the Einstein-Hilbert action. Computing with that continuous action is hard, so physicists often approximate spacetime with a lattice, a grid of flat pieces. The Regge action is the standard discrete stand-in: it sums angles around the lattice hinges. The classical question is whether the lattice sum converges to the continuous action as the grid spacing shrinks to zero. This library addresses that question, but only in a structural, conditional sense.

The library, called Track 1.B/1.C Structural, is part of a machine-checked collection of formal theorems. It proves two conditional statements. First, under a named geometric-residual hypothesis, the absolute difference between the Regge action and the Einstein-Hilbert action is bounded by a constant times the lattice spacing, so the difference tends to zero as spacing shrinks. Second, under the Schläfli identity at every vertex, a discrete version of the contracted second Bianchi identity holds at every vertex. The Bianchi identity is a key consistency condition in general relativity, expressing a kind of conservation of curvature.

In Recognition Science, the framework models physical structure from a discrete ledger of recognition events. Within that framework, this library provides a structural witness: it shows that the two pieces, Regge-EH convergence and discrete Bianchi, are consistent with the framework's master theorem hypothesis. The witness uses canonical examples, a flat substrate where both actions are zero, and a trivial triangulation. These examples are non-vacuous, meaning they genuinely inhabit the hypothesis structure, but they are not the physically interesting curved cases.

The unconditional versions remain open. Proving the geometric residual bound for a real curved triangulation, and proving the Schläfli identity for a physical lattice, are future work. What the library establishes is a clean formal scaffold: the kinematic content is in place, the hypotheses are named, and the logical structure is verified. It is a bridge with its span drawn but its piers not yet sunk.

/-- 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)

/-- **Inhabitant for the master theorem hypothesis input**
`RegEHContinuumAndBianchi` (from `Gravity.MasterTheorem`, Session 97).
This witness uses the structural Props for Regge-EH convergence and
discrete Bianchi, with canonical witnesses providing non-vacuous
inhabitation. -/
def regEHContinuumAndBianchiWitness :
    Gravity.MasterTheorem.RegEHContinuumAndBianchi where
  regge_to_einstein_hilbert_continuum := regge_eh_continuum_structural_prop
  regge_holds := regge_eh_continuum_canonical_witness
  discrete_bianchi_contracted := discrete_bianchi_structural_prop
  bianchi_holds := discrete_bianchi_canonical_witness

What this page does not claim

The library does not prove unconditional convergence for a physical curved triangulation. The library does not prove the Schläfli identity for a specific physical lattice. The library does not establish that the framework's discrete ledger reproduces all of 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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