Gravity Track1 Bcstructural Regge Eh Continuum Canonical Witness

The structural witness

General relativity describes gravity as the curvature of spacetime, and its standard equation, the Einstein-Hilbert action, is written in the language of continuous geometry. But many approaches to quantum gravity start from a discrete picture, where spacetime is a lattice or a triangulated mesh. The Regge action is the discrete analogue: it assigns a value to such a mesh based on its edge lengths and angles. A central question is whether the discrete picture converges to the continuous one as the mesh spacing shrinks to zero. In the Recognition Science framework, this question is formalized as the Regge-EH continuum convergence property.

The declaration regge_eh_continuum_canonical_witness is a theorem in the framework's machine-checked library of formal theorems. It establishes this convergence property for a specific, deliberately simple case: the flat substrate. On a flat substrate, both the abstract Regge action and the abstract Einstein-Hilbert action are defined to be zero for any lattice spacing. The theorem states that for any sequence of spacings tending to zero, the absolute difference between these two actions tends to zero. Since both are always zero, the difference is identically zero, and the convergence is trivial. This is a structural witness: it shows that the framework's hypothesis structure is not empty, that there exists at least one configuration satisfying the convergence condition.

The same module also contains a companion witness for the discrete Bianchi identity, a conservation law for curvature. This witness uses a trivial two-point triangulation that satisfies the Schläfli identity, a geometric relation between volume changes and angle changes. Together, these two witnesses inhabit the master theorem hypothesis structure, meaning they satisfy the formal requirements for the larger theorem that discrete gravity converges to continuous gravity. The combined statement, track1BC_one_statement, asserts that both structural properties hold and that the master hypothesis is inhabited.

What this declaration does not claim is equally important. It does not establish the unconditional convergence of Regge to Einstein-Hilbert for any physical triangulation. That would require an actual geometric residual estimate, a bound on how much the discrete action differs from the continuous one as a function of spacing, for a realistic mesh. The flat substrate bypasses this entirely because the difference is exactly zero. Similarly, the Bianchi witness does not establish the Schläfli identity for a physical triangulation; it only shows that a trivial example exists. The unconditional versions of both tracks remain future work, requiring multi-session geometric proofs in the simplicial-geometry tooling.

In plain terms, the declaration establishes that the framework's formal structure for the discrete-to-continuous bridge is coherent and has at least one valid instance. It is a proof of consistency, not a proof of physical content. The real work, establishing the convergence and the Bianchi identity for a realistic spacetime mesh, is still open. This witness is a checkpoint that the machinery works, not a demonstration that gravity itself behaves as expected.

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

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

/-- Abstract Regge action as a function of lattice spacing. The flat
substrate has Regge action zero for any spacing (the spacing argument
is intentionally unused in this canonical-witness form). -/
def abstract_regge_action (_spacing : ℝ) : ℝ := 0

/-- Abstract Einstein-Hilbert action. The flat substrate has zero
EH action (the spacing argument is intentionally unused). -/
def abstract_eh_action (_spacing : ℝ) : ℝ := 0

/-- 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

What this page does not claim

This declaration does not establish that discrete gravity converges to continuous gravity for any realistic spacetime mesh. This declaration does not establish the Schläfli identity for a physical triangulation; it only shows a trivial example exists. This declaration does not demonstrate that the Einstein-Hilbert action itself is derived from recognition, only that a formal bridge exists for a flat case.

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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