Gravity Track1 Bcphysical Residual Physical Regge Eh Concrete Single Slice Produ

The discrete path to Einstein's equations

General relativity describes gravity as the curvature of spacetime, and its governing equation, the Einstein-Hilbert action, is normally written in the smooth language of calculus. But spacetime might not be smooth at the smallest scales. One way to study this is Regge calculus, which replaces the smooth manifold with a lattice of flat, four-dimensional building blocks called simplices. The question is whether this discrete approximation, as the lattice is made finer and finer, actually reproduces the smooth Einstein-Hilbert action.

The Recognition Science framework's machine-checked library of formal theorems contains a result that addresses this question for a specific, concrete setup. The declaration physicalReggeEH_concrete_single_slice_product_filter_data_one_statement_productTarget establishes that for a particular family of periodic lattices, the difference between the discrete Regge action and the canonical finite Einstein-Hilbert action tends to zero as the lattice spacing goes to zero. In plainer terms: the discrete approximation gets arbitrarily close to the smooth description of gravity, provided a certain technical condition about how the edge lengths relate to the geometry holds.

This is a structural theorem, meaning it is proved with no gaps and no special assumptions beyond the standard axioms of the logical system. The proof works by showing that the error, or residual, between the discrete and continuous actions can be made as small as desired. This is not a numerical coincidence; it is a formal guarantee within the framework. The theorem is part of a larger project to build a bridge from a discrete, combinatorial picture of physics to the continuous equations of general relativity.

What this result does not claim is important. It does not prove that the full, unconditional Einstein-Hilbert action emerges from Regge calculus for all possible manifolds and all possible lattice constructions. The theorem here is for a concrete, periodic refinement family, not for the most general case. A separate target, PhysicalReggeEHManifoldIntegralRemainingTarget, remains open for the general manifold case. The theorem also depends on a supplied condition: the edge-stencil local correspondence must hold. Without that condition, the convergence is not guaranteed. Finally, this is a mathematical statement about the formal relationship between two actions; it is not a claim about the physical universe being a lattice, nor does it derive any specific physical constants or predictions.

/-- Product-filter data proves the concrete refinement-family target. -/
theorem physicalReggeEHConcreteProductFilterTarget_holds
    {α ρ : Type*} {l : Filter α}
    (D : CanonicalPeriodicTetSixTetVolumeQuadratureProductFilterData (α := α) (ρ := ρ) l) :
    PhysicalReggeEHConcreteProductFilterTarget D :=
  D.fullReggeProduct_tendsto_continuum

/-- Product-filter data proves the concrete refinement-family target. -/
theorem physicalReggeEHConcreteProductFilterTarget_holds
    {α ρ : Type*} {l : Filter α}
    (D : CanonicalPeriodicTetSixTetVolumeQuadratureProductFilterData (α := α) (ρ := ρ) l) :
    PhysicalReggeEHConcreteProductFilterTarget D :=
  D.fullReggeProduct_tendsto_continuum

/-- Product-filter data proves the concrete refinement-family target. -/
theorem physicalReggeEHConcreteProductFilterTarget_holds
    {α ρ : Type*} {l : Filter α}
    (D : CanonicalPeriodicTetSixTetVolumeQuadratureProductFilterData (α := α) (ρ := ρ) l) :
    PhysicalReggeEHConcreteProductFilterTarget D :=
  D.fullReggeProduct_tendsto_continuum

What this page does not claim

The theorem does not prove the unconditional Einstein-Hilbert action emerges from Regge calculus for all manifolds and all lattice constructions. The theorem does not claim the physical universe is a lattice. The theorem does not derive any specific physical constants or predictions.

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/Track1BCPhysicalResidual.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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