Gravity Track1 Bcphysical Residual

The residual

In numerical relativity, Regge calculus approximates spacetime as a patchwork of flat tetrahedra. The Einstein-Hilbert action, the integral that governs gravity, is then a sum over these pieces. The difference between the discrete sum and the true continuum integral is called the residual. Track 1.B-PHY is a named upgrade in the Recognition Science framework's machine-checked library of formal theorems. It proves that, under specific conditions, this residual tends to zero as the tetrahedral grid is refined.

The central result is a convergence theorem. The framework establishes that the normalized, fully nonlinear Regge finite aggregates converge to the canonical finite Einstein-Hilbert/Dirichlet action. The proof relies on a local correspondence between the discrete edge stencils and the continuum derivatives. When this correspondence holds, the difference between the discrete and continuum actions shrinks to zero along any refinement family. The theorem also shows the same local correspondence powers the bridge from finite sums to the continuum limit, provided a Riemann-sum identification is supplied.

This is a structural theorem, not a numerical one. It guarantees the limit exists and equals the right value, but it does not compute the rate of convergence. The proof is fully formalized with zero axioms beyond the standard ones, meaning the logic is machine-checked. The framework also packages a Bianchi identity result, the discrete analogue of the contracted Bianchi identities that ensure energy-momentum conservation in general relativity.

What remains open is the unconditional manifold-level statement. The current theorem requires the local correspondence and a Riemann-sum identification as inputs. The target is to prove the limit on a concrete periodic Freudenthal refinement family without these extra assumptions. This is a stated goal, not a proved result. The framework names this as the remaining target for the full Einstein-Hilbert theorem on manifolds.

In plain terms, this framework is a rigorous checkpoint. It shows that the discrete approximation to gravity's action is not just plausible but provably correct in the limit, under stated conditions. It closes a specific gap in the derivation of general relativity. The reader can now see that the path from discrete Regge calculus to the continuum Einstein-Hilbert action is formally verified, with the remaining work clearly marked.

/-- 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

theorem physicalReggeEHContinuumMasterProp_holds
    {α ρ : Type*} {l : Filter α}
    (D : CanonicalPeriodicTetSixTetVolumeQuadratureProductFilterData (α := α) (ρ := ρ) l) :
    physicalReggeEHContinuumMasterProp D :=
  physicalReggeEHConcreteProductFilterTarget_holds D

/-- The concrete slice target holds for every slice of a six-tet volume
quadrature refinement family. -/
theorem physicalReggeEHConcreteRefinementFamilySliceTarget_holds
    {α ρ : Type*} {l : Filter α}
    (F : CanonicalPeriodicTetSixTetVolumeQuadratureRefinementFamily l ρ) :
    PhysicalReggeEHConcreteRefinementFamilySliceTarget F :=
  fun r => physicalReggeEHConcreteSliceLimitWeightTarget_holds (F.slice r)

theorem physicalSchlafliBianchiMasterProp_holds (V B : Type) [Fintype B] :
    physicalSchlafliBianchiMasterProp V B :=
  Geometry.DiscreteBianchi.discrete_bianchi_contracted_from_schlafli

What this page does not claim

This framework does not prove the unconditional manifold Einstein-Hilbert theorem without the stated local correspondence and Riemann-sum identification. This framework does not compute a numerical rate of convergence for the residual. This framework does not claim the discrete Bianchi identity is equivalent to the full set of continuum Bianchi identities.

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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Derived articles

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