Gravity Master Theorem Unconditional Concrete Physical Reg Ehcontinuum Prop Hold

The continuum limit claim

In physics, the Einstein-Hilbert action (the integral that defines general relativity) is written in the language of smooth continuous spacetime. But many approaches to quantum gravity start from discrete building blocks, small chunks of spacetime, and try to show that when the chunks shrink, the smooth theory re-emerges. That recovery is called a continuum limit. The declaration concretePhysicalRegEHContinuumProp_holds (a theorem in the framework's machine-checked library of formal theorems) asserts that one such recovery works: on a specific periodic six-tet cubic torus, the discrete Regge action converges to the continuum Einstein-Hilbert integral.

Regge calculus, named after Tullio Regge in 1961, replaces smooth spacetime with flat pieces glued along edges, like a paper model. The Regge action is the sum over all hinges of the area times the deficit angle, the amount by which the flat pieces fail to meet flatly. The theorem states that for any product-filter refinement data on the canonical periodic six-tet cubic torus, the normalized full nonlinear Regge aggregate converges to the supplied continuum Einstein-Hilbert integral on the product filter. In plain language: as the lattice gets finer in a controlled way, the discrete sum approaches the smooth integral it is meant to approximate.

The theorem is unconditional, meaning it does not take the convergence as an assumption. It builds witnesses for the five inputs that an older conditional master theorem accepted as arguments, supplying a canonical zero-argument route through it. The theorem also installs a companion result: for any vertex and bond types, every Schläfli-satisfying Regge datum obeys the contracted discrete Bianchi identity at every vertex, a discrete analogue of a key identity in differential geometry.

In Recognition Science, this result is one piece of a larger master theorem that also covers amplitude linearity, page curve derived properties, and stochastic gravitational wave signals distinct from inflation. The closure status theorem explicitly records what remains open: full physical closure is false, with d2 quadrature, general triangulation, tensor TT recovery, Lorentzian causal triangulations, boundary GHY, and echo mechanism all marked as open targets. The convergence is proved only on the canonical periodic six-tet cubic torus, not on arbitrary triangulations.

theorem concretePhysicalRegEHContinuumProp_holds :
    concretePhysicalRegEHContinuumProp :=
  fun D => Track1BCPhysicalResidual.physicalReggeEHConcreteProductFilterTarget_holds D

theorem concretePhysicalBianchiProp_holds :
    concretePhysicalBianchiProp := by
  intro V B _
  exact Track1BCPhysicalResidual.physicalSchlafliBianchiMasterProp_holds V B

/-- The current zero-argument master assembly must not be cited as full
physical closure. -/
theorem closureStatus_unconditional_not_full_physical_closure :
    closureStatus_unconditional.theorem_built_witnesses_installed = true ∧
    closureStatus_unconditional.full_physical_closure = false :=
  ⟨rfl, rfl⟩

What this page does not claim

The theorem does not prove convergence on arbitrary triangulations, only on the canonical periodic six-tet cubic torus. The theorem does not establish full physical closure of the gravity master theorem; several targets remain open. The theorem does not claim that the discrete model reproduces all predictions of general relativity, only the continuum limit of the action.

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/MasterTheoremUnconditional.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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