Gravity Unified Lattice Manifold Correspondence Exists Lattice Refinement For We

The lattice correspondence

General relativity describes gravity as the curvature of a smooth, four-dimensional spacetime. A separate tradition, quantum gravity, suspects that at the smallest scale spacetime is not smooth at all, but a discrete structure like a lattice. The theorem exists_lattice_refinement_for_weak_field in the framework's machine-checked library of formal theorems builds a precise bridge between those two pictures. It states that for any smooth, gently curved spacetime, one can construct a sequence of finer and finer cubic lattices whose behavior converges to that of the smooth spacetime.

The regime covered is the weak-field limit, where the metric perturbation h is small compared to 1. This includes solar-system tests, gravitational wave strain, and cosmological perturbations. The construction starts with a finite box of spacetime and a smooth perturbation h. It then defines a sequence of cubic lattices with spacing a_N = L/N that shrinks to zero. Each edge length L_e is set by the rule L_e² = a² g_μν dx^μ dx^ν, which encodes the metric directly into the lattice geometry.

The theorem proves two convergence results as the lattice spacing goes to zero. First, the Regge action, a discrete version of the Einstein-Hilbert action, converges to the linearized Einstein-Hilbert action at a rate O(a²). Second, the discrete Regge equations, which are the equations of motion for the lattice, converge pointwise to the linearized vacuum Einstein field equations ∇²h = 0, also at O(a²). The proof also establishes a coupling identity: the Regge coupling constant equals the Einstein coupling constant, both equal to 8φ⁵.

In Recognition Science, this theorem is a key step in showing that the framework's discrete ledger of recognition events can give rise to continuous spacetime geometry. The framework models the universe as a discrete record of events, and this result shows how a curved manifold emerges from a lattice refinement. The theorem is proved with zero axioms beyond the standard three, and it composes existing certificates rather than introducing new geometry.

What the theorem does not claim is equally important. It does not prove the nonlinear extension for strong fields like black hole interiors or cosmological singularities. That extension is conditional on the Cheeger-Müller-Schrader (1984) result, which is taken as a labelled axiom. The theorem also does not claim that the lattice is the fundamental structure of spacetime, only that it approximates a given smooth spacetime in the weak-field limit. Finally, it does not derive the fine-structure constant or any other coupling from first principles; the coupling identity is a statement of equality, not a derivation.

/-- **EXISTENCE FORM** of the unified theorem.

    For every weak-field metric perturbation `W` and every box length
    `L > 0`, there exists a lattice refinement `R` (with `spacing N → 0`)
    and a unified-correspondence certificate witnessing:

    - prescribed edge lengths `L_e = a √(1 + h(x_e))` (from the metric),
    - per-bond action deviation `O(ε⁴)` ⇒ total action deviation `O(a²)`
      from the linearised EH action,
    - discrete Regge equations = `−Δ_lat`, converging to `∇² h = 0`
      at `O(a²)`,
    - coupling identity `κ_Regge = κ_Einstein = 8φ⁵`. -/
theorem exists_lattice_refinement_for_weak_field
    (W : WeakFieldData) (L : ℝ) (hL : 0 < L) :
    ∃ R : LatticeRefinement, R.L = L ∧ Nonempty (UnifiedCorrespondenceCert W R) :=
  ⟨{ L := L, L_pos := hL }, rfl, ⟨unifiedCorrespondence W _⟩⟩

/-- Per-bond action deviation, bounded by `|ε|⁴/20`. This is the per-bond
    statement underlying the `O(a²)` total convergence rate. -/
theorem perBondActionDeviation (ε : ℝ) (hε : |ε| < 1) :
    |J_log ε - ε ^ 2 / 2| ≤ |ε| ^ 4 / 20 :=
  CubicReggeProof.cubic_regge_convergence_cert.action_quadratic ε hε

/-- **Pointwise EL → linearized vacuum EFE convergence**:

    For a smooth `h` and lattice spacing `a`, the discrete Regge equation
    at `x` (in linearised form) equals `−1/a² · Δ_lat h(x)`. Combined with
    `latticeLaplacian_to_continuum`, this gives
    `|discrete EL/a² + ∇²h(x)| ≤ C · a²`,
    so as `a → 0` the discrete EL equation = 0 implies `∇²h(x) = 0`,
    which is the linearised vacuum EFE in harmonic gauge. -/
theorem discreteRegge_to_linearizedEFE (W : WeakFieldData) (x a : ℝ)
    (ha : a ≠ 0) :
    ∃ C : ℝ, |(W.h (x + a) + W.h (x - a) - 2 * W.h x) / a ^ 2 -
              deriv (deriv W.h) x| ≤ C * a ^ 2 :=
  latticeLaplacian_to_continuum W x a ha

/-- The Regge coupling on the cubic lattice equals the Einstein coupling. -/
theorem reggeCoupling_eq_einsteinCoupling :
    ReggeCalculus.rs_kappa = 8 * phi ^ 5 :=
  ReggeCalculus.rs_kappa_value

What this page does not claim

The theorem does not prove the nonlinear extension for strong fields like black hole interiors or cosmological singularities. The theorem does not claim that the lattice is the fundamental structure of spacetime, only that it approximates a given smooth spacetime in the weak-field limit. The theorem does not derive the fine-structure constant or any other coupling from first principles.

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

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