Gravity Unified Lattice Manifold Correspondence Discrete Regge To Linearized Efe

The lattice-to-continuum bridge

General relativity describes gravity as the curvature of a smooth four-dimensional spacetime. The Einstein field equations (EFE) relate that curvature to the distribution of matter and energy. For weak gravitational fields, where the metric is nearly flat, these equations simplify to a linear form. The declaration discreteRegge_to_linearizedEFE in the Recognition Science library establishes a precise bridge: a discrete lattice of edge lengths, when refined to zero spacing, converges to the solutions of those linearized equations.

The setup is concrete. Start with a smooth metric perturbation h on a finite box, so the full metric is g = η + h, with |h| much smaller than 1. Build a sequence of cubic lattices with spacing a_N = L/N, and assign each edge a length L_e = a_N · √(1 + h(x_e)), exactly the rule that the squared edge length equals a² times the metric component along that edge. The theorem proves two convergences as N → ∞. First, the Regge action, a discrete sum over lattice simplices, converges to the linearized Einstein-Hilbert action at a rate O(a_N²). Second, the discrete Regge equations, δS_Regge/δL_e = 0, converge pointwise to the linearized vacuum EFE, which in this setting is the wave equation ∇²h(x) = 0. The rate is again O(a_N²), uniformly in N.

The proof is fully machine-checked, with zero gaps and zero added axioms. It composes existing certificates: the edge-length rule from Regge calculus, the action convergence from the cubic Regge proof, the lattice Laplacian convergence to the continuum Laplacian, and the coupling identity that the Regge coupling equals the Einstein coupling, both equal to 8φ⁵. The theorem itself is a short statement: for any weak-field data and any nonzero lattice spacing a, there exists a constant C such that the absolute difference between the discrete second difference and the true second derivative is bounded by C·a². This is the core estimate that drives the convergence.

In Recognition Science, this result is part of a larger program: the framework models physical spacetime as emerging from a discrete ledger of recognition events, and this theorem shows that the discrete description reproduces the continuum physics of weak-field gravity. The linearized regime covers solar-system tests, gravitational wave strain, and cosmological perturbations. The nonlinear extension, for strong fields like black hole interiors, is provided separately as a conditional certificate, depending on the Cheeger–Müller–Schrader convergence result from 1984, which the library takes as a labelled axiom.

What the theorem does not claim is equally important. It does not prove that the nonlinear Einstein equations emerge from the lattice without external assumptions; that remains conditional. It does not derive the value of the gravitational constant from first principles; the coupling identity is a theorem, but the numerical value 8φ⁵ is a consequence of the framework's definitions. And it does not claim that the lattice is physically real; it is a mathematical approximation scheme whose convergence is what is proved.

/-- **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

/-- **MAIN THEOREM**: the unified correspondence certificate holds for
    every weak-field input `W` and every lattice refinement `R`.

    Zero `sorry`, zero new axioms. Each field is supplied by an existing
    proved certificate; this theorem just bundles them. -/
theorem unifiedCorrespondence
    (W : WeakFieldData) (R : LatticeRefinement) :
    UnifiedCorrespondenceCert W R where
  refinement_dense := R.spacing_eventually_small
  edges_positive := fun N hN x_e =>
    prescribedEdgeLength_pos W (R.spacing_pos hN) x_e
  edge_length_rule := fun N x_e =>
    prescribedEdgeLength_sq W (R.spacing N) x_e
  action_per_bond := perBondActionDeviation
  el_is_lattice_laplacian := fun D f x =>
    discreteRegge_eq_neg_lattice_laplacian f x
  el_continuum_limit := fun x N hN =>
    latticeLaplacian_to_continuum W x (R.spacing N) (R.spacing_ne_zero hN)
  coupling_identity := by
    have h1 : ReggeCalculus.rs_kappa = 8 * phi ^ 5 :=
      ReggeCalculus.rs_kappa_value
    have h2 : Constants.kappa_einstein = 8 * phi ^ (5 : ℝ) :=
      Constants.kappa_einstein_eq
    have h3 : phi ^ (5 : ℝ) = phi ^ (5 : ℕ) := by
      rw [show (5 : ℝ) = ((5 : ℕ) : ℝ) by norm_num, Real.rpow_natCast]
    rw [h1, h2, h3]
  coupling_closed_form :=
    ⟨ReggeCalculus.rs_kappa_value, Constants.kappa_einstein_eq⟩
  coupling_positive :=
    ⟨ReggeCalculus.rs_kappa_pos, Constants.kappa_einstein_pos⟩
  flat_baseline := ReggeCalculus.cubic_lattice_flat
  metric_positive := W.one_plus_h_pos

/-- 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 nonlinear Einstein equations emerge from the lattice without external assumptions. The gravitational constant is derived from first principles independent of the framework's definitions. The lattice is physically real rather than a mathematical approximation scheme.

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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