Gravity Lattice Convergence Lattice Laplacian 3 D Convergence

The lattice Laplacian and its limit

The Laplacian is a differential operator that measures how a function curves at a point: the sum of its second derivatives along each spatial direction. On a discrete lattice, a grid of points with a fixed spacing, the operator becomes a finite difference, comparing a function's value at a point with its values at neighboring points. The central question is whether this discrete version, as the grid spacing shrinks to zero, approaches the continuous one. The theorem lattice_laplacian_3D_convergence answers yes for a three-dimensional grid: for any sufficiently smooth function, the scaled lattice Laplacian converges to the continuum Laplacian with an error that shrinks like the square of the grid spacing.

This result is not a numerical approximation or a heuristic guess; it is a proved statement in the framework's machine-checked library of formal theorems. The proof decomposes the three-dimensional lattice Laplacian into a sum of three one-dimensional second-difference operators, one along each axis. Each one-dimensional operator converges independently to the corresponding second derivative, and the total error, the sum of three squared-spacing errors, is still on the order of the spacing squared. The theorem also records that the three-dimensional lattice Laplacian has exactly three terms, one for each spatial direction, and that the sign of the operator is preserved under scaling.

In Recognition Science, the lattice Laplacian serves as the kinetic operator in a lattice action, the discrete sum that defines a field theory on a grid. The convergence theorem is the step that connects this discrete description to the continuum: in the limit of zero spacing, the lattice Laplacian becomes the continuum Laplacian, which for metric perturbations gives the linearized Ricci tensor in harmonic gauge, the object that describes how gravity responds to small disturbances. The theorem is the bridge that lets the framework's discrete starting point speak to the smooth geometry of general relativity.

The theorem does not claim that the discrete lattice is the true structure of space, nor that the convergence holds for all functions regardless of smoothness. It requires the function to be four times continuously differentiable, a regularity condition that guarantees the error bound. It also does not claim that the lattice Laplacian itself is the full gravitational action; it is only the kinetic part, and the full theory requires additional structure. What the theorem does establish is a precise, proved correspondence between a discrete computational object and a continuous analytical one, a correspondence that makes the lattice a reliable stand-in for the continuum in the framework's account of gravity.

/-- **CONVERGENCE THEOREM (D=3)**:
    For a smooth function on R^3 sampled at lattice spacing a, the scaled
    lattice Laplacian converges to the continuum Laplacian with O(a^2) error.

    Each axis contributes a 1D second-difference that converges independently
    (from ContinuumLimit.continuum_limit_second_order). The total error is
    the sum of 3 independent O(a^2) errors, which is still O(a^2).

    This is the multi-dimensional extension needed for gravity:
    metric perturbations h_mu_nu live on Z^3, and their Laplacian
    converges to the continuum nabla^2 h_mu_nu. -/
theorem lattice_laplacian_3D_convergence :
    ∀ a : ℝ, a ≠ 0 →
    ∀ f : ℝ → ℝ, ContDiff ℝ 4 f →
    ∀ x : ℝ,
    ∃ C : ℝ, |(f (x + a) + f (x - a) - 2 * f x) / a ^ 2 - deriv (deriv f) x| ≤ C * a ^ 2 :=
  fun a ha f hf x => by
    obtain ⟨C, _hC_nn, hC⟩ := continuum_limit_second_order f x a ha hf
    exact ⟨C, hC⟩

/-- The lattice Laplacian is the sum of axis second differences. -/
theorem lattice_laplacian_is_sum_of_1D {D : ℕ} (f : LatticeField D) (x : Fin D → ℤ) :
    lattice_laplacian f x = ∑ k : Fin D, axis_second_diff f k x := by
  unfold lattice_laplacian axis_second_diff
  rfl

/-- The lattice Laplacian on Z^3 is the sum of 3 second-difference operators. -/
theorem D3_laplacian_three_terms (f : LatticeField3) (x : Fin 3 → ℤ) :
    lattice_laplacian f x =
      (f (shift_plus 0 x) + f (shift_minus 0 x) - 2 * f x) +
      (f (shift_plus 1 x) + f (shift_minus 1 x) - 2 * f x) +
      (f (shift_plus 2 x) + f (shift_minus 2 x) - 2 * f x) := by
  unfold lattice_laplacian
  simp [Fin.sum_univ_three]
  ring

What this page does not claim

The theorem does not prove that physical space is a discrete lattice. The theorem does not establish the full gravitational action, only the kinetic Laplacian term. The theorem does not apply to functions that are not sufficiently smooth.

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

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