Gravity Lattice Convergence Jcost Neighbor Approximation 3 D

The neighbor approximation

The Laplacian is the operator that measures how a function curves at a point, the sum of its second derivatives along each direction. In three dimensions, it is the natural extension of the one-dimensional second difference, and it appears throughout physics as the kinetic term in wave and field equations. On a discrete lattice, the Laplacian is built from differences between a point and its nearest neighbors along each axis. The Recognition Science declaration jcost_neighbor_approximation_3D establishes a precise bound for how a particular cost formula, derived from the framework's recognition cost, approximates this lattice Laplacian on the three-dimensional integer lattice Z³.

The theorem states that for a field on Z³ whose values at neighboring points differ from the central value by less than one, the absolute difference between the neighbor cost and the sum of squared differences along each axis is bounded by a sum of fourth powers of those differences, divided by 20. In plain terms, the cost formula matches the lattice Laplacian to fourth order in the small differences. The error term is a sum over the three axes of terms like |f(x+a) - f(x)|⁴/20, which is much smaller than the leading quadratic terms when the differences are small. This is a formal statement, checked by a machine, that the cost function carries the same local structure as the Laplacian in three dimensions.

The result is a specialization of a more general one-dimensional statement, and it relies on the fact that the three-dimensional lattice Laplacian decomposes as a sum of three independent one-dimensional second-difference operators along each axis. The declaration also builds on a separate convergence theorem showing that the scaled lattice Laplacian converges to the continuum Laplacian with error of order a² as the lattice spacing a goes to zero. Together these pieces form a bridge from the discrete cost to the continuum operator that appears in linearized gravity, where metric perturbations satisfy an equation involving the Laplacian in harmonic gauge.

In Recognition Science, this is one step in a larger program that derives physical structure from a single cost function. The theorem does not claim that gravity itself is derived, nor that the full Einstein equations follow from this approximation. It establishes only a local, quantitative relationship between the cost and the Laplacian on a lattice, under a small-difference condition. The physical interpretation of this bridge, and its role in a complete theory of gravity, remains a target for further work.

/-- The J-cost neighbor sum on Z^3 approximates the lattice Laplacian
    to O(eps^4) accuracy. This is the D=3 specialization of
    ContinuumLimit.jcost_gives_laplacian_structure. -/
theorem jcost_neighbor_approximation_3D (f : LatticeField3) (x : Fin 3 → ℤ)
    (h_small : ∀ k : Fin 3,
      |f (shift_plus k x) - f x| < 1 ∧
      |f (shift_minus k x) - f x| < 1) :
    |neighbor_cost f x -
      ∑ k : Fin 3, ((f (shift_plus k x) - f x) ^ 2 / 2 +
                     (f (shift_minus k x) - f x) ^ 2 / 2)| ≤
    ∑ k : Fin 3, (|f (shift_plus k x) - f x| ^ 4 / 20 +
                   |f (shift_minus k x) - f x| ^ 4 / 20) :=
  jcost_gives_laplacian_structure f x h_small

/-- 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

/-- **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⟩

What this page does not claim

The theorem does not derive the Einstein field equations or any full theory of gravity. The theorem does not apply to fields with differences larger than one between neighboring points. The result does not establish the physical recognition-to-linking bridge that would connect the cost to spacetime geometry.

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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