IndisputableMonolith.Gravity.LatticeConvergence
Establishes that the discrete Laplacian of J-cost dynamics on the cubic lattice ℤ³ converges, at second order, to the continuum Laplacian as the lattice spacing vanishes. Gravity and continuum-limit arguments cite it when replacing abstract Regge axioms by an RS-native lattice calculation. The argument factors the 3D operator into three 1D second differences, scales by the mesh, and reads off the O(h²) remainder.
claimOn the cubic lattice $\mathbb{Z}^3$ with spacing $h>0$, the scaled discrete Laplacian of a smooth field $f$ equals $\Delta f + O(h^2)$ pointwise. In particular, neighbor differences of the J-cost reproduce the continuum second-derivative structure forced by $D=3$ spatial dimensions.
background
Recognition Science works from discrete J-cost dynamics on a lattice and must recover smooth field equations in the long-wavelength limit. The companion continuum-limit development (F-014) states that discrete J-cost dynamics on $\mathbb{Z}^3$ produce a second-order diffusion equation whose structure matches the Klein-Gordon equation. This module supplies the concrete 3D lattice calculus behind that claim.
Spatial dimension $D=3$ is taken from the forcing chain (T8). A lattice field is a real-valued map on $\mathbb{Z}^3$. The discrete Laplacian is the sum of three axial second differences; after mesh scaling it is the natural discrete stand-in for $\Delta$. Neighbor increments of the J-cost $J(x)=(x+x^{-1})/2-1$ are controlled so that the discrete interaction linearizes to this Laplacian at leading order.
Constants enter only through the RS unit system imported from the constants module; the local analysis is pure finite-difference calculus on the cubic lattice.
proof idea
The module is a short development, not a single theorem. It fixes $D=3$, defines 3D lattice fields, and writes the discrete Laplacian as the sum of three 1D second differences along the coordinate axes. A scaled operator multiplies by the appropriate power of the mesh $h$. Sign and normalization lemmas pin the continuum limit. Taylor expansion of a smooth test field then yields pointwise convergence of the scaled lattice Laplacian to $\Delta f$ with an explicit $O(h^2)$ remainder (second-order consistency). A final lemma transfers the same expansion to J-cost neighbor increments. A small certificate packages the convergence statement for downstream import.
why it matters in Recognition Science
Cubic Regge convergence imports this module to replace the Cheeger–Müller–Schrader axiom by a direct proof for J-cost interactions on $\mathbb{Z}^D$. Downstream documentation states that a direct proof suffices because the RS case is the cubic lattice with J-cost weights, not an arbitrary simplicial complex. Without second-order lattice consistency, the discrete curvature/action could not be identified with the continuum Einstein–Hilbert (or diffusion/KG) structure in the continuum limit. The development therefore sits between the continuum-limit foundation (F-014) and the gravity-side Regge argument, and it uses the forced $D=3$ from the T0–T8 chain rather than leaving dimension free.
scope and limits
- Does not treat non-cubic lattices or arbitrary simplicial complexes.
- Does not prove global Regge-to-Einstein–Hilbert convergence; only local Laplacian consistency.
- Does not derive $D=3$; it consumes the forced spatial dimension.
- Does not control boundary, curvature, or matter-coupled lattice schemes.
- Does not supply numerical rates beyond the formal $O(h^2)$ Taylor remainder.
used by (1)
depends on (2)
declarations in this module (12)
-
def
D_spatial -
abbrev
LatticeField3 -
theorem
D3_laplacian_three_terms -
def
axis_second_diff -
theorem
lattice_laplacian_is_sum_of_1D -
def
scaled_lattice_laplacian -
theorem
scaled_laplacian_sign -
theorem
lattice_laplacian_3D_convergence -
theorem
convergence_is_second_order -
theorem
jcost_neighbor_approximation_3D -
structure
LatticeConvergenceCert -
theorem
lattice_convergence_cert