sample_grid
This definition supplies a placeholder that generates sample points on a uniform 4D grid for discrete spacetime integration given a volume element and point count. It would be referenced by implementers building numerical field actions or Einstein-Hilbert integrals inside the relativity module. The body is a stub returning the empty list, with comments noting a planned shift to adaptive quadrature.
claimGiven a spacetime volume element equipped with positive grid spacing and a natural number $n$, the function returns a list of 4-tuples of real numbers intended to sample the integral over the 4-volume with measure $d^4x$ times the metric factor.
background
The module supplies discrete approximations to volume integrals on spacetime using the square-root metric determinant measure. VolumeElement is the structure holding a positive real grid spacing that controls the discrete step size for such approximations. The surrounding setting is a scaffolded implementation of relativistic field theory inside Recognition Science, where full continuous integration via Mathlib measure theory remains future work.
proof idea
One-line wrapper that returns the empty list as a placeholder. Inline comments describe the intended uniform grid over a 4-cube and flag the future replacement by adaptive quadrature.
why it matters in Recognition Science
The definition provides the sampling primitive required by sibling integration routines such as integrate_scalar and einstein_hilbert_action. It fills the discrete-integration slot in the relativity module while the framework awaits a measure-theoretic replacement, consistent with the Recognition Science pattern of starting from discrete J-cost structures before continuum limits.
scope and limits
- Does not compute any weighted integral or quadrature rule.
- Does not incorporate the actual metric determinant or curvature.
- Does not generate boundary-aware or adaptive point sets.
- Does not depend on upstream ledger or spectral structures for its output.
formal statement (Lean)
24def sample_grid (vol : VolumeElement) (n_points : ℕ) : List (Fin 4 → ℝ) :=
proof body
Definition body.
25 -- Simplified: n_points^4 grid over [0, L]^4
26 -- Full version would use adaptive quadrature
27 [] -- Placeholder
28
29/-- Square root of minus the metric determinant.
30 STATUS: SCAFFOLD — Uses constant 1 as placeholder. -/