sample_grid
plain-language theorem explainer
Defines a placeholder function that accepts a volume element and natural number of points then returns a list of 4-tuples in real four-space for discrete spacetime sampling. Researchers building numerical field integrals within the Recognition Science relativity layer would cite this definition. The body is a stub that returns the empty list while recording the intended uniform grid construction.
Claim. Given a volume element $d^4x$ equipped with metric measure $√(-g)$ and a natural number $n$, the map returns a list of points in $ℝ^4$ intended for discrete integration over the spacetime domain.
background
VolumeElement is the structure that packages a positive real grid spacing together with its positivity proof; it stands for the discrete proxy to the volume form $d^4x$ with measure $√(-g)$ (see sibling definition at line 19). The module implements volume integration on spacetime using this measure and is explicitly marked as a scaffold that substitutes discrete point sampling for full Mathlib measure theory. Upstream structures supply the surrounding Recognition Science setting of J-cost convexity, ledger factorization, and spectral emergence of gauge content, yet the present definition isolates only the local sampling primitive.
proof idea
The definition is realized directly as a placeholder that returns the empty list, accompanied by inline comments describing a future uniform $n^4$ grid over $[0,L]^4$ and noting that a complete version would employ adaptive quadrature.
why it matters
The definition supplies the discrete sampling step required to assemble scalar, kinetic, and Einstein-Hilbert actions inside the relativity fields module. It therefore supports the continuum-bridge constructions that appear in sibling declarations such as integrate_scalar and einstein_hilbert_action. It touches the open question of passing from the discrete simplicial ledger and phi-ladder to a usable continuum integration measure.
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