radialInv
plain-language theorem explainer
Radial inverse supplies the function 1 over spatial radius to the power n on four-vectors. Researchers deriving Laplacians or directional derivatives of gravitational potentials cite it for explicit radial calculations. The definition is a direct one-line algebraic expression using the spatial radius.
Claim. For natural number $n$ and four-vector $x$, define $f(n,x) := 1/r(x)^n$ where $r(x) = (x_1^2 + x_2^2 + x_3^2)^{1/2}$.
background
The Derivatives module builds calculus operations on coordinate space using standard basis vectors for directional derivatives. Spatial radius extracts the Euclidean length from the three spatial components of a four-vector, excluding time. This definition relies directly on that radius computation. Upstream structures cover J-cost convexity from PhiForcingDerived and spectral emergence of gauge content, situating the work inside the Recognition Science framework.
proof idea
The definition is a one-line wrapper that applies the spatial radius function raised to power n before taking the reciprocal.
why it matters
This definition supplies the core function for downstream differentiability theorems and Laplacian identities in the same module, including laplacian_radialInv_n and partialDeriv_v2_radialInv. It closes specific Mathlib calculus axioms for radial potentials. The construction aligns with the three spatial dimensions from the unified forcing chain and supports potential theory in the Recognition framework.
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