Minimum-Volume Central Slabs of the Regular Octahedron

We determine the minimum volume of the intersection of a regular octahedron with a centered slab of every prescribed width, including all equality directions. For the octahedron with vertices +/-e_i and slab half-width t, the minimizing normal is diagonal below a single threshold tau = 0.4740349423895888... and is a coordinate normal above it, until the slab contains the whole body. The threshold is given exactly by a quadratic over Q(sqrt(3)). The analytic proof derives explicit chamber formulas, reduces the optimization to normals with two equal coordinates, and excludes interior extrema using sign-monotone derivatives and a polynomial resultant. This answers only the three-dimensional minimum-direction part of AIM Fourier-Convex Problem 23(d), associated with UnsolvedMath record AIM-CONVEX_GEOMETRY-0045. Higher-dimensional cross-polytopes, maximum volumes, cube slabs and negative spherical moments remain outside its scope. The release includes the manuscript, LaTeX source, 31 exact symbolic checks and a verification report. This AI-assisted, self-audited preprint is unrefereed; absolute priority and proof-assistant formalization are not claimed.