The Coordinate-Space Geometry of the Tetrad Frame

The tetrad (vierbein) is the local orthonormal frame on which the coupling of gravity to fermionic matter depends, yet its coordinate-space geometry has not, to our knowledge, been examined. Deriving four oriented half-space inequalities from the spatial sector of a single orthonormal tetrad, the 2^4 = 16 possible sign combinations yield exactly two bounded regions. The bounded pair form a stellated octahedron whose boundary decomposes into sixteen independent channels. No dual frame is introduced; the orientation-reversed tetrahedral sector emerges as the unique second bounded region of the same single-tetrad construction. All structural constants of the resulting geometry reduce to two integers: the number of tetrahedral faces K = 4 and the spatial dimension D = 3. A numerical rendering at four million sample points confirms the derived structure.