The Fine-Structure Constant and the Octahedron–Tetrahedron Asymmetry of the Tetrahedral-Octahedral Lattice

In the tetrahedral-octahedral tessellation of space (TOT) — the gapless filling of three-dimensional space by regular octahedra and tetrahedra, geometrically identical to the FCC lattice and to Fuller's octet truss — there is an exact, one-sided asymmetry between the two cells: the circumscribed sphere of an octahedron reaches beyond the center of each adjacent tetrahedron by Δ = (2√2 − √6)/4 ≈ 0.0947343 (in units of the edge), whereas the circumscribed sphere of a tetrahedron reaches the center of each adjacent octahedron exactly, with no excess. Distributing Δ over the 13 nodes of the first closed coordination cluster of the lattice (a node and its 12 nearest neighbors) gives a pure number with no adjustable parameter, α_geom = (2√2 − √6)/52 = 1/137.226, within 0.14% of the measured fine-structure constant, 1/137.036. An independent second trace of α appears in the doubling ladder of the same lattice counted from the Planck length, into which no electromagnetic quantity is built: the systematic offset of that ladder from the proton's Compton wavelength, multiplied by 3, gives 1/α ≈ 136.80 ± 0.6. The measured value lies between the two traces. We state the result as a hypothesis with an explicit open problem — the operation "asymmetry per node" is postulated, not derived, and the 0.14% gap is unexplained — and give a falsifiable consequence for Newton's constant G. Book, interactive 3D model of the lattice and the open laboratory: https://theoryof.space