The A₄ Obstruction and Icosahedral Resolution: Why Tetrahedral Symmetry Fails on Periodic Lattices but Succeeds on Quasicrystals

Paper 1 in this series proved that the alternating group A₄ — the rotation symmetry of the regular tetrahedron — cannot act on the cycle space of the FCC lattice despite the tetrahedral geometry of the primitive cell. Two of the twelve 3-cycles map bonds outside the fundamental domain of T³, and only the Klein four-group V₄ survives. Paper 7 proved that the full icosahedral group I ≅ A₅ does act on the bond cohomology H¹ ≅ Z⁵ of the icosahedral quasicrystal without obstruction. This paper explains the contrast through the A_{N-1} root lattice framework of Paper 10. The FCC cohomology lattice is A₃; cubic periodicity obstructs the 3-cycles of A₄ because 120° rotations are incompatible with the cubic torus. The icosahedral QC cohomology lattice is A₅; the icosahedral group acts by permuting coordinates of Z⁶, which is always compatible with the periodic torus (Z/LZ)⁶. The quasicrystal is the minimal structure in which the obstructed symmetry is realized: N = 6 is the first value where Alt(N−1) is simultaneously the natural point group and acts without periodic obstruction. The passage from FCC to icosahedral QC is algebraically the root lattice embedding A₃ ↪ A₅. Eleventh paper in a series; companion papers at DOI 10.5281/zenodo.18793518, 10.5281/zenodo.18926651, 10.5281/zenodo.19027249.