Universal Bond Cohomology for Cut-and-Project Quasicrystals: An Elementary Derivation via Lattice Homology

An elementary proof that the first cohomology of the bond graph of any cut-and-project quasicrystal from Z^N with full coordination shell satisfies H^1 ≅ Z^{N-1}, reproducing the known result of Forrest–Hunton–Kellendonk (2002) without inverse limits, groupoid C*-algebras, or pattern-equivariant cohomology. The proof uses only lattice homology and the geometry of the acceptance window: the N coordinate windings on the torus T^N generate H_1 ≅ Z^N, the acceptance strip removes the diagonal direction (1,…,1) whose perpendicular projection exits the window, and the remaining N−1 generators are independent. The cohomology lattice is identified canonically as the root lattice A_{N-1} = Z^N ∩ {x_1+…+x_N = 0}, carrying S_N symmetry. The Gram matrix (N−1)I_N − J_N of the simplex vertices has Lorentz signature (N−1, 1) and coincides with the generalized Descartes quadratic form for N mutually tangent hyperspheres, connecting bond cohomology directly to Apollonian sphere packing geometry. Verified computationally on four structure types: Fibonacci chain (N=2, H^1=Z), octagonal tiling (N=4, H^1=Z^3), Penrose tiling (N=5, H^1=Z^4), and icosahedral quasicrystal (N=6, H^1=Z^5) — 8/8 tests passing. Tenth paper in a series; companion papers at DOI 10.5281/zenodo.18793518, 10.5281/zenodo.18828073, 10.5281/zenodo.18926651, 10.5281/zenodo.18928463.