ALGEBRAIC RESOLUTION — Volume II Algebraic Crystallography: The Diffraction Revelation

The algebraic crystal H₀(A₃) — the truncated octahedron with 24 vertices, 36 edges, and coordination number 3 — has been accessed through algebraic manipulation (touch), intuitive perception (feel), and acoustic resonance (hear) in previous volumes. This volume establishes the fourth and final sensory channel: coherent light diffraction (sight).

We compute the complete diffraction theory of the crystal from its 24 vertex coordinates, which are all permutations of (0, ±1, ±2) in ℤ³. The structure factor F(q) = Σⱼ exp(iq·rⱼ) is computed exactly — no approximation, no fitting, no free parameters.

The central discovery: viewed along the [111] axis — the ℤ₃ symmetry axis of the monoid — the diffraction pattern exhibits perfect hexagonal symmetry, directly manifesting the cyclic automorphism σ: a → b → c → a. Phase vortices in arg(F(q)) carry topological charge matching the quasi-Hopf structure.

All 35 algebraic assertions are formally verified in Lean 4. The experimental correspondence with the Goler pilot study (2025) maps 9 out of 9 observations to algebraic predictions with zero parameter fitting.

Keywords: algebraic crystallography, diffraction, truncated octahedron, 0-Hecke monoid, ℤ₃ symmetry, structure factor, Lean 4, formal verification

Lean 4 proof file: ARVolII_Crystallography.lean — 35 theorems, all verified by native_decide