OMEGA NAUGHT. Paper IX: Exact Recovery on the 600-Cell

We study exact recovery for a quantized linear network on the 120 vertices of the regular 600-cell. Let A be its specified adjacency matrix, M = 13I + A, and T(d) = floor(339949771344778 Md / (25 × 10^15)) for d in Z^120. Although T is not injective, we prove that the reduced output together with the 120 predecessor residues modulo eight determines every integer input uniquely. The witness occupies exactly 360 bits, or 45 packed bytes. The proof combines an exact positive-definiteness certificate for M − 9I with integer separation and the strict difference bound for equal quantization cells. An exact inverse bounds each coordinate to an interval of width less than twelve; fixing its residue leaves at most two candidates, giving a complete finite decoder. Backward induction extends recovery to every finite trajectory when its terminal state and ordered witnesses are retained. We specify coordinate order, byte packing, portable integer interchange, and history framing, and provide reproducible exact checks. The 360-bit minimum for independently representing the complete residue vector is distinguished from the minimum of an arbitrary recovery witness conditioned on the output. A separate lattice-coset and geometric argument gives 307 ≤ ℓ* ≤ 352 for that latter problem. Its exact optimum, efficient general decoding, and application advantages remain open.