Block Permutation Routing on Ramanujan Hypergraphs for Fault-Tolerant Quantum Computing

We analyze permutation routing of rigid blocks representing surface code patches of d_C² atoms on a reconfigurable lattice with hypergraph transformations. For a hypergraph H, code distance d_C, s=d_C², number of blocks N_L, and guard distance g, we show the block routing number rt_B(H, s, g) = Θ\(d_C log N_L\). A spectral analysis of the quotient graph Q\(G_cl(H\), B) (blocks as supervertices) shows that the spectral ratio β_Q < 1 is preserved in the high-connectivity regime. Negative association of block permutations and congestion bounds are used for random intermediate configurations. Serialization establishes that each quotient routing phase requires O\(d_C\) physical sub-steps due to the block footprint width. A lower bound rt_B = Ω\(d_C log N_L\) follows from combining the spectral lower bound on quotient phases with the traversal cost per phase. We include error model analysis grounded in recent experimental results, syndrome extraction protocols (stop-and-correct, rolling active fault-tolerant (AFT) measurement, and adaptive deformation), and integration with lattice surgery compilation via the Litinski protocol. Composition with the correlated-decoding scheme reduces syndrome-extraction overhead from O\(d_C\) to O(1) per correction window, leaving routing as the leading-order contributor to the integrated O\(d_C log N_L\) depth. Spectral inheritance is organized in a hierarchy: exact (Haemers interlacing on equitable partitions), perturbative (Weyl bounds for near-equitable partitions, a practically relevant case for surface-code patches), and universal (higher-order Cheeger). Methods extend directly to QCCD trapped-ion architectures under the same regime condition, with junction crossings replacing AOD transports as the elementary single-hop translation.