Hierarchical Clifford Mean-Fields for Scalable Geometric Deep Learning\\on Multi-Cluster Particle Systems

Global mean-field aggregation is a cornerstone of linear-complexity geometric deep learning, yet we show it \emph{provably} loses per-subsystem geometric information: when two particle clusters rotate in opposite directions, the global Clifford mean-field cancels to near-zero regardless of feature richness. We introduce \textbf{Octree HC-Net}, a multi-resolution spatial aggregation framework in $\Cl(3,0)$ that replaces the single global mean with hierarchical per-cell Clifford means computed via an octree decomposition. Each particle interacts with its own cell's mean multivector at every resolution level through outer-product interactions, with learnable level weights that adapt to task structure. On a diagnostic rotation-detection task where global aggregation achieves 50\% (random chance), hierarchical aggregation achieves 97\% accuracy with only 26K--337K parameters. Octree HC-Net scales as $\mathcal{O}(N)$ with exponent $\alpha = 0.89$, matching flat mean-field speed (54\,ms vs.\ 52\,ms at $N{=}5{,}000$) while using $10.7\times$ less time and $6.3\times$ less memory than kNN-based alternatives. Across all four experiments, hierarchical spatial aggregation strictly dominates both global pooling and local message-passing approaches. We further demonstrate two applications of the octree hierarchy: (i)~a \emph{learned Fast Multipole Method} (FMM) that connects the spatial hierarchy to the classical P2M$\to$M2M$\to$M2L$\to$L2L$\to$L2P framework, beating flat mean-fields by 10\% on charged-particle systems; and (ii)~a \emph{Clifford Renormalization Group} (C-RG) that treats per-cell Clifford means as renormalized beads, achieving 100\% rotation/chirality classification where scalar coarse-graining achieves 50\% (chance), with 16.9$\times$ better dynamics accuracy and $3\times$ faster, $10\times$ less memory at $N{=}1{,}000$.