The Hypergraph Storage Modification Machine: A Computational Model for Signed Directed Hypergraphs

We introduce the Hypergraph Storage Modification Machine (HSMM), a computational model that extends Schönhage's Storage Modification Machine (SMM) from directed graphs to signed-incidence directed hypergraphs. The HSMM operates on a dynamic hypergraph via six structural primitives (Nav, NewV, NewE, Att, Det, MovC) plus control instructions (Brc, Halt), with the full runtime state captured in a 9-tuple configuration whose data component is a labelled signed directed hypergraph (SDH). We prove three main results: (1) every SMM program embeds syntactically into the binary-deterministic fragment of the HSMM; (2) the HSMM is Turing-complete; (3) under the HSMM access model with materialised incidence views, label-directed lookup in k-ary relations is O(1) versus Ω(k) in the SMM. These results are demonstrated via two worked examples in the strict base ISA: a Turing-machine step simulation and the construction of a 3-ary GMDH quadratic-neuron hyperedge. An FPGA execution engine targeting Xilinx Zynq UltraScale+ is sketched, with analytical throughput estimates of ~1.5×10⁸ instructions/s at 300 MHz. A Python reference simulator is publicly available at https://github.com/kyberszittya/hsmm.