The Program Semantic Graph (PSG) introduced in prior work on Dimensional Type Systems and Deterministic Memory Management encodes compilation-relevant properties as binary edge relations between computation nodes. This representation is adequate for scalar and tensor computations, but becomes structurally insufficient for two classes of problems central to heterogeneous compute: tile co-location and routing constraints in spatial dataflow architectures, which are inherently multi-way; and geometric algebra computation, where graded multi-way products cannot be faithfully represented as sequences of binary operations without loss of algebraic identity. This paper introduces the Program Hypergraph (PHG) as a principled generalization of the PSG that promotes binary edges to hyperedges of arbitrary arity. We demonstrate that grade in Clifford algebra is a natural dimension axis within the existing DTS abelian group framework, that grade inference derives geometric product sparsity eliminating the primary performance objection to Clifford algebra neural networks without manual specialization, and that the k-simplex structure of mesh topology is a direct instance of the hyperedge formalism. We assess the existing geometric algebra library ecosystem, identify the consistent type-theoretic gap that no current system addresses, and show that the PHG closes it within the Fidelity compilation framework. The result is a compilation framework where geometric correctness, memory placement, numerical precision selection, and hardware partitioning are jointly derivable from a single graph structure exposed as interactive design-time feedback.

翻译：先前关于维度类型系统和确定性内存管理的工作中引入的程序语义图（PSG），将编译相关属性编码为计算节点之间的二元边关系。这种表示适用于标量和张量计算，但对于异构计算中的两类核心问题而言，其在结构上存在不足：其一为空间数据流架构中的分块共置与路由约束（本质上具有多元性），其二为几何代数计算——其分级多元乘积无法在不损失代数恒等式的前提下被忠实地表示为二元操作序列。本文提出程序超图（PHG），作为PSG的理论化泛化，将二元边提升至任意元数的超边。我们证明：克里福德代数中的分级是现有DTS阿贝尔群框架内的自然维度轴；分级推理能推导几何乘积的稀疏性，从而无需手动特化即可消除克里福德代数神经网络的主要性能障碍；网格拓扑的k-单纯复形结构是超边形式体系的直接实例。我们评估了现有几何代数库生态系统，识别出当前系统均未能解决的、一致的类型理论缺口，并证明PHG在Fidelity编译框架内填补了这一缺口。最终得到的编译框架中，几何正确性、内存布局、数值精度选择及硬件划分均可从同一图结构（以交互式设计时反馈形式呈现）中联合推导得出。