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：先前工作在维度类型系统和确定性内存管理中提出的程序语义图（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，但对于异构计算中两类核心问题在结构上存在不足：空间数据流架构中的瓦片共置与路由约束（其本质是多路关系），以及几何代数计算（其中分级多路乘积无法在不损失代数恒等性的情况下忠实地表示为二元操作序列）。本文提出程序超图（PHG）作为PSG的原则性推广，将二元边提升为任意元数的超边。我们证明了克利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，分级推断可推导几何乘积稀疏性从而无需手动特化即可消除克利福德代数神经网络的主要性能障碍，并且网格拓扑的k-单纯形结构是超边形式主义的直接实例。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解决的持续类型论缺口，并展示了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、数值精度选择和硬件划分均可从单个以交互式设计时反馈形式暴露的图结构中联合推导。