Geometric algebra implementations suffer from a density problem. The geometric product of two general multivectors in n dimensions involves 2ⁿ × 2ⁿ component multiplications. Most are zero for any specific pair of input grades—a bivector times a vector doesn't produce all 2ⁿ components—but the compiler doesn't know that without hand-written grade specializations.
Haynes promotes the binary edges of Program Semantic Graphs to hyperedges of arbitrary arity, creating a Program Hypergraph that encodes algebraic structure directly in the type system. Grade in Clifford algebra integrates into a dimensional type framework. The result: grade inference eliminates the density problem automatically.
The compiler can derive which components of a multivector product are guaranteed zero from the types of the operands, without manual specialization per grade combination. A bivector-vector product is sparse not because someone wrote a specialized function, but because the type system knows what a bivector is.
More striking: the same hypergraph structure simultaneously derives geometric correctness, memory placement, numerical precision selection, and hardware partitioning. Multiple compiler concerns that are usually handled by separate passes—correctness checking here, optimization there, allocation elsewhere—collapse into a single graph query. When the type system encodes the algebraic structure rather than just the data shape, these concerns are not independent.
The key insight is that optimization and correctness are the same derivation when the types carry enough algebraic information. The sparsity pattern is a correctness property (these components must be zero) and an optimization opportunity (don't compute them). The type system makes both observations at once.