We present a compilation framework in which dimensional type annotations persist through multi-stage MLIR lowering, enabling the compiler to jointly resolve numeric representation selection and deterministic memory management as coeffect properties of a single program semantic graph (PSG). Dimensional inference determines value ranges; value ranges determine representation selection; representation selection determines word width and memory footprint; and memory footprint, combined with escape classification, determines allocation strategy and cross-target transfer fidelity. The Dimensional Type System (DTS) extends Hindley-Milner unification with constraints drawn from finitely generated abelian groups, yielding inference that is decidable in polynomial time, complete, and principal. Where conventional systems erase dimensional annotations before code generation, DTS carries them as compilation metadata through each lowering stage, making them available where representation and memory placement decisions occur. Deterministic Memory Management (DMM), formalized as a coeffect discipline within the same graph, unifies escape analysis and memory placement with the dimensional framework. Escape analysis classifies value lifetimes into four categories (stack-scoped, closure-captured, return-escaping, byref-escaping), each mapping to a verified allocation strategy. We identify implications for auto-differentiation: the dimensional algebra is closed under the chain rule, and forward-mode gradient computation exhibits a coeffect signature that the framework can verify. The practical consequence is a development environment where escape diagnostics, allocation strategy, representation fidelity, and cache locality estimation are design-time views over the compilation graph.

翻译：我们提出一种编译框架，在该框架中，维度类型注解在多阶段MLIR lowering过程中持续存在，使编译器能够将数值表示选择与确定性内存管理作为单一程序语义图（PSG）的协效应属性共同解析。维度推理确定值域；值域确定表示选择；表示选择确定字宽与内存占用；内存占用结合逃逸分类确定分配策略及跨目标传输保真度。维度类型系统（DTS）将Hindley-Milner泛化与有限生成阿贝尔群约束相结合，实现多项式时间可判定、完备且主类型的推理。传统系统在代码生成前擦除维度注解，而DTS将其作为编译元数据贯穿每个lowering阶段，使其在表示与内存布局决策点可用。确定性内存管理（DMM）在同一图中形式化为协效应规约，将逃逸分析与内存布局统一到维度框架中。逃逸分析将值生命周期划分为四类（栈作用域、闭包捕获、返回逃逸、引用逃逸），每类对应已验证的分配策略。我们指出其对自动微分的意义：维度代数在链式法则下封闭，且前向模式梯度计算呈现框架可验证的协效应签名。实际结果是构建了一个开发环境，其中逃逸诊断、分配策略、表示保真度与缓存局部性估计成为编译图上的设计时视图。