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Recent advances in large language models and programmatic CAD have significantly improved Text-to-CAD generation for individual parts. However, production-ready mechanical assembly generation remains largely unsolved. Unlike single-part modeling, assemblies require coordinated reasoning over multiple components, functional interfaces, assembly relations, engineering principles, and physical consistency. Consequently, directly generating executable CAD code is insufficient for constructing mechanically valid and reusable assemblies. We present AssemCAD, an axiom-grounded framework for production-ready CAD assembly generation from natural language. Instead of representing an assembly as monolithic CAD code, AssemCAD first constructs an axiomatic Assembly Specification consisting of typed parts, geometry-backed ports, executable mates, and engineering axioms. Each assembly relation is explicitly grounded in one or more engineering principles, making the resulting specification interpretable, reusable, and verifiable. To realize this specification, AssemCAD introduces a port- and mate-based CAD assembly library that executes symbolic assembly relations through deterministic mate transformations and validates declared interfaces using concrete B-Rep geometric evidence. Built on this representation and library, AssemCAD further supports on-demand synthesis of reusable parametric component factories for both standard and open-world geometries. Experiments on AssemBench show that AssemCAD substantially improves assembly preservation and physical validity over code-centric CAD generation baselines, while generalizing across different foundation-model backbones. By combining axiom-grounded assembly reasoning with deterministic geometric execution, AssemCAD extends Text-to-CAD from isolated part generation toward production-ready mechanical assembly design.