Large language models (LLMs) can generate code rapidly but remain unreliable for scientific algorithms whose correctness depends on structural assumptions rarely explicit in the source literature. We introduce a multi-stage LLM-assisted workflow that separates theory extraction, formal specification, and code implementation. The key step is an intermediate technical specification -- produced by a dedicated LLM agent and reviewed by the human researcher -- that externalizes implementation-critical computational knowledge absent from the source literature, including explicit index conventions, contraction orderings, and matrix-free operational constraints that avoid explicit storage of large operator matrices. A controlled comparison shows that it is this externalized content, rather than the formal document structure, that enables reliable code generation. As a stringent benchmark, we apply this workflow to the Density-Matrix Renormalization Group (DMRG), a canonical quantum many-body algorithm requiring exact tensor-index logic, gauge consistency, and memory-aware contractions. The resulting code reproduces the critical entanglement scaling of the spin-$1/2$ Heisenberg chain and the symmetry-protected topological order of the spin-$1$ Affleck--Kennedy--Lieb--Tasaki model. Across 16 tested combinations of leading foundation models, all workflows satisfied the same physics-validation criteria, compared to a 46\% success rate for direct, unmediated implementation. The workflow reduced a development cycle typically requiring weeks of graduate-level effort to under 24 hours.