Text-to-SQL systems built on large language models still struggle in real deployment, where users underspecify queries while databases enforce rigid schemas. The gap is rarely lexical; it lies in implicit knowledge—hidden joins, derived formulas, and domain-specific filters—that natural language (NL) omits but SQL must encode. We observe that this knowledge is not missing: every historical (NL, SQL) pair already records it, with NL stating the goal and SQL stating the steps. Treating the two as dual views of one program, we propose KODE-SQL, a framework with three agents: KODE-Miner aligns NL and SQL in a shared graph space and distills the unmatched regions into reusable templates; KODE-Solver retrieves these templates to generate schema-grounded SQL under a rule-then-retrieval policy; KODE-Critic verifies each candidate on the same graph and feeds successful cases back to the knowledge base. KODE-SQL improves execution accuracy by up to 11.1 points on the knowledge-heavy ScienceBenchmark, remains competitive on BIRD and Spider, and yields consistent gains (+4–7 points) when plugged into existing solvers—validating both its standalone strength and its portability as a knowledge module.