Most legal analytics systems try to predict lawsuit outcomes by estimating probabilities-how likely a plaintiff is to win, settle, or lose based on past cases. This paper addresses a different and often more important question: whether a lawsuit can legally succeed at all. We introduce the concept of structural unwinnability, where a case is not merely unlikely to win, but impossible to win under the law, regardless of evidence, advocacy, or judicial discretion. We present a deterministic framework that models litigation as a system governed by legal rules and constraints. In this framework, a lawsuit can succeed only if at least one legally admissible path leads to a valid winning outcome. When no such path exists, the lawsuit is provably unwinnable. We show that many common legal barriers-such as missed filing deadlines, lack of jurisdiction, missing claim elements, or statutory limits on remedies-create absolute barriers that eliminate all winning outcomes. We demonstrate why probabilistic models, including Markov-based approaches, cannot reliably certify these impossibility conditions. Probability can estimate how often outcomes occur, but it cannot prove that an outcome cannot occur. In contrast, deterministic analysis can produce explicit, checkable proofs showing why a lawsuit must fail. We formalize these proofs mathematically and introduce the idea of proof-carrying litigation risk: deterministic certificates that demonstrate when legal success is impossible. This approach enables stronger compliance, clearer audits, and defensible legal risk elimination rather than mere risk estimation.