AI coding assistants are now used to generate production code in security-sensitive domains, yet the exploitability of their outputs remains unquantified. We address this gap with Broken by Default: a formal verification study of 3,500 code artifacts generated by seven widely-deployed LLMs across 500 security-critical prompts (five CWE categories, 100 prompts each). Each artifact is subjected to the Z3 SMT solver via the COBALT analysis pipeline, producing mathematical satisfiability witnesses rather than pattern-based heuristics. Across all models, 55.8% of artifacts contain at least one COBALT-identified vulnerability; of these, 1,055 are formally proven via Z3 satisfiability witnesses. GPT-4o leads at 62.4% (grade F); Gemini 2.5 Flash performs best at 48.4% (grade D). No model achieves a grade better than D. Six of seven representative findings are confirmed with runtime crashes under GCC AddressSanitizer. Three auxiliary experiments show: (1) explicit security instructions reduce the mean rate by only 4 points; (2) six industry tools combined miss 97.8% of Z3-proven findings; and (3) models identify their own vulnerable outputs 78.7% of the time in review mode yet generate them at 55.8% by default.