# Symbolic Verification

When a general-purpose model writes a geometry proof, it first retrieves structurally similar solved problems as examples, and a symbolic verifier then points out errors round by round. On 50 SAT-level Euclidean geometry problems from FormalGeo-7k, the full pipeline reaches 68%–96% proof accuracy, against 10%–44% for baselines that use neither analogical retrieval nor verifier feedback. Under the same verifier feedback and inference budget, analogical guidance alone lifts GPT-5 from 88% to 96% and o1 from 52% to 80%.

When a general-purpose model writes a geometry proof, it first retrieves structurally similar solved problems as examples, and a symbolic verifier then points out errors round by round. On 50 SAT-level Euclidean geometry problems from FormalGeo-7k, the full pipeline reaches 68%–96% proof accuracy, against 10%–44% for baselines that use neither analogical retrieval nor verifier feedback. Under the same verifier feedback and inference budget, analogical guidance alone lifts GPT-5 from 88% to 96% and o1 from 52% to 80%.

For planning under hard constraints, a new preference-optimization objective replaces the binary preference signal with a continuous violation margin computed by a deterministic symbolic verifier and scaled by violation severity, while a lexicographic objective keeps hard constraints separate from soft preferences. Direct preference optimization previously gave every violation the same training signal and was prone to length and style bias when preference pairs came from different model families. A companion synthesis pipeline builds preference pairs from procedural constraint profiles and minimal-edit distillation; the 8B model reaches 89.2% pass rate on TravelPlanner and 93.4% schedule rate on NaturalPlan, and beats GPT-4o by 9.2 points on unseen PlanBench Blocksworld.

For planning under hard constraints, a new preference-optimization objective replaces the binary preference signal with a continuous violation margin computed by a deterministic symbolic verifier and scaled by violation severity, while a lexicographic objective keeps hard constraints separate from soft preferences. Direct preference optimization previously gave every violation the same training signal and was prone to length and style bias when preference pairs came from different model families. A companion synthesis pipeline builds preference pairs from procedural constraint profiles and minimal-edit distillation; the 8B model reaches 89.2% pass rate on TravelPlanner and 93.4% schedule rate on NaturalPlan, and beats GPT-4o by 9.2 points on unseen PlanBench Blocksworld.