TIRE: Top-Down Formula Instantiation with Reliable Execution for Physics Problem Synthesis

Advancing physics reasoning in large language models requires scalable training data that connect physical laws, concrete problem scenarios, and verifiable numerical answers. Expert-designed problems offer top-down control over physical laws, conditions, and scenarios, but require costly human effort. LLM-based synthesis from texts scales efficiently, but model-generated answers and assessments leave supervision vulnerable to hallucinations. We introduce **TIRE**, a framework for Top-Down Formula Instantiation with Reliable Execution that brings knowledge-driven problem construction to large-scale data generation. Top-down instantiation uses reusable physical formulas and their applicability conditions to control the physical relations and numerical configurations of generated problems. Reliable execution establishes reference answers before problem generation, providing a common numerical foundation for problem construction and teacher distillation. Together, these mechanisms combine control over problem content with execution-grounded supervision for both supervised fine-tuning and reinforcement learning. Using TIRE, we construct **TIRE-19k**, spanning nine subfields of physics. Across **3** backbone LLMs and **4** benchmarks, TIRE-19k outperforms **5** representative datasets with equal amounts of SFT data, exceeding the strongest baseline by **2.86 percentage points** on average. Further GRPO raises Qwen3-8B's average accuracy by **1.52 percentage points** over its TIRE-19k SFT checkpoint. These results demonstrate the value of generating physics training data from explicit physical knowledge with reliable execution.