LaplaceReason: A Certified Benchmark and Continuous Dynamic Systems Reasoning Suite for Frontier Large Language Models

Abstract: Frontier Large Language Models (LLMs) such as DeepSeek-R1, OpenAI o1/o3, and Qwen-2.5-Math have achieved near-human proficiency on discrete reasoning benchmarks (e.g., GSM8K, MATH). However, they systematically exhibit catastrophic hallucination, severe phase lag, and structural divergence when evaluated on continuous dynamic systems, multiscale stiff ordinary differential equations (ODEs), and high-order operational calculus. Traditional synthetic data pipelines fail to provide reliable supervision for these domains because conventional computer algebra solvers suffer from Wilkinson's catastrophic conditioning when factorizing high-order characteristic polynomials (K > 18). In this work, we present LaplaceReason, the first certified, massive reasoning benchmark designed to evaluate and train frontier LLMs on continuous multi-physics and dynamic systems. Ground truth solutions and step-by-step reasoning trajectories are certified via a deterministic, high-precision operational oracle guaranteeing machine-precision residual bounds (||R||_inf < 10^-14) across arbitrary orders without numerical root-finding. The benchmark spans seven fundamental task families:1. High-order rational dynamics (K from 2 to 60)2. Multiscale stiff ODEs (stiffness ratio S > 100x)3. Real-time randomized SVD with universal Gavish-Donoho noise rank thresholding4. 3D tomographic C-scan reconstruction5. Sub-millisecond zero-lag kinematic trajectory extrapolation6. Matrix-pencil modal extraction without Fourier spectral leakage7. Cascaded passive RLC ladder networks Deterministic Reinforcement Learning Verifier:The package provides an exact, rule-based reward model (laplace_cot_verifier) offering zero-inference-cost reward modeling for Group Relative Policy Optimization (GRPO). Open Community Benchmark & Commercial Data:- Hugging Face Open Community Split (1,750 verified samples, >165 research downloads):https://huggingface.co/datasets/pabloenrique/LaplaceReason-Continuous-Dynamics - Commercial Production Partitions & Bespoke Synthesis (Lemon Squeezy):https://rootfreelaplace.lemonsqueezy.com