arXiv:2502.03325v2 Announce Type: replace-cross Abstract: Large language models (LLMs) such as DeepSeek-R1 have achieved remarkable performance across diverse reasoning tasks. To uncover the principles that govern their behaviour, we introduce the Electronic Circuit Principles (ECP), which maps inference-time learning (ITL) onto a semantic electromotive force and inference-time reasoning (ITR) onto a resistive network governed by Ohm's and Faraday's laws. This circuit-based modelling yields closed-form predictions of task performance and reveals how modular prompt components interact to shape accuracy. We validated ECP on 70,000 samples spanning 350 reasoning tasks and 9 advanced LLMs, observing a about 60% improvement in Pearson correlation relative to the conventional inference-time scaling law. Moreover, ECP explains the efficacy of 15 established prompting strategies and directs the development of new modular interventions that exceed the median score of the top 80% of participants in both the International Olympiad in Informatics and the International Mathematical Olympiad. By grounding LLM reasoning in electronic-circuit principles, ECP provides a rigorous framework for predicting performance and optimising modular components.