A new reasoning method, CRAM, learns an explicit energy function from solved puzzles and uses it to guide search. In tests on two challenging logic‑puzzle collections, CRAM solved 99.996 % of Sudoku‑Extreme test puzzles and every Maze‑Unique puzzle within 42 ms or 132 ms of training on a single GPU. The trained energy never returns a wrong answer; instead it may simply run out of the allotted search budget. CRAM achieves these results at least 1,000× faster than traditional loop‑based models.

Compared with sophisticated looped baselines, CRAM reaches the highest accuracy while using at least 282× fewer inference floating‑point operations per puzzle. This efficiency comes from early stopping: the energy value of a board signals when a satisfactory solution has been found, allowing the search to terminate as soon as the energy falls below a learned threshold. The method therefore scales its compute to the difficulty of each puzzle, unlike fixed‑budget looped approaches.

These successes illustrate that an energy‑based perspective can compactly encode constraint‑based reasoning from data and enable rapid, efficient solutions—though the current energy form may not directly generalize to more complex reasoning tasks.

(Figure 2: energy‑based inference cost vs accuracy)