Can a network reason by pushing back against broken rules?

Rules that push back

What if a network could solve a problem by making unresolved contradictions increasingly difficult to maintain?

In The Lagrangian Brain: A Constraint Mechanics of Mind and Machine, Yuhang He and Si Wu propose a mathematical framework in which constraints actively steer a network’s changing state. They connect this framework to brain circuitry and test computational implementations on Sudoku, mazes and sequences of permutations.

The work is a preprint. Its evidence consists of mathematical arguments and computer experiments, not new measurements in people or animals.

A corrective signal with memory

The central ingredient is a Lagrange multiplier: a variable that determines how strongly a constraint pushes against the system. In the proposed network, that variable accumulates violations. If an incompatible pattern persists, the corrective signal grows until the network changes.

This differs from assigning a broken rule a fixed penalty. A fixed penalty can be outweighed by other influences; an accumulating multiplier keeps increasing the pressure.

For Sudoku, the network represents each cell with competing units for its possible digits. Some constraints require a group’s activity to sum to one; others prohibit particular pairs from being active together. Modules update in turn, selecting the unit with the strongest input while the multipliers evolve.

The authors describe this as reasoning through the network’s own dynamics, without gradient-based training or a separate backtracking procedure. It still requires computation over time: difficult puzzles can demand many updates.

What the Sudoku result actually means

For the main Sudoku benchmark, the system counted patterns across 1,000 solved training grids. This recovered the 243 group constraints and 7,290 prohibited pairs used by its representation of Sudoku.

On all 422,786 Sudoku-Extreme test puzzles, the authors report 422,759 solutions within a budget of two million module updates per puzzle—approximately 99.99%. The remaining 27 were solved when allowed to run longer. Thus, the claim of solving the entire set includes runs exceeding the original budget.

Removing the integrating multipliers reduced performance sharply. On a 1,000-puzzle sample, the version without them solved none, while versions retaining only one of the two multiplier types achieved intermediate results.

A separate learning experiment alternated free-running activity, called “sleep,” with training examples, called “waking.” Patterns generated during sleep gained inhibition; patterns present in lawful examples lost it. After 60 cycles, the system had stored 98.6% of Sudoku’s prohibited pairs. A subsequent consolidation step completed the group constraints. This is distinct from the counting procedure used for the main benchmark.

Planning and remembering a sequence

In mazes, activity spread outward from the goal. Each reached location retained a changing signal that allowed the system to trace a route back from the start. The authors report shortest paths on all tested Maze-Hard examples and on larger perfect mazes, including grids with 64 times the training area.

That result needs a qualification: training fitted only the ordering of directions for ties and the choice to launch the spreading activity at the goal. It was not unrestricted learning of a planning algorithm.

For sequence tracking, the system learned allowed transitions between a running state, an input symbol and the next state. It then applied those transitions repeatedly. It achieved exact results on tested strings up to length 1,000 after training on lengths up to 40. On the five-object permutation task, the tested Transformer baselines failed to produce an entirely correct string at lengths of 100 or more. This comparison concerns those models and experiments, not every possible Transformer system.

A brain hypothesis, not a brain demonstration

The authors propose that inhibitory neurons, adaptation, synapses and feedback could carry different kinds of multipliers. Their differing response speeds could connect constraint enforcement to rhythms, learning and planning.

These proposed biological roles remain hypotheses. Among the paper’s testable predictions is that slowing a multiplier-carrying mechanism should shift its associated rhythm in a specific way.

The mathematical distinctions matter too. Under the stated assumptions, a rule-violating state cannot be a resting point; that does not by itself guarantee a fast route to a solution. The successful benchmarks also use structured, task-specific representations, and the published comparison scores do not establish equal computational cost.

The narrower conclusion is substantial enough: adaptive constraint signals can support strong performance on these structured tasks. Whether they offer a general account of intelligence—or describe how living brains work—remains unresolved.