Stellar Colosseum Moves Beyond Single-Prompt Proofs

Stellar Colosseum shifts the burden of mathematical research from single-pass model generation to a multi-agent, feedback-driven harness. Standard language models struggle with extended research problems because they treat proofs as a flat, linear token stream. When a model relies on its internal context to maintain coherence across thousands of tokens, a single hallucination early in the chain invalidates the entire structure. Colosseum instead decomposes a proof into interdependent subproblems, managing them as discrete logical units that the system can verify and regenerate independently.

The Mechanism of Managed Proofs

Rather than assuming a rigid hierarchy, the system monitors model confidence scores across sampled path trajectories to determine when a problem path is mature enough for decomposition. The model uses overlapping random-sample tree aggregation to generate multiple concurrent completion variants for a single step. The system evaluates these candidates by calculating the probability mass of the output against the formal proof constraints, selecting the variant with the highest consistency across the sampled nodes before pruning lower-probability branches. This prevents the model from committing to a single hallucinated path early in the derivation.

To manage these segments, the system breaks the high-level objective into local goal specifications by invoking a secondary agent to translate a natural language query into a series of bounded constraints. These constraints act as input and output templates for each node. When generating a node, the system injects these constraints as a system prompt, essentially forcing the model to complete the logical segment such that the output satisfies the required variables and premises defined for that sub-goal.

This structure is maintained as a Directed Acyclic Graph (DAG) where each node represents a logical proposition and edges represent the dependency of one node upon the premises established by its parent. When a node is generated, the system records the input premises as parent pointers, allowing the harness to trace the dependency chain backwards from any conclusion.

Verification and Falsification

When a section is generated, the system passes the candidate logic snippet to a secondary model instance for falsification. This reviewer receives the candidate node, the formal problem specification, and the required dependencies. The reviewer is tasked with identifying logical gaps, boundary-case failures, or invalid assumptions. If the reviewer identifies an error, the harness traverses the DAG to identify all downstream nodes that consumed the output of the flawed segment and invalidates that entire subtree. The system then triggers a re-generation of the affected nodes while keeping the verified, upstream dependencies intact.

Consider a proof of the pigeonhole principle: the system breaks this into nodes for the definition of the set, the mapping function, and the contradiction check. If the reviewer finds the mapping function violates the injective property, it triggers a rebuild of that node without requiring the model to re-derive the definition of the set. This modular approach prevents cascading logical errors that occur when a model drifts from its initial premises.

Performance and Scaling

On the TCS-Bench dataset, which extracts problems from sources including FOCS, STOC, SODA, and JMLR, this modular approach achieved 71.0% accuracy. In separate evaluations, the proof-oriented pipeline with execution feedback solved 218 out of 222 problems in the Codeforces dataset.

Integration into existing workflows requires accounting for the complexity of managing a dynamic DAG. The harness struggles with non-hierarchical, densely connected problems, as it requires strict parent-child task decomposition. It is currently unclear how the system handles problems where the dependency structure is not clear until the proof is nearly complete, or how it scales when the number of concurrent agents competing for system context begins to saturate the available inference budget.