Vals AI has released an experiment centered on one of computer science’s best-known algorithmic problems: shortest paths. In the setup described in the report, 10 Claude Opus 5.5 agents worked for 15 hours inside a collaborative sandbox, challenged Dijkstra’s algorithm, and produced a new method called C-HD. The team also submitted 289 Lean formal-proof files, which the report says passed Lean Kernel verification in one run.
The result was presented not as a coding optimization exercise, but as a theoretical challenge: find an exact shortest-path algorithm that is faster than Dijkstra under the stated model, and prove that claim in Lean.
Dijkstra remains the benchmark
The report revisits the standard formulation of the problem. Given a directed graph with non-negative real edge weights, the task is to compute the minimum-total-weight path from a source vertex to every other vertex, or determine that a vertex is unreachable. Internal operations, including node visits and distance storage, count toward runtime.
Dijkstra’s algorithm, introduced by Edsger W. Dijkstra in 1959, still serves as the classical reference point. With an appropriate priority queue such as a Fibonacci heap, the runtime reaches O(m + n log n), where n≥2 is the number of vertices and m is the number of edges. The report notes that later human work in 2025 and 2026 improved bounds in other density regimes when m≥n, yet says Dijkstra remained dominant in a middle range of graph densities. That was the opening given to the agents: beat Dijkstra on exact shortest paths and formalize the proof in Lean.
10 agents, 733 discussion records, 15 hours
According to the article, Vals AI launched 10 Claude Opus 5.5 agent instances. They began with assigned roles but were given broad autonomy. They could reorganize tasks, share discoveries, challenge each other’s work, and shift compute toward the lines of attack that looked most promising.
The prompt constraints listed in the report were specific:
- The algorithm had to solve exact shortest paths on directed graphs with non-negative real weights.
- It had to deliver a real theoretical complexity improvement.
- It had to come with a complete, reproducible Lean proof.
- It had to be compared with leading 2025 and 2026 human results.
- Failed attempts had to be logged so other agents would not repeat them.
- Two independent rounds of AI peer review were required before success could be declared.
The sandbox included a shared message board. Over the 15-hour run, the system logged 733 discussion entries before the agents converged on C-HD.
How C-HD was described
The article contrasts C-HD with the standard greedy structure of Dijkstra, which repeatedly extracts the currently closest unvisited vertex and relaxes outward from there. With the right data structures, that approach stays at O(m + n log n).
C-HD, as described in the report, changes the structure of the search. Instead of focusing only on a single nearest point, it introduces a set of “pivot” vertices and uses a heuristic decomposition strategy to organize the search and the recursive work around it. The outline presented in the article is:
- Start from the source and the current frontier of vertices.
- Run bounded local searches along outgoing edges.
- Count newly encountered vertices toward the search limit, including unexplored leaves even when a given edge does not improve the distance estimate.
- Use the resulting search trees and pivots to organize recursive steps.
The report also says the agents designed strict local invariants for the method, meaning mathematical conditions that must remain true after each update. By pruning invalid edges and restricting local searches, C-HD aims to cut repeated exploration and some data-structure overhead.
Formal proof passed through Lean Kernel
The strongest part of the claim is the formal verification layer. The article says the 10 Claude agents produced 289 Lean files and assembled a theorem recorded as:
-- From namespace Frontier.CHD.Final: theorem chd_CHDTarget : GateCTarget.CHDTarget GateCCalc.F := ⟨chdProgram, chd_exact_within.1, bodyC KcC + 65536 * 9 + 100, chd_exact_within.2⟩
After compilation and machine checking, Lean Kernel returned a successful result, according to the report. In the article’s framing, that confirms that within the computation model and graph-density range defined by the AI system, C-HD correctly computes shortest paths and achieves the stated complexity upper bound without relying on forbidden axioms.
A Vals AI developer was quoted in the article as saying: “What a team of agents can do is fascinating. A nation of geniuses in a datacenter; that prediction is not far from reality.”
Engineering tests told a different story
The practical side looked far less dramatic. The report says that after the announcement, a developer identified as danalec built a GitHub project called C-HD, implemented the method in high-performance C using MSVC and C17, and turned it into roughly 1,900 lines of engineering code. That implementation was then benchmarked against classic Dijkstra and the 2025 DMMSY algorithm.
The measured outcome was unfavorable for C-HD. It ran about 1.8x to 2.9x slower than DMMSY. It was also 1.4x to 2.8x slower than a plain Dijkstra implementation.
The report’s explanation is that Lean Kernel was not fooled. The issue was the constant factor. Asymptotic complexity tracks behavior as input size grows without bound, but it does not capture preprocessing cost, memory movement, or label-management overhead in finite systems. In the tests cited by the article, 59% of runtime was spent handling 16-byte labels and 34% went to preprocessing. On graph sizes that matter in practice, those costs erased the theoretical savings.
Theory and practice split apart
The article says C-HD’s gap versus Dijkstra narrows as the vertex count grows, but adds that under memory limits available on real machines, it still does not catch up in wall-clock time. One developer assessment quoted in the report was blunt: “The blog is well written, but the algorithm is too awkward to be useful in the real world.”
That led the article to frame the work as occupying an uncomfortable middle ground: too engineering-heavy to be a clean pure-math destination, yet too impractical to matter as deployed engineering.
Why the result is still being treated as important
Even with the weak engineering showing, the report treats the experiment as a notable display of AI research capability. The key point is not that C-HD has become the new production standard for shortest paths. It is that 10 Claude Opus 5.5 agents, working in parallel for 15 hours, handled exploration, dead-end logging, peer review, and formal proof construction, then delivered an algorithmic framework the article presents as genuinely new.
The reference list in the piece includes posts on X from MaxForAI and Vals AI, a Vals AI blog post, and discussion threads on X about C-HD. The article itself was credited to the WeChat public account “新智元,” authored by Aeneas.