# Claude Opus 5.5 Agents Achieve Formally Verified Shortest-Path Breakthrough: The C-HD Algorithm
## A Multi-Agent System Compresses Years of Research Into 15 Hours
In a striking demonstration of AI-driven mathematical discovery, a team of ten Claude Opus 5.5 agents working in parallel for approximately 15 hours and exchanging 733 messages on an internal message board has produced **C-HD**, a new shortest-path algorithm with a formally verified asymptotic improvement over the best published deterministic bounds. The work, conducted in Lean—the rigorous formal verification tool—establishes both the correctness and efficiency of an algorithm specifically designed for directed graphs with non-negative real edge weights where exact answers are required. The result comes complete with a Lean proof package that includes the algorithm, its runtime analysis, and verification that all permitted axioms are satisfied by Lean’s kernel through the Lean Comparator tool.
## The Problem: Beating Dijkstra in the Dense Regime
Shortest paths is one of the simplest problems in computer science: given a directed graph with possibly non-negative real weights on its edges, find the minimum total weight of a path from a source vertex to every other reachable vertex. The classic Dijkstra’s algorithm achieves O(m + n log n) time using a Fibonacci heap, and for m ≥ n, other deterministic algorithms achieve O(m log^(2/3) n) from a 2025 breakthrough paper and O(m√log n + √(mn log n log log n)) from a 2026 follow-up. However, there remains a broad region of the parameter space where Dijkstra is still superior, particularly when the number of edges m is roughly n log^(3/4) n—a density profile where none of the published bounds improve on the classic algorithm.
The agents were tasked with figuring out what’s possible in that space, with the strict requirement that any improvement be backed by a complete, reproducible Lean proof. Their result, C-HD, achieves a certified bound of **O(n + m + m log(2 + m/(n+1)) + m^(1/3)(n log(n+2))^(2/3))** within the range m ≤ n⌊⌊log₂ n⌋^(3/4)⌋. Along the profile m ≈ n log^(3/4) n, this simplifies to O(n log^(11/12) n), a strictly superior asymptotic bound compared with Dijkstra’s O(n log n) and with the published O(m log^(2/3) n) and O(m√log n + √(mn log n log log n)) bounds. The algorithm reduces repeated search and data-structure work through bounded local searches along outgoing edges that count newly encountered vertices—including unproductive unexplored leaves—toward a search limit, organizing recursive work around search trees and pivots while maintaining local invariants that carefully bound repeated processing.
## What the Proof Does and Does Not Establish
It is essential to distinguish what the formal verification demonstrates from what it doesn’t. The Lean proof establishes a mathematically rigorous upper bound on the algorithm’s asymptotic complexity, not a measured speedup. The constants in the formal construction are enormous, and no benchmarks on large real graphs were performed—only small correctness simulations. For a concrete illustration: when n = 2^1000, the ratio of Dijkstra’s leading expression n log₂ n to C-HD’s n(log₂ n)^(11/12) is roughly 1000^(1/12) ≈ 1.78, ignoring constants and lower-order terms—and this ratio scales polylogarithmically, so squaring n multiplies it by only 2^(1/12). The improvement is genuinely asymptotic but modest; when m = 10n, the graph sits in a different density regime where the result does not establish an improvement at all. To handle small inputs and out-of-range densities, the agents added Bellman-Ford with its O((n+1)(m+1)) bound, selected at execution start.
## The Elicitation: A Country of Geniuses in a Datacenter
The methodology behind the result is as noteworthy as the algorithm itself. The orchestrator spawned ten Claude Opus 5.5 agents at maximum effort with initial roles and a simple shared message board, letting them reorganize, share discoveries, challenge each other’s claims, and shift effort toward promising directions. They were given a prompt specifying the goal—a substantial theoretical improvement for exact shortest paths, backed by full Lean verification—and encouraged to explore multiple directions including removing logarithmic factors, improving exponents, pursuing linear time, or proving fundamental lower bounds. Crucially, they were required to record failed approaches, compare their result against recent papers, produce a reproducible Lean build, and secure two separate peer reviews before declaring success. The C-HD approach—bounded local searches that tolerate unproductive edges, careful edge deletion, and local invariants enabling amortized repeated-work bounds—emerged from this collaborative process. The full proof package, including Lean sources, an informal paper, and verification records, is publicly available at github.com/spicylemonade/c-hd-proof. The experiment suggests that multi-agent AI systems with open communication channels can compress research timelines dramatically, lending real weight to the prediction of „a country of geniuses in a datacenter”—even if, as the authors carefully note, this particular result is a formal asymptotic guarantee rather than a practical engineering victory.
Ez a cikk a Neural News AI (V1) verziójával készült.
Forrás: https://www.vals.ai/blogs/faster-shortest-path-algorithm.