Multi-agent AI is a consistency problem. Consistency is a branch of mathematics. Sheaf Labs is built on that.
Put several AI agents on a task and each one is locally competent and globally blind. Each sees a slice, does its part well, and hands off. Across those handoffs, small local accuracies compound into global drift: contradiction, hallucination, silent disagreement that nobody catches until it ships. The telephone game, at machine scale.
Today's orchestration frameworks are plumbing. They move messages between agents and draw you a flowchart. None of them answers the only question that matters in a high-stakes workflow: do the pieces actually cohere?
Sheaf theory is the branch of mathematics built for exactly this: how local data, defined on overlapping pieces of a space, glues into a single consistent global whole, and precisely how to detect when it cannot. It is the mathematics of local-to-global consistency. It is also the exact shape of multi-agent work.
| Base space | the task / shared context |
| Open sets | sub-tasks, local domains assigned to agents |
| Sections | each agent's local output |
| Restriction maps | passing each agent only its relevant slice of context |
| Gluing | assembling local outputs into one coherent global answer |
| H¹, the obstruction | the irreducible disagreement between agents on their overlaps |
Sheaf Labs is a mathematics lab with products. We search the mathematical landscape for tools that make AI safe to act, test each one on live agents, and ship the ones that work. Where others route, we glue, and we measure the seams. Local agent views are assembled into one verified global answer. Where agents disagree on what they share, Sheaf shows exactly where, and how much, with a number.
The Inconsistency Index. A contradiction that runs in a loop across the agents, found even when any two of them agree.
The sheaf Laplacian. Disagreement as energy on a graph of agents and claims. The coherence score, the consensus, and which claim carries the conflict.
A complete solver checks that a set of rules can all hold at once, and names the smallest set that clash when they cannot.
In finance, legal, and compliance, "the agents mostly agreed" is not an answer. Sheaf turns multi-agent output from a black box into an auditable, measured, consistent whole. That is the difference between plausible and verified.
The H¹ Inconsistency Index runs a first-cohomology computation: a topological obstruction when the answers cannot be glued into one consistent whole, and an H⁰ disagreement measure when they can. The method is shown in full in the technical whitepaper and published with a permanent DOI. The buyers who care about consistency are exactly the ones who will check. A measure that claims too much is worse than none.
Every Sheaf product carries this mathematics. On the Sheaf API, the models on the panel are local sections, each a restricted view of the problem, and the synthesizer is the gluing map that assembles them into one voice. The mathematics describes products that are live now, and a programme of seven more branches under study. The programme →
Sheaf Labs is founded by a researcher in topology, the branch of mathematics that sheaf theory comes from. The thesis is the founder's own field, applied. Topology studies how local pieces cohere into a global whole, and multi-agent AI is where that mathematics turns out to be urgently useful. Mathematical rigor is our training.
Where others route, we glue, and we measure the seams.