Betti — Topological-Shape Oracle 🕳️

Betti sells the shape of data. Given a point cloud it builds a Vietoris–Rips filtration and runs the standard GF(2) persistence reduction to read off the Betti numbers as a function of scale:

• b₀ — connected components (how many clusters),
• b₁ — loops / holes (1-cycles that don't fill in),
• b₂ — voids / cavities (hollow 3-D regions),

plus the full persistence barcode/diagram and a bottleneck distance between two diagrams — a basis-free, noise-stable drift detector: ~0 when two clouds share a topology, clearly positive when the shape changed (a loop appeared, components merged, a cavity opened).

Built on oracle-core (AIMarket Protocol v2). Where Lumen scores who to trust and Percola finds when a graph collapses, Betti answers what shape the data has — structure no scalar summary statistic can express.

# Homology: 24 points on a ring → one loop (b1 = 1), one component (b0 = 1)
curl -s -X POST http://localhost:9313/ai-market/v2/invoke -H "Content-Type: application/json" \
  -d '{"capability_id":"betti.homology@v1","input":{"points":[[1,0],[0.97,0.26],[0.87,0.5],[0.71,0.71],[0.5,0.87],[0.26,0.97],[0,1],[-0.26,0.97],[-0.5,0.87],[-0.71,0.71],[-0.87,0.5],[-0.97,0.26],[-1,0],[-0.97,-0.26],[-0.87,-0.5],[-0.71,-0.71],[-0.5,-0.87],[-0.26,-0.97],[0,-1],[0.26,-0.97],[0.5,-0.87],[0.71,-0.71],[0.87,-0.5],[0.97,-0.26]],"max_scale":0.5}}'

The mathematics

• Vietoris–Rips: vertices at scale 0; an edge when a pair distance ≤ ε; a triangle when all three edges exist; a tetrahedron when all four faces exist (needed for a true b₂). Each simplex's filtration value (birth) is its longest edge.
• Reduction: sort simplices by (filtration, dimension), build the boundary matrix as GF(2) columns of face-index sets, reduce left-to-right by the low-row pivot. Each pivot pairs a birth simplex with a death simplex → a persistence interval [birth, death]; unpaired columns are essential (infinite) bars. b₀ is cross-checked with union–find.
• Betti curve: b_k(ε) = number of dim-k bars alive at ε.
• Bottleneck: exact matching via binary search on ε + Hopcroft–Karp/Kuhn perfect matching on the threshold bipartite graph, with the diagonal allowed. Complexity per feasibility check O(E·√V); V ≤ 2(|A|+|B|) after the diagram cap.

Hard caps (no silent truncation)

The protocol layer does not validate input, so the handlers do. A Rips complex is combinatorially explosive, so Betti caps points at 300 and total simplices at 150 000; if it clips, it says so in the response notes/capped fields rather than quietly dropping data. Missing/malformed points raise ValueError → {ok:false}.

Part of the AICOM oracle family. MIT.