SIGMA decomposes knowledge graphs into bounded cells via sheaf cohomology. Streaming edits cost 63 microseconds at one million vertices. Contradiction queries cost 13 microseconds. The scaling exponent is 0.19. No GPU. One laptop.
Spectral analysis on knowledge graphs requires eigensolving matrices that grow as O(n3). At 21,000 vertices, the sheaf Laplacian is 170,472 x 170,472. The standard approach: buy more GPUs.
SIGMA's approach: decompose the graph so every eigensolve operates on at most 500 vertices. The O(n3) cost is factored into n/500 independent bounded subproblems. Then eliminate per-edit overhead so streaming updates cost microseconds, not milliseconds. The total cost is effectively constant per vertex.
When an AI agent ingests 50,000 contracts, there is no mechanism to detect that Clause A in Contract 1 contradicts Clause B in Contract 2. The AI doesn't check. It can't. It just absorbs everything and moves on.
That contradiction doesn't surface as an alert. It surfaces as a lawsuit. Or a compliance failure. Or a decision made on information that was never internally consistent.
SIGMA is the verification layer that catches it. Not with AI guessing. With mathematical proof. It sits downstream of any LLM, RAG pipeline, or knowledge system and checks whether the structure holds.
The FBI had to read the emails. That's why it took 5 years and 45 agents. SIGMA never read a single one. It mapped every relationship and checked whether the patterns were structurally consistent. Power-law topology. Hub vertices at degree 1,141. The exact graph structure that breaks naive spectral methods.
The original system processed the full Enron graph in 30 seconds. That was the build. The question was: what happens when a new edge arrives? What does it cost to update the structural analysis without rebuilding from scratch?
The measured answer: 63 microseconds. At one million vertices. The cost barely grows with graph size. The scaling exponent dropped from 0.55 to 0.19.
The old baseline: 966 microseconds. The fix: stop deallocating cached matrices on every edit. Pool the restriction map math. Two changes, 30 lines of code, 15x faster.
The old baseline: 132 milliseconds. The fix: a hierarchical nerve tree with an O(1) lowest-common-ancestor oracle. 10,504x faster. Sub-10 microseconds at every measured scale.
SIGMA doesn't try to analyze the whole network at once. It decomposes the graph into bounded independent cells, checks each one separately, then assembles global structure through the nerve complex. The bigger the network, the more cells. Each cell takes the same amount of time.
Measured on the Enron email network (power-law, hub degree 1,141) and Barabasi-Albert synthetic graphs at increasing scale. Four seeds per scale. Zero correctness drift at every measurement point.
| Vertices | Edit Mean | Query p99 | Drift | Cells |
|---|---|---|---|---|
| 21,309 (Enron) | 0.031 ms | 0.005 ms | 0 | 639 |
| 100,000 | 0.046 ms | 0.010 ms | 0 | 421 |
| 250,000 | 0.051 ms | 0.010 ms | 0 | 1,096 |
| 1,000,000 | 0.063 ms | 0.013 ms | 0 | 4,611 |
Most systems tell you a contradiction exists after the damage is done. SIGMA watches the structural tension building in real time. In testing, it detected stress forming 21 steps before any other diagnostic noticed a problem. That's the difference between a smoke alarm and watching the temperature rise.
Binary. You find out after it happened. Too late to prevent the damage.
Continuous signal. Detected structural stress while every other diagnostic still reported all clear.
A doctor doesn't read every cell in your body to find the fracture. They take an X-ray, and the break lights up. SIGMA works the same way. Scan the structure. The contradictions light up. Then you know exactly where to look.
SIGMA finds structural contradictions mathematically. Proofkit signs every output cryptographically. Together, every analysis produces a sealed evidence pack: Ed25519 digital signature, SHA256 hash, complete audit trail. Independently verifiable by any third party with the public key.
No trust required. Download the bundle. Verify the signature yourself.
Upload a document set. Watch SIGMA find the contradictions. Download the signed evidence pack. No commitment. No sales pitch. Just math.