Invariant Research

What is SIGMA?

SIGMA is Invariant Research's structural verification engine. It models mutable AI, legal, compliance, and scientific state as constraint-bearing structures and checks whether updates preserve coherence before they are committed. SIGMA is not a language model and does not generate answers. It is the verification substrate underneath SATYA, designed to maintain structural checks efficiently as state changes.

The Problem SIGMA Solves

AI systems, knowledge graphs, agent state, and scientific models change constantly. When one piece changes, global coherence can break silently.

The classical approach to detecting structural inconsistencies in a graph requires recomputing cohomology from scratch after each edit. For a graph with n vertices, that computation costs O(n^3) per edit through coboundary factorization. For a stream of m edits, the total cost is O(m * n^3). At scale, this is prohibitive.

SIGMA solves this by decomposing the graph into bounded cells via spectral bisection, so every eigensolve operates on at most a fixed number of vertices. Under a bounded local geometry assumption (bounded cell size, stalk dimension, and nerve degree), each vertex insertion, edge insertion, or restriction map update touches only a bounded set of local coboundary blocks. The result is amortized edit processing that is constant in n.

35 µsper streaming edit at V=5M
13 µsper query at V=1M
0.19scaling exponent
0GPU, 0 parameters
How SIGMA Works

Cellular sheaf cohomology as a verification primitive.

SIGMA equips a graph (or cellular complex) with a cellular sheaf: data spaces (stalks) attached to each vertex and edge, connected by restriction maps that encode how local data should relate across the graph. The first cohomology group H^1 measures global inconsistency. When H^1 is zero, local data assignments are globally reconcilable. When H^1 is nonzero, structural contradictions exist, and SIGMA localizes them.

The key insight is that H^1 serves as a contradiction signal, while H^0 (the zeroth cohomology, which the broader sheaf ML community has primarily optimized for) measures global agreement. SIGMA focuses on H^1 because verification is about detecting failures, not confirming consensus.

Step 1 Decompose Fiedler spectral bisection divides the graph into bounded cells. At 21,309 vertices (Enron scale), SIGMA produces 446 cells with at most 500 vertices each.
Step 2 Stream New edges and vertices route to their cell in microseconds. Restriction maps come from a pre-computed pool. Per-edit cost is amortized O(1) in the graph size.
Step 3 Query Hierarchical nerve tree with O(1) LCA oracle. Contradiction queries resolve in 13 microseconds at 1 million vertices. Single array lookup.
What SIGMA Does Not Do

SIGMA is not a generative model.

SIGMA does not generate text, produce predictions, or estimate probabilities. It verifies structural consistency. It cannot hallucinate because there is no generative model in the verification loop. This is deliberate: the verifier must be categorically different from the system it verifies.

SIGMA does not replace your LLMs, RAG pipelines, or agent frameworks. It sits underneath them as a deterministic proof layer. Your AI systems generate and retrieve. SIGMA checks whether the output is structurally consistent with its evidence before the output becomes action.

Benchmarks

Verified at scale. Signed artifacts. Zero drift.

SIGMA's benchmark results come from signed artifacts (pinned Ed25519 issuer key) with correctness receipts at 1M and 5M vertices. All results are deterministic and reproducible: zero correctness drift across seeds, across runs, and across scales.

At 5 million vertices (25,473 cells, 0.50 MB RestrictionStore), SIGMA achieved 35 microseconds median per streaming edit with assembled-cohomology drift equal to zero. The scaling exponent of 0.19 means doubling the graph increases per-edit cost by 13%.

Full benchmark details Read the papers
Frequently Asked Questions

Common questions about SIGMA.

What is SIGMA?
SIGMA is Invariant Research's streaming structural verification engine. It models mutable state as constraint-bearing structures using cellular sheaf cohomology and checks whether updates preserve coherence before they are committed. SIGMA is the verification substrate underneath SATYA.
What is cellular sheaf cohomology?
Cellular sheaf cohomology is a mathematical framework from algebraic topology that attaches data (stalks) to the cells of a complex and uses restriction maps to enforce consistency between neighboring cells. SIGMA uses the first cohomology group (H^1) as a contradiction signal: when H^1 is nonzero, the local data assignments cannot be reconciled globally, indicating a structural inconsistency.
How fast is SIGMA?
SIGMA achieves 35 microseconds median per streaming edit at 5 million vertices, with a scaling exponent of 0.19. Contradiction queries resolve in 13 microseconds at 1 million vertices. These numbers come from signed benchmark artifacts with zero correctness drift.
Does SIGMA use machine learning?
No. SIGMA uses zero learned parameters, zero GPU, and zero training data. The verification is a mathematical computation, not a statistical inference. This is deliberate: a verifier with no generative model in its loop cannot hallucinate.
What is the relationship between SIGMA and SATYA?
SIGMA is the engine; SATYA is the product layer. SIGMA maintains structural coherence checks as state changes. SATYA applies those checks to specific workflows (legal, compliance, protein, agents) and handles domain translation, source retrieval, constraint construction, and receipt generation.

See the engine in action.

Review the benchmark data, read the published papers, or try SATYA on your own files.

View benchmarks Read the papers Download SATYA