SIGMA Performance

We contained O(n3).
Then we made edits nearly free.

SIGMA decomposes knowledge graphs into bounded cells via sheaf cohomology. Streaming edits cost 35 microseconds at five million vertices. The scaling exponent is 0.19. No GPU. Single CPU core.

These are the engine numbers behind every Invariant product.
Generated is not verified. Same engine. Different evidence. Same receipt.

35 usper edit (V=5M)
13 usper query (V=1M)
0.19scaling exponent
0GPU required
The Insight

The cube didn't disappear. It got imprisoned inside a constant.

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.

The Proof

Enron. A real power-law graph, fully decomposed.

SIGMA never read a single email. It mapped every relationship and checked whether the patterns could all hold at once. Enron-scale public email topology, analyzed on one laptop, no GPU.

SIGMA Enron Decomposition (Run D, 4 seeds)

3-core vertices21,309
Edges166,039
Sheaf Laplacian170,472 square
Bounded cells446
Seeds4, bit-identical
HardwareNo GPU
4x
seeds, identical results. Zero correctness drift across seeds 42, 137, 2718, 31415. The partition structure depends only on graph topology. Deterministic. Reproducible. Every time.
April 17, 2026

The streaming breakthrough.

The measured answer: 35 microseconds per edit at five million vertices. The scaling exponent dropped from 0.55 to 0.19.

Edit Path

35 us
median per streaming edit at V = 5,000,000

Faster than V=1M. Streaming-from-zero architecture eliminates cold-start overhead.

Query Path

13 us
per contradiction query at V = 1,000,000

10,504x faster than baseline. Hierarchical nerve tree with O(1) LCA oracle.

Scaling exponent: 0.19 (R2 = 0.975)
Doubling the graph increases per-edit cost by 13%.
Architecture

Cellular Sheaf Decomposition

01
DecomposeFiedler spectral bisection. 21,309 vertices become 446 bounded cells. Each cell holds at most 500 vertices.
02
StreamNew edges route to their cell in microseconds. Restriction maps from a pre-computed pool. 35 us median at V=5M.
03
QueryHierarchical nerve tree with O(1) LCA oracle. 13 microseconds at V=1M. Single array lookup.
Scale Validation

Flat across two orders of magnitude.

VerticesEdit MeanQuery p99DriftCells
100,0000.046 ms0.010 ms0421
250,0000.051 ms0.010 ms01,096
1,000,0000.063 ms0.013 ms04,611
5,000,0000.035 ms--025,473
V grew 50x. Edit cost dropped. Streaming-from-zero beats cold-start at scale.
Early Warning

See it coming before it breaks.

Traditional Detection

Contradiction:
Yes / No

Binary. You find out after it happened.

SIGMA Spectral Tracking

21-cycle
lead time

Continuous signal. Detected stress while every other diagnostic still reported all clear.