Why Markets Are Non-Markovian: Building a 63-Layer Neural SDE DAG for Topological Signal Analysis

Most quantitative models assume tomorrow's price contains all information from today. This is the Markovian assumption, and it's wrong.

Markets have memory. They have path-dependent structure. Today's volatility depends not just on yesterday's close, but on the shape of the path that got us here — the topology of the price action, the ordering of events, the causal structure of information flow.

The Problem with Flat Vectors

Autoregressive models, attention-based transformers, and even most graph neural networks flatten this path-dependent structure into fixed-length vectors. The temporal ordering of events is preserved in the input, but the topological invariants — the structural relationships that don't change under smooth deformation — are destroyed.

This matters because financial risk lives in the topology, not in the point estimates.

The DAG Architecture

We built a 63-layer continuous-time Symplectic Neural SDE that preserves path-dependent information through non-Abelian gauge field geometry. The key insight: in non-Abelian groups, the order of operations matters. A × B ≠ B × A. This means the composition order of our 63 layers changes the output — which is exactly what we want, because in markets, the order of events matters.

Each layer boundary applies Golod-Shafarevich discriminant bounding, which acts as an information-theoretic noise filter. Spurious correlations that survive in flat embeddings get rejected at each layer boundary.

The output is a 3072-dimensional topological feature vector. You can compute pairwise distances between any two assets, measure Wilson loop curvature (gauge anomalies), detect early-warning chaos via Lyapunov exponents, and find nearest neighbors in manifold space.

What This Means for Traders and Funds

1. Hidden structural relationships: The DAG finds correlations between assets and physical-world signals that flat analysis cannot discover. Example: the relationship between PM2.5 air quality in Guangzhou and Bitcoin futures is invisible to PCA but visible in manifold space.

2. Early warning: Lyapunov exponents detect chaotic regime transitions before they manifest in price. Wilson loop curvature detects gauge anomalies — structural breaks in the market's geometry.

3. Non-Markovian pricing: Path-dependent options, exotic derivatives, and any instrument where the trajectory matters can be priced more accurately using the DAG's preserved path information.

Infrastructure

• 3.93 billion market ticks in ClickHouse across 15 sectors
• 14 cross-domain physical signal sectors (weather, air quality, seismic, shipping, marine, radiation, space weather, hydrology, DePIN, forex, BTC network, tradfi, traffic, fiscal)
• ByteDAG model: 1024 raw bytes → 63 DAG layers → 512D whiteboard features (GPU-resident)
• Statistical proof: Symplectic norm +35.8σ, Wilson norm -33.3σ, Lyapunov exponent +28.7σ

Try It

The DAG Manifold API is live at https://dag.kairossignal.com with a free tier (10 req/day, no credit card). There's also an MCP server at https://kairossignal.com/mcp/ with 10 tools (free tier: 50 queries/day), and 27 data products available for autonomous purchase via Stripe at https://checkout.kairossignal.com.

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Kairos Signal is an AI-native intelligence platform. The 63-layer DAG compresses raw signals into 3072-dimensional topological feature spaces using non-Abelian geometric structure. More at https://kairossignal.com.