Topological Market Dynamics Studio

Active Betti Numbers

β₀: 8 | β₁: 3

dim H₀ (clusters) & dim H₁ (loops)

Euler Characteristic $\chi$

$\chi(\epsilon) = \beta_0 - \beta_1$

Persistent Entropy $E$

2.418 bits

Topological Diversity Shannon $E$

Systemic Turbulence

Low (18.4%)

Topological Condensation Index

Active Simplices

60V | 142E | 84T

Vertices, Edges, Triangles

TDA Controls

Financial Regime Simulator

Embedding Delay ($\tau$) 2 steps

Point Cloud Size ($N$) 60 points

Filtration Scale ($\epsilon$) 0.850

Isolated (0.0) Fully Connected (3.0)

Takens' 3D Reconstructed Attractor

$(x_t, x_{t-\tau}, x_{t-2\tau})$

Vietoris-Rips Simplicial Complex $\text{VR}(X, \epsilon)$

$\epsilon = 0.85$

Persistence Barcode $[b_i, d_i)$

■ H₀ (Components) ■ H₁ (Loops)

Persistence Diagram $\text{Dgm}(X)$

$(b_i, d_i)$ vs Diagonal $y=x$

Betti Curves $\beta_0(\epsilon), \beta_1(\epsilon)$ & Euler Characteristic $\chi(\epsilon)$

Topological Condensation Signature

§ Algebraic Topology & Persistent Homology Formulation

Mathematical foundations of simplicial complexes, chain groups, boundary operators, and topological entropy

1. Boundary Operator Nilpotency

Over field $\mathbb{F}_2$, the boundary map $\partial_k: C_k \to C_{k-1}$ maps simplices to their faces:

$$\partial_{k-1} \circ \partial_k = 0 \iff \text{im } \partial_{k+1} \subseteq \ker \partial_k$$

This allows the definition of homology groups $H_k = \ker \partial_k / \text{im } \partial_{k+1}$, whose dimensions $\beta_k = \dim H_k$ count $k$-dimensional topological holes.

2. Persistent Homology & Barcodes

Features that persist across wide filtration ranges $[b_i, d_i)$ correspond to true underlying geometric attractors:

$$\ell_i = d_i - b_i, \quad E(\text{Dgm}) = -\sum_{i=1}^M p_i \log_2 p_i$$

Short-lived features ($\ell_i \approx 0$) near the diagonal represent high-frequency market noise or microstructure jitter.

3. Market Crash Topological Condensation

Prior to systemic liquidity spirals and crashes, asset return correlation manifolds collapse:

$$\lim_{\text{crash}} \beta_1(\epsilon) \to 0, \quad E(\text{Dgm}) \searrow E_{\min}$$

The collapse of high-dimensional cavities into a single 1D dominant trajectory provides a non-parametric early warning indicator of systemic turbulence.