Topology-Based Machine Learning and Regime Identification in Stochastic, Heavy-Tailed Financial Time Series

Abstract

Classic machine learning and regime identification methods applied to financial time series lack theoretical guarantees and exhibit systematic failure modes: heavy-tails invalidate moment-based geometry, rendering distances and centroids dominated by extremes or unstable; jumps violate smoothness, destabilizing local regressions, kernel methods, and gradient-based learning; and non-stationarity disrupts neighborhood relations, so distances in classical feature spaces no longer reflect meaningful proximity. To address these challenges, we propose a topology-based machine-learning framework grounded on probabilistic reconstruction of state-space geometry, which replaces moment- and smoothness-dependent representations with deformation-stable summaries of state-space geometry, preserving neighborhoods, adjacency, and topology. The finite-sample validity of homeomorphic state-space reconstruction, required for topology-based machine learning, is assessed through numerical studies on synthetic data with heavy tails, jumps, and known ground-truth regimes. Further diagnostics of local invertibility and bounded geometric distortion quantify when embedding windows are consistent with local diffeomorphic behavior, enabling metric-sensitive, geometry-aware learning. Clustering of Hilbert-space summaries accurately recovers underlying market tail-risk regimes with robust results across selected filtrations. Temporal, feature-space, and cluster-label null tests confirm that topology-based clustering captures genuine topological structure rather than noise or artifacts, and encodes temporal dependencies at local, mesoscopic, and network levels associated with market regimes.

1. Introduction

Machine learning methods perform well when data are naturally geometric and assumptions about distances, neighborhood relations, and smoothness hold, as in images, speech, or sensor data. In financial time series, these geometric assumptions often fail: distances may not reflect true proximity, nearby points may not share task-relevant properties, and feature representations may distort or destroy the underlying geometry. Hidden Markov Models (HMMs) and Gaussian Mixture Models (GMMs) [1,2] are widely used for market regime detection but rely on summary statistics (mean, variances, correlations, quantiles) that obscure the underlying geometry and fail under heavy tails, jumps, and non-stationarity as means do not converge, covariances fluctuate erratically, and distances become dominated by extremes. Wasserstein-based K-means methods preserve distributional geometry [3,4] but still lack guarantees that clustering boundaries correspond to true regime dynamics when finite-moment assumptions fail, and inherit sensitivity to cluster compactness and centroid placement in heavy-tailed environments [5,6]. Consequently, distances in conventional feature spaces are algebraic rather than geometric, and downstream learning may reflect representation artifacts rather than true system dynamics.

Persistence homology offers an alternative to classic machine learning methods, providing multiscale, deformation-stable summaries of geometry that are robust to noise [7,8], free of parametric assumptions, and mitigate the curse of dimensionality [9]. Applied to delay-embedded time series when diffeomorphic conditions hold, it preserves neighborhoods and relative distances in the reconstructed state space, making distances between topological summaries meaningful for machine learning applications. However, financial time series are stochastically forced systems exhibiting heavy tails, power-law decay, and volatility clustering [10,11,12,13,14], which violate Takens’ delay embedding theorem [15], compromising its guarantees.

As a result, many applications of topological data analysis in finance bypass Takens’ delay embedding and heuristically apply persistence homology to point clouds constructed on rolling windows of multivariate returns [16], correlation matrices [17,18,19], or k-nearest-neighbor networks [20]. The resulting topological summaries are interpreted as indicators of evolving market states rather than reconstructions of an underlying geometry [16]. Variations include heuristic calibration of Takens’ delay parameters [21] and fitting price bubble dynamics to quasi-periodic processes with a finite-time critical point [22]. Figure 1 illustrates this approach for the four US indices between 1981 and 2025. Spikes of $L^2$-norms $H_1$ are observed before the dot-com, Lehmann, and COVID-19 crashes.

Empirical applications of this heuristic approach show that changes in topological features—such as $L^1$ or $L^2$-norms of $H_1$—can provide early warning signals of financial market crises [16,23,24,25], identify critical market transitions [26,27,28], characterize tail-risk geometry [29,30], and improve realized volatility forecasting using PH-based models [29]. However, because these point clouds are purely statistical objects, they do not encode the dynamical flow, and the resulting neighborhoods, distances, and clusters lack a principled geometric interpretation required for regime identification and machine learning.

In this context, Gidea et al. [31] introduced the notion of a slowly varying instantaneous attractor, so that financial dynamics over short windows can be approximated as deterministic. Under this assumption, Takens’ delay embeddings are expected to yield a locally stable and geometrically meaningful reconstruction of state-space geometry, so that neighborhoods, distances, and clustering outcomes are directly interpretable and useful for machine learning tasks. In practice, the assumption of local deterministic dynamics often fails in financial time series. Delay embeddings may still yield empirically meaningful geometric structure, but without theoretical justification from local determinism, direct machine learning interpretability is compromised: clusters may reflect noise, finite-sample effects, or arbitrary projections rather than true dynamical regimes.

Building on extensions of Takens’ theorem to stochastically forced systems due to Stark et al. [32,33], we develop a framework under which calibrated delay embeddings can be both theoretically and empirically justified for clustering, regime detection, and machine learning of heavy-tailed, stochastic financial time series. Under appropriate embedding dimension and smoothness conditions on the stochastic forcing, the stochastic embedding theorems of Stark et al. [32] guarantee that the delay-embedding defines a $C^r$ homeomorphism: it preserves neighborhood relations, proximity structure, and topological invariants, providing a theoretical reference framework for topology-based clustering and regime identification under which delay embeddings preserve topological structure along stochastic trajectories. Under stronger smoothness assumptions, the delay map is locally diffeomorphic for almost every noise realization, preserving local metric and differential structure up to bounded distortion, thus justifying metric-sensitive machine-learning techniques.

However, these guarantees are asymptotic and probabilistic; therefore, they may not hold in finite samples of noisy financial time series. Because neither the true system state nor the underlying map is observable, this work conducts a numerical study to systematically evaluate through quantitative diagnostics whether the delay embeddings from such samples exhibit the observable consequences of $C^r$ homeomorphism or local diffeomorphic reconstruction. Accordingly, we do not interpret these results as establishing the validity of the embedding, but rather as motivating diagnostics targeting their observable consequences. To enable rigorous testing in a setting where the true system dynamics are known, we generate synthetic price trajectories exhibiting jumps, heavy tails, and known ground-truth regimes, construct their corresponding delay-embedded point clouds, and apply these diagnostics.

The numerical experiments further assess whether clustering of topological features derived from these embeddings captures genuine temporal dependence and regime structure. To this end, we compute compact, Lipschitz-stable functional summaries of persistent homology—L²-norms of $H_0$ and $H_1$ [34,35]—and apply a heavy-tailed K-means algorithm [6]. To distinguish genuine topological and temporal structure from noise and finite-sample artifacts, we compare the resulting filtration and clustering outcomes against pointwise, temporal, and cluster-label null models extending the universal topological randomness results of Bobrowski and Skraba [36]. Significant separation relative to the temporal nulls—in metrics such as Total Accuracy or Silhouette score—provides evidence that preserved neighborhood structure is captured in the embedding and topological features, while feature-space and economic validation evaluate whether clusters capture genuine topological and distributional differences.

The proposed framework extends recent methods based on network-based representations of attractors for change-point detection [37] and zigzag persistence on temporal networks for identifying periodic or chaotic transitions [38] to regime detection and clustering of stochastically forced, heavy-tailed time series. Unlike these methods, which rely on discretized phase-space states or network snapshots relative to a baseline—well suited for deterministic, low-noise systems with well-defined trajectories but scaling poorly—our temporal topological network tracks the probabilistic temporal evolution of local geometry directly across temporal scales, using stable and computationally efficient persistence-based functional summaries, grounded on rigorous probabilistic guarantees of homeomorphic and diffeomorphic stochastic reconstruction.

2. Theoretical Framework

This section presents the mathematical and methodological foundations of a topology-based machine-learning framework for identifying market regimes in heavy-tailed financial time series. In this context, we address the fundamental challenges of applying topology-based methods to financial time series, formulate a heuristic topological model of temporal dependence for regime evaluation, and introduce a hierarchy of geometric and temporal topological null models designed to discriminate genuine temporal topological organization (market regimes) from noise.

2.1. Delay Embedding of Stochastically Forced Time Series

Time series of asset prices $S_t$ and log-returns $r_t$ are realistically modeled as stochastically forced dynamical systems

$$d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t+S_{t-}d{Y}_t$$

(1)

where $\mu S_{t}dt$ is a deterministic drift and $\sigma S_{t}d{W}_t+S_{t-}d{Y}_t$ introduces stochastic forcing through Brownian fluctuations and discontinuous jump dynamics. Here $d{W}_t$ denote the increments of a standard Brownian Motion and $d{Y}_t$ represents the jump increment at time $t$. The corresponding log-returns process satisfies

$$r_t=\left(\mu -{\displaystyle \frac{1}{2}}{\sigma }^2\right)dt+\sigma \Delta W_t+{\Delta Y}_t$$

(2)

Following Stark et al. [32], delay-coordinate embeddings of stochastically forced dynamical systems can recover the underlying state geometry provided the stochastic forcing enters through a finite-dimensional, smooth family of maps.

Formally, let $M$ be a compact manifold of dimension $m\ge 1$, and let $N$ be a compact manifold of dimension $n$. Let $f:M\times N\to M$ be a $C^r$ map with $r\ge 1.$ Let

$$\omega =\left({\eta }_0, {\eta }_1,\dots .,\right) \in N^{\mathbb{Z}}$$

(3)

denote a bi-infinite sequence of forcing parameters, and define the associated skew-product system

$$F\left(x,\omega \right)=\left(f_{{\eta }_0}\left(x\right),\sigma \omega \right)$$

(4)

where $\sigma :N^{\mathbb{Z}}\to N^{\mathbb{Z}}$ denotes the left shift on sequences. Given a scalar observable $\phi \in C^r\left(M\times N,\mathbb{R}\right)$, the fiber-wise delay-coordinate map along a fixed noise realization $\omega$ is defined by

$${\Phi }_{f,\phi .\omega }\left(x\right)=\left(\phi \left(x\right), \phi \left(f_{{\eta }_0}\left(x\right)\right),\phi \left(f_{{\eta }_1}°f_{{\eta }_0}\left(x\right)\right),\dots ..,\phi \left(f_{{\eta }_{d-1}}°\dots ..°f_{{\eta }_0}\left(x\right)\right)\right)$$

(5)

where $d$ is the embedding dimension and $f_k\left(x\right)=f\left(x,{\omega }_k\right)$.

Theorem 5 of Stark–Broomhead–Davies–Huke [32]: if $d\ge 2m+1$ and $r\ge 1$, where $m= dim (M$), then there exists a residual set of pairs $\left(\phi ,f\right)\in C^r\left(M\times N,\mathbb{R}\right) \times C^r\left(M\times N,M\right)$ such that, for an open dense set of noise realizations $\omega \in N^{\mathbb{Z}}$, the fiberwise delay-coordinate map ${\Phi }_{f,\phi ,\varpi }:M\to {\mathbb{R}}^d$ is a $C^r$ embedding, i.e., an injective $C^r$ map that is a homeomorphism onto its image.

Crucially, Theorem 5 imposes no invertibility assumptions on the fiber maps $f_{\eta }$; contractions, expansions, and folds are all permitted. The embedding dimension $d$ is independent of the forcing dimension $n$, reflecting that the infinite-dimensional forcing sequence $\omega$ enter only through a finite-dimensional smooth family of maps $\left\{f_{\eta }:\eta \in N\right\}$. As a result, the classical Takens dimension bound remains valid, and the underlying state manifold is reconstructed almost surely along stochastic trajectories. The reconstruction is therefore fiberwise and probabilistic rather than a deterministic recovery of a single global attractor.

Under these conditions, delay embeddings preserve neighborhood relations and topological invariants of the state-space, providing a rigorous foundation for topological summaries and adjacency-based representations that rely on local connectivity rather than smoothness or metric information, including clustering and manifold-based learning. Temporal variations in these topological features therefore encode dependence structure and multiscale temporal dynamics.

While Theorem 5 guarantees the existence of topological embedding almost surely, it does not ensure that such a structure is detectable, or statistically stable in finite-sample delay embeddings constructed from noisy, heavy-tailed data. Since neither the true state $x_t\in M$ nor the stochastic map $f_{\eta }\left(x_t\right)$ are observable from finite samples, empirical validation must be performed on the observable consequences of $C^r$ embeddings. Accordingly, we assess consistency with Theorem 5 through diagnostics of local injectivity, noise regularity, smoothness, finite-dimensional forcing, and Lipschitz stability of reconstructed topological summaries derived from using the scalar observations $y_t=\phi \left(x_t\right)$ without claiming to verify the regularity assumptions of the theorem [32]. The formal definition and detailed formulation of the diagnostics are provided in Appendix A, with implementation described in Algorithm A1.

Theorem 6 of Stark–Broomhead–Davies–Huke [32]: Suppose that the fiber maps are parameterized by a smooth family $p\in C^r\left(N,{Diff}^r\left(M\right)\right)$ so that each realization $f_{\eta }=p\left(\eta \right):M\to M$ is a $C^r$ diffeomorphism. Let the observable be state-dependent only, $\phi \in C^r\left(M,\mathbb{R}\right)$, and assume the forcing variables ${\omega }_k$ are i.i.d. with a common distribution $\mu$, where $\mu$ is absolutely continuous with respect to Lebesgue measure on $\mathbb{N}.$

Then, for a generic pair $\left(p,\phi \right)$, the delay-coordinate map ${\Phi }_{f,\phi ,\omega }$ is a $C^r$ embedding for a full-measure set of noise realizations $\omega \in N^{\mathbb{Z}}$. Moreover, the embedding varies smoothly with respect to the noise realization, yielding a family of delay maps that is locally diffeomorphic almost surely along the stochastic trajectory.

Consequently, the reconstructed dynamics are locally one-to-one and smoothly invertible along almost every finite observation window. Local geometric and differential structure—such as distances, angles, and Jacobians—is preserved up to bounded distortion. These guarantees justify the use of metric-sensitive and geometry-aware machine learning techniques, while remaining inherently local and probabilistic. Since neither the true state $x_t\in M$ nor the underlying stochastic map $f_{\eta }\left(x_t\right)$ is directly observable in finite samples, empirical verification is naturally formulated in terms of the observable consequences of local $C^r$ diffeomorphism through Jacobian-based diagnostics of local invertibility, bounded local distortion, and smooth variation across time. The formal formulation of the diagnostics is provided in Appendix B, with implementation described in Algorithm A2.

Figure 2 illustrates both the feasibility of delay-coordinate reconstruction from scalar data under stochastic forcing and how local diffeomorphic structure can be empirically assessed in practice through Jacobian-based diagnostics. The delay-coordinate reconstruction uses an embedding dimension d = 7 satisfying the sufficient conditions for almost-sure local $C^r$ diffeomorphic embedding under smooth stochastic forcing. Window-wise Jacobian diagnostics of local injectivity, bounded distortion, and smooth variation over time indicate that the observable geometric consequences of local $C^r$ diffeomorphism hold in approximately 90% of windows, indicating that despite stochastic forcing and finite sampling, the reconstructed dynamics preserve local geometric structure along most stochastic trajectories.

2.2. Temporal Topological Networks

Building on Theorem 5 [32], which guarantees almost-sure topological embedding under stochastic forcing, we summarize the evolving structure of the reconstructed state-space using stable topological functionals. For each sliding window $W_t$ we define a scalar functional:

$$H_t=F\left(D_t\right)$$

(6)

where $D_t$ denotes the persistence diagram computed from the delay-embedded point cloud in the window $t$ and $F$ is a stable map on persistence diagrams. Two natural instantiations of $F$ are considered. The first is the diagram-space total persistence

$$H_t^{PD}={\sum }_{\left(b,d\right)\in D_t^{\left(0\right)}}(d-b)+{\sum }_{\left(b,d\right)\in D_t^{\left(1\right)}}\left(d-b\right)$$

(7)

that corresponds to the sum of the lifetimes of all features in the $H_0$ and $H_1$ persistence diagrams. Alternatively, following Bubenik [34], persistence diagrams can be mapped to persistence landscapes, forming the Hilbert space $L^2$-norm instantiation

$$H_t^{L_2}={‖{\lambda }^0\left(D_t\right)‖}_2+{‖{\lambda }^1\left(D_t\right)‖}_2$$

(8)

which supports standard geometric distances and similarity kernels while retaining multiscale topological information. The $L^2$-norm induces a detrended Hilbert-space measure of topological change analog of Wasserstein distances ${\varphi }_t^{wd}=\left|H_t-{\overline{H}}_{t-w}\right|$, where ${\overline{H}}_{t-w}$ is the mean over a rolling window of length $w$.

Both functionals are Lipschitz continuous with respect to Wasserstein distances [34], stable under homeomorphic delay embeddings [32], yielding compact time series ${\left\{H_t\right\}}_{t=1}^T$ that captures the evolution of the reconstructed geometry, preserving adjacency, similarity, and neighborhood relations inherited from the underlying state space. Under the stronger local diffeomorphic reconstruction conditions of Theorem 6 [32], the Hilbert-space $L^2$ formulation further admits metric-based comparisons and geometry-aware learning.

Each functional summary $H_t$ provides a compact descriptor of the instantaneous topological state of the reconstructed point cloud at time t. Collectively, the sequence ${\left\{H_t\right\}}_{t=1}^T$ encodes the temporal evolution of the system’s topology. To capture temporal organization, we construct a temporal topological network

$$V=\left\{v_1, v_2,\dots v_T\right\}$$

(9)

where each node $v_i$ represents one such instantaneous topological state at time t. Edges $e_{t,s}$ between nodes encode similarity between pairs of topological states at rime $t$ and $s$:

$$e_{t,s}=f\left(D_t,D_s\right)$$

(10)

where f is a diagram-level and geometry-aware distance such as $L^2$ distances in the Hilbert space formulation. In the diagram-space formulation, we use Wasserstein distances:

$$d_{t,s}^{Wass}=W\left(D_t, D_s\right)$$

(11)

measuring the magnitude of topological change between times t and s. Collecting these distances yields the matrix

$$D^{Wass}={\left[d_{t,s}^{Wass}\right]}_{t,s=1}^T$$

(12)

which provides a visual map of similarity over time. Figure 3 shows this matrix for $H_0$ and for $H_1$, directly representing the edges of the temporal topological network. The dark diagonal reflects minimal Wasserstein distances between consecutive windows, indicating topological similarity and temporal continuity, whereas off-diagonal light regions reveal significant topological changes between windows.

Distances are converted into topological similarities via an exponential kernel

$${sim}_{diag}\left(t,s\right)=exp\left(-{\displaystyle \frac{d_{t,s}^{Wass}}{{max}_{i,j}{d}_{i,j}^{Wass}}}\right) \in (0,\left.1\right]$$

(13)

which preserves ordering and ensures boundedness in $⌈0, 1⌉$. To incorporate geometric scale information of the reconstructed local geometry, we compute the minimum spanning tree (MST) length $L_t$ for each point cloud and define a scale-based similarity:

$${sim}_{MST}\left(t,s\right)= exp\left(-{\displaystyle \frac{\left|L_t-L_s\right|}{\sigma \left(L\right)}}\right), \sigma \left(L\right)=std\left({\left\{L_t\right\}}_{t=1}^T\right)$$

(14)

This prevents geometrically rescaled but topologically similar states from being treated as temporally coherent, while $\sigma \left(L\right)$ ensures the kernel is scale invariant. Edges $e_{t,s}$ exist for $s>t$, with $s-t\le T_{max}$ if the resulting similarity is non-negligible. Thus, edge weights $w_{t,s}$ are defined as follows:

$$w_{t,s}=g\left(d_{t,s},L_t,L_s\right)\in (0, 1]$$

(15)

where g is a similarity functional acting on both topological and geometric information. In diagram-space,

$$w_{t,s}=sim\left(t,s\right)=exp\left(-{\displaystyle \frac{W\left(D_t,D_s\right)}{\max W}}\right) exp\left(-{\displaystyle \frac{\left|L_t-L_s\right|}{\sigma \left(L\right)}}\right)$$

(16)

while in Hilbert-space,

$$w_{t,s}=sim\left(t,s\right)=exp\left(-{\displaystyle \frac{d_{t,s}^{PL}}{{\sigma }_{PL}}}\right)exp\left(-{\displaystyle \frac{\left|L_t-L_s\right|}{\sigma \left(L\right)}}\right)$$

(17)

where $d_{t,s}^{PL}$ is the instantaneous pairwise $L^2$ distance between persistence landscapes embeddings. Large weights $w_{t,s}$ indicate topological similarity and geometric coherence, while small weights signal regime transitions, structural changes, or temporal deformation.

Thus, Equations (9)–(17) specify the quantitative metrics that operationalize the temporal topological network. Figure 4 illustrates the metrics for heavy-tailed financial time series. Geometric scale $L_t$ is derived from the minimum spanning tree (MST) of a point cloud, while the similarity matrix $w_{t,s}$ combines diagram-level Wasserstein distances with the geometric MST scales. High values indicate windows that are topologically and geometrically coherent, whereas low values indicate windows with greater dissimilarity in topological and geometric structure, implying regime shifts.

2.3. Clustering as a Statistical Operator on Temporal Topology

The temporal topological network provides the foundation for clustering and regime detection under the probabilistic guarantees of Theorems 5 and 6. Let $\mathcal{M}$ denote the underlying state-space manifold, and ${\mathbb{R}}^d$ the delay-embedding space induced by $\Phi :\mathcal{M}\to {\mathbb{R}}^d$. By Theorem 5 [32], $\Phi$ is a $C^r$ embedding (i.e., homeomorphism onto its image) along almost every stochastic trajectory, preserving the topological structure of the underlying state space. In particular, $\Phi$ preserves neighborhoods and adjacency relations:

$$x \in n_M\left(y\right)\iff \Phi \left(x\right)\in n_{{\mathbb{R}}^d}\left(\Phi \left(y\right)\right)$$

(18)

where $n_M\left(y\right)$ denotes the neighborhood of $y$ on the manifold and $n_{{\mathbb{R}}^d}\left(\Phi \left(y\right)\right)$ denotes the corresponding neighborhood in the embedding space.

Let $V=\left\{v_1, v_2,\dots v_T\right\}$ denote time-indexed nodes associated with topological summaries $H_t$, each computed on a window ${\mathcal{W}}_t$. A regime is defined as a temporally contiguous subset $C_k\subset V$ for which the reconstructed local geometry exhibits consistent adjacency relations along the observed stochastic trajectory. Nodes $v_t\in C_k$ are mutually adjacent, in the sense of strong topological similarity, while nodes in different regimes show weaker adjacency. This is illustrated in Figure 5, which show how spectral embedding of the k-NN similarity graph and the thresholded temporal connectivity reveal underlying regimes and transitions in the temporal geometric structure of a heavy-tailed process.

The coherence within a regime can be quantified via adjacency-weighted averages:

$$\mathrm{C}\mathrm{o}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e} \left(C_k\right)={\displaystyle \frac{1}{\left|C_k\right|\left(\left|C_k\right|-1\right)}} {\sum }_{\begin{array}{c}t,s \in C_k, \\ t\ne s\end{array}}{w}_{t,s}$$

(19)

where $w_{t,s}\in \left[0, 1\right]$ encodes combined topological and geometric similarity (e.g., Wasserstein distances between persistence diagrams and MST-based scaling). Similarly, the separation between the two regimes $C_k$ and $C_{\mathcal{l}}$ is as follows:

$$\mathrm{S}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \left(C_k, C_{\mathcal{l}}\right)=1- {\displaystyle \frac{1}{\left|C_k\right|\left|C_{\mathcal{l}}\right|}}{\sum }_{t\in C_k,s\in C_{\mathcal{l}}}{w}_{t,s}$$

(20)

Therefore, clustering acts as a statistical operator on the temporal topological network, partitioning the trajectory into subsets that maximize internal adjacency and coherence while maintaining separation between regimes. Crucially, this construction relies solely on the preservation of neighborhood structure, adjacency relations, and topological invariants guaranteed by Theorem 5 [32].

Within this framework, two complementary clustering approaches are admissible: (i) geometry- and topology-aware clustering of similarities $w_{t,s}$, using network-based methods such as spectral clustering [39,40] or weighted edge clustering [41]; and (ii) node-level clustering of functional topological summaries $H_t$, treating distances between summaries as statistical dissimilarities between stable topological states and operating solely on the topological summaries, including vector-based methods (K-means) and clustering using Wasserstein (or bottleneck) distances [42,43,44], interpreted as intrinsic dissimilarities between node-associated persistence diagrams.

For sequences of persistence diagrams, Wasserstein or bottleneck distances provide a principled measure of topological deformation between reconstructed neighborhoods, forming natural similarity metrics for clustering and regime detection. These distances are defined intrinsically on the space of persistence diagrams, independent of the ambient state-space metric, and thus compare node-level functional topological summaries. For homology dimension $k$, the $p$-Wasserstein distance between persistence diagrams $D_t$ and $D_s$ is

$$W_p^k\left(D_t,D_s\right)={\left({inf}_{\gamma :D_t\to D_s}{\sum }_{x\in D_t}{‖x-\gamma \left(x\right)‖}_{\infty }^p\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}, p\ge 1$$

(21)

with $p=\infty$ corresponding to the bottleneck distance. To enable clustering in a linear space, persistence diagrams may be embedded into a Hilbert-space via persistence landscapes. Following Bubenik [34], persistence landscapes admit a natural $L^2$ metric

$$d_{t,s}^k={‖{\lambda }^k\left(D_t\right)-{\lambda }^k\left(D_s\right)‖}_2$$

(22)

which quantifies the dissimilarity of topological features in homology dimension $k$. For multiples homology dimensions $k=0, 1,\dots ,K$, the combined $L^2$ distance is defined as follows:

$$d_{t,s}=\sqrt{{\sum }_{k=0}^K{‖{\lambda }^k\left(D_t\right)-{\lambda }^k\left(D_s\right)‖}_2^2}$$

(23)

Each node $v_t$ encodes the topological state at time $t$ as a vector of Hilbert-space functionals:

$$v_t=\left(H_0\left(t\right),H_1\left(t\right),{\varphi }_t^{wd}\left(t\right)\right)$$

(24)

summarizing the local topology of the reconstructed trajectory.

Thus, clustering in the Hilbert space of topological summaries naturally groups nodes with similar topological neighborhoods, while regime transitions correspond to sustained increases in distances $d_{t,s}$, reflecting topological deformation rather than geometric displacement. In this setting, clustering outcomes are directly interpretable in terms of stable adjacency relations and topological change, consistent with the probabilistic guarantees of Theorem 5 [32].

If the observable consequences of local $C^r$ diffeomorphism hold for a sufficiently large proportion of embeddings, intrinsic and embedded distances satisfy

$$d_M\left(x,y\right)\approx d_{{\mathbb{R}}^d}\left(\Phi \left(x\right),\Phi \left(y\right)\right)$$

(25)

up to smooth, bounded distortion. Consequently, local neighborhood geometry and tangent structure are preserved to first order, elevating the temporal topological network from a pure topological graph to a geometry-aware approximation of the manifold’s intrinsic temporal organization, thereby expanding the class of statistical operators that can legitimately act on it.

In this setting, similarity weights $w_{t,s}$ admit metric interpretability as approximations of intrinsic distances between reconstructed neighborhoods. This enables geometry-aware clustering techniques, including diffusion-based methods, spectral clustering with meaningful kernels, tangent-space approximations, and kernel-based methods, complementing the topology-based approaches.

2.4. Temporal Scales of Topological Organization

Financial time series exhibit persistent temporal organization at multiple scales. Short-range dependence reflects significant correlations at small lags ℓ, whereas long-range dependence decays slowly according to a power law [12,45]:

$$\rho \left(\mathcal{l}\right) ~{\mathcal{l}}^{-\gamma }, 0<\gamma <1$$

implying long memory [46,47] and the absence of a characteristic correlation horizon [47].

Under conditions of homeomorphic reconstruction [32], financial time series multiscale temporal dependence is encoded probabilistically in the evolving stochastic trajectory of node-level topological functionals $H_t$, pairwise distances $d_{t,s}$, similarity measures $e_{t,s}$, and geometry-aware edge weights $w_{t,s}$, which integrates topological similarity (via persistence distances) with geometric coherence (via MST lengths $L_t$).

Figure 6 illustrates this framework for a stochastically forced, heavy-tailed process with regime shifts. MST lengths $L_t$ summarizes local geometric evolution, while similarities $w_{t,s}$ combine diagram-based Wasserstein distances and geometric rescaling. High similarity indicates regime persistence, sharp drops signal topological deformation or geometric rescaling, and sustained plateaus reveal mesoscopic temporal organization.

Moreover, the topological and geometrical quantities associated with nodes, edges, and weights encode distinct scales of temporal organization, thereby defining the structure of a multiscale topological temporal network:

• Local scale: Short-range dependence appears as fine-scale fluctuations in $H_t$, and low-lag autocorrelation of deviation signals as ${\varphi }_{t,s}^{wd}$ for small $\left|t-s\right|$. Although computed on short windows, $H_t$ aggregates topological information across landscapes, providing mesoscopic resolution at a local time scale.
• Mesoscopic scale: Intermediate- and long-range dependence governing regime persistence appears as a sustained elevation in $H_t$, persistent bursts in ${\varphi }_t^{wd}$, and densely connected sub-graphs in $w_{t,s}$ linking windows with similar reconstructed geometry.
• Global scale is captured by network-level statistics derived from $w_{t,s}$, including cluster coherence, cross-regime separation, tail dependence, and graph-based metrics.

This framework provides a principled, probabilistic basis for regime detection at different scales of the temporal organization in stochastic, heavy-tailed financial time series. By explicitly linking topological and geometric quantities to distinct temporal scales, it guides the selection of appropriate statistics and null models for inference, enabling rigorous tests of whether observed temporal structure reflects genuine dynamics rather than noise. In this way, the multiscale topological temporal network becomes a falsifiable, heuristic construct grounded in the embedding guarantees of Stark et al. [32], suitable for interpretable machine learning applications.

2.5. Topological Null-Models

As highlighted in Section 2.1, delay-coordinate reconstruction guarantees provided by Theorems 5 and 6 of Stark et al. [32] are inherently probabilistic, holding almost surely or window-wise along the observed trajectory. This implies that reconstructed topological features from stochastically forced systems should be interpreted as probabilistic manifestations of structure. Thus, they cannot be distinguished from artifacts induced by random fluctuations, finite sampling, or noise without a principled benchmark since neither the true system state nor the underlying map is observable.

To assess topological randomness rigorously, explicit topological Mull models are required. In this context, Bobrowski and Skraba [36] established that persistence diagrams arising from random point clouds converge to a universal distribution, independent of sampling distribution, geometry, or curvature. This result provides a principled benchmark against which observed topological features can be tested.

Let ${dgm}_k$ denote a persistence diagram in homological dimension k, decomposed conceptually as follows:

$${dgm}_k={dgm}_k^S\cup {dgm}_k^N$$

(26)

where ${dgm}_k^S$ captures latent topological features inherent to the data and ${dgm}_k^N$ consists of cycles arising from random point-cloud geometry. For a persistence point p = (birthday (p), death (p)), the persistence ratio of $p\in {dgm}_k$ is defined as follows:

$$\pi \left(p\right)={\displaystyle \frac{death \left(p\right)}{birth \left(p\right)}}$$

(27)

Asymptotically, persistence ratios arising from noise grow polylogarithmically with the sample size n,

$$\pi \left(p\right)\approx o\left({\left(\log n\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$k$}\right.}\right)$$

(28)

whereas ratios associated with genuine geometric features scale polynomially,

$$\pi \left(p\right)\approx \Theta \left(n^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$d$}\right.}\right)$$

(29)

with d the intrinsic dimension of the underlying manifold. This separation provides a theoretical basis for distinguishing noise-induced from structure-induced topological features.

To obtain a universal limiting law, a double-logarithmic transformation is applied:

$$\mathcal{l}\left(p\right)=A\log \log \left(\pi \left(p\right)\right)+B$$

(30)

Here, A depends on the filtration and homology dimension (e.g., for Čech complexes, A = 1, for Rips complexes, A = 1/2), and B is a location constant chosen to center the transformed data. Under the geometric null, the ℓ-values of noise-induced cycles converge to a left-skewed Gumbel distribution, providing a universal reference for a rigorous per-cycle hypothesis testing of structure.

This geometric null model is point-wise and spatial: it characterizes randomness in point-cloud geometry within individual windows but is agnostic to temporal ordering and therefore cannot assess whether topological features are organized in time. Such tests are well-suited to classification tasks in domains where objects of interest are spatially indexed, including cellular organization, biology, morphology, or cancer genomics. However, financial time series are high-dimensional and noisy, so that meaningful topological structure does not manifest primarily at the level of isolated persistence diagrams but in the temporal organization, dependence, and persistence of topological features along the observed trajectory. In this setting, pointwise geometric tests are insufficient, motivating the introduction of temporal topological null models to assess for genuine temporal structure.

2.6. Temporal Topological Null-Models

To extend topological null-models from individual diagrams to time-evolving features, we consider sequences of diagrams ${\left\{D_t\right\}}_{t=1}^T$ together with diagram-level functional $H_t=h\left(D_t\right)$ (e.g., $L^2$-norms of persistence landscapes, persistence weighted distances, etc.).

The temporal topological null model is defined by a random permutation $\pi \in S_T$ of the functional sequence:

$${\left\{H_t^{\tau }\right\}}_{t=1}^T= {\left\{H_{\pi \left(t\right)}\right\}}_{t=1}^T \pi \in S_T$$

(31)

where $H_t^{\pi }$ denotes the permuted sequence induced by $\pi$. By construction, this null model preserves the full geometry of each persistence diagram $D_t$, the diagram-level functional values $H_t=h\left(D_t\right)$ derived from each diagram, and the exact empirical marginal distribution of all topological and geometric features. At the same time, it destroys all temporal organization, including the temporal order of the functionals, serial correlations and regime structures, temporal memory encoded in the ordering of diagrams, and cross-cycle dependencies among functionals.

Significance is evaluated empirically via $N_{surr}$ permuted sequences:

$$p_{emp}= {\displaystyle \frac{1+\sum _{j=1}^{N_{surr}}1\left(D_j\ge D_{obs}\right)}{1+N_{surr}}}$$

(32)

where $D_j$ denotes the statistic computed on the j-th permuted sequence and $1\left(\cdot \right)$ is the indicator function. This framework applies to any arbitrary statistic $D\left(\left\{H_t\right\}\right)$ and requires no assumptions on Gaussianity, stationarity, or finite moments, making it suitable for heavy-tailed, heteroskedastic, and non-stationary financial series. Implementation details and executable pseudocode are provided in Algorithm S7 (Supplementary Materials).

Figure 7 illustrates how the temporal topological model provides an appropriate benchmark for randomness in time-indexed topological features. For a Gaussian i.i.d., the z-scored deviation functional ${\varphi }_t^{wd}$ exhibits no temporal organization; autocorrelations at all lags remain within the null envelope, consistent with the absence of memory or regime structure. In contrast, for a regime-shifting, fat-tailed process, autocorrelations of both signed and absolute deviations lie well outside the null bounds across a wide range of lags, revealing persistent temporal dependence and regime structure that cannot be attributed to marginal geometric variability alone.

By removing dependence at all temporal scales (local, mesoscopic, global) while preserving within-window geometry, the temporal null provides a principled benchmark for detecting meaningful multiscale temporal organization. However, statistical coherence alone does not guarantee economic interpretability or relevance. We therefore introduce a cluster label null to test whether regime assignments correspond to persistent market regimes rather than arbitrary labeling, with implementation details and pseudocode provided in Algorithm S12 (Supplementary Materials).

Together, this framework defines three nested layers of inference:

(1)

Geometric structure of individual windows;

(2)

Temporal organization of evolving topological features;

(3)

Economic significance of clusters.

This hierarchy ensures that candidate regimes are both statistically robust and economically interpretable, linking the conceptual distinction between local, mesoscopic, and global temporal scales to actionable market-relevant structures. This approach complements existing inference methods in topological data analysis. While Robinson and Turner [48] use permutation-based null hypothesis tests on pair-wise distances between persistence, our approach targets temporal dependences and regime structures within a single evolving sequence of persistence diagrams.

3. Materials and Methods

We conduct a numerical study, formulated as a controlled machine-learning experiment on a synthetic dataset, to evaluate whether delay-embedded topological functional features recover genuine regime structure in stochastic, fat-tailed, and non-stationary financial time series with known ground-truth regimes. Specifically, using this controlled testbed, we validate whether the delay embeddings exhibit the observable consequences of probabilistic homeomorphic or diffeomorphic reconstruction, as well as the validity of clustering based on topological functionals.

Apparent structure in clustered topological summaries can arise from three distinct and independent sources of randomness: (i) geometric randomness within windows, arising from finite-sample point-cloud effects; (ii) temporal randomness across windows, arising from loss of adjacency and neighborhood structure; and (iii) regime assignment randomness, reflecting artifacts of the clustering procedure. To distinguish genuine structure and dynamical regimes from artifacts of finite sampling, reconstruction, or clustering, we explicitly test all three through the hierarchy of null hypotheses:

(1)

Geometric null: Topological features arise from random point-cloud geometry within windows.

(2)

Temporal null: Observed structure is indistinguishable from random temporal ordering of features.

(3)

Clustering label null: Regime assignments arise from random labeling rather than persistent structure.

Rejection of both the temporal null hypothesis and the cluster-label null hypothesis is required to establish that clustering reflects genuine market regime structure. This hierarchy grounds machine-learning inference in interpretable changes in reconstructed geometry.

3.1. Synthetic Data Generation

Synthetic price paths simulate alternating bull and bear regimes of deterministic duration. Prices follow a regime-dependent geometric Brownian motion (GBM) with fat-tailed innovations and jumps, aggregated into open–high–low–close (OHLC) bars preserving regime structure. The synthetic price process is instantiated once under a fixed random seed to ensure reproducibility. Full executable pseudocode is provided in Algorithm S1 (Supplementary Materials).

3.2. Tests of Local Homeomorphic Delay-Embeddings

Log-returns are partitioned into overlapping windows and embedded using delay-coordinates. Embeddings are assessed by empirical diagnostics designed to test the observable consequences of local homeomorphic embedding guarantees consistent with Theorem 5 of Stark et al. [32]. Specifically, we assess the following:

• Injectivity: Stability of persistence diagrams, landscapes, and Fill-factor criteria.
• Noise regularity: Smooth variation in delay vectors, autocorrelation, and Mutual Information.
• Finite-dimensional forcing: Inter-window autocorrelation differences.
• Lipschitz stability of topological features: Bottleneck and Hilbert-space distances.

Appendix A and Algorithm A1 provide full methodological and implementation details of the diagnostics. Algorithm S2 in Supplementary Materials offers an executable-style pseudocode.

3.3. Jacobian-Based Diagnostics of Local Diffeomorphic Embedding

We assess the window-wise, local diffeomorphic validity of the reconstructed delay-embeddings using Jacobian-based diagnostics derived from the observable consequences of Theorem 6 of Stark et al. [32]. Specifically, we evaluate the following:

• Local invertibility: Non-singularity of the embedding Jacobians.
• Bounded distortion: Local bi-Lipschitz behavior quantified via Jacobian condition-number bounds.
• Smooth variation: Temporal stability of differential quantities across windows, ensuring that the local linearization does not fluctuate erratically across the window.

Windows satisfying all three criteria are considered locally invertible (hence locally one-to-one), geometrically non-degenerate, and differentially stable, consistent with $C^r$ diffeomorphic reconstruction. Full implementation details are provided in Appendix B and Algorithm A2. Algorithm S3 in Supplementary Materials offers an executable-style pseudocode.

3.4. Computation of Topological Functionals

Topological features are extracted from OHLC price representations, using two filtrations:

(1)

Vietoris–Rips (VR), yields $H_0$ and $H_1$ persistence; computed using the giotto-tda library version 0.6.2 in Python version 3.10.15;

(2)

One-Pass K-Clusters (1PKF) [49]: A graph-based filtration, which by construction yields $H_0$ persistence only.

VR captures fine-scale geometric deformation of the reconstructed trajectory, while the 1PFK emphasizes the emergence and persistence of statistically meaningful clusters by suppressing small or noisy components. Consistency across both pipelines supports robustness to filtration choice.

Persistence diagrams are mapped to Hilbert-space functionals, with local fluctuations summarized by rolling deviation functionals capturing bursts of topological activity, heteroskedasticity, and departures from i.i.d. assumptions. Algorithm S4 in Supplementary Materials provides an executable-style pseudocode of 1PFK-filtration.

3.5. Unsupervised Learning

Heavy-tailed K-means [6] is applied to time-indexed feature vectors. Centroids minimize an α-powered dispersion functional relative to an isotropic α-stable reference distribution, ensuring stable centroid updates under heavy tails (Gaussian mean for α = 2; Cauchy for α = 1). To ensure deterministic and reproducible clustering, centroids are initialized using evenly spaced feature percentiles rather than random seeding. Tail-risk-aware reweighting assigns more pull to extreme points in centroids updates, which emphasizes time windows where topological features are most extreme. Full executable pseudocode of heavy-tailed K-means is provided in Algorithm S5 (Supplementary Materials).

3.6. Geometric Null Hypothesis

To rigorously assess topological structure, we apply Bobrowski and Skraba’s [36] point-wise geometric null test to the persistence diagrams obtained from both VR and 1PFK-filtrations. For each persistence point, the persistence ratio is transformed into a $\mathcal{l}$-value and compared against the universal left-skewed Gumbel distribution to compute a p-value. Points with p-values below a significant threshold (with and without Bonferroni correction) are classified as signal, forming signal diagrams. To reduce computational cost, a 10% random subset of diagrams may be used. The fraction of significant points and their mean p-value are used to classify the dataset as either consistent with random topology or exhibiting genuine topological structure. Executable-style pseudocode is provided in Algorithm S6 (Supplementary Materials).

3.7. Temporal Topological Null Hypothesis Test

We test whether the observed clustering reflects statistically significant temporal topological structure rather than random temporal fluctuations. Let ${\mathit{A}}_{\mathit{i}\mathit{j}}$ denote the observed adjacency-level statistics computed from the time-indexed topological features:

$$A_{ij}=\left\{\begin{array}{c}{\rho }_{ij} \mathrm{s}\mathrm{h}\mathrm{o}\mathrm{r}\mathrm{t}-\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e} \mathrm{d}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{s} \left(\mathrm{a}\mathrm{u}\mathrm{t}\mathrm{o}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\right) \\ {\kappa }_{ij} \mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{s}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{i}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}-\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{c}\mathrm{o}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e} \\ {\lambda }_{L,ij} \mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}-\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r} \mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r} \mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l} \mathrm{d}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e} \end{array}\right.$$

For each metric, a temporal null distribution is generated by applying random permutations $\mathit{\pi }$ to the time indices of the feature sequence:

$${\left\{{\mathit{A}}_{\mathit{i}\mathit{j}}^{\mathit{\pi }}\right\}}_{\mathit{\pi }=1}^{{\mathit{N}}_{\mathit{s}\mathit{u}\mathit{r}}}$$

where ${\mathit{N}}_{\mathit{s}\mathit{u}\mathit{r}}$ is the number of surrogate sequences. This temporal null preserves the marginal geometric structure while destroying temporal ordering, serial correlations, and regime-dependent structure.

The clustering algorithm $\mathit{C}\left(\cdot \right)$ is applied to each surrogate realization, yielding surrogate datasets ${\left\{{\mathit{X}}^{\mathit{\pi }}\right\}}_{\mathit{\pi }}^{{\mathit{N}}_{\mathit{s}\mathit{u}\mathit{r}\mathit{r}}}$. Significance is assessed by comparing the observed adjacency metric of observed clusters ${\mathit{A}}_{\mathit{i}\mathit{j}}\left(\mathit{C}\left(\mathit{X}\right)\right)$ to its null distribution ${\mathit{A}}_{\mathit{i}\mathit{j}}\left(\mathit{C}\left({\mathit{X}}^{\mathit{\pi }}\right)\right)$, yielding empirical p-values:

$${\mathit{p}}_{\mathit{e}\mathit{m}\mathit{p}}={\displaystyle \frac{1+\sum _{\mathit{j}=1}^{{\mathit{N}}_{\mathit{s}\mathit{u}\mathit{r}}}1\left({\mathit{A}}_{\mathit{i}\mathit{j}}^{\mathit{\pi }}\left(\mathit{C}\left({\mathit{X}}^{\mathit{\pi }}\right)\right)\ge {\mathit{A}}_{\mathit{i}\mathit{j}}\left(\mathit{C}\left(\mathit{X}\right)\right)\right)}{1+{\mathit{N}}_{\mathit{s}\mathit{u}\mathit{r}}}}$$

(33)

Metrics used for ${\mathit{\kappa }}_{\mathit{i}\mathit{j}}$ include the following:

• Ground-truth clustering metrics: Total Accuracy (TA), Bear-Recall (BR), and Adjusted Rand Index (ARI) [50,51].
• Unsupervised cluster separation and coherence metrics: Silhouette [52], Calinski-Harabasz (CH) [53], Davies Bouldin Index (DB) [54], Maximum Mean Discrepancy (MMD) [55] measure intra-cluster coherence and inter-cluster separation; computed using Python’s scikit-learn library version 1.3.2 [56].

The temporal null construction and validation are detailed in Algorithm S7, while metric-specific implementations are provided in Algorithm S8 (autocorrelation tests ${\mathit{\rho }}_{\mathit{i}\mathit{j}}$), Algorithms S9 and S10 (clustering metrics tests ${\mathit{\kappa }}_{\mathit{i}\mathit{j}}$), and Algorithm S11 (tail dependence tests ${\mathit{\lambda }}_{\mathit{L},\mathit{i}\mathit{j}}$). Monte Carlo experiments confirm correct Type 1 error control under i.i.d. heavy-tailed nulls and non-trivial power against structured temporal alternatives.

3.8. Topological Cluster-Label Null Hypothesis Tests

To test whether clusters encode genuine topological and economically meaningful structure, we construct a label-permutation null model. Let $C_t$ denote the cluster label associated with window-level features $w_t$. Under the null, labels are randomly permuted:

$$\left(w_t, C_t\right)\to \left(w_t,\pi \left(C_t\right)\right), \pi \in S_k$$

(34)

This procedure preserves all window-level features $w_t$ while destroying any information contained in the original cluster assignments. Test statistics are computed on both $C_t$ and $\pi \left(C_t\right).$ Let $T_{obs}$ denote the observed statistic computed on $C_t$ and ${\left\{T_j^{\pi }\right\}}_{\pi =1}^{N_{perm}}$ the statistic computed on $N_{perm}$ permuted labels sequences. The empirical p-value is given by the following:

$$p={\displaystyle \frac{1+{\sum }_{j=1}^{N_{perm}}1\left(T_j^{\pi }>T_{obs}\right)}{1+N_{perm}}}$$

(35)

Statistics T include Kruskal–Wallis, Mann–Whitney, Levene, Kolmogorov–Smirnov, Anderson–Darling, and a distribution-free permutation test. Two complementary null hypotheses are evaluated:

(1)

Feature-Space or Topological Separability: Rejection of the null indicates that clusters arise from genuine changes in Hilbert-space-embedded topological functionals.

(2)

Economic Distinguishability: Rejection of the null indicates that clusters correspond to distinct market regimes and tail-risk patterns, rather than arbitrary temporal partitions.

The cluster-label null construction and validation are detailed in Algorithm S12. Executable-style pseudocode of the feature-space and economics distinguishability test is provided in Algorithm S13 in Supplementary Materials. Monte Carlo experiments using Algorithm S12 confirm correct Type 1 error control under i.i.d. heavy-tailed nulls and non-trivial power against structured temporal alternatives.

3.9. Left-Tail Power-Law Fitting and Distributional Analysis of Cluster Tails

For each cluster, left-tail behavior is analyzed using power-law fitting via the Clauset–Shalizi–Newman [57] framework, adapted to include the Anderson–Darling test for increased sensitivity to tail behavior. For each cluster, we estimate the left-tail scaling exponent $\alpha$ and the tail cut-off $x_{min}$, goodness-of-fit p-values using Kolmogorov–Smirnov and Anderson–Darling tests, and likelihood-ratio tests comparing power-law fits to alternative distributions (log-normal, exponential). Executable-style pseudocode of power-law fitting tests is provided in Algorithm S14 (Supplementary Materials).

4. Results of the Numerical Studies

Synthetic price trajectories with known regime shifts and heavy-tailed dynamics were generated to evaluate homeomorphic and diffeomorphic delay-coordinate reconstruction. We also assess whether Hilbert-space summaries encode geometric or temporal dependence, and whether topology-based clustering genuinely recovers the underlying regimes compared to temporal and cluster-label nulls.

4.1. Ground-Truth Synthetic Price Trajectories

Ground-truth bull and bear regimes were simulated with heavy-tailed dynamics and jumps, producing synthetic price trajectories at 5 min resolution over 2 years (39,212 observations). The simulation parameters are summarized in

Table 1. Parameters for synthetic price trajectories with ground-truth regimes.

Regime	Process	Parameter	Value 1
Bull	Fat-tail	True or False	True = Student t
	Drift	${\mu }_{bull}$	0.07
	Volatility	${\sigma }_{bull}$	0.20
	Deg. Freedom	${\nu }_{bull}$	20
	Jump Intensity	${\lambda }_{bull}$	0.5
	Jump Mean, std.	${\mu }_Y^{bull},{\sigma }_Y^{bull}$	0.01,0.01
Bear	Fat-tail	True or False	True = Student t
	Drift	${\mu }_{bear}$	−0.15
	Volatility	${\sigma }_{bear}$	0.30
	Deg. Freedom	${\nu }_{bear}$	5
	Jump Intensity	${\lambda }_{bear}$	2.5
	Jump Mean, std.	${\mu }_Y^{bull},{\sigma }_Y^{bull}$	−0.02,0.01

¹ Annualized value.

. Figure 8 shows the resulting price trajectory and log-returns corresponding for a single realization generated under a fixed random seed to ensure reproducibility. Periods corresponding to the ground-truth bear regime are highlighted in light blue and marked with the letter “B”.

The provided dataset for reproducibility includes latent $S_t$ price, time stamps, OHLC prices, log-returns, and ground-truth bull/bear regime labels for each timestamp.

4.2. Local Homeomorphism Tests for Delay Embeddings

We evaluated whether the reconstructed delay-coordinate maps satisfy the observable consequences of a homeomorphic embedding of Theorem 5 of Stark et al. [32]. Since neither the true state space nor the embedding map is observable, we test necessary local conditions across 39,291 sliding windows using four diagnostics: (i) injectivity across embedding dimensions, (ii) temporal coherence and noise stability, (iii) local stationarity of the stochastic flow, and (iv) Lipschitz stability of persistence-based summaries.

These diagnostics are evaluated across a range of candidate embedding parameters (embedding dimension $d$, delay $\tau$). These parameters are not treated as tunable hyperparameters; rather, they are selected based on reconstruction diagnostics. Evaluating them over a range of candidate values serves a dual purpose: (i) identifying admissible parameter regions consistent with valid delay reconstruction, and (ii) verifying that the observed topological and statistical properties are not confined to a single calibration.

Accordingly, the results provide an implicit sensitivity analysis, distinguishing parameter regions where the observable consequences of embedding theory are satisfied from those where reconstruction degrades.

4.2.1. Local Injectivity

Following Appendix A.3, injectivity was assessed via stabilization of topological summaries across candidate embedding dimensions $d=3,\dots , 10$, using Wasserstein distances between persistence diagrams, $L^2$ distances between persistence landscapes, and the Fill-factor across rolling windows and homology degrees $k=\mathrm{0,1}$ (

Table 2. Local injectivity test: Wasserstein distances consecutive persistence diagrams across embedding dimensions $d=3,\dots ,10$.

	Wasserstein Distances Across Embedding Dimensions
Homology	d = 3	d = 4	d = 5	d = 6	d = 7	d = 8	d = 9	d = 10
$H_0$	0.002788	0.002493	0.002318	0.00228	0.002355	0.002528	0.002754	0.003023
$H_1$	0.000411	0.000505	0.000563	0.000588	0.0006	0.000588	0.00058	0.000563

and

Table 3. Local injectivity test: $L^2$ distances consecutive persistence landscapes across embedding dimensions $d=3,\dots ,10$.

	${\mathit{L}}^2$ Distances Across Embedding Dimensions
Homology	d = 3	d = 4	d = 5	d = 6	d = 7	d = 8	d = 9	d = 10
$H_0$	0.002601	0.00282	0.002361	0.00174	0.002152	0.002265	0.001906	0.001577
$H_1$	0.000248	0.000305	0.000328	0.000409	0.00031	0.000259	0.000334	0.000287

). Wasserstein and $L^2$ distances remain uniformly small $\left(O\left({10}^{-3}\right)\right)$, well below the tolerance level $\epsilon =0.1$, with no systematic drift as $d$ increases. No material changes in $H_0$ or $H_1$ topology occur as additional coordinates are added. Importantly, the absence of systematic drift across embedding dimensions $d=3,\dots ,10$ indicates that the reconstructed topology is stable across a range of admissible dimensions, rather than being specific to a single choice.

Fill-factors for $d=3$ exhibit a mean normalized volume $\overline{V}=\mathrm{e}\mathrm{x}\mathrm{p}(-5.3055)\approx 0.0055$ (std. ~0.0041), indicating non-degenerate local neighborhood volumes and absence of geometric collapsing. These results, illustrated in Figure 9, support injective reconstruction and preservation of neighborhood structure.

4.2.2. Noise Regularity and Smoothness

Temporal smoothness was assessed via window-to-window variation $‖X_t-X_{t-1}‖$ for the candidate delays $\tau =1,\dots 10$. Mean variations remain near 3.6 × 10⁻³ with stable upper quantiles across all delays (

Table 4. Smoothness measured as window-to-window variation $‖X_t-X_{t-1}‖$.

	Window-to-Window Variation
$\mathbf{Delay} \mathit{\tau }$	Mean	Max	90th perc.	95th perc.	99th perc.
1	0.003566	0.028987	0.006046	0.007162	0.010146
2	0.003651	0.021807	0.005936	0.006948	0.009584
3	0.003653	0.022596	0.00593	0.006932	0.009563
4	0.003653	0.024183	0.00594	0.006932	0.009482
5	0.003652	0.021622	0.005939	0.006912	0.0096
6	0.003647	0.022339	0.005927	0.006961	0.009628
7	0.003652	0.022091	0.005932	0.006928	0.009602
8	0.003653	0.024198	0.005936	0.006911	0.009646
9	0.003653	0.023394	0.005919	0.006961	0.009653
10	0.003651	0.022713	0.005934	0.006938	0.00968

), indicating that the delay-embeddings evolve smoothly, consistent with Lipschitz-continuous behavior.

To operationalize $‖X_t-X_{t-1}‖ <\delta$, the smoothness scale $\delta$ was determined empirically using both autocorrelation decay and Mutual Information estimates (histogram-based and Kraskov estimators). Autocorrelation results (Figure 10a) are near zero for all $\tau =1-10$, with no clear $1/e$ monotone decay, no clear first minimum, or a meaningful first root, indicating $\tau =1$, is the minimal non-redundant delay. Kraskov Mutual Information (Figure 10b) remains tightly between −0.2407 and −0.2486 across delays, with the normalized Mutual Information fluctuating closely around 1.0.

The stability of smoothness diagnostics across delays $\tau =1, \dots , 10$, together with the absence of a pronounced minimum in Mutual Information, indicates that the embedding is not sensitive to variations in delays within this range.

4.2.3. Finite Dimensional Forcing, Low Dimensional Flow

Low-dimensional flow and local stationarity were assessed by monitoring the short-lag autocorrelation function (ACF) across consecutive sliding windows. A window was classified as locally stationary if the change in ACF satisfied $\left|\Delta ACF\left(\mathcal{l}\right)\right|\le {\delta }_{ACF}=0.15, \mathcal{l}\in \left[1, 5\right]$. Approximately 94–95% of the windows satisfy this criterion (

Table 5. Autocorrelation function (ACF) test results for monitored lag $\mathcal{l}\in \left[1, 5\right]$: Percentage of windows satisfying $\left|\Delta ACF\left(\mathcal{l}\right)\right|\le {\delta }_{ACF}=0.15$.

	Percentage of Windows |ΔACF| ≤ 0.15
Criteria	Lag 1	Lag 2	Lag 3	Lag 4	Lag 5
% Windows |ΔACF| ≤ 0.15	94.45	94.45	94.67	94.73	94.81

), indicating that the stochastic dynamics are locally stable and effectively low-dimensional over short intervals. These results support finite-dimensional forcing and smooth local evolution of delay coordinates along typical trajectories.

4.2.4. Lipschitz Stability of Topological Summaries

Persistence-based summaries were monitored using rolling bottleneck distances $W_{\infty }\left(D_t,D_{t-1}\right)$ and $L^2$ distances in Hilbert space. Following Cohen–Steiner et al. [58], rolling bottleneck distances $W_{\infty }\left(D_t,D_{t-1}\right)$ for $H_0$ and $H_1$ remain uniformly small across windows (means: $4.45\times {10}^{-4}$,$1.22\times {10}^{-4}$ and maxima: $7.8\times {10}^{-3}$, $1\times {10}^{-3}$) with occasional spikes coinciding with regime transitions. These excursions indicate bounded perturbations rather than structural instability (Figure 11a). $L^2$ distances between persistence landscapes exhibit similar pattern of results while being computationally more economical, solving a linear problem rather than an optimal transport problem (Figure 11b).

Similarly, $L^2$ distances between persistence landscapes $\lambda \left(D_t\right)$ and $\lambda \left(D_{t-1}\right)$ remain low (mean: $7.32\times {10}^{-4}$, $1.22\times {10}^{-4}$ and maxima: $2.98\times {10}^{-2}$, $1.81\times {10}^{-3}$), indicating smooth temporal evolution at the functional level (Figure 11b). Together, these results provide empirical evidence of Lipschitz continuous evolution of persistence-based summaries under time-local perturbations of the delay embeddings.

4.2.5. Summary of Local Validity

Overall, injectivity, noise regularity, finite-dimensional forcing, and Lipschitz stability diagnostics support the local homeomorphic validity of the delay embeddings. Neighborhood structure, adjacency relations, topological invariants, and smooth local evolution are preserved over finite windows, consistent with Theorem 5 [32]. This provides empirical evidence consistent with conditions under which persistence-based clustering and regime identification are expected to be meaningful under jumps, heavy tails, and regime shifts.

4.3. Jacobian-Based Test of Local Diffeomorphism

We next evaluate whether the reconstructed delay-coordinate maps satisfy the observable differential consequences of diffeomorphic embeddings implied by Theorem 6 of Stark et al. [32], for the synthetic stochastic, heavy-tailed, and regime-shifting price dynamics. Since the true dynamics are unobservable, testing is conducted through window-wise diagnostics evaluating (i) local invertibility, (ii) bounded geometric distortion, and (iii) smooth temporal variation in the Jacobian field associated with the reconstructed dynamics.

Table 6. Window-wise Jacobian diagnostics of observables Theorem 6.

Diagnostic	Windows $\mathit{p}>\mathit{\eta }$	$\mathbf{Mean} \mathit{p}$/Window	Window $\mathit{p}<\mathit{\eta }$	Rate $\mathit{p}<\mathit{\eta }$	Mean Per Window
$\mathrm{Local} \mathrm{invertibility} (\mathrm{full} \mathrm{rank}) {\sigma }_{min}\left(J_t\right)>{\epsilon }_J$	1.000	0.9982	13	0.0003	0.3439
$\mathrm{Bounded} \mathrm{distortion} \kappa \left(J_t\right)<K_{max}$$, K_{max}=500$	0.997	0.9952	125	0.003	31.7410
$C^1$$\mathrm{smoothness} Var\left({\sigma }_{min}\left(J_t\right)\right)<{\delta }_J$$, {\delta }_J=1.0$	1.000	-	-	-	0.0496
All conditions jointly	0.997	-	-	-

summarizes the results of the probabilistic, window-wise diagnostics.

Local invertibility, assessed via the minimum singular value ${\sigma }_{min}\left(J_t\right)>{\epsilon }_J$, is satisfied essentially by all windows, with a pass rate of 1.000 for the probabilistic threshold $p>\eta$ ($\eta =0.90)$. Lipschitz bounded distortion, quantified by the Jacobian condition number $\kappa \left(J_t\right)<K_{max}=500$, holds for 99.7% of windows. Temporal $C^1$ smoothness, assessed through the temporal variation in the minimum singular value, is satisfied by all windows. Joint enforcement of all three conditions yields a pass rate of 99.7%, with violations of invertibility or bounded distortion well below 1%, indicating that the probabilistic thresholds required by Theorem 6 are satisfied almost surely.

Overall, the Jacobian-based diagnostics provide numerical evidence that the delay-coordinate behaves as a locally diffeomorphic family: it preserves local invertibility, exhibits uniformly bounded local geometric distortion, and varies smoothly over time. Consequently, local metric structure—distances, angles, and neighborhood geometry—is not excessively distorted, supporting the applicability of metric-sensitive machine learning methods, including distance-based clustering and kernel methods.

4.4. Computation of Topological Functionals

Persistence diagrams were computed for all rolling windows using both the Vietoris–Rips (VR) and One-Pass K-Cluster Filtration (1PKF) pipelines, yielding 39,291 persistence diagrams per filtration. Functional summaries in $L^2\left(\mathbb{R}\right)$ were constructed for VR ($H_0$, $H_1$) and 1PFK ($H_0$). To quantify temporal variation in topological structure, the rolling deviation functional ${\varphi }_t^{wd\left(k\right)}$ was computed on $L_2\left(H_1\right)$ time series, instantiated at k = 1. Figure 12 reports the resulting trajectories of topological functional summaries for the VR-filtration.

4.5. Geometric Null Hypothesis Test

Geometric significance was assessed using Bobrowski and Skraba’s [36] topological null model on the full ensemble of persistence diagrams (39,291 diagrams) and a 10% random subsample (4000 diagrams). This test evaluates whether individual birth–death pairs arise from genuine geometric structure or are statistically indistinguishable from noise, using the universal distribution of the persistence ratio $\mathcal{l}$ as a benchmark.

For the VR-filtration, mean p-values across all settings are well above the rejection threshold $p\le \alpha =0.1$, with average values of approximately 0.67 for $H_0$ and 0.56 for $H_1$ (

Table 7. Geometric Null Test for VR-filtration $H_0$ and $H_1$ homology dimensions (full data ensemble and 10% sample).

	${\mathit{H}}_0$	${\mathit{H}}_1$	${\mathit{H}}_0$	${\mathit{H}}_1$
Ensemble	Full	Full	Sample	Sample
Number diagrams tested	39,291	39,291	4000	4000
Total ℓ-values computed	667,947	114,785	68,000	11,772
Mean p-value	0.6704	0.5557	0.5704	0.5555
Signal points α = 0.1 (uncorrected)	0	133	0	9
% Signal points (uncorrected)	0%	0.1%	0%	0.1%
Signal points α = 0.1 (Bonferroni)	0	0	0	0
% Signal points (Bonferroni)	0%	0%	0%	0%

). After Bonferroni correction, no persistence points are identified as significant in either homology dimension. Uncorrected tests yield at most ~0.1% signal points for $H_1$. Results for $H_0$ persistence diagrams from 1PFK-filtration are qualitatively identical, with a mean p-value ~0.57, well above the rejection threshold $p\le \alpha =0.1$. Figure 13 shows that the empirical distribution of $\mathcal{l}$-values closely match the left-Gumbel distribution predicted under the null. Overall, these findings indicate that individual persistence points are largely indistinguishable from geometric noise.

4.6. Temporal Null: Global Autocorrelation Test

Temporal organization was tested via autocorrelation tests applied to the rolling deviation functional ${\varphi }_t^{wd\left(k\right)}$, using $N_{surr}=500$ surrogate realizations generated under the temporal null model. Tests were conducted for both VR-filtration using $L^2\left(H_1\right)$ and 1PFK-filtration using $L^2\left(H_0\right)$. For both filtrations, absolute deviations ${a\varphi }_t^{wd\left(k\right)}$ are consistent with the null $\left(p=1.0\right)$, while signed deviation functional ${s\varphi }_t^{wd\left(k\right)}$ shows a small but statistically significant mean bias $\left(p=0.002\right)$, large $T_{max}$ and $T_{sq}$, and the large number of FDR-significant lags (

Table 8. Temporal topology null test: autocorrelation test VR-filtration.

Functional	Statistic	Observed	Surrogate Mean *	Surrogate Std. *	z-Score	p-Value	Significant Lags	Statistical Significance
$\mathrm{Signed} {s\varphi }_t^{wd\left(k\right)}$	Mean	1.36 × 10−7	−1.38 × 10−9	±2.07 × 10−8	6.64	0.0020		Yes
	$T_{max}$	0.7950	-	-	-	0.0020	$\left[1,\dots ,44\right]$	Yes
	$T_{sq}$	2.1823	-	-	-	0.0020		Yes
$\mathrm{Abs}. {a\varphi }_t^{wd\left(k\right)}$	Mean	1.55 × 10−4	1.69 × 10−4	±1.64 × 10−7	−80.6	1.000		No
	$T_{max}$	0.7361	-	-	-	0.0020	$\left[1,\dots ,50\right]$	Yes
	$T_{sq}$	1.6291	-	-	-	0.0020		Yes

* Surrogate distributions obtained via $N_{surr}=500.$

and

Table 9. Temporal topology null test: autocorrelation test 1PFK-filtration.

Functional	Statistic	Observed	Surrogate Mean *	Surrogate Std. *	z-Score	p-Value	Significant Lags	Statistical Significance
$\mathrm{Signed} {s\varphi }_t^{wd\left(k\right)}$	Mean	1.19 × 10−7	−3.090 × 10−9	3.27 × 10−8	3.74	0.002		Yes
	$T_{max}$	0.9035	-	-	-	0.002	$\left[1,\dots ,47\right]$	Yes
	$T_{sq}$	4.4984	-	-	-	0.002		Yes
$\mathrm{Abs}. {a\varphi }_t^{wd\left(k\right)}$	Mean	1.53 × 10−4	2.251 × 10−4	2.41 × 10−7	299.79	1.000		No
	$T_{max}$	0.9031	-	-	-	0.002	$\left[1,\dots ,50\right]$	Yes
	$T_{sq}$	6.0798	-	-	-	0.002		Yes

* Surrogate distributions obtained via $N_{surr}=500.$

).

This combination of results indicates that, although topological deviations are modest in size, their temporal ordering is highly non-random. In particular, the persistence of significant autocorrelation across extended lag ranges reveals coherent temporal organization in the evolution of topological features. These findings provide strong evidence for structures, regime-dependent topological dynamics that cannot be explained by random temporal reshuffling alone.

4.7. Unsupervised Learning: Heavy-Tailed K-Means Clustering

Heavy-tailed K-means was applied to time series of topological functionals to assess whether regime structure can be recovered in a fully unsupervised manner. For the VR-filtration, clustering was applied to $L^2\left(H_0\right), L^2\left(H_1\right)$ and the rolling deviation functional ${\varphi }_t^{wd\left(k\right)}$. For the 1PFK-filtration, clustering was applied to $L^2\left(H_0\right)$. Figure 14 and Figure 15 display the synthetic price trajectory with ground truth regimes, together with the resulting cluster assignments obtained from heavy-tailed K-means. Despite operating solely on topological summaries, the clustering recovers a clear segmentation of the time series that aligns closely with the underlying regime structure. In particular, Cluster 1 corresponds predominantly to the ground-truth bear-market periods, while Cluster 0 aligns with the ground-truth bull-market regimes.

Descriptive statistics of the clusters (

Table 10. VR-filtration: Descriptive statistics of clusters extracted using heavy-tailed K-means.

	Windows	Mean Return 5 min	Vol. /Std.	Ann. Mean Return	Ann. Vol	Skew	Kurtosis
Cluster 0	26,543	5.093 × 10−7	0.001445	0.01	0.201096	−0.004068	0.341958
Cluster 1	12,728	−1.711 × 10−5	0.00195	−0.3362	0.274687	−0.003642	4.538515

and

Table 11. 1PFK-filtration: Descriptive statistics of clusters extracted using heavy-tailed K-means.

	Windows	Mean Return 5 min	Vol. /Std.	Ann. Mean Return	Ann. Vol	Skew	Kurtosis
Cluster 0	25,047	−0.000004	0.001400	−0.0786	0.196630	0.004704	2.416410
Cluster 1	14,245	−0.000008	0.001952	−0.1573	0.273702	−0.015169	4.981497

) show clear regime differentiation. For both filtrations, Cluster 1—aligned with bear regime—exhibits higher volatility and substantially higher kurtosis, while Cluster 0 displays near-Gaussian tails. These characteristics are consistent with the known construction of the synthetic data, which embeds regime-dependent changes in tail thickness and volatility.

4.8. Temporal Null Hypothesis Tests

We evaluate the statistical significance of the extracted clusters using three temporal null tests: (i) ground-truth alignment, (ii) unsupervised cluster-quality measures independent of labels, and (iii) tail dependence structure.

4.8.1. Ground-Truth Alignment

Cluster assignments were mapped to price points and compared with known ground-truth regimes using Total Accuracy (TA), Bear Recall (BR), and the Adjusted Rand Index (ARI). For both VR- and 1PFK-filtrations, all metrics significantly exceed their null expectations (p = 0.000,

Table 12. Temporal topology null test: ground-truth metrics.

	VR-Filtration			1PFK-Filtration
Metrics	Observed	Mean Null	p-Value	Observed	Mean Null	p-Value
Total Accuracy (TA)	0.8026	0.5708	0.0000	0.7545	0.5549	0.0000
Bear-Recall (BR)	0.7149	0.3231	0.0000	0.6951	0.3630	0.0000
Adjusted Rand Index (ARI)	0.3593	0.0001	0.0000	0.2501	0.0000	0.0000

). In particular, ARI values are far above the null mean $\left(~0.001\right)$, indicating strong, non-random alignment between topological clusters and true regime labels. Overall, these results indicate that the unsupervised, topology-based clustering reliably recovers the latent regime structure of the synthetic process, well beyond what is expected under temporal randomization.

4.8.2. Unsupervised Clustering Metrics

Independent of ground truth, cluster quality was assessed using Silhouette (S), Calinski–Harabasz (CH), Davies–Bouldin (DB), and Maximum Mean Discrepancy (MMD). Inter-cluster separation metrics (S, CH, DB, between-cluster MMD) are all highly significant relative to the temporal null (p = 0.000,

Table 13. Temporal topology null hypothesis test: unsupervised clustering metrics.

	VR-Filtration			1PFK-Filtration
Metrics	Observed	Mean Null	p-Value	Observed	Mean Null	p-Value
Silhouette (S)	0.5348	−0.0003	0.000	0.5472	−0.0005	0.000
Calinski–Harabasz (CH)	24,187.0	0.9391	0.000	28,823.3	0.9412	0.000
Davies–Bouldin Index (DB)	0.7553	570.32	0.000	0.7186	643.32	0.002
MMD between cluster separation	0.70709	0.0093	0.000	0.7596	0.0092	0.000
MMD within-cluster 0 self-similarity	0.00148	0.0093	0.5374	0.00121	0.0092	0.5391
MMD within-cluster 1 self-similarity	0.00160	0.0093	0.4609	0.00156	0.0092	0.4534

), while within-cluster MMD values are small and statistically indistinguishable from the null. Together, these results indicate strong regime separation combined with internally coherent cluster structure, implying that clustering reflects genuine temporal organization rather than random partitioning.

4.8.3. Upper- and Lower-Tail Dependence

Upper- and lower-tail dependence coefficients were compared against temporal null surrogate distributions. For the VR-filtration, clusters exhibit reduced lower-tail dependence relative to the null, indicating strong separation in downside risk, while upper-tail differences remain statistically indistinguishable from the null (

Table 14. Temporal topology null hypothesis test: lower tail-dependence.

	VR-Filtration			1PFK-Filtration
Metrics	Observed	Mean Null	p-Value	Observed	Mean Null	p-Value
Upper Tail-Dependence	0.034	0.0249	0.7850	0.0183	0.0249	0.4930
Lower Tail-Dependence	0.0127	0.025	0.1730	0.0230	0.0256	0.3300

). The 1PFK-filtration shows a qualitatively similar but weaker pattern. These results indicate that cluster separation is primarily driven by extreme left-tail behavior. Moreover, the reduced left-tail dependence relative to the null is incompatible with random temporal partitioning and therefore provides evidence that the clustering recovers genuine structure.

4.9. Cluster-Label Null Hypothesis Tests

We next assess whether the clusters are intrinsically meaningful (i.e., encode genuine regime information) rather than artifacts of the clustering or incidental temporal partitioning using a cluster-label null model. The tests focus on three complementary effects: feature-space separability, economic relevance, and empirical calibration.

4.9.1. Feature-Space Validation

Across both filtrations, non-parametric and distributional tests (Kruskal–Wallis, Mann–Whitney U, Kolmogorov–Smirnov, Anderson–Darling) strongly reject the cluster-label null ($p=0.001,$ 

Table 15. Feature-space statistical tests compared to the temporal null values.

		VR-Filtration			1PFK-Filtration
Tests	Metric	Observed	Mean Null	p-Value	Observed	Mean Null	p-Value
Kruskal–Wallis	L2_H0	25,007	0.9950	0.0010	27,241	0.9847	0.0010
Kruskal–Wallis	L2_H1	223.62	0.9636	0.0010	-	-	-
Kruskal–Wallis	${\varphi }_t^{wd}\left(1\right)$	504.01	1.0003	0.0010	-	-	-
Mann–Whitney U	L2_H0	1.69 × 108	8.39 × 105	0.0010	1.78 × 108	0.86 × 1010	0.0010
Mann–Whitney U	L2_H1	1.57 × 107	8.45 × 105	0.0010	-	-	-
Mann–Whitney U	${\varphi }_t^{wd}\left(1\right)$	2.36 × 107	8.68 × 105	0.0010	-	-	-
Kolmogorov–Smirnov	L2_H0	0.9990	0.0094	0.0010	1.000	0.0090	0.0010
Kolmogorov–Smirnov	L2_H1	0.1046	0.0093	0.0010	-	-	-
Kolmogorov–Smirnov	${\varphi }_t^{wd}\left(1\right)$	0.1196	0.0094	0.0010	-	-	-
Anderson–Darling	L2_H0	18,421.4	−0.0182	0.0010	19,029	−0.0212	0.0010
Anderson–Darling	L2_H1	313.89	0.0170	0.0010	-	-	-
Anderson–Darling	${\varphi }_t^{wd}\left(1\right)$	408.87	−0.0153	0.0010	-	-	-

). Thus, the clusters correspond to statistically distinct distributions of persistence-based summaries, indicating that they occupy disjoint regions in the reconstructed topological feature-space. This separation is robust across selected filtrations, persists across homology degrees, and is not driven solely by higher-order topological features, implying that the extracted clusters reflect genuine differences in the underlying stochastic dynamics rather than artifacts of label assignment or clustering geometry.

4.9.2. Economic Meaning Validation

We assessed whether the extracted clusters encode economically meaningful regimes using returns and volatility statistics against the cluster-label null. Location-based tests for returns (Kruskal–Wallis, Mann–Whitney) do not reject the null, indicating no systematic mean-return differences between clusters (

Table 16. Economic meaning statistical tests compared to the temporal null values.

		VR-Filtration			1PFK-Filtration
Tests	Metric	Observed	Mean Null	p-Value	Observed	Mean Null	p-Value
Kruskal–Wallis	Return	0.3691	1.0037	0.5774	0.3449	1.0173	0.8362
Mann–Whitney U	Return	638,142	837,865	0.5495	201,857	875,962	0.8691
Kolmogorov–Smirnov	Return	0.0490	0.0091	0.0010	0.0422	0.0090	0.0010
Anderson–Darling	Return	122.15	0.0163	0.0010	118.47	−0.0099	0.0010
Levene	Vol roll.	5646.48	1.1666	0.0010	4,008,94	1.0378	0.0010
Levene	Vol ann.	5648.48	0.9749	0.0010	4,008.94	1.0041	0.0010

).

In contrast, distribution-sensitive tests (Kolmogorov–Smirnov, Anderson–Darling) strongly reject the null $\left(p=0.001\right)$, indicating substantial differences in distributional shape and tail behavior. Volatility tests (Levene) also reject the null for both rolling and annualized volatility, indicating the clusters correspond to distinct volatility and tail-risk regimes.

Taken together, these results show that the clustering isolates risk regimes characterized by changes in dispersion and tail behavior rather than average returns, consistent with economically meaningful regime differentiation in heavy-tailed financial systems.

4.9.3. Empirical Calibration of Cluster-Label Tests

Empirical size and power analysis confirm that feature-space and distribution-sensitive economic tests achieve the correct size $\left(~0\right)$ and maximal power $\left(~1\right)$, whereas location-based tests exhibit zero power, consistent with the absence of mean effects (

Table 17. Empirical size and power summary across filtrations.

Test Class	Metric Type	VR-Filtration	1PFK-Filtration	Interpretation
Feature-Space	$L^2\left(H_0\right),L^2\left(H_1\right),$	Size = 0	Size = 0	Strong, filtration robust
		Power = 1	Power = 1	Geometric separation
Feature-Space	${\varphi }_t^{wd}\left(1\right)$	Size = 0	Size = 0	Persistent topological
		Power = 1	Power = 1	discrimination
Economic (Returns)	Kruskal	Size controlled	Size controlled	No systematic mean return
	Mann–Whitney	Power = 0	Power = 0	differences
Economic (Returns)	KS, AD	Size = 0	Size = 0	Regime-dependent distributional
		Power = 1	Power = 1	and tail behavior
Economic (Volatility)	Levene	Size = 0	Size = 0	Distinct volatility regimes
		Power = 1	Power = 1

). These results demonstrate that the clustering is robust across selected filtrations, and encodes genuine temporal–topological and economic structure, rather than reflecting random assignment or artifacts of the clustering procedure.

4.10. Left-Tail Power-Law Fitting and Distributional Analysis of Cluster Tails

Left-tail behavior of clusters returns was analyzed using the Clauset–Shalizi–Newman framework [57]. For the VR-filtration, Cluster 1 exhibits a substantially heavier left-tail $\left(\alpha =4.58\right)$ than Cluster 0 $\left(\alpha =7.44\right)$ indicating a much higher frequency of extreme negative returns (

Table 18. Power-law fit results for the left tails of VR-filtration clusters.

Test Class	$\mathit{\alpha }$	${\mathit{x}}_{\mathit{m}\mathit{i}\mathit{n}}$	KS Statistic	KS p-Value	AD Statistic	Ad p-Value	Power-Law vs. Exp (R2)	Power-Law vs. Logn (R2)
Cluster 0	7.4438	0.034	0.0430	0.577	1.146	0.290	−4.1133	−3.9712
							p = 0.016	p = 0.060
Cluster 1	4.5813	0.039	0.0269	0.953	0.441	0.803	5.9846	−0.9089
							p = 0.223	p = 0.406

). Goodness-of-fit tests support a power-law model for both clusters, though likelihood-ratio tests suggest Cluster 0 marginally favors a lognormal or exponential alternative, while Cluster 1 is well described by a power law.

For the 1PFK-filtration, results are consistent: Cluster 1 again displays a much heavier left-tail $\left(\alpha =4.95\right)$ compared to Cluster 0 $\left(\alpha =12.65\right)$ (

Table 19. Power-law fit results for the left tails of 1PFK-filtration clusters.

Test Class	$\mathit{\alpha }$	${\mathit{x}}_{\mathit{m}\mathit{i}\mathit{n}}$	KS Statistic	KS p-Value	AD Statistic	AD p-Value	Power-Law vs. Exp (R2)	Power-Law vs. Logn (R2)
Cluster 0	12.6486	0.035	0.0568	0.647	1.1284	0.257	−1.738	−2.741
							p = 0.018	p = 0.183
Cluster 1	4.9498	0.040	0.0234	0.963	0.3322	0.890	8.924	−0.422
							p = 0.098	p = 0.583

). Poer-law fits are validated by the KS and AD tests, with likelihood-ratio comparisons favoring power-law behavior. Figure 16 illustrates the left-tail CCDFs and fitted models.

5. Discussion

In deterministic dynamical systems, Takens’ theorem guarantees that delay embeddings faithfully reconstruct the global geometry of the underlying attractor. However, for stochastic time series, such guarantees do not hold a priori: stochastic forcing, heavy tails, and regime shifts violate the assumptions required by classical embedding theory. Stark et al. [32,33] extended embedding theory to stochastically forced systems, demonstrating that delay embeddings can reconstruct the underlying state geometry almost surely along stochastic trajectories, provided the stochastic forcing enters through a finite-dimensional, smooth family of maps, the embedding dimension satisfies $d\ge 2 \mathrm{d}\mathrm{i}\mathrm{m}\left(M\right)+1$ and the smoothness order satisfies $r\ge 1$.

Under these conditions, the delay-coordinate map is a $C^r$ embedding, i.e., an injective $C^r$ map that is a homeomorphism onto its image, yielding trajectory-wise probabilistic reconstruction of state-space geometry that preserves topological invariants, neighborhood structure, and adjacency relations. Since $C^r$ regularity cannot be directly verified from finite observations, we assessed through numerical studies whether synthetically generated financial time series—designed to mirror stylized facts of financial markets such as heavy tails, jumps, regime shifts—exhibit the observable consequences of $C^r$ embeddings in reconstruction space, rather than the unknown true flow.

These diagnostics indicate that delay-coordinates extracted from stochastic, heavy-tailed time series exhibit observable consequences consistent with $C^r$ embeddings, thus neighborhood structure, adjacency relations, and topological invariants are preserved, providing justification for the use of adjacency- and topology-based clustering, as well as other machine-learning applications that rely on neighborhood structure. Jacobian-based window-wise tests further show stable local invertibility, bounded geometric distortion, and smooth temporal variation, consistent with the trajectory-wise guarantees of Stark et al. [32] (Theorem 6). This confirms that distances, angles, and local metric structures are preserved sufficiently for the use of metric-sensitive machine learning approaches, including topology-based clustering and regime detection with theoretical guarantees.

At the same time, windows that fail Theorem 5 or 6 diagnostics reveal the framework’s natural failure modes. In such cases, delay embedding does not reliably preserve topology or neighborhood relations, and local geometric distortions become significant. Consequently, clustering or other machine learning procedures relying on adjacency, distances, or topological invariants should not be applied to these windows. This provides an explicit and falsifiable criterion for determining when the methods are valid, and when extreme jumps, short-lived regimes, infinite variance innovations, or abrupt structural changes compromise reconstruction.

Topological features derived from these embeddings provide a principled summary of the evolving stochastic flow. Locally, persistence diagrams are largely indistinguishable from noise, but low-lag autocorrelations reveal short-term temporal structure that propagates to mesoscopic and network scales, so that Hilbert-space-embedded functionals display persistence temporal dependencies, cross-cluster coherence, and clear separation in extreme left-tail behavior. Thus, local temporal ordering gives rise to higher-level mesoscopic and network structure, independent of higher-order homology dimensions.

Cluster-label null tests of feature-space separation and economic distinguishability confirm that the detected clusters encode intrinsic temporal and topological structure, rather than random assignments or artifacts of a particular clustering algorithm. Specifically, the observed clusters correspond to statistically distinct distributions of persistence-based summaries, occupying disjoint regions in the reconstructed topological feature-space. Distribution-sensitive economic tests further reveal distinct tail behavior and volatility regimes, while location-based tests confirm the absence of systematic mean-return differences consistent with the stochastic nature of the price process. Clustering outcomes are robust across both VR- and 1PFK-filtrations for recovery of genuine regime structure.

Taken together, these findings indicate that topological functionals computed on locally homeomorphic delay-embeddings from stochastic, heavy-tailed time series provide valid, dynamically interpretable features for adjacency- and topology-based machine-learning applications. They capture intrinsic temporal organization, encode economically meaningful distinctions, and provide a computationally efficient framework that bridges geometric interpretability and practical machine learning.

However, this numerical analysis is conducted on a single, carefully controlled synthetic environment. While this allows precise evaluation against known ground truth, it does not establish robustness across alternative stochastic specifications. Extending the framework to multiple data-generating processes, including variations in jump intensity, tail behavior, and regime persistence, is an important direction for future work.

Moreover, the present framework does not normalize the topological summaries as scale itself carries meaningful information: periods of elevated market activity simultaneously amplify connected components and loop structures, contributing to the temporal signal. Normalization would partially remove this effect, potentially obscuring the relevant dynamics. However, distinguishing scale-driven and structure-driven components—e.g., through normalization or orthogonalization—remains an open and relevant research question and a promising direction for further research.

A related direction for future research is to assess the role of operational thresholds and parameter choices, particularly for the Jacobian-based diagnostics. While the current framework identifies admissible regions for valid reconstruction through diagnostic pass criteria, a systematic mapping of parameter regions to diagnostic outcomes would provide a stronger characterization of robustness and boundary behavior. Developing such explicit sensitivity mappings would further strengthen the interpretability and practical implementation of the methodology.

This topology-based methodology serves as a blueprint for applications to empirical financial data and extensions to supervised learning, metric-sensitive tools (e.g., kernel methods, gradient flows), and geometry-aware approaches (e.g., manifold learning, geometric deep learning) using labeled regimes.