Does the signal's trajectory trace a persistent loop in reconstructed phase space — the signature of a smooth, simple limit cycle?
Delay-embeds the 1-D signal into phase space via a Takens reconstruction (delay τ = a quarter of the dominant FFT period, the classic clean-loop choice; points spread over the whole orbit), then runs a Vietoris-Rips filtration with ripser: imagine inflating a ball around every point and tracking when loops (H1 cycles) are born and then filled in. A smooth periodic signal reconstructs to a clean circle (S¹), so one loop is born early and dies late — high persistence. Noise, chaos, and symbolic streams have no smooth closed orbit, so any loops they form are short-lived filtration noise. Phase-space reconstruction is only meaningful for continuous-valued signals, so a low-cardinality (binary/symbolic) stream — a finite lattice whose "loops" are an artifact — returns 0 below 64 distinct values.
Persistence (death − birth) of the single most robust 1-cycle in the Vietoris-Rips filtration of the delay embedding, normalized to [0, 1]. A positive-control-backed detector (metric_roles, positive control = Sine Wave): smooth limit cycles top it — Clipped Sine (0.95), Sine Wave (0.92), Damped Pendulum (0.89), Triangle Wave (0.79), Lotka-Volterra (0.73), Van der Pol (0.71) — because each traces a clean closed orbit. Symbolic, genomic, and chaotic sources sit at the zero floor (57 of 298 sources read 0). Cross-domain by construction (it fires on limit cycles in both signal-synthesis and continuous-flows), so it ranks only mid-pack on the domain-grouped F-stat despite being a clean detector. Max |r| = 0.73 (Fisher Information, cross-geometry); survives both dedup passes. (Before 2026-06-05 this geometry emitted six aggregate persistence statistics from a lag-1 embed on the first 100 samples — which read symbolic-lattice cardinality, not topology, and ranked symbolic streams at the top; that artifact has been removed.)
| Source | Origin | Value |
|---|---|---|
| Clipped Sine | signal-synthesis | 0.9546 |
| Sine Wave | signal-synthesis | 0.9163 |
| Damped Pendulum | continuous-flows | 0.8862 |
| ··· | ||
| DNA SARS-CoV-2 | genomic | 0.0000 |
| DNA Dog | genomic | 0.0000 |
| DNA Chimp | genomic | 0.0000 |
Persistent Homology is the atlas's limit-cycle detector. It uniquely flags signals whose phase portrait is a single smooth loop — pure and clipped sinusoids, relaxation oscillators (Van der Pol), predator-prey cycles (Lotka-Volterra) — and stays at zero for anything without a closed orbit. It is the one lens that separates a smooth periodic orbit from a chaotic or stochastic trajectory by topology rather than by spectrum.