Within a single cycle, does the signal rise faster than it falls (or vice versa) — temporal shape asymmetry, not harmonic content?
The companion to Isochronicity in the oscillator lens: where Isochronicity measures dynamics across cycles, this measures the shape within one. After a DC-only detrend, positive-going zero crossings mark cycles; per cycle it takes t_rise = |argmax − argmin| and t_fall = T − t_rise, and reports the median of |t_rise − t_fall| / T. A sawtooth (slow ramp, fast reset) scores high; a sine, triangle, or square wave is symmetric in this sense and scores ~0 — by anti-pattern rule 5 the name describes the measured quantity (temporal asymmetry), not "relaxation-oscillator detector."
Median per-cycle rise/fall asymmetry. Highest on ramp-and-reset shapes (Sawtooth 1.00, Tank Drain Cascade 0.99); 0 on symbolic streams with no cycles (Thue-Morse, Kolakoski 0.00). On binary or sparse-spike signals the same readout captures spike-position asymmetry (Champernowne 0.97, Copeland-Erdős 0.89) — broader than the name implies, documented after a 2026-05-28 probe.
| Source | Origin | Value |
|---|---|---|
| Sawtooth Wave | signal-synthesis | 0.9960 |
| Tank Drain Cascade | self-organized-criticality | 0.9946 |
| Champernowne | number-theory | 0.9714 |
| ··· | ||
| Thue-Morse | symbolic-dynamics | 0.0000 |
| Kolakoski Sequence | symbolic-dynamics | 0.0000 |
| Van der Pol Oscillator | continuous-flows | 0.0006 |
Separates ramp-like, asymmetric-cycle waveforms from symmetric ones (sine, triangle, square). Paired with Isochronicity, it completes the within-cycle / across-cycle decomposition of oscillatory shape.