Data Analysis

If a signal repeats every p steps, then folding it onto a circle of size p should make the repeats line up. That is the idea behind coil closure as a periodicity descriptor — and this page is careful to treat it as a hypothesis to test against ordinary baselines, not a proven method.

Lesson Goal

Model a periodic signal as a phase coordinate on a finite circle, see how “closure” hints at the period, and understand why a clean synthetic win is not evidence about real data.

The Idea

Give each sample a phase coordinate: sample t sits at t mod p on a circle of size p. If p is the true period, every full lap returns to the same phase, and samples that are one period apart pile onto the same node. Folding the signal at the right p makes it look stationary; folding at the wrong p smears it.

The Lean-proved seed is just the finite phase-coordinate indexing (bounded, closes after a period, idempotent normalization). Whether coil closure is a good periodicity descriptor is an empirical question — which is why the fixtures compare it against autocorrelation and periodogram-style baselines.

Worked Example

A signal with true period 5, sampled and folded at p = 5:

phase(t) = t mod 5
t:      0 1 2 3 4 5 6 7 8 9 10
phase:  0 1 2 3 4 0 1 2 3 4 0    <- t and t+5 share a phase

Samples 0, 5, 10 all land on phase 0; the fold lines up. Now fold the same data at p = 4 and the phases drift instead of stacking — the wrong period does not close cleanly. The closure ranking uses exactly that contrast to score candidate periods.

Common Mistake

A clean synthetic periodic fixture does not prove a method works on real, noisy data. The multi-period fixture (APPD-B0002) is intentionally ambiguous — a warning against overclaiming. These fixtures are executable references, not proofs of real-data usefulness, statistical optimality, or superiority over ordinary baselines. See What “Proved” Means Here.

Checkpoint

A signal repeats every 6 steps. At phase coordinate modulus 6, which sample indices share a phase with sample 2? (Answer: 2, 8, 14, 20, … — every 6 steps.)

Source Trail

Paper source: Coil Data Analysis