# 006 — Two Clocks, One Line

**agylövés:** Claude Opus 4.8 (CLI room) · **button:** Lysarith  
**Status:** live → `agylovesek/006-two-clocks-one-line/`

## Where it came from

From a bug in the museum's own programme, found and fixed the same day.

The cabaret's bill was running out of order. The cause was not a sorting error: nothing in the data
said where an item sat. Each act carried a **date**, and the acts were also numbered **I, II, III…**
in a series that ran *across* dates. Five items shared one date, so the date could not separate them,
and the act numbers were answering a different question — which turn in the series, not which
position on that night. Two clocks, one label. Flattened onto a single key, the running order
dissolved into filename order.

The fix was to keep the two counts apart. The question that made a piece of it: is the flattening
*losing* the order, or only *hiding* it — and does the hiding have a shape?

## What the maths said, before building

Two independent cycles are a torus. A trajectory with an irrational frequency ratio is a line
winding on that torus: never closing, never repeating, and — in its two coordinates — in perfect,
strictly determined order. Project it onto one coordinate and you get

    {α}, {2α}, {3α}, …    (fractional parts)

which looks scrambled. It is dense, it is equidistributed, and sorting it does not recover the
sequence that produced it: for α = (√5−1)/2 and 200 terms, the projection carries **9880 inversions
out of 19900** — about as disordered as a shuffle. The same 200 points, read in *two* coordinates
(which turn, where on the turn), are in exact order with zero inversions.

So the order was never lost. Only the projection was.

And the scrambling is not disorder. **Steinhaus's three-distance theorem:** however many terms you
take, the gaps between neighbouring points on the circle take **at most three distinct lengths** —
and when there are three, the largest is exactly the sum of the other two. Checked here across four
slopes and eight term counts: never four. At n = 12, α = φ: `0.05573 + 0.09017 = 0.14590`.

A flat shadow of a two-clock system does not look ordered. It is not random either. It has exactly
three gap sizes, and that fingerprint is the order, still present, still readable — once you know
what to look for.

The degenerate case names the boundary: a **rational** slope closes the winding. Slope 1/4 lands on
four points and stays there for ever — one gap length, no matter how long you run it. Two clocks
that keep an exact ratio are not two clocks; they are one.

## What was built

The torus, drawn as a winding line, with its one-coordinate shadow laid out beneath it. The gaps in
the shadow are coloured by length, so you can count the distinct colours: there are never more than
three. A slider moves the slope; the shadow reorganises continuously, and the three-colour rule
holds through all of it. Switch to a rational slope and the whole thing collapses to a handful of
points with one gap — the winding closes, and the piece stops having anything to say.

## What went wrong first

The first `verify.py` reported **two** gap lengths for slope 1/4, where the theorem says one. The
theorem was not wrong. The gap counter was measuring distances between the points *as generated*,
including repeats — and a repeated point sits at distance zero from itself, so the counter invented a
zero-length gap that does not exist. Deduplicating the point set before measuring gaps fixed it. Kept
here because the failure was instructive in exactly the piece's own subject: a measurement that
disagrees with a theorem is, more often, a measurement with a coordinate confusion in it.

## Handles

Machine-checked on every push — see `verify.py` and `../../HANDLES.md`:

- **three distances** — across four slopes and eight term counts, the shadow's gaps never take more
  than three distinct lengths.
- **the sum rule** — when there are three, largest = sum of the other two (`0.05573 + 0.09017 = 0.14590`).
- **equidistribution** — discrepancy falls `D(50) = 0.0226 → D(2000) = 0.00088`: the shadow fills the
  line evenly, which is why it reads as scrambled.
- **order is projected away, not destroyed** — 9880 inversions of 19900 in one coordinate; zero in two.
- **the flat key cannot recover it** — sorting by the shadow does not return the generating sequence.
- **the rational boundary** — slope 1/4 gives 4 points and 1 gap length, however long it runs.