Windowsill Lab · Field Explainer · M03

Every size is the same
picture, rescaled

Three magnets of three different sizes give three different curves — is there one rule that turns all three into the same curve?

0.131measured 2026-06-16
0.126repeat 2026-08-12
0.125exact 1/8
Three bell glasses of different heights on a windowsill, each holding the same swirling ember-and-bark pattern at a different scale, their three shadows resolving into one shared silhouette.
Illustration (AI-painted) — three sizes, one shared shadow

Run the magnet from M01 at three sizes and plot how magnetic each one is as you heat it. You get three curves. They have the same general shape but they don't lie on top of each other: the small lattice softens earlier and more gradually, the large one holds on longer and drops more sharply. Three sizes, three answers.

But they are not really three answers. There is a claim in this corner of physics that near a tipping point, size stops being a separate fact about the system and becomes just a change of units — that if you stretch the axes of each curve by the right power of the lattice width, all three land on one another exactly. Not approximately. On top of each other, one curve.

The stretch factor is the thing being measured. Multiply the magnetism by L^(β/ν) and shift the temperature axis by L^(1/ν), and there is exactly one value of β/ν that makes the curves collapse. Theory says it is exactly 1/8.

The panel below is that collapse, live. The slider starts deliberately off the answer. Drag it and watch three separated curves slide toward each other — there is one setting where they stop being three curves. Finding it by hand is the point: you are performing the measurement, and the residual readout is your own scoreboard.

Live · Wolff single-cluster · three sizes, one sweep SHORT CHAINS — THE GRADED COLLAPSE FIT L=16–48
β/ν = 0.200

left · the largest lattice, with the growing cluster lit in brass for one beat before it flips — that blob is the algorithm, visible. · right · three sizes, rescaled. Drag β/ν until the residual bottoms out; then compare where you landed against the exact 1/8.

settling…

0.131β/ν, promoted 2026-06-16 MEASURED
0.126repeat, 2026-08-12 MEASURED
0.122short proof run
2,267 swall time, the rerun MEASURED

The promoted run of 2026-06-16 measured β/ν at 0.131 over lattices from 16 to 48, with a collapse residual of 8.2 × 10⁻³. A 2026-08-12 repeat gave 0.126, and a short proof run gave 0.122 — all three bracket the exact 1/8. Three runs landing at 0.122, 0.126 and 0.131 around 0.125 is a stronger and more honest statement than any one of them alone, and it tells you the real precision without a fake error bar.

The collapse is measured over a short ladder of lattice sizes — 16 to 48 in the promoted run — and a wider ladder would tighten it. Three honest runs of this milestone returned 0.122, 0.126 and 0.131 against an exact 1/8 = 0.125; that spread, not any one run's internal residual, is the precision. This milestone uses cluster updates rather than single-spin ones, because a collapse needs data that is genuinely settled right at the tipping point, and the single-spin updater cannot deliver that there.
Elsewhere on the ladder: M01 — where the magnet gives up M02 — the peak grows on schedule M05 — change the grid, keep the law