The experiment whose
right answer is nothing
How do you prove that something never happens — that a magnet has no tipping point at all, at any temperature above absolute zero?
Give every site an arrow that can point anywhere in three-dimensional space, spread those arrows over a flat two-dimensional sheet, and a theorem forbids the obvious outcome. The sheet can never settle into a globally aligned state at any temperature above zero. Not "it aligns weakly" — it does not align at all.
The previous rung had the same prohibition and escaped through a loophole: its needles lived on a circle, and a circle admits vortices, so it got a transition of a different kind. An arrow on a sphere has no such loophole — a sphere has no way to trap a stable whirl. So this system has no ordering transition and no vortex transition. It has nothing. Nothing is the correct answer, and the job is to show it convincingly.
That is harder than it sounds, because on any lattice small enough to actually run, the arrows do look aligned. At a fixed cool temperature a 16 × 16 patch reports an average arrow of respectable length. Read that single number and you would happily announce a magnet. The trick is that the number is not a property of the material — it is a property of how small your sample was.
So the measurement is a ladder. Run the identical system at the identical temperature at three sizes and watch the average arrow shrink: 0.478, then 0.282, then 0.142, each step roughly halving. Meanwhile the energy per site stays put at about −1.15 at every size — the receipt that these are not three different states of matter, just three different amounts of room for the order to drain into. Extrapolate and the alignment goes to zero. The panel below runs all three sizes side by side, at one shared temperature, so you can watch the biggest one wash out.
left · the same experiment at three sizes, all at the same temperature, colour taken from the arrow's direction. At T = 0.7 the raw field is rough site to site — neighbours agree only about 57 % of the way, which the energy per site of −1.15 out of a possible −2 is telling you — so the panel colours by the local mean direction over a 3 × 3 block by default. That is a viewing aid and nothing else; toggle smooth the view off to see the unsmoothed field, and note that the measurement never touches either. Smoothed, every panel shows the same slow drifting eddies, and the biggest panel's average colour is the palest of the three: local agreement does not add up to global agreement · right · the running mean of the alignment |⟨S⟩| against sweep number, one trace per size, each settling on a visibly lower plateau. The inset plots those plateaux against 1/L: a line aimed at the origin. There is no temperature control on this panel on purpose — heat is not the variable, and a slider would invite you to hunt for a transition that does not exist.
Those are the promoted run — 2026-06-25, fixed T = 0.7, 20 000 sweeps per
size. An independent repeat on 2026-08-13, 285 seconds on CPU, reproduced the
same conclusion with third-digit differences: 0.473 → 0.290 → 0.142, step ratios
×0.613 and ×0.489, a slope against 1/L of +6.895, and energies per site of −1.162,
−1.150 and −1.148 at the three sizes — flat, as required. The gate the check actually
asserts is not a number at all but a shape: monotone_decreasing: true,
with a positive slope against 1/L.
The loophole this rung closes is the subject of the one before it: M08 — a transition with nothing to point at →