Windowsill Lab · Field Explainer · M04

How much heat
the sheet drinks

If you measure the tipping point a second time with a completely different instrument, does it land in the same place?

2.275heat peak
2.280magnetism peak
2.2692exact
A shallow brass pan on a windowsill holding a sheet of scale-mail threaded with glowing seams between agreeing patches, heat shimmer rising off it.
Illustration (AI-painted) — the glowing seams between agreeing patches

M01 found the magnet's tipping point by watching its magnetism. That is one instrument. There is another, and it doesn't look at magnetism at all.

Push a little heat into the sheet and its temperature goes up a little. How much heat it takes is the specific heat — the same quantity that makes a pan of water slow to boil. Near the tipping point the sheet becomes unusually thirsty: energy poured in goes into rearranging patches of agreement rather than into raising the temperature, so the specific heat spikes. You can measure that spike without ever asking which way the needles point.

So there are two needles on this experiment, and they are pointing at the same thing from different directions. They are not unrelated quantities — both are second derivatives of the same underlying free energy — but they are separately measured, and a finite lattice does not force them to peak at exactly the same place.

Press sweep below and the panel walks the temperature from 2.0 to 2.6, painting both curves as it goes. Watch two humps rise over the same spot, and watch the two peak markers place themselves. Switch the left panel to the energy view while it runs: below the transition it is nearly dark, at the peak it is a blaze of shifting domain walls, above it settles into uniform static.

Live · square-lattice Ising · C = N(⟨E²⟩−⟨E⟩²)/T² SMALL LATTICE — THE REAL RUN USED L=128

left · the sheet by local energy — bright where a site disagrees with its neighbours, so the picture is the web of domain walls rather than the spins. · right · two needles: specific heat (moss) and susceptibility (ember), each scaled to its own maximum so they can be compared. Peak markers are placed by the run, not typed in.

idle — press sweep

2.275T_c from the heat peak MEASURED
2.280T_c from the magnetism peak MEASURED
0.2 %relative error, heat peak MEASURED
33 swall time on GPU MEASURED

The graded run — 25 temperatures across [2.0, 2.6], 40,000 sweeps each, on a lattice 128 across — put the heat peak at 2.275 and the magnetism peak at 2.280. The exact answer is 2.2692. Two separately measured quantities agreeing to within a couple of thousandths, neither told the answer, is the instrument checking itself.

What is calibrated here is the location of the peak, not its height. The exact theory predicts a logarithmic divergence with amplitude A = (2/π)(2/T_c)² ≈ 0.495, and a finite lattice cannot resolve that — so the graded claim is where the peak sits, full stop. Both peaks land slightly above the exact 2.2692 because the lattice is finite; that shift is expected. And the two curves are not independent in the deep sense — specific heat and susceptibility are both second derivatives of the same free energy — but they are separately measured, and on a finite lattice they need not peak at exactly the same temperature. Here they came out 2.275 and 2.280.
Elsewhere on the ladder: M01 — the first needle M02 — the peak grows on schedule M10 — two needles, one of them blind