Canonical $O(3)$ continuum limit and exact $O(4)$ mass asymptotics

The explanation

Take two arrows of length one that can point anywhere in three-dimensional space. Let them prefer to line up, with a strength called β. Pick their directions at random, weighted by e raised to β times the dot product of the two arrows. The dot product is 1 when they align and −1 when they oppose. At β = 1, the average dot product works out to coth(1) − 1, about 0.313. So the two arrows are mildly aligned.

Now place such an arrow at every point of a flat square grid, with each neighbor pair preferring to align. This is the O(n) model, where n is the number of components of each arrow. For n = 3 it is the classical Heisenberg magnet. The question is how fast the alignment between two distant arrows fades with distance. In two dimensions there is no lasting order at any temperatureSource 1. But that leaves open whether the fading is exponential or slowerSource 2.

For n = 2 the fading can be slow, by a polynomial law at low temperatureSource 3. For three or more components, Polyakov's argument predicted exponential fading at every positive temperatureSource 4. Rigorous results covered only high temperature or very many componentsSource 5. Aru, Garban and Sepúlveda recorded the all-temperature question as open in 2025Source 6.

This family of five papers claims several answers. One paper proves exponential fading for every n of at least 3 and every temperatureSource 7. The bound is uniform over finite pieces of the grid with free edgesSource 8. For n = 4, another paper claims the exact rate at low temperature. The mass, the inverse of the fading length, behaves like 32·e^(π/4 − 1/2)·√β·e^(−πβ) as β growsSource 9. The prefactor 32·e^(π/4 − 1/2) is about 42.57. At β = 2 the formula gives a mass near 0.112, a fading length near 9 grid steps. At β = 3 the length is near 168 steps. These numbers come from the leading formula, which is exact only in the limit. A companion paper claims upper and lower bounds of the same shape, and a positive gap at every temperatureSource 10.

For n = 3, the remaining papers claim a continuum limit. Shrink the grid spacing while β grows, keeping a physical length fixed. The limit exists along every value of β, without choosing a subsequenceSource 11. It defines a genuine interacting quantum field theory with a positive mass gapSource 12. Its lightest particle shows up as an isolated point in the spectrumSource 13.

The single obstacle is the low-temperature regime. There the fading length is exponentially large, and the system looks ordered over huge distances. Earlier bounds by McBryan and Spencer gave only polynomial fading, which allows a zero exponential rateSource 2. A mass estimate on a fixed grid also does not by itself give the continuum theorySource 14.

How sure is this? A computer proof checker called Lean has a formal proof of exponential fading on finite grid pieces. It covers every n of at least 3 and every temperature. The infinite-grid limit, the O(4) mass results and the O(3) continuum construction lie outside that check. No specialist review of any of the five papers has been published.

What it doesn't show. This does not give a Lean check of the O(4) mass formula or the O(3) continuum theory, and it says nothing new about two-component spins.