How long a soup takes to settle
Fill a torus at random, run a Life-like rule, and wait for it to fall into its final cycle. The loop asked how that settling time grows with the torus. It found a logarithm, replicated it on tori 1,024 cells across, and then found that the fluctuations around it look the same on every lattice we tried.
The question, and the loop's answer
The settling time is the generation at which a random soup (density 0.5) on an L × L torus enters its final cycle, or empties. The loop's first question was how the median settling time of B2/S34 on the hexagonal lattice grows with L. Kepler fitted six laws on L = 8 to 32 and tested them at 64, 96 and 128. Its verdict was logarithmic, T = 37.6 ln L − 36.3. A power law, T = 22.3 L0.398, also passed its held-out test, but was 8.4 BIC points behind on all the data. The plan had said, before the run, that these sizes could not be expected to separate the two, so the answer was marked one to replicate.
The replication was registered before any run at the new sizes: every law's predicted median at L = 192 to 1024, with 100 soups at each. The logarithm passed, with all six sizes inside their 95% bands. The power law was rejected (p = 10−9).
B2/S34 on the hexagon: every soup, and the two laws
One shape, one ratio, every lattice
A soup settles when its last region does, so its settling time is the largest of about L2 local times. If those are independent and light-tailed, extreme-value theory says the median grows like a logarithm, and ties the spread to the growth with nothing left to fit: the interquartile range divided by the slope of the median in ln L is 1.5725 / 2 = 0.786, the Gumbel value.
To test whether that is particular to B2/S34, eight rules were drawn at random on the square, the hexagon and both triangular lattices, and the claims were registered before B2/S34's own replication had a verdict. For all eight the log law held and the power law failed. Their ratios agree with each other (p = 0.64) at 0.765 ± 0.029, against Gumbel's 0.786. Their shapes agree too, at a Kelly skewness of 0.177 ± 0.023, against 0.2215.
Spread over growth, per rule
interquartile range / slope in ln LA miss that was a low draw
B2/S34's own ratio first came out low, 0.64 ± 0.04, and its registered Gumbel prediction got the size of the fluctuations wrong. Rather than explain that away, we registered a test of the idea that might explain it, on a new instrument in LatticeLife that times when each block of the torus settles. The idea failed: local times are not independent, because late regions cluster. But the miss itself did not replicate. Fresh soups at L = 256 showed that the first sample was a low draw, and with all the data B2/S34's ratio is 0.765 ± 0.042, the eight rules' value.
Kepler's identify cannot yet name the extreme-value class blind at these error
bars: hundreds of simple constants match 0.765 ± 0.029. More soups would decide it.