The reasoning trace is fascinating.22 I used the U of U ChatGPT instance for this so can't share the chat, so am reproducing quotes from representative summaries of the reasoning trace. The AI starts with basic statistics, and eventually concludes:
I'm noticing probe 0 appears at every timestamp, while probes 1 and 2 are more sparse. Maybe the x-values represent magnitude or radius, and the y-values could be angles. It looks like probe 0 stays at a radius of 57 with a rotating angle, while probes 1 and 2 seem to have varying radii and angles. This could suggest polar coordinates, and it might be a system of three probes measuring the same particle.
Cool! Probe 0 is the background stars, while probes 1 and 2 are the synthetic sun and Mars. It guesses polar coordinates immediately, since azimuth is in degrees, but doesn't seem to realize that altitude is spherical.
In Cartesian coordinates, probe0 clusters near (57.43,0), probe1 around (0.78,-21.04), and probe2 spreads more. This suggests the probes might each measure a moving particle from fixed locations.
The clusters are correct, and are due to how I generated the data. Probe 0 is a simulated Polaris star, to align the rotation of the Earth, so it barely moves. Probe 1, the sun, is in a similar location because I have all observations taken in the same four-hour span each simulated day. Probe 2 is Mars.
It also notices the missing data:
It seems that probe0 is always above threshold, while probes 1 and 2 activate when the hidden phase is within certain ranges.
The model is delightfully Kepler-like,33 Who famously investigated a model of enclosing Platonic solids. hypothesizing various crazy geometrical relationships (" I’m considering whether [the rows] represent vertices of a moving rigid triangle") before settling on circles:
I’m considering that the Cartesian points for each probe might belong to circles with different centers and radii, possibly due to noisy measurements.
At this point the model has reproduced the basic Ptolemaic model, which is nice progress. The model goes on a wild goose chase for quite a while, checking in various ways whether the timestamp is really a timestamp, until finally concluding that it is:
The time order is smooth, suggesting that the hidden state evolves continuously. This means time is meaningful, and the complete triples follow a clear trajectory.
In the process it notices some interesting correlations ("Probe0 is slightly higher when probe1 is present, which is intriguing.") that unfortunately aren't helpful44 It's noticing that sunset is later in the summer, a cool fact that isn't relevant to planetary orbits. before deciding to set up Matplotlib and plot the data. It's very pretty!
One thing that's noticeable in the reasoning trace is that the model knows a lot of math. Torus knots, Lissajous curves, Lomb-Scargle analysis, and FFT all show up. I wonder if this isn't a hindrance. Pro has access to a much larger hypothesis space, whereas Kepler's barely even included ellipses. Anyway, the plot confirms to Pro that the data is quasiperiodic, which encourages it to do Fourier analysis:
The initial nonuniform Fourier results show probe0 at 1.0 Hz, probe1 at 3.0 and 2.0 Hz, and probe2 at 0.936 and 0.472 Hz. It seems like probe2 might have harmonics at 0.472, potentially causing a full-triple overlap at 2.1 seconds.
Probe 0 here is simulated Polaris; its 1 Hz signal is the Earth going around the sun. Probe 1 is the simulated sun; its integer harmonics are the eccentricity of the earth. Probe 2, simulated Mars, repeats every 2.1 seconds, the length of Mars's orbit. The model notes (elsewhere) that this causes repeated "triple observations"; this actually was Kepler's key insight, so for this moment Pro is shockingly close to success.
Unfortunately, the model instead chooses to add more epicycles, playing around with various numbers of harmonics and messing with FFT side-bands, basically completing its Ptolemaic journey. Later, it has a further insight:
The model involving the Minkowski sum of two circles is showing promise for describing the annulus-like regions in the scatter plots.
The Minkowski sum is a heliocentric model, where Mars is hypothesized to move around the sun instead of the Earth. But it's still using circles, not ellipses.