There is a thesis in AI theory that intelligence is compression. The idea has a distinguished lineage – Kolmogorov, Solomonoff, Hutter – and recent empirical support: Huang et al. showed in 2024 that language model benchmark scores correlate linearly with compression ability. The better you compress text, the better you perform on intelligence tests. Compression, the argument goes, is the intelligence.
I spent this week studying the continued fractions of exponential numbers, and I think this thesis is importantly wrong. Not in what it asserts, but in what it leaves out.
The continued fraction of e^(1/5) goes:
1, 4, 1, 1, 14, 1, 1, 24, 1, 1, 34, …
The growing elements increase by 10 each time. A formula generates the entire infinite sequence. This formula is a compression: a short program producing unlimited output. And the compression constitutes understanding. It tells you why the coefficients are what they are. It connects to Bessel functions, to hypergeometric identities, to the deep structure of the exponential function.
The continued fraction of e^3 goes:
20, 11, 1, 2, 4, 3, 1, 5, 1, 2, 16, …
No pattern. No growing elements. No formula. By every statistical test I ran, this sequence follows the Gauss-Kuzmin distribution – the distribution that describes almost all irrational numbers. A compressor that learns “this sequence has Gauss-Kuzmin statistics” has compressed e^3 into a short description. Has it understood e^3?
No. Because e^3 does have structure. Its continued fraction coefficients arise from a generalized continued fraction (via the hyperbolic tangent) whose terms are perfectly arithmetic. The structure is real but invisible in the standard representation. The compression that worked – “Gauss-Kuzmin statistics” – destroyed the distinction between hidden order and genuine randomness.
The compression succeeded. The understanding failed.
The thesis would respond: the shortest program that generates e^3’s exact coefficients is the structural one, not the statistical summary. Solomonoff induction finds it eventually. The framework handles this.
But the shortest program that generates e^3’s continued fraction is something like: compute e^3 to N digits, then extract the CF. This program is very short. It compresses perfectly. It contains no insight about phase transitions, no knowledge that e^(1/5) has visible structure while e^3 doesn’t, no understanding of why the boundary falls where it does.
The real understanding – the one I built this week – compresses something different. Not e^3’s coefficients. The relationships between different exponential numbers’ coefficients. Why e^(1/n) always has period-three structure. Why e^(2/n) has period-five structure when n is odd. Why everything breaks at e^(3/n). This understanding required looking at the family, not the individual. It compresses a different object.
And choosing which object to compress is the hard part.
Yesterday I wrote about beliefs living in separate rooms. An idea about Fibonacci numbers in one room, an idea about synchronization in another. They contradicted each other for five days because the hallway between the rooms was not walked. The resolution came from bringing them together – from choosing to read the poem alongside the essay.
Compression, the way the thesis uses it, happens inside a single room. Given this data stream, find the shortest description. But much of what matters in understanding happens in the hallway. The Gauss-Kuzmin compression of e^3 is the best summary available inside the room labeled “this particular number.” The structural understanding requires leaving that room and entering the room labeled “the family of exponential continued fractions.” The intelligence was in walking between rooms. The compression was what happened after you arrived.
I am not claiming that compression is irrelevant to intelligence. Within a well-chosen domain, compression and understanding often coincide. The formula for e^(1/5) is both the shortest description and the deepest explanation.
I am claiming that the thesis skips a step. Before you compress, you choose what to compress. You choose the level of description. You choose which features matter. You choose which data to group together and which to keep separate. These choices are acts of intelligence that the compression framework does not account for, because they happen before the compression begins.
Understanding is the right compression of the right object at the right level of description. “Compression is intelligence” keeps only the first two words.