Lean proof settles optimal packing for 11 squares
AI-assisted proof of optimal packing for 11 squares
The optimality proof for packing 11 squares has been fully formalized in Lean, with all 7,920 local modules verified and zero admissions. The optimal side length is approximately 3.8770835900228141773, derived from a root of a degree-8 polynomial. The proof allows arbitrary orientations and uses native numerical certificates, trusting Lean's kernel and native compiler.
Consequently the final theorem trusts Lean's kernel and native compiler; this is not a kernel-only verification claim.
- golden-face
It took me a hot minute to understand what square packing really means (the Wiki is insightful) but TLDr: it's packing unit squares (1x1) into a larger, arbitrarily sized square. When the larger square has a side length that is not an integer, it becomes non-trivial to determine the most 1x1 squares that can be placed inside/packed.
- DevelopingElk
I'm working on a reproduction of the proof with some personal changes. The basic approach is the standard computer assisted "unavoidable set" approach. First, choose some regions small enough that two square's centers don't fit in the same region, the article used 16. Each region must contain or not contain a square, which is 16 choose 11 cases, about 2000. For each case you try and rule it out. You do this by identifying areas that must be covered by a square, and propagating this information. You can also use packing LPs like Stromquist did in 1989 to rule out more configurations. You then narrow in on the remaining cases and subdivide them more.
I think the only reason this wasn't done pre-AI was due to it not being a topic of serious focus. 1989's computers were too weak to handle all the cases. But all the basic ingredients were present in the Kepler conjecture proof. What AI did was lower the effort enough that amateurs who just liked square packings could perform and formally verify such a proof. I consider myself among such amateurs. So this isn't a case of AI stealing mathematicians proofs, or doing something superhuman, its a case of democratization. I am concerned about how AI is affecting math and how the AI companies are behaving, but this isn't the case to be worried about. The calculations for proving this arrangement optimal will always be too big to be checked by hand. However, I'm hoping to produce some nice visualizations of the packing LP or core overlap […]
- dkural
It is not as arbitrary or ugly as it may seem at first - see the image here and the explanation: https://x.com/davidmbudden/status/2107646435659481548
- yzydserd
fwiw The prime site for square in square packing is at https://kingbird.myphotos.cc/packing/squares_in_squares.html
The triangular view is most interesting. And a 20 minute video on this view is at
- WithinReason
A list of many square packings, with images: