A Revision of the Proof of the Kepler Conjecture

Abstract

The Kepler conjecture asserts that no packing of congruent balls in threedimensional Euclidean space has density greater than that of the face-centered cubic packing. The original proof, announced in 1998 and published in 2006, is long and complex. The process of revision and review did not end with the publication of the proof. This article summarizes the current status of a long-term initiative to reorganize the original proof into a more transparent form and to provide a greater level of certification of the correctness of the computer code and other details of the proof. A final part of this article lists errata in the original proof of the Kepler conjecture.

Monte Carlo simulations of densely-packed athermal polymers in the bulk and under confinement

“…The established value is about 13.5% lower than the corresponding one for regular packings of non-overlapping hard spheres (<p reg ~ 0.7404). The later value, initially conjectured by Kepler and proven recently by Hales (Hales, 2005;Hales et al, 2010), corresponds to the packing density of the face-centered cubic (fee) lattice. While no such proof exists for the corresponding densest limit of random sphere packings the concept of the maximally random jammed (MRJ) state provides a precise mathematical and geometrical definition of the aforementioned state (Donev et al, 2005a(Donev et al, , 2005bTorquato et al, 2000).…”

Some Reflections on Mathematics and Its Relation to Computer Science

“…The fourcolor theorem, for instance, has been formalized in this way by Gonthier [18] and also Hales and others are now working for some years on the so-called FlySpeck project which aims at formalizing the computer-assisted proof of Kepler's conjecture [19]. The reason why people like Gonthier and Hales are working on such proofs is rooted in the uncertainty associated with lengthy computerassisted proofs.…”

“…The reason why people like Gonthier and Hales are working on such proofs is rooted in the uncertainty associated with lengthy computerassisted proofs. By means of formalization, it is believed that possible errors are excluded since every single logical step is (supposedly) verified by the proof assistant [19]:…”

“…Also here it seems precarious to believe that, in the future, mathematical knowledge will become completely formalized in some computer-based system. Besides the fact that there is a problem here of infinite regress, indeed, "who will control the controller ", 15 , "[i]t is a large labor-intensive undertaking to transform a traditional proof into a formal proof " [19]. Hence, just as one can wonder whether your average mathematician will really bother about non-rigorous mathematicsà la Zeilberger, one can wonder whether he/she will be bothered about formalizing their results.…”

Beyond Provable Security Verifiable IND-CCA Security of OAEP

“…Initially developed by logicians to experiment with the expressive power of their foundational formalisms, proof assistants are now emerging as a mature technology that can be used effectively for verifying intricate mathematical proofs, such as the Four Color theorem [16] or the Kepler conjecture [18,19], or complex software systems, such as operating systems [21], virtual machines [22] and optimizing compilers [24]. In the realm of cryptography, proof assistants have been used to formally verify secrecy and authenticity properties of protocols [26].…”