This paper constitutes the official published account of the now completed Flyspeck project and describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants.Expand

We correct Miyabe's proof of van Lambalgen's Theorem for truth-table Schnorr randomness (which we will call uniformly relative Schnorr randomness). An immediate corollary is one direction of van… Expand

Abstract Let $ \mathbb{B} $ be a $p$-uniformly convex Banach space, with $p\geq 2$. Let $T$ be a linear operator on $ \mathbb{B} $, and let ${A}_{n} x$ denote the ergodic average $(1/ n){\mathop{\sum… Expand

This dissertation develops connections between algorithmic randomness and computable analysis. In the first part, it is shown that computable randomness can be defined robustly on all computable… Expand

PACT is proposed, a general methodology for extracting abundant self-supervised data from kernel-level proof terms for co-training alongside the usual tactic prediction objective and applied to Lean, an interactive proof assistant which hosts some of the most sophisticated formalized mathematics to date.Expand

The paper contains a new type of randomness—endomorphism randomness)—which the author hopes will shed light on the open question of whether Kolmogorov‐Loveland randomness is equivalent to Martin‐Löf randomness.Expand

Doob's martingale convergence theorem for computable continuous time martingales on Brownian motion is studied, showing that a point is Doob random if its tail is computably random in a certain sense.Expand

A slight strengthening of Tao's theorem is proved, and it is shown that when the first bound is given by a continuous functional, the bound in the conclusion can be computed by a recursion along the tree of unsecured sequences.Expand

This work establishes basic computable equivalences, and shows that $L^1$ representations contain fundamentally more computable information than the other representations, but that $0'$ suffices to move between computable such representations.Expand