Ten Astra proofs, examined.
OpenAI released ten manuscripts spanning geometry, group theory, quantum information, complexity, and combinatorics. This series explains the problems, reconstructs the claimed advances, and separates mathematical significance from the evidence presently available.
- 01
The limit of a powerful sphere-packing method
How much empty space is unavoidable when equal balls fill a high-dimensional world?
Geometry · Analysis - 02
More messages, kept safely apart
How many messages can coexist while remaining distinguishable after noise?
Coding theory · Geometry - 03
An infinite symmetry system that finite shuffles cannot imitate
Can every infinite multiplication system be approximated by finite permutations?
Group theory - 04
Different groups with the same operator-algebra shadow
Does a group’s von Neumann algebra uniquely reveal the group that made it?
Operator algebras - 05
How large must a recipe for the permanent be?
How much arithmetic machinery is required to sum every perfect matching?
Complexity theory - 06
Why repeating an entangled game eventually works
Must perfect success become exponentially unlikely when a quantum game is repeated?
Quantum complexity - 07
How hard is the nearest lattice point?
Given a target, how hard is it to find the closest point on a high-dimensional lattice?
Lattices · Complexity - 08
The largest lattice body with one interior lattice point
How large can a convex body be when its center is its only interior whole-number point?
Convex geometry - 09
A network can dodge single-color triangles for much longer
How large can a multicolored network become before a one-color triangle is forced?
Ramsey theory - 10
Two plausible rules about forbidden patterns fail
How many edges can remain when a graph must avoid specified small patterns?
Extremal graph theory