# The Isostatic Conjecture

## TL;DR

This paper investigates the properties of jammed disk packings with varying radii, establishing minimal contact conditions and unique equilibrium stress dimensions, while exploring connections to circle packing theory and addressing a density conjecture.

## Contribution

It proves that generic jammed disk packings have minimal contacts and a single equilibrium stress dimension, and discusses related circle packing results and a counterexample to a density conjecture.

## Key findings

- Jammed packings have minimal contacts.
- There is only one dimension of equilibrium stresses.
- Counterexample to a density conjecture by Nazarov.

## Abstract

We show that a jammed packing of disks with generic radii, in a generic container, is such that the minimal number of contacts occurs and there is only one dimension of equilibrium stresses. We also point out some connections to packings with different radii and results in the theory of circle packings whose graph forms a triangulation of a given topological surface. We also point out a counterexample, due to F. Nazarov, to a previous conjecture that that triangulated packings with fixed numbers of disks with fixed numbers of disks for each radius claiming that such packings were the most dense.