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Predicting the densest random disc packing fraction is an unsolved paradigmproblem relevant to a number of disciplines and technologies. One difficulty isthat it is ill-defined without setting a criterion for the disorder. Another isthat the density depends on the packing protocol and the multitude of possibleprotocol parameters has so far hindered a general solution. A new approach isproposed here. After formulating a well-posed form of the generalprotocol-independent problem for planar packings of discs, a systematiccriterion is proposed to avoid crystalline hexagonal order as well as furthertopological order. The highest possible random packing fraction is then derivedexactly: $\phi_{RCP}=0.852525...$. The solution is based on the cell orderdistribution that is shown to: (i) yield directly the packing fraction; (ii)parameterise all possible packing protocols; (iii) make it possible to defineand limit all topological disorder. The method is further useful for predictingthe highest packing fraction in specific protocols, which is illustrated for afamily of simply-sheared packings that generate maximum-entropy cell orderdistributions.

The disc random packing problem: a disorder criterion and an explicit solution