Tree-region bounds for hard-sphere Mayer coefficients
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For hard-sphere Mayer coefficients of order k≥2, seek a representation of k! times the coefficient magnitude as a sum of no-broken-circuit region volumes over spanning trees in every positive integer dimension. In dimension three the target tree bound is (17/32)^ν(T)κ^(k−1), with κ the unit-ball volume and ν(T) the largest disjoint packing of ordered forks a<b<c with tree edges ab and ac.
Why it matters and what counts as progress
Why it matters: It links combinatorial cancellation in the hard-sphere expansion to measurable geometry and explicit bounds on its coefficients.
What counts as progress: Establish the measurable tree-volume identity or improve the explicit three-dimensional suppression factor while retaining the stated fork statistic.
How the problem developed
- 1 Oct 1963 · Foundational result [1]
Oliver Penrose Developed convergence bounds for Mayer fugacity expansions, establishing an early analytic framework for their coefficients.
- 29 Aug 2026 · Reported result [2]
Hard-sphere NBC project authors Reported an exact no-broken-circuit region identity for hard-sphere Mayer cluster integrals and a dimension-three tree-region upper bound.
AI contribution
AI-assisted Lean proofs give a hard-sphere volume identity and bound
For k≥2 in any positive integer dimension, the magnitude of the kth hard-sphere Mayer coefficient, multiplied by k!, equals a sum of no-broken-circuit region volumes over spanning trees. In dimension three, each tree-region volume is at most (17/32)^ν(T)κ^(k−1), where κ is unit-ball volume and ν(T) is the largest packing of disjoint ordered forks a<b<c with tree edges ab and ac.
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AI-assisted Lean proofs give a hard-sphere volume identity and bound