Prove the conjectured quadratic-logarithmic factorial defect

Establish that there exists an absolute constant C>0 such that the factorial defect D(n)=log(n!)−log g(n) satisfies lim_{n→∞}D(n)/(log n)^2=C, equivalently that g(n)=n! exp(−(C+o(1))(log n)^2).

Background

The authors formulate a conjecture about the precise asymptotic distance between g(n), the maximum number of facets of an n-dimensional 0/1-polytope, and the factorial scale n!. The conjecture predicts that the defect is asymptotic to a positive constant times (log n)2, which would place the true growth of g(n) closer to the paper’s construction than to the previously known upper bound of order (n−2)!.

References

We dare to formulate the following deliberately provocative conjecture.

\begin{conjecture}\label{conj:factorial-defect} There exists an absolute constant $C>0$ such that

\lim_{n\to\infty} \frac{\log(n!)-\log g(n)}{(\log n)2} =C.

Equivalently,

g(n)

n!\exp\left(-(C+o(1))(\log n)2\right)

n!\,n{-(C+o(1))\log n}.

\end{conjecture}

— Almost factorial many facets for 0/1-polytopes  (2608.27247 - Castillo et al., 27 Aug 2026) in Section 5, Final remarks, Conjecture \ref{conj:factorial-defect}