A rational completely positive matrix with no rational nonnegative factor

An explicit integer matrix of order $444$ and rank seven is constructed that
is completely positive over the reals but has no nonnegative rational Gram
factor of any finite width. All its entries are strictly positive, and its
real cp-rank is seven. The construction gives a negative answer to the
universal boundary-factorization question. The obstruction is a rational
polyhedral cone in $\mathbb R^7$ on which a trace-zero quadratic form is
nonnegative and has exactly seven irrational zero rays forming an orthonormal
basis. The arithmetic seed lies in $\mathbb Q(\sqrt[3]{2})$. Rational interval
calculations establish its signs, and exhaustive integer computation
establishes all $444$ facets of the cone. The complete matrix, a compact
integer representation, and a standard-library Python checker accompany the
proof.