Rational profile pairs approach any admissible real rate

Every rate achievable by a real stationary pair is achievable by an integral one.

Let $a, b$ be strictly positive real ten-class profiles with $b \in \mathcal N$ and $a - b \in Y$, the two-dimensional kernel of the marginal map spanned by the displayed vectors $\sigma$ and $\tau$. Then for every

there are strictly positive integral profiles $\beta, \beta^{*}$ with the same nine-grade marginals, $Q\beta^{*} = Q\beta$, whose normalised partner is again stationary, $\beta^{*}/D \in \mathcal N$, and which already beat $V$:

The point is that $\mathcal N$ is cut out by two binomial relations, $b_2 b_7^{2} = b_4 b_5 b_9$ and $b_3 b_7 b_8 = b_4 b_6 b_9$, both homogeneous of degree three. Binomial equations are solvable for one variable in terms of the others, so the positive rational points of $\mathcal N$ are dense in its positive real points — one approximates eight coordinates freely and defines the remaining two by the relations, which then hold exactly rather than approximately. Homogeneity means no normalisation is needed to stay on $\mathcal N$. Together with the fact that $Y$ is spanned by two integer vectors, both profiles can be produced directly as integers on a common marginal fibre.

This is what turns the integral form of Theorem 5.3 into the real one: the rate is continuous in the profile on the positive orthant, so a strict inequality at the real pair survives the approximation.

Formalization note. The construction is explicit. At scale $n$ one takes ceilings $u_i = \lceil n b_i \rceil$ of eight coordinates, multiplies through by $u_7^{2} u_8$ to clear denominators, and sets the second and third coordinates to $u_4u_5u_9u_8$ and $u_4u_6u_9u_7$; the two relations then hold as identities in $\mathbb N$. The companion profile is obtained by adding integer multiples of the two kernel vectors, which changes neither the marginals nor the class-weighted total.