The Frankl--Tokushige product conjectures for $r$-cross-intersecting families

We settle the uniform and biased product conjectures of Frankl and Tokushige for $r$-cross-intersecting families. Let $r\geq2$, let $0\leq k_i\leq(r-1)n/r$, and let $\mathcal{F}_i\subseteq\binom{[n]}{k_i}$ be $r$-cross-intersecting. We prove the sharp inequality $$\prod_{i=1}^r\frac{|\mathcal{F}_i|}{\binom{n}{k_i}}\leq \prod_{i=1}^r\frac{k_i}{n},$$ with equality attained by the corresponding levels of a common $1$-star. As a consequence, we obtain the analogous $p_i$-biased measure theorem for $0\leq p_i\leq(r-1)/r$, $$ \prod_{i=1}^rμ_{p_i}(\mathcal{F}_i)\leq \prod_{i=1}^r p_i.$$The main difficulty is that unequal parameters do not determine a single common target level; instead, the target levels $\ell_1,\ldots,\ell_r$ must satisfy $\sum_{i=1}^r \ell_i=(r-1)n$. We overcome this asymmetry in three steps. An ordered-partition coupling gives a sharp additive inequality for every such choice of target levels. A star-calibrated upper-shadow inequality relates the density of a family on its original level to the density of its upper shadow on a suitably chosen target level; it is proved by induction on $n$, with the induction step reduced to a two-point inequality. Finally, an analytic inequality shows that the resulting asymmetric additive estimate implies the required product bound. Perhaps surprisingly, the coupling captures all the combinatorial information of cross-intersection, reducing the remainder of the proof to an analytic argument.