  We develop a formalism to extract triple crossing symmetric positivity bounds
for effective field theories with multiple degrees of freedom, by making use of
$su$ symmetric dispersion relations supplemented with positivity of the partial
waves, $st$ null constraints and the generalized optical theorem. This
generalizes the convex cone approach to constrain the $s^2$ coefficient space
to higher orders. Optimal positive bounds can be extracted by semi-definite
programs with a continuous decision variable, compared with linear programs for
the case of a single field. As an example, we explicitly compute the positivity
constraints on bi-scalar theories, and find all the Wilson coefficients can be
constrained in a finite region, including the coefficients with odd powers of
$s$, which are absent in the single scalar case.
