  We review the effective field theory (EFT) bootstrap by formulating it as an
infinite-dimensional semidefinite program (SDP), built from the crossing
symmetric sum rules and the S-matrix primal ansatz. We apply the program to
study the large-$N$ chiral perturbation theory ($\chi$PT) and observe excellent
convergence of EFT bounds between the dual (rule-out) and primal (rule-in)
methods. This convergence aligns with the predictions of duality theory in SDP,
enabling us to analyze the bound states and resonances in the ultra-violet (UV)
spectrum. Furthermore, we incorporate the upper bound of unitarity to uniformly
constrain the EFT space from the UV scale $M$ using the primal method, thereby
confirming the consistency of the large-$N$ expansion. In the end, we translate
the large-$N$ $\chi$PT bounds to constrain the higher derivative corrections of
holographic QCD models.
