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A<sc>bstract</sc> 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-Nchiral perturbation theory (χPT) 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 scaleMusing the primal method, thereby confirming the consistency of the large-Nexpansion. In the end, we translate the large-N χPT bounds to constrain the higher derivative corrections of holographic QCD models.more » « less
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