Bootstrap AdS Veneziano Amplitude with Arbitrary Kaluza-Klein Modes

We present a derivation of the first curvature correction to the AdS Veneziano amplitude for arbitrary Kaluza-Klein (KK) modes, using a bootstrap approach based on the world-sheet representation and AdS$\times$S formalism. Our results establish a universal formula for the first order curvature correction without considering any low-lying KK configurations. We give new predictions for Wilson coefficients in the low-energy expansion. In the high-energy regime, the amplitude exhibits a universal exponent independent of the external KK charges, providing a coherent picture of AdS stringy amplitudes in different backgrounds.