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In this paper, we explore the open string amplitude's dual role as aspace-time S-matrix and a 1D CFT correlation function. We pursue thiscorrespondence in two directions. First, beginning with a general diskintegrand dressed with a Koba-Nielsen factor, we demonstrate that exchangesymmetry for the factorization residue of the amplitude forces the integrand tobe expandable on SL(2,R) conformal blocks. Furthermore, positivity constraintsassociated with unitarity imply the SL(2,R) blocks must come in linearcombinations for which the Virasoro block emerges at the "kink" in the space ofsolutions. In other words, Virasoro symmetry arises at the boundary ofconsistent factorization. Next, we consider the low energy EFT description,where unitarity manifests as the EFThedron in which the couplings must live.The existence of a worldsheet description implies, through the Koba-Nielsenfactor, monodromy relations which impose algebraic identities amongst the EFTcouplings. We demonstrate at finite derivative order that the intersection ofthe "monodromy plane" and the EFThedron carves out a tiny island for thecouplings, which continues to shrink as the derivative order is increased. Atthe eighth derivative order, on a three-dimensional monodromy plane, theintersection fixes the width of this island to around 1.5$\%$ (of $\zeta(3)$)and 0.2$\%$ (of $\zeta(5)$) with respect to the super string answer. This leadsus to conjecture that the four-point open superstring amplitude can becompletely determined by the geometry of the intersection of the monodromyplane and the EFThedron.

Carving out the Space of Open-String S-matrix