  In this paper, we explore the open string amplitude's dual role as a
space-time S-matrix and a 1D CFT correlation function. We pursue this
correspondence in two directions. First, beginning with a general disk
integrand dressed with a Koba-Nielsen factor, we demonstrate that exchange
symmetry for the factorization residue of the amplitude forces the integrand to
be expandable on SL(2,R) conformal blocks. Furthermore, positivity constraints
associated with unitarity imply the SL(2,R) blocks must come in linear
combinations for which the Virasoro block emerges at the "kink" in the space of
solutions. In other words, Virasoro symmetry arises at the boundary of
consistent 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-Nielsen
factor, monodromy relations which impose algebraic identities amongst the EFT
couplings. We demonstrate at finite derivative order that the intersection of
the "monodromy plane" and the EFThedron carves out a tiny island for the
couplings, which continues to shrink as the derivative order is increased. At
the eighth derivative order, on a three-dimensional monodromy plane, the
intersection 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 leads
us to conjecture that the four-point open superstring amplitude can be
completely determined by the geometry of the intersection of the monodromy
plane and the EFThedron.
