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Abstract We compare various ways of decomposing and decompactifying the string field theory vertex and analyze the relations between them. We formulate axioms for the octagon and show how it can be glued to reproduce the decompactified pp-wave SFT vertex which in turn can be glued to recover the exact finite volume pp-wave Neumann coefficients. The gluing is performed by resumming multiple wrapping corrections. We observe important nontrivial contributions at the multiple wrapping level which are crucial for obtaining the exact results.