Uniqueness and Analytic Structures of Bosonic String Effective Amplitudes

We revisit the zero-transcendentality sector of bosonic string effective amplitudes with spin-1 external states, conjectured to correspond to a mass-deformed $(DF)^2$ theory, known as the $(DF)^2{+}\text{YM}$ theory. Imposing gauge invariance, locality, and cyclicity under minimal assumptions uniquely fixes a set of dimension-raising operators and leads to a recursive construction of amplitudes from Yang-Mills amplitudes in the $\alpha'{\to}0$ limit. At finite $\alpha'$, certain derivative operators dressed with gauge invariant and $\alpha'$-dependent factors, what we call $\textit{inverse operators}$, reconstruct the full bosonic string effective amplitudes, yielding compact expressions that universally factorize into tachyon-pole coefficients times Yang-Mills-Scalar amplitudes. This structure holds at arbitrary multiplicity and also extends to the amplitudes of the pure $(DF)^2$, $(DF)^2{+}\phi^{3}$ and $(DF)^2{+}\text{YM}{+}\phi^{3}$ theories.