EFT (String) Tower Building

We develop a bottom-up framework to reconstruct the asymptotic spectrum of light towers in four-dimensional $\mathcal{N}=1$ effective field theories directly from their K\"ahler potential. Our construction uses the Integral Scaling Relation, which relates the mass scales of towers becoming light at infinite distance to the tensions of EFT strings, together with the Emergent String Conjecture applied recursively upon decompactification. These conditions organize the tower scaling vectors into a lattice generated by the EFT-string vectors and select the globally consistent tower polytopes. When applied to K\"ahler potentials arising from string compactifications, our algorithm precisely reproduces the known arrangements of towers and duality frames. We then classify the admissible polytopes for asymptotic K\"ahler potentials $K\sim-\log P(s)$, with $P(s)$ a homogeneous polynomial of degree at most seven, and show that certain apparently consistent K\"ahler potentials are incompatible with the assumed quantum-gravity constraints. For general polynomials, additional restrictions arise from gluing the tower arrangements across different growth sectors. Remarkably, every tower polytope allowed by our reconstruction can be obtained as a concrete slice of the polytope associated with M-theory on a Joyce $G_2$-manifold. This provides evidence for a form of string universality in which EFT strings act as the fundamental building blocks of the UV tower structure in the EFT perturbative regimes.