Emergence of a Primordial Vacuum Lattice from Bosonic String Field Theory

The emergence of macroscopic spacetime geometry from a fundamental quantum framework remains a central challenge in string theory. In this work, we investigate the non-perturbative vacuum structure of Bosonic String Field Theory (SFT) to derive a microscopic foundation for a geometric background. Utilizing Witten's cubic open string action alongside Effective Field Theory (EFT) techniques, we demonstrate that tachyon condensation mathematically necessitates the formation of a structurally locked, primordial geometric lattice ($\Psi_{\text{lattice}}$) rather than a featureless vacuum. We provide an analytical formulation and robust computational evidence demonstrating that, within the local Wilsonian EFT approximation, the heavy-mode backreaction structurally locks this crystalline phase into a stable topological attractor. Furthermore, a spectral analysis of the residual fluctuations reveals the exact emergence of a strictly massless vector field, physically trapped on the solitonic defects of the lattice. Exploiting the shifted BRST cohomology, we algebraically prove that this interaction is protected by an exact macroscopic $\mathrm{U}(1)$ gauge invariance (Ward-Takahashi identity). Crucially, the acoustic Nambu-Goldstone phonons of this continuous translation-breaking lattice dynamically act as Stückelberg compensators. These modes physically dress the intrinsic SFT tensor excitations to kinematically assemble the exact 5-degree-of-freedom composite multiplet required for a macroscopic massive spin-2 field. By subjecting this emergent kinematic structure to standard macroscopic EFT unitarity constraints (the ``No-Ghost'' conditions), we reveal that the action uniquely and inevitably collapses into the ghost-free Fierz-Pauli theory. By strictly confining our analysis to the bosonic sector, this framework offers a rigorous derivation of a rigid geometric substratum and its emergent gauge connection, providing the precise microscopic kinematic scaffolding from which macroscopic massive gravity dynamically emerges.