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 and level truncation techniques, we demonstrate that tachyon condensation can dynamically favor the formation of a stable, primordial geometric lattice ($\Psi_{\text{lattice}}$) rather than a featureless vacuum. We provide analytical formulation and numerical validation of the thermodynamic and energetic stability of this lattice configuration. Furthermore, a spectral analysis of the residual fluctuations around this new lattice background reveals the natural emergence of a massless vector field. We show that this vector interaction is protected by a topologically induced $\mathrm{U}(1)$ gauge invariance, intrinsically generated by the lattice symmetries. By strictly confining our analysis to the bosonic sector, this framework offers a rigorous, ab initio derivation of an effective geometric substratum and an emergent gauge interaction directly from string microphysics, paving the way for consistent geometric Effective Field Theories (EFTs).