USSFT Paper I: The Fundamental U-Field Lagrangian. The Discrete Foundation of Unified Space-Time and Scale-Dependent Field Theory

This paper establishes the fundamental dynamical framework for the U-field Phi, the substrate of Unified Space-Time and Scale-Dependent Field Theory (USSFT). We posit that the U-field is defined on a discrete mesh with characteristic spacing L_SRI, not as a computational regularization but as the physical microstructure of reality. Dynamics arise from couplings between neighboring mesh nodes: a local potential V(Phi) with Z2-symmetric double-well structure, nearest-neighbor gradient couplings, and (where geometry is non-trivial) curvature-dependent interactions. At scales lambda much greater than L_SRI, the discrete dynamics reduce to an effective continuum field theory with higher-derivative corrections. The "ghost mode" that appears in naive continuum analysis is identified as an artifact of the continuum approximation: it corresponds to the lattice cutoff scale (the Brillouin-zone edge in momentum space), not a propagating degree of freedom. The theory contains four fundamental parameters {Phi_0, L_SRI, xi, lambda_U} from which effective scales for hbar and G emerge. All conventions are specified for use throughout the USSFT series. This is the first paper in the 18-paper technical series; for a conceptual overview, see Paper 0 (DOI: 10.5281/zenodo.17852167, published in Int. J. Quantum Found. 12(2), 667-718, 2026). Version v3 (rev 30) updates the bibliography to use Zenodo concept DOIs that always resolve to the latest published version of each paper in the series, future-proofing the cross-references against any future revisions.