The Standard Model of Particle Physics is currently axiomatic, relying on experimentally determined parameters for mass, coupling constants, and symmetry breaking potentials. The Selection-Stitch Model (SSM) proposes a background-independent, discrete vacuum geome- try based on a saturated tetrahedral lattice with coordination number K = 12. In this paper, we demonstrate that the fundamental Lagrangian of the Standard Model is the emergent continuum limit of this discrete geometry. We systematically derive: (1) The Klein-Gordon scalar sector from lattice tension; (2) The Dirac spinor interaction from topological braid defects, explicitly resolving the fermion doubling problem via non-bipartite symplectic topol- ogy; and (3) Yang-Mills gauge fields from stitch preservation requirements. Furthermore, we provide two falsifiable numerical predictions. First, using the integer topology of the unit cell (Surface 108 / Volume 1728), we derive a theoretical Higgs self-coupling of λ = 0.125, predicting a Higgs mass of 123.11 GeV (within 1.6% of experiment). Second, interpreting the event horizon as a lattice saturation boundary, we predict Gravitational Wave Echoes with a characteristic time delay of ∆t≈0.27s for a 60M⊙ black hole merger.