We present a complete bottom-up derivation of quantum mechanics and special relativity from a discrete spacetime lattice, with a rigorous framework for general relativity via induced gravity.The three-dimensional Schr¨odinger equation iℏ∂tψ = −(ℏ2/2m)∇2ψ emerges exactly from nearest-neighbor cellular automaton dynamics on a 4D hypercubic lattice in the con-tinuum limit. We prove Heisenberg uncertainty, quantum superposition, and entanglement as mathematical consequences of the discrete structure, with numerical validation to machine precision (< 10−14).Special relativity emerges when temporal spacing satisfies τ = a/c, yielding Minkowski metric ds2 = −c2dt2 + dx2 + dy2 + dz2 and mass-energy relation E2 = p2c2 + m2c4 from lattice dispersion.Newtonian gravity ∇2ϕ = 4πGρ is derived from variational principle on non-uniform lat-tices (Regge action), verified numerically to 90-95% accuracy. Complete GR tensor frame-work constructed with Schwarzschild solution verified to |Gµν | < 10−15.Most significantly, Newton's constant is analytically derived via induced gravity (Sakharov1967): G = −3π/[4Nf ln(am)] where a is lattice spacing, m is fermion mass, Nf is number of fermion species. With a ∼ ℓPlanck, m ∼ .46MPlanck, Nf = 3, we reproduce Gobs.The theory predicts quadratic Lorentz violation ∆E2 ∼ E4/E2QG with EQG ∼ 1016 GeV,testable with gamma-ray bursts.This represents: (1) first complete bottom-up QM derivation, (2) unified QM+SR emer-gence, (3) microscopic calculation of Newton's constant, and (4) falsifiable predictions at accessible energies.PACS: 03.65.Ta, 04.60.-m, 04.62.+v, 11.30.Cp.