Lie Algebras from the Killing Metric: Gauge Symmetries as Emergent Consequences of Relational Geometry

We demonstrate that simple compact Lie algebras—the mathematical structure underlying allgauge symmetries in the Standard Model—emerge from a single variational principle on the Killingmetric, without imposing any algebraic axioms. Starting from random matrices with no algebraicstructure, the minimization of Killing metric tension TK = ∥K − Ktarget∥2 produces exact Lie algebras with 100 % success rate across all classical and exceptional families (SU(N), SO(N), Sp(2N),G2, F4, E6) and non-compact families (sl(2,R), so(3,1), su(2,2)). The algebraic structure (commutation relations, Jacobi identities, Casimir operators) emerges automatically—it is never imposed. AKilling-only ablation confirms that the non-commutativity tension is redundant: the Killing metricalone contains sufficient information to generate all algebraic structure. A density phase transitionreveals that algebraic closure is a step function: for SU(4) with 15 generators, closure is 100 % atfull density and exactly 0 % at any partial density. The Lorentz algebra so(3,1) emerges from thesame principle with indefinite Killing target, with signature (3+, 3−) exactly. Spacetime is not thearena—it is the pattern of non-compact directions in the emergent Killing form. Gravity emergesfrom entanglement thermodynamics following Jacobson (1995): the same relational density ρ thatcontrols the gauge symmetry phase transition also controls the emergence of spacetime curvature,with gauge forces and gravity emerging simultaneously at ρc ≈ 1. The framework predicts that“dark matter” is not a particle but the footprint of the Levi-Civita connection in regions of intermediate relational density, yielding testable predictions (filamentary topology, cored profiles) consistentwith recent ultrafaint dwarf galaxy observations.