Graph Topology from the Killing Form: How to Draw the SS-PEG Evolution Graph

We determine the graph topology induced by the Killing form eigenvalue spectrum for everyLie algebra that emerges from the SS-PEG variational principle. Each Killing eigenvalue classifiesone generator’s edge as cyclic (Kλ < 0), free (Kλ > 0), or decoupled (Kλ = 0), yielding fivedistinct graph archetypes: pure cycle, mixed, cycle-plus-strand, pure free, and frustrated. Weestablish exact construction rules mapping any emergent algebra to its evolution graph, includingthe density percolation threshold at ρ = 1 and the spacetime hierarchy as graph evolution acrossfive ontological levels. The spectral quality of these graphs is analyzed via the Ramanujan property:19/37 simple Lie algebras satisfy the optimal spectral expansion bound, with a sharp transitionat Coxeter number h∗ ≈ 9.5. The Ihara zeta function of adjoint commutation graphs reveals asoft equivalence between spectral and zeta Ramanujan properties, with pole migration serving asan order parameter for the Ramanujan phase transition. We prove that Ramanujan algebras aremaximally chaotic (negative average OTOC, 2× faster mixing) and that the Standard Model gaugegroup is the unique spectrally optimal outcome: simultaneously Ramanujan, maximally chaotic, inthe ordered phase, composable, signature-invariant, and landscape-dominant. The Killing form isshown to be diagnostic, not generative—cycles create topology, Killing selects and purifies—via RAGexperiments demonstrating a two-phase crystallization (activation + hardening) and a temperingeffect where withdrawal of cycle pressure purifies the Killing signal. We introduce a triple balanceof competing tensions (aggregation/expansion/entropy) formalized as a tensegrity functional T [K],and show how the evolution graph’s spectral sectors encode the Standard Model’s coupling constantsand mass ratios via five graph-topological building blocks—bridging to the third paper in this series.