Theoretical Derivation of the Activation Invariant ξ_c from O(6) Symmetry — Paper 2

We derive the activation invariant ξ and its universal threshold ξ_c from symmetry-preserving O(6) geometry. Using the guardrail set {2π, 1/π, 1/6, π/(2√3)} we construct a dimensionless strain coordinate ξ that maps heterogeneous observables (spectral gradients, scattering cross sections, rotational shear, dispersion metrics, weak-lensing amplitudes) onto a single response axis R(ξ). We show that any physically admissible activation family with a single inflection produces an OFF→ON transition at the curvature flip R''(ξ_c)=0; cross-domain datasets place ξ_c≈0.56. We prove ξ’s uniqueness under loop honesty, Hilbert causality, orthogonality fairness, and packing economy, and we provide worked examples, a falsifiability protocol, and predictions spanning atomic to cosmic scales.