We develop a complete mathematical framework for phylogenetic evolution grounded in Operational Geometry (OpGeom). Species emerge as equivalence classes of operational sequences threading through morphospace along memory gradients. Phylogenetic trees map species threading paths across billions of years, with bifurcations representing relational transformations rather than mere temporal events. We extend classical cladistics to incorporate merging (endosymbiosis, hybridization), reversibility (anamnesis), process primacy (species as pressure-induced fixed points), and relational bifurcation paths. Key predictions: convergence follows beauty gradients, living fossils occupy transcendental fixed points, adaptive radiations release accumulated potential, and evolutionary reversals proceed via dormant trait reactivation rather than de novo evolution. The framework unifies temporal and relational perspectives, revealing bifurcation paths as encodings of ecological, morphological, and behavioral relationship changes.