Emergence of Classical Constants from a Minimal Recursive Triadic Framework

Five classical mathematical constants (π/4, the golden ratio φ, Euler’s number e, √2, and ln 2) are presented as limits of a single family of three-variable recurrences on a triple (N, D, C). The variables have fixed functional roles: negotiation (running state), definition/limitation (constraint), and contribution/integration (accumulation). Each branch is specified by an initial seed triple and a traversal rule for how (D, C) evolve. The five branches fall into three traversal classes: Advancing (Class A: external step), Self-redefined (Class B: internal transformation), and Fixed (Class C: constant parameters). The paper does not claim new series identities. Its contribution is structural unification: one three-role grammar from which all five classical mechanisms descend, together with proved structural results (diagonal invariance, swap symmetry, perfect-square denominators, offset consistency, parametric families, and uniqueness of the Class A update form under an explicit axiom set). The π/4 branch is set apart by seeds {1, 3, 5} and an external step Δ = 4. The other four branches use seeds from {0, 1, 2}. Two open conjectures address whether these are the only named limits at low seed complexity within a bounded rule class. A reference Python implementation and convergence data are included. Type: Mathematical preprint (structural unification / discrete dynamical systems).Version: 1.4 (12 July 2026).