This paper is the third in a sequence. The first paper motivates the existence of a first directed distinction. The second paper proves that the balanced-ternary alphabet {-1, 0, +1} is the unique minimal integer-valued state space capable of representing a directed transition intrinsically, without an external sign convention. Starting from that established alphabet, this paper asks what mathematical structure appears when the substrate is carried through successive structural and analytical completion demands. Under explicit requirements such as independent generators, metric comparison, symmetry-preserving operators, refinement, rotation, recurrence, and summation, a sequential ladder of constants emerges. The paper derives: i, √2, √3, √5, φ, e, π, ln 2, ln 3, ζ(2), ζ(3), γ, Catalan’s constant G, and the lemniscate constant ϖ. Each result depends only on structure or completion machinery already established. No physical interpretation is assumed or required. The claim is limited to this: once the stated succession of structural and analytical demands is made explicit, this particular derivation ladder is ordered, coherent, and non-arbitrary, and each constant enters at the earliest stage permitted by the available machinery.