Chapter 4: Orbit-Seeded Generation and the No-Jump Principle

Chapters and established the signature Σ_τ = (α, π, γ, η, ω, ρ, <) and three kernel axioms: K1 (strict order on the generators), K2 (ω absorbs ρ), and K5 (ω is unreachable from below). We now introduce the two axioms that govern how ρ generates new objects from the non-ω generators. K3 (Orbit-Seeded Generation) declares that each generator g ∈ {α, π, γ, η} seeds an infinite orbit ray O_g = {g, ρ(g), ρ²(g), …}. K4 (No-Jump / Cover) asserts that ρ acts as the immediate successor within each ray — there is no gap, no skipping, no alternate path from ρ^n(g) to ρ^n+1(g). From K4, we derive that ρ is injective on each orbit ray (Proposition [prop:rho-injective]), a result that will be essential for establishing the countability and disjointness of orbits in Part II.