Gamma Run

Both clocks are the real formulae, integrated live. The Earth clock runs t = d/v: past about 0.9c it barely improves, because you cannot beat light by much. The ship clock runs τ = t/γ with γ = 1/√(1 − v²/c²) — and γ climbs a wall near c, so the last few hundredths of the lever are worth years of your life while buying almost no calendar time. That asymmetry is the entire game, and the entire point of lesson 27.3.

The relay contract uses relativistic velocity addition: u = (v + u′)/(1 + vu′/c²). Throw the packet forward at 0.60c from a ship already doing 0.85c and it leaves at 0.96c — not 1.45c. No lever position, and no relay chain, ever crosses the dashed line at c (lesson 27.4).

The twin contract is the twin paradox with a scoreboard: you and the rival connect the same two events — departure and docking — along different worldlines, and the one who spends more of the trip at high γ ages less. There is nothing paradoxical left once you both stand on the station deck comparing logbooks (lesson 27.5).