The ring is the numbers 0…n−1 with n prime. Start the token at 1 and keep multiplying by g mod n — each turn of the crank is one ×g, and the token walks the cyclic group ⟨g⟩ hidden inside the units.
Some g are primitive roots: their orbit visits every non-zero station, one grand cycle. Others trap the token in a short subcycle — and its length always divides the group order n−1. That is Lagrange's theorem, felt as a crank.
The lock pairs a public exponent e with its partner d = e⁻¹ mod (n−1). Scramble a letter by xᵉ, then unscramble by xᵈ: because xᵉᵈ ≡ x (Fermat), it lands home exactly. Pick an e that shares a factor with n−1 and no d exists — the message is lost.
That one identity, run on 600-digit primes, is RSA. This little brass ring is the whole idea at toy scale.
The orrery jammed mid-turn. The lesson continues without it — everything you have already done is safe.