Are cooling and a clock one mechanism?

Speculative — an open question, not a coordinate.

The question. That cooling (driving a system toward its ground state) and a clock (a monotone counter of time) are one mechanism rather than two.

Discriminator (the gate). Name the shared theorem: a common structure — a monotone under one flow (entropy production, a Lyapunov function, Landauer's bound tying erasure to a thermodynamic cost) — that makes a clock a cooling process or the reverse, versus a metaphor pairing "time" and "temperature." Filed neighbours: the problem of time [[CASE-05]], the exponent-status line [[CASE-19]], and the error-signature sort lab/runs/020. Kept open and explicitly speculative.