The thermodynamic floor of writing a clock's record: faithful time stamps cost at least a logarithm of the accuracy

Clocks need irreversibility twice: to tick, and to leave a record that a tick happened. The cost of the record is currently disputed: an experiment reports a readout dissipation nine orders of magnitude above that of the clockwork, and a comment argues that readout costs are not fundamental. Neither gives a lower bound. We derive one for an idealised but explicit readout model. First, in a collision model in which every moment is written into fresh environment units, the entropy production splits exactly into the distinguishability of successive recorded moments plus the correlation still shared with the system; the Bures length of the record then obeys a horse–carrot type bound Nmom ≤(2/π)√nΣ, which sharpens for uncorrelated records by a constant c∗∈[π2/(16 ln 2),1) that we bound analytically. Second, for ticks time-stamped into equilibrium memory units by local binary tests, we prove that a faithful record (missing at most a fraction 1/N of the ticks and adding at most 1/N false ticks per tick) at a resolution of μ slots per period costs at least (1 − 1N) ln[N(μ−1)] −h(1/N) in units of kB per tick. If the slot grid stays incommensurate with the clock period (a quantisation constant bounded uniformly in N), a first-order error budget shows that a read-out accuracy N requires μ ≳√N, so the cost grows as 3/2 ln N to leading order; Monte-Carlo-verified records follow this scaling. Writing a reliable record of time is therefore never free in this model, but its cost is only logarithmic in the accuracy, about 5–9 kB per tick forN ∼102–103. The cost of the reference time base itself is not included.