The Toll at the Bench: Landauer's Price, Weighed in Four Laboratories

Every toll in physics is a claim that something cannot be had for free, and most tolls live on paper for decades before anyone pays one in public. Landauer's toll is the exception, and this paper is its receipt book. In 1961 Rolf Landauer stated the price of forgetting: to erase one bit of information — to take a memory that could be in either of two states and force it into one — a device must pay its surroundings at least k_B T ln 2 of heat, about 2.87 × 10^{-21} joules at room temperature — 0.0179 of an electron-volt. The claim sat in foundations arguments for fifty years, the smallest invoice in physics, until the laboratories learned to weigh zeptojoules. Since 2012 three different physical systems have paid the toll under measurement — a colloidal bead in a light trap, a bead in a feedback trap with a control run that paid nothing, and a nanomagnetic bit of the kind that actually stores the world's data — and a fourth family of experiments, the bench demons, has run the same ledger in reverse, converting information back into work at the same rate; and the books close: the mean erasure heat approaches k_B T ln 2 from above as the erasure is slowed, in no measurement to date settling below it beyond its uncertainty, and the excess over the toll is itself a priced, finite-time quantity that falls as the protocol slows. This paper prices the toll's ledger in both directions — erasure paying heat out, information engines taking work back — and prints the verdict thermodynamics has earned and rarely gets: a bound with laboratory receipts. It also prints, in the same breath, what the receipts do not cover: the toll governs the mean, not the single event; it prices erasure, not computation; and the demon's books balance only when information itself is entered as a line item, which is the toll's real content. Grade: PAID AND RECEIPTED — THE LEDGER CLOSES AT THE TOLL. This paper stands outside my corpus's vacuum ledger; it neither spends that ledger nor repairs it — though it notes, once and at portrait grade only, that the one entropy toll with lab receipts and the one my cosmology keeps meeting at the bounce are the same shape of law: entropy is never deleted, only transferred, and the transfer always has a price.