The Unattainable Degree: Absolute Zero's Price List

Every other address in physics is reached by going a distance. Absolute zero is reached by paying a price, and the price is the subject of this paper. I keep the third law of thermodynamics the way I keep all my books: as a ledger with two columns — what the law charges, and what the laboratories have actually paid. The charge, in its modern form, is no longer a proverb. Nernst's unattainability principle of 1912 — no process reaches zero kelvin in a finite number of steps or a finite time — was derived and quantified by Masanes and Oppenheim in 2017: for any cooling process at all, reaching exactly zero requires an infinite bath volume or an infinite worst-case work fluctuation, and with a radiation bath in three dimensions the lowest reachable temperature falls no faster than the inverse seventh power of the cooling time. Carried to the records, that scaling prints a price per decade: each factor of ten colder costs at least 1.39 times the time, or 1.78 times the work fluctuation at fixed bath volume, or ten times the bath volume at fixed work — finite at every step, and infinite in sum at zero itself. The payments on record are kept here in two separate currencies, because mixing them is the oldest error in this field: an effective expansion temperature of 38 picokelvin, paid by a falling rubidium condensate in Bremen in 2021, and a nuclear-spin temperature of 100 picokelvin, paid by two grams of rhodium in Helsinki — neither of them a bulk equilibrium temperature, and I print neither as one. Absolute zero is the one address in physics with a price list instead of a distance: every decade colder costs a printed multiple, and the receipts show the field paying it. Grade: PRICE LIST STANDS — FINITE PER DECADE, INFINITE AT ZERO. This paper stands outside my work on the vacuum catastrophe; it draws nothing from that work and repairs nothing in it, and it touches no claim of that program.