Below Zero, Audited: The Negative-Temperature Fork

In January 2013 a gas of potassium atoms in Munich was driven into a state its makers described as a negative absolute temperature. A year later Dunkel and Hilbert answered that no such thing can exist, because the thermometer was built from the wrong definition of entropy; a year after that Frenkel and Warren answered that the objection's own entropy fails the oldest test in thermodynamics. This paper audits the dispute and finds it is not an experimental dispute: both sides agree on every measurement. They disagree about which count of states deserves the name entropy. The Boltzmann (surface) entropy's slope turns negative past the middle of a bounded spectrum, so its temperature goes below zero; the Gibbs (volume) entropy's slope never does. Each fork's price is printed in full: what Boltzmann's entropy buys (the zeroth law, direct contact with population thermometry, the lattice experiment's fitted T = -2.2 J/kB) and costs (the Dunkel-Hilbert consistency, equipartition, and small-system charges); what Gibbs's entropy buys (exact consistency at all sizes, equipartition, positivity) and costs (failure to mark equilibrium or heat-flow direction, and a temperature diverging exponentially with system size in inverted states). A two-level toy computed by exact enumeration at N = 100 shows the fork happening: T_B = -1.18 where T_G = +43,157 at 70 percent excitation. The bookkeeper's position: a quantity whose sign depends on a definitional choice must be spent with the definition attached. Grade: FORK STANDS - DEFINITIONAL, BOTH PRICES PRINTED. This paper stands outside my work on the vacuum catastrophe; it draws nothing from that work and repairs nothing in it.