A Boltzmann Brain on a Maxwell Demon: Distribution Selection, Finite Reservoirs and the de Sitter Price of Fluctuation-Assigned Records

We distil a Maxwell demon to three ingredients and a ledger. The ingredients are a world variable X, a record M carrying mutual information I = I(X; M ), and a feedback rule λ(M ). The ledger says the extracted work is at most kBT I, and producing the record costs at least kBT I of dissipation. We assign a Boltzmann brain to this demon as the case in which the record is paid for in probability rather than dissipation. By Sanov’s theorem, a fluctuation that correlates a record with its world at I nats has probability e−I . This is the Boltzmann factor of the information’s free energy kBT I. Veridical fluctuated memories are therefore suppressed exponentially relative to false ones, and a fluctuated record performs exactly as a shuffled-record null. A Gaussian skeleton is used only for bookkeeping. The physics is carried by selecting among three distributions, labelled by their density-of-states exponent k: Boltzmann (configurational, k = 1), Maxwell–Boltzmann (volume-sampled kinetic energy, k = 3/2) and the Maxwellian flux seen at a door (k = 2). Power-optimal capture sizes follow s Q(k, s) and sit at s∗ = 1, 1.30 and 1.62 kBT . Tail capture wins only on Landauer-ideal efficiency and loses once detector cost is counted. The de Sitter static patch enters as a finite reservoir. To second order in interior mass, its cosmological horizon behaves as a bath whose effective heat capacity equals its entropy, SdS ≈ 3 × 10122 kB. It also imposes a hard Nariai cutoff, a first-law price e−E/kBTGH (about e−3×1069 per kilogram), and a measurement bandwidth starved to the Hubble rate. Pre-registered null schedules on surrogate hosts cross check these possibilities. Demons beat information-removal nulls on every host. Door sampling beats volume sampling at sorting thresholds of 1.5 kBT and above, and loses at 0.5 kBT . Finite reservoirs suppress tail capture and cut it off, and the effect vanishes only as the reservoir becomes infinite. The result is a bedrock of possibilities and exclusions for implementing a Maxwell demon with a Boltzmann brain.