Completed Observations and the Boltzmann-Brain Problem: A Causal-Relational Analysis

The Boltzmann-brain problem concerns the possible cosmological dominance of observers generated by rare fluctuations over observers arising from ordinary nonequilibrium histories. A causal-relational closure (CRC) formulation separates the existence of an observer-like configuration, the physical initiation of an observational episode, and completion of an observational relation under an admissible future contract. The resulting framework does not modify quantum probabilities or prescribe a cosmological measure. Several conditional results follow. Exact future-equivalence reduction preserves the full distribution of completed-event counts whenever counting protocols belong to the contract; descriptive compression therefore cannot remove genuine events. Stationarity alone neither establishes observer production nor excludes recorded transitions. Finite completion times enter cutoff counts through a Laplace transform of the delay distribution and cannot generically annihilate a positive production channel. An explicit formation–completion–loss model separates completion probabilities from production rates. For a specified non-recycling cosmological cohort, integrated completed-observation yields are controlled by the competition between physical-volume growth, vacuum depletion, and the age dependence of the event rate. These results sharpen necessary tests for proposed resolutions while leaving observer formation dynamics and cosmological measure selection as independent physical inputs. No independent observer-production rate or general resolution of the Boltzmann-brain problem is assumed.