When Does de Sitter Space Produce Boltzmann Brains? Metastability, Recurrence, and Quantum-State Capacity

The Boltzmann-brain problem contains a formation question logically prior to cosmological population counting: does an asymptotic de Sitter vacuum autonomously generate completed observational episodes at a nonzero rate? Causal-Relational Closure (CRC) separates observer-ready configurations, dynamical observation, and completed records, allowing that question to be stated without identifying static projection weight with event production. Three routes often invoked to motivate nonzero production are examined. First, false-vacuum metastability does not determine transition probability into an independently specified observer-ready sector; even the com-plete survival amplitude fixes only the spectral measure of the metastable preparation. Second, finite-dimensional quantum recurrence guarantees repeated access only to sectors contained in the dynamically accessible cyclic subspace. If a ready sector and a finite-duration completion history are accessible, both recur arbitrarily late, but recurrence alone does not supply an objective event interpretation. Third, finite Gibbons–Hawking entropy does not imply finite Hilbert-space dimension. An explicit infinite-rank state on an infinite-dimensional Hilbert space realizes any prescribed finite von Neumann entropy, while a finite-dimensional conclusion follows only after an additional capacity postulate identifying de Sitter entropy with the maximum entropy of the complete Type-I state space. Type-II1 static-patch constructions provide a physically relevant infinite-dimensional alternative. The resulting classification isolates the remaining Boltzmann-brain question as a structural property of quantum de Sitter dynamics rather than a consequence of metastability, recurrence, or semiclassical entropy alone