Tunnelling of a closed quantum universe in unimodular and scalar-field time: identical barrier transmission, opposite eventual fates

We study a closed Friedmann universe with a massless scalar field phi and a unimodular cosmological constant lambda in canonical quantum cosmology, and compare the two global internal clocks used in the flat case by Gielen and Menéndez-Pidal: unimodular time t and the scalar field phi. Spatial curvature creates a potential barrier, present when k2λ2 < 8 (k the scalar momentum; units 8πG = ℏ = 1, unit comoving volume), between a recollapsing small universe and an ever-expanding one. (i) Both clock theories share one stationary Wheeler-DeWitt equation, in which λ plays the role of ℏ at fixed q = k2λ2. Since each clock's asymptotic probability flux is the Wronskian current times a constant, the barrier transmission T(q, λ) is by construction the same function in both clocks. We compute it from 10-172 to 0.50, with an exact flat-case gate, convergence checks and a WKB comparison. (ii) We argue that for states with the same distribution of the two conserved quantities, the tunnelling probability per encounter with the barrier is the same in both clocks. States matched instead by holding different variables sharp sample T differently, by a ratio r(q) that, semiclassically, does not depend on λ and grows like q-1/2 deep under the barrier; we compute it. (iii) Assuming the relevant spectra are absolutely continuous, the eventual fates are opposite: in unimodular time every state in the λ > 0 continuum eventually escapes and expands forever, while in scalar time every positive-frequency state eventually ends at the singularity. This follows from a RAGE-type argument and the endpoint structure of the two operators, which curvature does not change, so it is inherited from the flat case. (iv) The scalar-clock boundary condition at large volume must use the closed de Sitter WKB phase as its reference instead of the flat-case phase (the standard phase, not a new result); its expansion adds a term growing like v1/3. The results are largely structural; the paper states which are inherited and which are specific to the closed model. The source package contains scripts that reproduce every number, the pre-registration with its hashes, two amendments, and summaries of three blind reviews.