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

We study a closed Friedmann universe with a massless scalar field φ and a unimodular cosmological constant λ in canonical quantum cosmology, and compare two of the three clocks used in the flat case by Gielen and Menéndez-Pidal, the two that remain global in the closed model: unimodular time t and the scalar field φ. 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 (a structural equality that also holds in the flat case). 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 ln T over ranges that, for equal fractional spreads, differ by a factor 1 + r(q), where r(q), 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 (not proved here), 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; fixing such a boundary condition by a semiclassical phase is not new (Feinberg and Peleg 1995). Its expansion adds a term growing like v1/3. The results are largely structural; the paper states which are structural, which are inherited from the flat case, and which (the values of T and r) are specific to the closed model. The source package contains scripts that reproduce every number, a pre-registration with self-recorded hashes (not externally registered; it was hashed after only a classical gate had run), two amendments (the pre-registered hypothesis tests were not run), and summaries of the plan review and of four review passes, all by independent Claude (AI) instances. Computations, code and drafting were done with Claude. Note on versions. Version 2 supersedes version 1, which stays available. It follows an independent post-publication review in which the equations were re-derived and every number reproduced with an independent solver; no number or computation changed. Corrected: the description of Feinberg and Peleg (1995), which version 1 placed among “dust and scalar-field models” studied “with one clock at a time”; “the two global internal clocks used in the flat case”; the attribution of the endpoint table; the process wording (“hashed before any result”). Added: credit to Feinberg and Peleg, Gryb and Thébault, and Pawłowski and Ashtekar; earlier closed-universe tunnelling work; the spectral assumption in the title; and the fact that the pre-registration had gated the headline on two tests that were not run. Full list in the paper's Note on versions.