Everpresent Lambda Model 2 and DESI DR2: rare fits and a Bayes factor favouring LambdaCDM

In Model 2 of everpresent Λ (Zwane, Afshordi and Sorkin, ZAS) the dark energy fraction is Ωde = tanh Ω̂, with Ω̂ a Gaussian random function of ln a with amplitude α and coherence parameter μ. We found no test of it by other authors against BAO data since ZAS's own in 2018 (an exploratory check in the author's earlier note is superseded here; Sec. 1 of the paper). We draw 40,000 histories at each of 25 points of an (α, μ) grid and confront them with DESI DR2 BAO and Pantheon+ supernovae, refitting ωm and ωb (BBN prior) for each history. The protocol was committed before the formal runs (our own time stamps) and reviewed by a separate AI instance; after that review and pilot runs, DESI+Pantheon+ replaced DESI+Pantheon+ plus the CMB acoustic scale as the primary combination. Changes made after the first results (the sound horizon, the ωm range, the Bayes-factor computation and the final optimiser) are disclosed. Results. Model 2 is excluded by our pre-set rule at 6 grid points. At 15 points some histories come within Δχ² ≤ 4 of ΛCDM, at most 21 of 40,000. At ZAS's best-fit point, 178 of 400,000 histories do (4.5×10−4); adding the CMB acoustic scale θ* leaves 7. The fitting histories have H0 from 37 to 168 km/s/Mpc (median 57). Averaged over histories, the Bayes factor favours ΛCDM at every grid point, by about 37:1 at the least unfavourable one. That figure is the largest of 15 noisy estimates, uncertain by a factor of about 2 to 4 (bootstrap 95%: 17:1 to 142:1), and flagged unreliable because its effective sample size is below 10. The estimate with by far the largest effective sample size (55, against at most 12 on the grid), from 400,000 histories at ZAS's best-fit point, gives about 63:1. The priors are not matched: a wider h prior for ΛCDM than our [0.5, 0.9] would lower these odds, and restricting Model 2 to H0 in [50, 90] would raise them. A 2018 statement. ZAS wrote that at their best fit there is a 78% chance that Ωde is negative somewhere in 1.5 < z < 3. Under their printed model we find 57% (35% given Ωde ≈ 0.7 today). Of the readings we tried, only a process in redshift with μ doubled (not registered) gives about 78%. Their Fourier-space formula implies the doubled μ. Caveats. The CMB enters only through θ*. One registered control failed as registered: for ΛCDM the sampling path shifts θ* by only 0.028σ, but the control was evaluated where θ* is 8.7σ from the Planck value, which turns that shift into Δχ² = 0.49. The recombination treatment is a hydrogen-only three-level atom. This is a short data note. Code, run outputs, the protocol with its amendments and the review records are included. Version 2 is a text revision after a post-publication review by two independent AI (Claude) referees. No run or decision changed; three small diagnostics were run with deposited code on deposited outputs (the 400,000-history Bayes factor and its subset spread, an H0-filtered Bayes factor, and the failed control expressed as a θ* shift). It corrects the description of the protocol amendment, reinterprets the failed control, discloses an exploratory Model 2 check in the author's earlier Model 1 note, describes the earlier CMB test of Model 2 more accurately, and adds references; the Note on versions in the paper summarises the changes, and REVIEW_POSTPUB.md in the deposit lists every change. Version 1 (doi:10.5281/zenodo.23090683) stays available.