Pre-registration note: falsification conditions for Ω_Λ = tanh²(2√π/3) ahead of the Euclid DR1 cosmological results

This note fixes publicly, before the release of any cosmological product of Euclid Data Release 1, the conditions under which the framework Ω_Λ = tanh²(2√π/3) (anchor α = 2/27, Paper I, doi:10.5281/zenodo.19084839) and its running-vacuum extension (Paper II, doi:10.5281/zenodo.21595464) will be declared falsified, disfavoured, compatible or suspended. The arbiter is the first official Euclid DR1 cosmological-parameter analysis (LE3 products, expected mid-2027); the primary combination is Euclid + Planck CMB in flat ΛCDM, with a declared fallback ladder. Rules: R1 (static test, |Ω_Λ − 0.685529| ≥ 3σ → falsified) when the internal Euclid–CMB tension is below 2.5σ; R2 (dynamical test on w₀wₐ, with a single bounded suspension R2b passing the verdict to Euclid DR2 or DESI DR3) when it is above; R3 (running test: ν ∈ [0.02, 0.04] with the frozen Paper II pipeline). The note states that the Paper I v5 criterion has already formally triggered on DESI DR2+CMB (3.3σ), that the value is a retrodiction, and that the author waives any further discretionary suspension, change of combination, or reinterpretation of thresholds. A falsified outcome will be published as such on this channel.