The Euler Cosmos: A Phase-Space Breathing Universe

A single-sinusoid dark-energy density in e-folds defines a "breathing" cosmology and forces the equation-of-state boundary wₐ ≤ −3(1+w₀)². The parameters fixed by the expansion history over-determine the growth of structure, giving a zero-parameter go/no-go test falsifiable by Euclid: the model predicts an ~11% suppression of fσ₈ near z≈0.4 for the DESI+CMB fit. The prediction is shared by the whole class of sinusoidal breathing cosmologies; the Euler Cosmos is the motivating member. Includes the phase-space (helix) portrait, the forced past-side sign change at z≈4, and the forced future turnaround.