The Single Breath Euler Universe: The dark sector, gravity, and entropy as one rotational phase

This capstone of the Euler Cosmos series recasts the model so that a single uniform rotation in the plane is its generator rather than a lens laid over it. The dark-energy sector is carried by one first-order law: a complex density Z rotating at constant rate Ω_N, whose real part is the observed density — a cosine in e-folds — and whose conserved modulus is the amplitude a spatially flat, zero-energy universe holds fixed. Projected onto the equation of state, the rotation is a Riccati equation with the single parameter Ω_N, and the harmonic oscillator of the earlier papers is recovered as its linearization. The three conjugacy classes of the Möbius group sort into the breath, the cosmological constant (ΛCDM as the degenerate Ω_N → 0 limit), and an excluded region, so the exclusion boundary is a transition of the group. The one zero-energy (Hamiltonian) constraint is read twice: as an amount, its residual — one part in 10^122 — appears as dark energy, entropy, and gravity; as a motion, it is a single traversal from a low-entropy dawn to a crunch that cannot close. A dedicated section derives the rotation from fundamentals — it is the unique modulus-conserving, scale-free, linear dark sector — and evaluates, as a graded conjecture, whether the breath frequency Ω_N and the dark-matter acceleration scale a₀ are two faces of one complex de Sitter modulus (the weld pins a₀ = cH/2π at KSS saturation but leaves Ω_N free). Throughout, the parameters are the DESI DR2 calibration (H₀ = 62.75 km s⁻¹ Mpc⁻¹, Ω_m0 = 0.3605, Ω_N = 1.4908, φ = 0.8624), and every claim is graded — forced, fits, or conjecture. The model is falsifiable now and under test, with pre-registered blades for DESI DR3, Euclid, Rubin, Roman, and Gaia.