Gravitational Waves and Vector Dark Matter from a Coleman–Weinberg Dark U(1) Sector: Internal-Consistency Audit and Corrected Phase-Transition Analysis

We audit, and independently recompute, the central predictions of a classically scale-invariant dark U(1) model in which a complex scalar obtains its vacuum expectation value through the Coleman–Weinberg mechanism and the massive dark photon A′ is the dark-matter candidate. An earlier unpublished draft by the author claimed a strongly first-order phase transition with a LISA signal-to-noise ratio of order 200, anti-correlated with a vanishing direct-detection rate in the limit of zero Higgs portal. Ten issues are identified (B1–B10), five of them critical: an inconsistent relic-abundance line, critical temperatures that are too low by a factor 3–9 and have the wrong coupling dependence, gravitational-wave peak frequencies inconsistent with the draft's own critical temperatures, an unresolved thermal history at vanishing portal coupling, and the ordering of freeze-out and the transition. With the relic line corrected to v_φ ≈ 1.69 g_D TeV, we compute the one-loop finite-temperature effective potential with full thermal functions and daisy resummation, the O(3) bounce action by shooting, and the nucleation and percolation temperatures. For g_D ≤ 0.55 the transition does not complete: the bounce action stays above the nucleation threshold down to the QCD epoch (by a factor of three or more for g_D ≤ 0.4, while g_D = 0.5–0.55 miss it by only 10–50% and are marginal), so the draft's low-coupling benchmarks are not viable. For 0.6 ≲ g_D ≲ 1 the transition completes with strong supercooling (T_p/T_c ≈ 0.003–0.42), β/H ≈ 19–135 and peak frequencies of 0.2–6.3 mHz, inside the LISA band. At g_D = 1 a sound-wave estimate including finite-lifetime suppression gives Ω_GW h² ≈ 2.3×10⁻¹² at 6.3 mHz and an indicative three-year SNR of about 120. The draft's central falsifiable claim therefore survives, but only in a restricted coupling window, and subject to an open thermal-history problem: at λ_p = 0 the dark and visible sectors never equilibrate. Code and data are provided.