rho_DE(T) through the QCD crossover and a conservative bound on Omega_GW in the PTA band

EN: Dark energy is evolved through the QCD crossover, yielding an extremely small gravitational-wave signal far below PTA sensitivity. This manuscript is part of a series of 17 papers (A–S). For a compact self-contained guide to the full program, including epistemic levels, falsification roadmap, and all DOIs, see the program summary: https://doi.org/10.5281/zenodo.19218767 Abstract Within a framework where dark energy obeys rho_DE = kappa G_N mu_*^6, we calculate its thermal evolution rho_DE(T) around the QCD crossover using lattice inputs for the effective hadronic scale mu_*(T) (CLS E250). The hadronic transition produces an O(1) change in mu_* and therefore a rapid "turn-on" of rho_DE as one moves to low temperatures, but the absolute magnitude of rho_DE(T) is always negligible compared with radiation at T ≈ 0.1–0.2 GeV. By treating the change in rho_DE(T) in a deliberately optimistic way as a source of cosmological tensor modes and using standard gravitational-wave formulas for phase transitions, we obtain an upper bound h^2 Omega_GW^peak ≲ O(10^-73), more than ~60 orders of magnitude below the PTA level observed by NANOGrav. The characteristic frequency falls in the nHz range, but the amplitude is strongly suppressed by the ratio rho_DE/rho_rad