Light Relics from the Thermal Neutral-Pion Pole: A Cosmological Bound on the Invisible Width of the π⁰

A hot plasma maintains an equilibrium population of neutral pions, which can decay into light hidden-sector states. We bound the invisible width of the π⁰ from the energy such decays deposit in decoupled dark radiation. Using the inclusive decay source of the equilibrium pion gas, the tabulated degrees of freedom of the Standard Model plasma and the entropy factor F_s = 1 + (1/3) d ln g*s / d ln T, we find Γ_inv < 1.5 × 10⁻¹² eV, i.e. Br(π⁰ → inv) < 1.9 × 10⁻¹³, for ΔN_eff < 0.28 and a reheating temperature T_RH = 150 MeV, with a systematic uncertainty of about 5%. The energy yield peaks near 25 MeV and falls by six orders of magnitude between T_RH = 100 and 6 MeV; below T_RH ≈ 7.6 MeV the laboratory limit from NA62 is stronger. The bound assumes a fully invisible decay into stable, decoupled relics that are relativistic at recombination (m_X ≲ 0.05 eV). For the same channel and observational input, the heating of the bath by muon and pion annihilation after production makes the classic estimate of Ng too strong by about one third; for the semi-invisible channel we find Γ(π⁰ → γX)/Γ_γγ < 3.8 × 10⁻¹³. Contents of this record: (1) pion_pole_preprint.pdf, the manuscript; (2) pion_pole_code.zip, Python code (NumPy, SciPy, Matplotlib) implementing every equation of the paper, the input table of degrees of freedom from Husdal (2016), the script that generates the figures, and a test script that reproduces every number in the paper with stated tolerances (69 automated checks), with its expected output.