Axion Quality, Gravitational Instantons, and the Allowed Window for the QCD Axion

The Peccei–Quinn solution to the strong CP problem relies on a global symmetry, and quantum gravity is not expected to respect global symmetries. We quantify the resulting "axion quality" requirement across the full range of decay constants and combine it with quantum-gravity constraints to delimit the allowed window for the QCD axion. Starting from the two-flavour chiral potential, whose normalization we check against the lattice-informed next-to-leading-order result to 1% (χ^1/4=76.3 vs. 75.5 MeV), we derive in closed form the induced θ̄ generated by a Planck-suppressed operator of order n and solve for the minimum n consistent with the neutron electric dipole moment bound θ̄≲10^-10. The required order rises steeply and nonlinearly with the decay constant, from n≥9 at f_a=10^8 GeV to n≥14 at 10^12 GeV and n≥36 at 10^16 GeV, so that a protecting discrete symmetry must be at least Z_n of that order. We then observe that the quality requirement can be recast as a universal, f_a-independent bound on any Planck-scale instanton contributing with generic phase: its action must satisfy S≥202.7, a number fixed entirely by the ratio χ/M_Pl^4 and the nEDM limit. Combined with the axion Weak Gravity Conjecture in the form S≲ c_wh M_Pl/f_a, this yields an upper bound f_a≲1.2×10^16 c_wh GeV, complementary to and independent of the operator-counting analysis. Together with the SN1987A lower bound this closes the window to 4×10^8 GeV≲f_a≲10^16 GeV, or 0.5 neV≲m_a≲14 meV. We note that the protection burden is mildest precisely in the decade f_a∼10^9–10^11 GeV favoured by post-inflationary dark matter production, and that canonical string axions at f_a∼10^16–10^17 GeV sit at or beyond the quality boundary — a tension we argue is the sharpest phenomenological handle the Swampland program currently has on axion physics.