The Strong-CP Census: Every Solution Class for the Small Angle, Its Price, and What the Live Searches Have Already Done to It

Quantum chromodynamics allows one more term than nature uses. The term carries a dimensionless angle, θ̄, which can sit anywhere from 0 to 2π and is pushed below 10⁻¹⁰ by the neutron electric dipole moment. No symmetry of the Standard Model sets it there. This paper is a census of the ways physics has proposed to pay for that small number, built under the discipline of the charge census in this program: the candidate set closed in advance, each class given its mechanism, its producing test, its price, and its live status against the searches as they stand in October 2026. The closed set has six entries: the Peccei–Quinn axion in its KSVZ family, the same axion in its DFSZ family, the massless up quark, spontaneous CP violation of Nelson–Barr type, parity solutions of left–right type, and the null entry, fine-tuning accepted as a prior cost. The census's result is a class statement: of the six classes in the closed set, one is dead, two are fully alive and untested in most of their ground, two are alive but cornered, and one is the baseline every other entry must beat. No class in the set is confirmed, and no class except the massless up quark is dead. The axion families carry the only producing test that is running at design sensitivity today, and they are alive in most of their mass range because the search has covered a few narrow bands, not the range. The massless up quark is dead, killed by lattice QCD at 26 standard deviations. Nelson–Barr and parity are cornered by their own quality problems: each moves the small number into a place where loops or Planck-scale operators can move it back. The single measurement that would most change this table is a haloscope detection of the axion at a measured mass and coupling. It would select one family, close the question this census grades, and turn the other four live entries into history.