Two Three-Body Problems: Chaos, the Efimov Tower, and the Anomaly that Discriminates

"The three-body problem" names two entirely different objects. The classical gravitational problem — three point masses under Newtonian attraction — is the origin problem of chaos: non-integrable (Bruns, Poincaré), solvable only by a uselessly slow series (Sundman), marbled into KAM tori and a chaotic sea. The quantum resonant problem — three particles tuned to a two-body resonance — is the Efimov effect: universal, exactly characterized, experimentally confirmed, organized by a single number s₀ ≈ 1.00624 into a clean geometric tower. Same name, same body-count, opposite character. This paper argues that the Euler lens e^((λ+i)θ) discriminates the two by a single structural criterion — is the shared scaling symmetry anomalously broken to a discrete residue? — and that this illuminates the general problem: three-body hardness is not a property of the number three but of the absence of a governing scale-invariant fixed point. Where such a fixed point exists (Efimov, resonant), an anomaly mints a fixed imaginary exponent Δ = −1 + i s₀, the lens lights up, and the problem is solvable. Where it does not — the gravitational problem, and, revealingly, the Coulomb helium atom, which is quantum-clean yet lens-dark — there is no stationary exponent, the motion is chaotic, and the lens is silent: the same absence read two ways. Chemistry supplies the control: the helium atom and the helium trimer are two three-body problems in one element, and the lens brightens on the trimer's Efimov excited state while going dark on the atom. The contribution, as throughout the series, is nothing numerical: a discriminator, and the observation that classical chaos and lens-silence share one root.