The neutron–nucleus interaction in light few-body systems displays two longstanding anomalies unexplained by existing nuclear theory. First, the elastic scattering cross sections of 1H (20.4 b), 2H (3.39 b), and 3H (1.66 b) are essentially constant from 10−5 eV to 104 eV; since the de Broglie wavelength at thermal energies (∼1.8×106 fm) exceeds the Yukawa nuclear force range by six orders of magnitude, short-range interactions cannot account for this constancy. Second, the reaction cross sections of 3He, 6Li, 10B, and 7Be obey the 1/v law over nine decades of energy, yet their anomalously large effective interaction radii (272–1344 fm) and the complete absence of resonance parameters in the JENDL-5 library for these nuclides demonstrate that neither Breit-Wigner nor Lane-Lynn theory provides a microscopic foundation. We show that a long-range nuclear force V (r) = −NA/r4 resolves both anomalies within a single framework. The r−4 form is derived uniquely from the 1/v law by a contrapositive argument with no free parameters. The coefficient NA is determined from a single cross-section measurement with no free parameters (NA = 1.935×10−71 Jm4 for 3He). Two-dimensional time-dependent Schr¨odinger equation simulations reproduce both the 1/v reaction law and the energy-independent elastic scattering cross section simultaneously, in agreement with JENDL-5 within 14% for elastic scattering and 5.7% for absorption. The −NA/r4 potential operates at ranges of hundreds to thousands of femtometres, complementing Yukawa theory at short range and providing the microscopic origin of the non-resonant background left unspecified by Lane-Lynn theory. PACS numbers: 24.10.-i, 25.40.Dn, 27.10.+h, 24.50.+g