We report on a new measurement of the beam transverse single spin asymmetry in electron-proton elastic scattering, A ep ⊥ , at five beam energies from 315.1 MeV to 1508.4 MeV and at a scattering angle of 30 • < θ < 40 • . The covered Q 2 values are 0.032, 0.057, 0.082, 0.218, 0.613 (GeV/c) 2 . The measurement clearly indicates significant inelastic contributions to the two-photon-exchange (TPE) amplitude in the low-Q 2 kinematic region. No theoretical calculation is able to reproduce our result. Comparison with a calculation based on unitarity, which only takes into account elastic and πN inelastic intermediate states, suggests that there are other inelastic intermediate states such as ππN, KΛ and ηN. Covering a wide energy range, our new high-precision data provide a benchmark to study those intermediate states.PACS numbers: 13.60. Fz, 11.30.Er, 13.40.Gp As a probe of hadron structure, electron scattering has two advantages: the structurelessness of the electron and the smallness of the electromagnetic coupling (α ≈ 1/137). The small coupling allows to expand the scattering amplitude in powers of α and to interpret experiments within the one-photon-exchange (Born) approximation. This leading order approximation enables a straightforward extraction of the electromagnetic form factors with the Rosenbluth separation technique [1]. For a precise extraction of the form factors it is necessary to include higher order quantum corrections [2,3]. Importantly, most of those corrections do not alter the Rosenbluth formula in that they contribute an overall factor to the cross section.The contribution that is expected to break this pattern [4,5] is the two-photon-exchange (TPE) diagram depicted in Fig. 1. For a long time the TPE effects have eluded direct experimental searches [6][7][8]. The situation changed when a striking discrepancy between the Rosenbluth separation [9, 10] and the polarization transfer [11?-13] data on the proton form factor ratio µ p G E /G M was observed. To evaluate the TPE corrections one needs to model the doubly-virtual Compton scattering (VVCS) in the most general kinematics. This involves calculating the two-current correlator with inclusive hadronic intermediate states. The full account of the inclusive intermediate states contribution can be made in the limited nearforward kinematics [15]. Beyond the forward kinematics, it is only possible to account for the elastic [16][17][18][19] or the pion-nucleon (πN) [20] intermediate state contributions.The theoretical framework for calculating the TPE contributions plays an important role in evaluating the two-boson-exchange corrections to precision low-energy tests of the Standard Model (SM) in the electroweak sector. The proton polarizability contribution to the fine structure of light muonic atoms stems from the TPE diagram and is a substantial ingredient [21] in the proton radius puzzle, the 7σ discrepancy in the value of the proton charge radius extracted from hydrogen spectroscopy [22] and electron-proton (ep) scattering [23] on one hand, an...