Newton lets gravity bend a light ray (treat the photon as a fast corpuscle) by 0.87″ at the Sun's limb. Einstein says 1.75″ — exactly twice. Which does starlight actually follow, and where does the missing factor of two come from?
▶ Launch the interactive simulationA photon is a NULL geodesic of the Schwarzschild metric, which in the orbital plane is the exact orbit equation u'' + u = 3(GM/c²)u² (u = 1/r) — Binet's straight-line equation u'' + u = 0 plus one relativistic curvature term (no Newtonian 1/L² term: the photon's mass cancels, so light would be perfectly straight in flat space). Integrate that ODE with RK4 from closest approach out to infinity, measure the total turn 2φ_∞ − π, and read the deflection at the real solar values (GM☉, c, R☉). No deflection formula is coded. A Monte-Carlo 'eclipse plate' of background stars at random impact parameters, with 10% astrometric noise, recovers the limb deflection by least squares; a Newtonian corpuscle at speed c through −GM/r² is integrated with the same machinery for the rival.
the ray BENDS by 1.751″ at the limb — matching Einstein's 1916 prediction and Eddington's 1919 eclipse (and modern VLBI to 0.02%), recovered from the traced geodesic with no formula fed. The deflection follows δ = 4GM/(c²b): the log–log slope of δ vs b comes back −1 (grazing rays bend most, which is why a total eclipse was needed), and the tiny residual above 4GM/c²b is the genuine second-order GR term (it halves when b doubles). The decisive control is the Newtonian corpuscle: integrated the same way it bends by EXACTLY half — 0.876″, ratio 2.0000 — the Soldner/Newton value Eddington's plates ruled out. The missing factor of two is the curvature of space itself (the same 'extra' curvature that supplies 5/6 of Mercury's perihelion advance in ?world=schwarzschild). The on-screen bend is exaggerated ~10⁵× so it is visible; the printed 1.751″ and 0.876″ are the true traced values, not the drawn angle.