Eddington's Line, Derived and Crossed: The Luminosity Limit and Its Five-Hundred-Fold Violation, Priced

The Eddington limit is the second great derived number of stellar astrophysics: the luminosity at which radiation pushing out exactly cancels gravity pulling in, derived here in the open from the Thomson cross-section and Newton's constant — L_Edd = 1.26 × 10³⁸ erg/s per solar mass for pure hydrogen, 1.48 × 10³⁸ for a solar composition, computed from constants in my notebooks. Unlike the Chandrasekhar wall, this line is crossed in public, routinely, by a factor of hundreds: six pulsating ultraluminous X-ray sources are now known to be neutron stars — objects whose masses cap their Eddington luminosity near 2 × 10³⁸ erg/s — shining at up to 10⁴¹ erg/s. The flagship, NGC 5907 ULX1, peaks near five hundred times its Eddington luminosity (483× against the value derived here for a 1.4 M_☉ star; the discovery literature quotes ~500×). This paper treats the crossing the way the derivation demands: the limit rests on five assumptions — spherical symmetry, steady state, hydrostatic balance, Thomson opacity, no magnetic field — and the crossing sources break them one at a time, in identifiable ways, each with a price. The derivation is not wrong; it is a fence whose every post the violators had to pay to remove, and the prices are the physics of super-Eddington accretion. Grade: THE LINE IS DERIVED — ITS CROSSING IS PRICED, POST BY POST. This paper stands outside my work on the vacuum catastrophe; it draws nothing from that work and repairs nothing in it.