DoV = λαg The Generalized Density of Value Identity: Surface Productivity Across All Eddington Regimes

The identity DoV = αg — that surface productivity at the Eddington limit is identical to gravitational field intensity scaled by α = m₝c/σₜ = 7. 5401 × 10⁹ W·s²/m³ — was established in Handono (2026b) and verified at λ = 1. This paper derives and verifies the generalized form: DoVₐctual = λαg, where λ = Lₒbs/LEdd is the Eddington ratio of any accreting black hole. The identity is verified across six systems spanning eighteen orders of magnitude in λ, from Sgr A* (λ ≈ 1. 85 × 10⁻⁸) to ULX-regime black holes (λ ≈ 500), with 100% match and zero free parameters. We further demonstrate that the linear scaling is exact by algebraic necessity for black holes — not an approximation — because Aₑh scales with M² while LEdd scales with M, making λ the sole variable. We also compute the aggregate DoV of all accreting supermassive black holes in the observable universe and find it consistent with the generalized identity at cosmological scale. Two open questions remain: the validity of the Schwarzschild denominator for neutron star ULXs, and the existence of analogous λ-efficiency structure in non-gravitational productive domains.