The Atmosphere That Thins Away: Debye & Hückel's Linearised Electrolyte, Priced

Every ion in a solution travels with company. That is the picture Peter Debye and Erich Hückel gave the world in 1923, and it is a true picture as far as it goes: each ion gathers around itself an atmosphere of opposite charge, the atmosphere screens the ion's field, and from that one construction flows the first quantitative theory of electrolyte solutions — the limiting law, log γ± = −A|z₊z₋|√I, with the constant A computed from named constants alone, and a screening length, the Debye length, that a century of solution, colloid, and plasma physics still spends. I credit all of that at full strength in this paper, and then I price what the closed form stands on, because my census exists to price exactly that joint. The closed form exists only because a nonlinear equation was replaced by its small-potential truncation: the linearisation is valid where zeψ ≪ kT, and my bench computes that validity parameter on the theory's own grid — it crosses a quarter at I = 0.011303 mol/L, crosses unity at I = 0.180853 mol/L, and stands at 2.351459 at I = 1, where the linearised co-ion density is not merely wrong but negative. The construction rides, besides, on a bundle of premises — point charges, a structureless dielectric continuum carrying a measured bulk permittivity, full dissociation — and where the point-charge premise fails, the extended law repairs it with a fitted ion-size parameter the founding premise forbade, a dial that alone sweeps γ± by a factor of 2.409 at I = 1. None of this falsifies the limiting law inside its honest domain. It prices the domain, and it prices the spend outside it. Grade for this census entry, printed at full size: REASON FOUND / STILL TURNING — explicitly not a kill and not a falsification. The program headline stands unmoved: vacuum catastrophe NOT SOLVED / PARTIAL, producing-count 0.