The Contour Below the Pole: Landau's Damping Prescription, Priced

In 1946 Lev Landau found that a collisionless plasma damps its own oscillations — waves lose energy to the resonant particles moving near the phase speed, with no collisions anywhere in the equation. I credit that paper at full strength: Landau damping is real physics, confirmed in the laboratory within years, and it reorganised plasma physics — the resonant-particle picture, the sign rule, and the dielectric machinery every kinetic treatment since has used. My own bench re-returns that content at the declared Maxwellian exactly. Nothing in this paper falsifies the damping. What I evaluate is the step that manufactures the damping term inside the mathematics. The linearised Vlasov dielectric integral is singular at v = ω/k, and the paper resolves it by prescription: the contour is taken to pass below the pole — the Landau contour — justified by the initial-value problem's analytic continuation, not derived from the collisionless dynamics the equation itself states. My bench evaluates that prescription on declared inputs only. The integral's principal-value reading is purely real at all six declared phase speeds, and the Landau contour adds exactly the half-residue iπf₀′(ω/k) and nothing else. At the Re ε = 0 root for k = 0.250 (ω_r = 1.105662, f₀′(ζ) = −0.000100) one and the same integral returns γ = +0.000000 under the principal value, γ = +0.002167 with the contour above (growth), or γ = −0.002167 with the Landau contour below (damping); at k = 0.500 the prescription's term at the root is πf₀′(ζ) = −0.128063, Im ε = +0.512252, γ = −0.456520 in the weak-damping form applied formally. The damping's whole size and sign at this address are the prescription's contribution. Counted plainly: 1 integration prescription against 0 derivations in 1946. Grade for this census entry, printed at full size: **REASON FOUND / STILL TURNING** — PREMISE-ONLY, explicitly not a kill and not a falsification. The program headline stands unmoved: **vacuum catastrophe NOT SOLVED / PARTIAL, producing-count 0.**