2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/196183The negative mode of the Schwarzschild black hole is central to Euclidean quantum gravity around hot flat space and for the Gregory-Laflamme black string instability. Numerous gauges were employed in the past to analyze it. Here _the_ analytic derivation is found, based on postponing the gauge fixing, on the power of the action and on decoupling of non-dynamic fields. A broad-range generalization to perturbations around arbitrary co-homogeneity 1 geometries is discussed.16 pages, 1 figure. v2: Published version including an expanded introduction and minor changes. The title of the published version was changed due to an editorial request to "Gauge-eliminated derivation of the black hole negative mode"High Energy Physics - TheoryGeneral Relativity and Quantum Cosmologytext