A Soft-Threshold Operator as a Diagnostic Toolkit for Threshold and Decoherence Transitions

We present the genuine, defensible result of the Δ-Gate program as a tool, not a law: a single soft-threshold operator that makes the character of threshold and decoherence transitions determinable from data. Three things are claimed and no more. First, secured: the Δ-Gate measurement channel embeds exactly into a GKSL/Lindblad dephasing generator, recovering the Born statistics as the diagonal of the decohered state — that is, by decoherence, not collapse. Second, owned and new: a two-Fisher uncertainty relation relating the location-Fisher (entropy-arrow) and the parameter-Fisher (gate-sharpness) of any threshold density. Third, a canonical information-geometric framework (Fisher-information substrate, Fisher–Rao length, the 2D-Ising self-dual invariant, and the fluctuation–dissipation relation) that is correct textbook structure, honestly tagged as found, not produced. From these we assemble a diagnostic toolkit that determines a threshold's dynamical class, bifurcation type, and static-to-dynamic coupling. The limits are explicit; the universality claims of the original program are refuted separately in the companion audit (Paper II).