Power-Sum Deformation and Information-Dissipation Complexity: Sharp Low-Level Transition Theory and Four-Level Bounds

A power-sum deformation of a finite probability vector traces a one-parameter path through Rényi entropy, escort distributions and the cumulants of information content. We use this path to ask a structural question that is not visible from entropy values alone: how many times can information variance change direction as the deformation increasingly favours the most probable states? Writing the escort varentropy as V(t), its derivative is V′(t) = −κ₃(t), so transition counting becomes a zero problem for a structured exponential polynomial. We derive an exact remainder formula linking Shannon and Rényi entropy, global varentropy sum rules, and a complete two-level transition classification. For three distinct probability levels we prove the sharp bound Nₘₐₓ(3) = 3 and obtain an exact geometric-spacing bifurcation at λ = 4, with a complete zero/one/two/three-transition phase diagram. A pair–triple decomposition exposes the algebra controlling higher levels. For four distinct levels, a finite chamber certificate and variation-diminishing theory give Nₘₐₓ(4) ≤ 11, while outward-rounded interval arithmetic certifies an explicit five-transition example, hence 5 ≤ Nₘₐₓ(4) ≤ 11. Tail, scaling and collision analyses isolate the remaining exact four-level problem.