Complete monotonicity of Tsallis entropy under heat flow: a sharp order classification

AI-assisted, unrefereed candidate. The quadratic order is the unique positive Tsallis order with universal alternating heat-flow entropy derivatives. For each fixed nonexceptional order, a density-varying strictly log-concave family supplies high-derivative failures. Shannon counterexamples and quadratic positivity are inherited and attributed. A separate exact-arithmetic and analytic-transfer witness has order 3/2 and derivative 13; no log-concavity or least-order assertion is made for that witness.