Naor's question: does nontrivial Markov type force an equivalent uniformly smooth norm (superreflexivity)?

Claims that every real Banach space with Markov type $p$ for at least one $p>1$ admits an equivalent uniformly convex norm, hence is superreflexive and (by Pisier) admits an equivalent uniformly smooth norm. With the known converse (Naor-Peres-Schramm-Sheffield plus Pisier) this characterizes superreflexivity by nontrivial Markov type, answering Naor's question negatively. The proof builds, from a nonreflexive space finitely representable in a non-superreflexive one, a spreading norm and explicit reversible walks whose image has linear displacement. It does NOT address complex scalars separately, the exact Markov type exponent of a given space, or Ball's Markov type 2 questions for specific spaces.

Claims that every real Banach space with Markov type $p$ for at least one $p>1$ admits an equivalent uniformly convex norm, hence is superreflexive and (by Pisier) admits an equivalent uniformly smooth norm. With the known converse (Naor-Peres-Schramm-Sheffield plus Pisier) this characterizes superreflexivity by nontrivial Markov type, answering Naor's question negatively. The proof builds, from a nonreflexive space finitely representable in a non-superreflexive one, a spreading norm and explicit reversible walks whose image has linear displacement. It does NOT address complex scalars separately, the exact Markov type exponent of a given space, or Ball's Markov type 2 questions for specific spaces.