Unconditional tree index ($o(T_C(X))$)

Let $T_C(X)$ consist of finite normalized sequences for which every scalar combination and sign change satisfy $\Vert {\sum }_i{\epsilon }_i{a}_i{x}_i\Vert \le C\Vert {\sum }_i{a}_i{x}_i\Vert$. These inequalities give a closed tree on a Polish space when $X$ is separable. An infinite branch is an unconditional basic sequence. If there is no such sequence, each fixed-constant tree is well-founded and has countable rank of a well-founded tree. Consequently, finite unconditional configurations with one fixed constant and arbitrarily high countable ranks force an infinite unconditional sequence.