Schanuel Property SC_K

Establish the Schanuel property SC_K: Let K be a subfield of C of finite transcendence degree; show that for all tuples x ⊆ C, the inequality trd(e^x) + ld(x/K) − ld(x) ≥ −trd(K) holds.

Background

The paper uses a Schanuel-type property to obtain a lower bound for the predimension function needed in the Hrushovski construction over real closed fields. Specifically, the authors assume SC_K for K = Q(βi) to locate a finitely rcl-generated substructure B ensuring (\bar{R}, G) lies in the class C_B, which is essential for proving that the real field with a dense family of logarithmic spirals is a model of their theory.

While a special case is known (SC_K holds for K = Q(βi) when β is exponentially transcendental, by Bays–Kirby–Wilkie), the general SC_K remains a conjectural statement, and the paper explicitly frames it as a conjecture to support the construction.