Algebraic Dependence Number and Cardinality of Generating Iterated Function Systems

The paper proves Theorem 3.6 by combining a structural formula for gap lengths under SSC (Lemma 2.4 and its IFS corollary (2.2)) with the ratio-analysis lemmas (Definitions 2.6, 2.8; Lemmas 2.9–2.10), yielding R_GL(Fu)(θ) ⊂ A_Q*+ (GD-IFS) and, for IFS and small θ, XZ*+ ⊂ R_GL(K)(θ) ⊂ X_Q*+; taking logarithms gives the stated span inequalities and equalities for the algebraic (in)dependence number (Theorem 3.6 and identity (3.5)) . The candidate solution derives essentially the same conclusions via a direct, constructive argument: (i) localizes each gap length to a scaled inter-child distance and (ii) uses a pigeonhole/rational-span argument to show any geometric ratio in GL lies in span_Q log A; for the IFS case, it shows every contraction ratio occurs in R_GL(K)(θ) for sufficiently small θ, giving equality of spans. Minor caveats in Step 1 (the need to ignore a finite exceptional set of large thresholds and to use a bottleneck characterization of when an inter-child edge is critical) are acknowledged later in the candidate’s Claim 2 and do not affect the main conclusions. Hence both are correct, with the paper using ratio-analysis lemmas and the model giving a more constructive but compatible proof.