When K = [0, 1] Weak Separation Condition coincides Finite type Condition

The paper states and sketches a proof that, for similarity IFS on R with attractor K = [0,1], the Weak Separation Property (WSP) is equivalent to the finite neighbourhood/finite type condition (Theorem 1), working via net intervals Fa and neighbourhood sets Va(I) as defined in the preliminaries; both directions are addressed and the same core mechanism (endpoint separation at scale a and bounded neighbourhood types) underpins the argument . The candidate solution proves the same equivalence using a compact antichain/net-interval argument with an explicit “endpoint separation implies bounded neighbours” direction and, conversely, deducing a uniform lower bound on net-interval lengths from the finiteness of normalised neighbours to get WSP. While the paper’s proof outline contains some gaps (e.g., switching between two formulations of WSP without proof and an unsubstantiated multiplicity bound in the ⇐ direction), its main result and strategy align with the model’s, though the technical realisations differ; hence both are correct with different proofs.