Lemma 9. Fix an integer n≥1 . Let two paths have the same endpoint and weights λ>0 and μ≥0 . If ρ=μ/λ<1/n , then there is an endpoint w∈K^s admitting a path of weight 1/t for every integer 1≤t≤n .

Set α_t=(1/t-ρ)/(1-ρ) (1≤t≤n). These numbers satisfy 0<α_n<⋯<α_1=1 . Partition [0,1] at these points. For each resulting interval of length l>0 , apply to both paths the endomorphism of K defined by ξ_zξ_ √l/λ\,z . It preserves diagonal factors and multiplies every copy-factor weight by l/λ . The two new paths thus have the same endpoint and weights l and ρl . Concatenate the interval paths in increasing interval order. The resulting endpoint is independent of which of the two paths is chosen on each interval. For a given t , choose the larger weight on the intervals before α_t and the smaller weight on the remaining intervals. The total weight is α_t+ρ(1-α_t)=1t. The fixed order of concatenation gives one endpoint w for all t ; no commutation of factors is used.