On the common index jump theorem and further developments

The paper’s goal is to show that the single-path index recurrence statements (IR1")–(IR3") from CGG24 are already contained in the enhanced common index jump theorem of DLW16, and it does so by identifying the β-invariants with splitting numbers and by matching (IR1")–(IR3") term-by-term to [DLW16,(3.20)–(3.22)] and [DLW16,(3.42)–(3.43)] (see the restatement of (IR1")–(IR3") and definitions in the paper, and Propositions 2–4 establishing the coincidences). In particular: the paper restates (IR1")–(IR3") for a single path and connects μ± to the i,ν indices and the mean index μ̂, proving (IR1") via [DLW16,(3.42)–(3.43)] and (IR2")–(IR3") via [DLW16,(3.20)–(3.22)], using also β−(γ)=S−_{γ(1)}(1) and S+_{M}(1)=S−_{M}(1) identifications. These ingredients are explicitly shown in the uploaded note. Meanwhile, the model’s solution gives a direct Bott–Long/splitting-numbers and Diophantine-approximation argument for the same (IR1")–(IR3"): it constructs a no-carry window for roots of unity, defines a window-sum constant d independent of ℓ and of the sign ±, derives (IR2") and (IR3") by Bott bookkeeping and the one-sided jump at 1 encoded by β±, and obtains (IR1") from homogeneity of μ̂ plus the sandwich bound |μ±−μ̂|≤m. This route is standard and sound. Hence both the paper and the model correctly establish (IR1")–(IR3"); the approaches are different in emphasis (paper: reduction to DLW16 formulas and splitting numbers; model: direct Bott-sum/Diophantine window), so the appropriate verdict is that both are correct with different proofs. Key places in the paper where these steps are made explicit include the single-path reformulation of (IR1")–(IR3") and β± facts, the identification μ−=i and μ+=i+ν, and the matching of (IR2")–(IR3") to [DLW16,(3.20)–(3.22)] via the β–splitting-number bridge.