At an $\varepsilon$-safe cut of a $4$-AP-free enumeration of $\mathbb{Z}$, nothing before the earlier extreme has the extremes' $2$-adic valuation

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At an $\varepsilon$-safe cut of a $4$-AP-free enumeration of $\mathbb{Z}$, nothing before the earlier extreme has the extremes' $2$-adic valuation · Prove2Me