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Statement

Setting (Section 3, pp. 7-8). With $k_{K,\varepsilon}$ and $T_n=n(n+1)/2$ as on the Theorem 2.8 page, Lemma 3.1 (p. 7) writes $k_{K,\varepsilon}$ on the unit circle as a cosine series: for every real $\theta$,

$$e^{i\theta}k_{K,\varepsilon}(\varepsilon e^{2i\theta}) =2\sum_{n=0}^{\infty}\frac{\varepsilon^{T_n}}{K^{T_n}}\cos\bigl((2n+1)\theta\bigr),$$

so $\max_{\lvert z\rvert=1}\lvert k_{K,\varepsilon}(z)\rvert$ is the maximum of the modulus of the right side over $\theta\in[0,2\pi)$. The paper fixes the exact rationals $K_0=7137/2000$ and $\alpha_0=198074929/50000000$, puts $\varepsilon_0=e^{i\alpha_0}$, $k_0=k_{K_0,\varepsilon_0}$ and $A_0=\max_{\lvert z\rvert=1}\lvert k_0(z)\rvert$, and splits the series into the six-term truncation $P(\theta)$ (the terms $n=0,\ldots,5$) and the tail $R(\theta)$ (the terms $n\ge6$), so that $A_0=\max_{\theta\in[0,2\pi)}\lvert P(\theta)+R(\theta)\rvert$.

Lemma 3.2 (p. 8). For all real $\theta$, $\lvert R(\theta)\rvert\le2K_0^{-21}/(1-K_0^{-7})$.

Lemma 3.3 (p. 8). $P$ is $4$-Lipschitz on $\mathbb R$: $\lvert P(\theta)-P(\varphi)\rvert\le4\lvert\theta-\varphi\rvert$.

Lemma 3.4 (p. 8). With $M=2{,}000{,}000$ and $\theta_j=2\pi j/M$ for $j=0,1,\ldots,M-1$, $\max_{0\le j<M}\lvert P(\theta_j)\rvert\le1.709176398$. This is a computer-assisted certification in Arb ball arithmetic through python-flint; Appendix A (pp. 10-12) describes the procedure and records the program's output, and the code is in the second author's public repository ep513-arb-certification.

Proposition 3.5 (p. 8). $A_0<1.70919$.

Source. Yixin He and Quanyu Tang, "Generalizing the Clunie-Hayman construction in an Erdős maximum-term problem," arXiv:2602.12217v1 (12 February 2026), Section 3, pp. 7-9, and Appendix A, pp. 10-12; Proposition 3.5 is stated on p. 8 and proved on p. 9. The paper is recorded on its source card.

Read depth. Claims checked: Lemmas 3.1-3.4 and the proposition were read clause by clause on the printed pages, and the arithmetic of the proof on p. 9 was followed. The certification of Lemma 3.4 was not rerun for this page; the paper reports one run, with $M=2000000$, 90 decimal digits and one worker, returning a mesh maximum of about $1.7091763974$ (p. 11). Nothing here is independently reviewed.

Proof pointer

Page 9. Every $\theta$ lies within circular distance $\pi/M$ of a mesh point, so Lemmas 3.3 and 3.4 give $\max_\theta\lvert P(\theta)\rvert\le1.709176398+4\pi/2{,}000{,}000<1.709183$. Lemma 3.2 bounds the tail by less than $5.1\times10^{-12}$, and the two bounds add to less than $1.70919$. Lemma 3.2 uses $T_6=21$ and $T_{n+1}-T_n\ge7$ for $n\ge6$; Lemma 3.3 bounds $\lvert P'\rvert$ termwise using $K_0>3.5$.

Dependencies

Lemma 3.1 (p. 7), which uses the absolute convergence of the Laurent series from Lemma 2.1 (p. 3); the computer certification of Lemma 3.4 (Appendix A).

Bears on

• Problem 513: through Theorem 2.8, the bound gives $\beta(f_{K_0,\varepsilon_0})=1/A_0>100000/170919$, the lower bound $B>0.58507$ of Theorem 1.2. It concerns these parameters only and says nothing about an upper bound for $B$.