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Source. Section 6, pp. 8--10, with Lemma 6.1 and Theorem 6.2 on p. 9 and Remark 6.4 on p. 10, and Appendix A, pp. 10--14, of Venkata Siddharth Pendyala, Radial access for polynomial lemniscates: the cubic theorem and a degree-six obstruction, SSRN preprint (2026), doi:10.2139/ssrn.6850818, the edition named on the source card.

Statement

Admissible polynomials, safe radii, $\Delta$, $\mathcal B$ and $N^*$ are as on the Theorem 1.1 page.

The polynomial (pp. 8--9). Let

$$\rho=\frac{1999999}{2000000},\qquad S=(927,927,317,317,697,547),\qquad q_\rho(z)=\prod_{s\in S}\bigl(\rho+e^{-2\pi is/960}z\bigr),$$

the product running over $S$ with its repetitions, so that

$$q_\rho(z)=\bigl(\rho+e^{-2\pi i\,927/960}z\bigr)^2\bigl(\rho+e^{-2\pi i\,317/960}z\bigr)^2 \bigl(\rho+e^{-2\pi i\,697/960}z\bigr)\bigl(\rho+e^{-2\pi i\,547/960}z\bigr).$$

Its leading coefficient is $\prod_{s\in S}e^{-2\pi is/960}=e^{-2\pi i\,3732/960}=e^{-2\pi i\,71/80}$, so

$$P_\rho(z)=e^{2\pi i\,71/80}\,q_\rho(z)$$

is monic of degree six, and $|P_\rho|=|q_\rho|$. Its zeros are $-\rho e^{2\pi is/960}$ for $s\in S$: double at $s=927$ and $s=317$, simple at $s=697$ and $s=547$, all of modulus $\rho<1$; and $|P_\rho(0)|=\rho^6<1$. As fractions of a turn, their directions are $447/960$ and $797/960$ for the double zeros and $217/960$ and $67/960$ for the simple ones, the four directions Remark 6.4 names (p. 10). For $z=re^{2\pi iu}$ the paper's (6.1) is

$$|q_\rho(re^{2\pi iu})|^2=\prod_{s\in S}\Bigl(\rho^2+r^2+2\rho r\cos2\pi\bigl(u-\tfrac{s}{960}\bigr)\Bigr).$$

Lemma 6.1 (Three-radius certificate, p. 9). Let $R=\{1,1/300,439/1000\}$. For every $u\in[0,1]$,

$$\max_{r\in R}\lvert q_\rho(re^{2\pi iu})\rvert>1,$$

and, more quantitatively,

$$\min_{0\le u\le1}\ \max_{r\in R}\lvert q_\rho(re^{2\pi iu})\rvert^2>1+2.327036066010500056098356\cdot10^{-6}.$$

Theorem 6.2 (Strict degree-six radial obstruction, p. 9). There is a monic polynomial $P_\rho$ of degree six, with all its zeros in the open disk $\mathbb D$, such that

$$\min_{\theta\in\mathbb R}\ \max_{0\le r\le1}\lvert P_\rho(re^{i\theta})\rvert>1 .$$

Consequently $N^*\le6$. The polynomial is the $P_\rho$ above: every radius carries a point at distance $1$, $1/300$ or $439/1000$ from $0$ where $|P_\rho|>1$, so no radius is safe, $\mathcal B(P_\rho)>1$, and Proposition 2.2 gives $\Delta(P_\rho)>0$ (p. 10).

The certificate

Appendix A (pp. 10--12) states the certificate for Lemma 6.1 as a finite datum and checks it by rational inequalities.

• Intervals. $[0,1]$ is cut into $I_j=[j/8192,(j+1)/8192]$ for $0\le j<8192$, and each $I_j$ gets a radius $r_j\in R$ from Table 1 (p. 11), which the paper calls the certificate; no search or numerical optimization enters the check.
• Table 1, the blocks of $j$ with their radius: $0$ to $363$, radius $1$; $364$ to $795$, $439/1000$; $796$ to $1627$, $1$; $1628$ to $2058$, $439/1000$; $2059$ to $3317$, $1$; $3318$ to $4260$, $1/300$; $4261$ to $6353$, $1$; $6354$ to $7296$, $1/300$; $7297$ to $8191$, $1$. The outer radius $1$ blocks every direction except four short corridors centered at the zero directions; the middle radius $439/1000$ closes the corridors at $67/960$ and $217/960$ of the simple zeros, and the inner radius $1/300$ those at $447/960$ and $797/960$ of the double zeros (Remark 6.4, p. 10).
• Cosine bound. For a phase $s$ let $J_{j,s}=I_j-s/960$ and let $\delta_{j,s}$ be the distance from $J_{j,s}$ to $\mathbb Z+1/2$. If $\delta_{j,s}=0$ then $c_{j,s}=-1$; otherwise $c_{j,s}=\max\{-1,-U(2\pi_0\delta_{j,s})\}$, with $\pi_0=103993/33102<\pi$ and $U(x)=\sum_{k=0}^{10}(-1)^kx^{2k}/(2k)!+x^{21}/21!$. Each $c_{j,s}$ is rational, and $\cos2\pi(u-s/960)\ge c_{j,s}$ for $u\in I_j$, the paper's (A.1).
• Inequalities. With multiplicities $(m_{927},m_{317},m_{697},m_{547})=(2,2,1,1)$, let $M_j=\prod_{s\in\{927,317,697,547\}}\max\{0,\rho^2+r_j^2+2\rho r_jc_{j,s}\}^{m_s}-1$. By (6.1) and (A.1), $M_j>0$ gives $|q_\rho(r_je^{2\pi iu})|^2>1$ for all $u\in I_j$, so Lemma 6.1 reduces to the 8192 rational inequalities $M_j>0$.
• Margins. Table 2 (p. 12) lists the least $M_j$ in each block. The least of all is at $(j,r)=(3318,1/300)$, in block 6, an exact rational with a numerator of 1289 digits and a denominator of 1295 digits; the paper states that the smallest $M_j$ is at least $2.327036066010500056098356\cdot10^{-6}$. The other block minima run from about $2.6\cdot10^{-6}$ (block 8) to about $1.5\cdot10^{-2}$ (blocks 2 and 4). The paper calls its displayed decimals renderings of exact positive rationals.
• Checker. Pages 12--14 print a Wolfram Language program that checks that the blocks cover $j=0,\dots,8191$, computes every $M_j$ in exact rational arithmetic, stops if some $M_j\le0$, and prints the block minima with SHA-256 hashes of their exact values; floating point enters only the displayed decimals (p. 12). It was not run for this page.

Read depth. Claims checked: the construction, (6.1), Lemma 6.1, Theorem 6.2, Remark 6.4 and the definitions and Tables 1 and 2 of Appendix A were read clause by clause on the page images. Only the certificate's bookkeeping was recomputed here, in exact arithmetic and without the printed checker: the leading coefficient, the zero directions, which match Remark 6.4, and that the blocks of Table 1 cover $j=0,\dots,8191$ and give the radius $439/1000$ to the intervals containing $67/960$ and $217/960$ and the radius $1/300$ to those containing $447/960$ and $797/960$. The derivation of (A.1) was read and not checked, none of the 8192 inequalities was checked, and the printed checker was not run. The paper is an unrefereed preprint, and nothing here is independently reviewed.

Proof pointer

Theorem 6.2, pp. 9--10. $P_\rho$ is monic and admissible by construction. Since $|P_\rho|=|q_\rho|$, Lemma 6.1 gives $\max_{0\le r\le1}|P_\rho(re^{2\pi iu})|>1$ for every $u$, so no radius lies in $\{|P_\rho|\le1\}$; then $\mathcal B(P_\rho)>1$, and Proposition 2.2 gives $\Delta(P_\rho)>0$. Lemma 6.1 (p. 9) follows from the certificate above, since the intervals $I_j$ cover $[0,1]$. For (A.1) the paper uses that the cosine is symmetric about half-integers and decreasing on $[0,\pi]$, that $\pi_0<\pi$, and Taylor's theorem for the cosine with remainder at most $x^{21}/21!$ (pp. 10--11).

Dependencies

Lemma 6.1 (p. 9) with Appendix A (pp. 10--14), and the harder half of Proposition 2.2 (p. 2), stated on the Theorem 1.1 page: zero detour forces a safe radius.

Bears on

• Problem 1120: in the problem page's worst-case reading $S(n)$, the theorem gives $S(6)\ge L(P_\rho)>1$, so in degree six the trivial bound $1$ is not attained. It gives no lower bound for $L(P_\rho)-1$, does not say whether $L(P_\rho)$ is finite, and says nothing about other degrees.