Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections

Numerical estimates for the packing inequality

The circle-slice inequality can be relaxed to finitely many radii. The numerical values below compare that relaxation with specific configurations. They are historical optimisation measurements; only the directed rational bounds used in Section 4 enter its certificate.

At $a=1$ the constants of the failure inequalities have the rounded values $\delta \approx 0.4586751$, $D\approx 5.3770730$, and $d_{\mathrm{low}}\approx 2.1700770$; the identity $\cosh (D/2)=e^2$ is exact. The quantity being estimated is $\sum _j\lambda (d_j)$ for $k$ roots at this fixed area, not a Euclidean path length. The four columns compare the earlier bound

where $\lambda (d)=-\log \tanh (d/2)$ as in Section 4; the hyperbolic packing bound used for the earlier threshold $2/5$; the linear-programming value $M_{\mathrm{circ}}$ of the relaxation of Lemma 4.3; and the largest reported objective value at a numerically constructed configuration. The last column is a search result, not a certified feasible value or a proved optimum.

Both right-hand columns come from floating-point searches. In addition, $M_{\mathrm{circ}}$ uses a $2600$-point $d$-grid and a $1400$-point $r$-grid. Restricting the allowed $d$ values and checking only finitely many circle constraints have opposite effects on the feasible set. Without a separate continuous feasibility or dual certificate, the computed value is not a proved lower or upper bound for the continuous relaxation. At $k=2$, the grid value $0.31018$ differs from the analytic value, approximately $0.310338$. Their proximity supplies no rigorous bracket.

The certified lower bounds for $k$, rounded up to integers, are as follows at $a=1$.

In one explicit-Euler floating replica with step ${10}^{-3}$ and a geometric grid of $40$ initial areas from ${10}^{-6}$ to $1$, the largest computed time to reach the area bound falls from $0.89703$ with the earlier packing lower bound for $k$ to $0.65503$ with the certified circle-slice bounds, and to $0.63003$ if the relaxation’s linear-programming value replaces those bounds. Shrinking the relaxation by the worst measured slack factor $1.045$ moves it only to $0.60703$. These computations suggest that this particular relaxation may stop near $e^{-0.607}=0.545$; they do not prove that it does, and they exclude neither a stronger lower bound on the root count nor a larger certified range.

The search uses only ten or fourteen radii in its two settings and a coarse $a$-grid with upward rounding. The reported loss from the first restriction at $a=1$, $k=8$ is about one per cent; the reported rounding loss is comparable. These are empirical estimates, not certified error bounds relative to the unrestricted continuous optimum.

The continuous optimisation remains unevaluated: maximise $\sum _j\lambda (d_j)$ subject to $d_j\ge d_{\mathrm{low}}$ and $\sum _{j}w(d_j,r)\le \pi$ for every $r>0$, allowing the relaxed radial distributions used in the computation. The numerical optimiser spreads mass over several radii and suggests logarithmic growth in $k$. Neither that growth law nor its coefficient is proved. Evaluating the relaxation would clarify this packing argument; its effect on the final path threshold would still need to be established.