Question 1

Geometric Mapping to Rectangle-Packing Problems

Accepted Answer

The critical insight provided by Tao, which Aristotle subsequently formalized, was the realization that the integral defining the length of the lemniscate could be bounded through a transformation into a combinatorial rectangle-packing problem. By applying a conformal mapping from the complex plane to a specialized Riemann surface, Tao was able to represent the components of the lemniscate as boundaries of regions with constrained areas. This moved the problem out of the realm of pure analysis and into the territory of discrete geometry, where machine logic excels. Aristotle was tasked with proving that no arrangement of these regions could exceed the density threshold associated with the conjectured maximal length, a task that required checking an immense number of topological configurations.

Question 2

Handling High-Degree Polynomial Extremality

Accepted Answer

The second major hurdle in the EHP conjecture was the behavior of the lemniscate as the degree $n$ tends toward infinity. In high-degree regimes, the polynomial $P(z)$ becomes extremely sensitive to small perturbations in its roots, leading to chaotic fluctuations in the length of the level set $|P(z)| = 1$. Traditional manual proofs often falter here because the &#8220;error terms&#8221; in the asymptotic expansions become unmanageable. Aristotle solved this by implementing a symbolic-numeric hybrid approach within the Lean proof state. It used high-precision interval arithmetic to &#8220;prune&#8221; the search space, proving that certain classes of root distributions could never yield a maximal length, thereby allowing the formal logic to focus on a narrow set of candidate extremal configurations.

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