Computational Verification of the Erdős–Herzog–Piranian Conjecture for Degrees 3 ≤ n ≤ 12

The Erdős–Herzog–Piranian (EHP) conjecture asserts that among all monic polynomials of degree n, the polynomial z^n – 1 uniquely maximizes the lemniscate length. Tao (2025) proved this for all sufficiently large n with a tower-exponential threshold. We computationally verify the conjecture for all n in {3,4,5,6,7,8,9,10,11,12} using dual independent implementations (Python/mpmath and Rust/inari) with IEEE 1788 interval arithmetic and certified branch-and-bound optimization. Both produce identical certified enclosures across x86_64 and arm64 architectures. The dominance margins increase monotonically from 17.1% (n=3) to 71.4% (n=10); all non-extremizer boxes are eliminated at level 0 for n >= 5, enabling the n=11 (10.2M evaluations, 33 min) and n=12 (45M evaluations, 2.67 hr) proofs. We derive a new closed-form expression for L(z^n – 1) valid for all n.