feat(ErdosProblems): add Problem 114 (Erdős–Herzog–Piranian conjecture)

[bengoechea]

Summary

Adds FormalConjectures/ErdosProblems/114.lean — the Erdős–Herzog–Piranian (EHP) conjecture (#114) with a computationally certified small-n result.

• erdos_114: Full open conjecture (all n)
• erdos_114_small_n: Solved for 3 ≤ n ≤ 14 via IEEE 1788-rigorous interval arithmetic

Context

Moritz Firsching confirmed a standalone PR is welcome:
Zulip message

This PR is intentionally separate from #3422.

Certificates

Each n case is verified independently by branch-and-bound over the compact
parameter space of monic degree-n polynomials, with IEEE 1788-2015 certified
interval arithmetic bounds:

• n = 3–12: Python/mpmath
• n = 13–14: Rust/inari (MPFR-backed)

All results deposited with SHA-256 checksums:
doi:10.5281/zenodo.19322367

Definitions

Name	Description
levelCurveUnit p	Lemniscate {z ∈ ℂ : ‖p(z)‖ = 1}
arcLength p	1-dimensional Hausdorff measure of the level curve

Future work

• Uniqueness theorem (erdos_114_small_n_unique) as a follow-up
• Extension to n = 15+ as compute budget allows

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[Smetalo]

Is this a duplicate of #3708?

[bengoechea]

Not a duplicate — #3708 was the first attempt but had CLA issues and a messy commit history, so I closed it and opened this one clean. Same content, cleaner PR. #3708 can be ignored.

[bengoechea]

For context on what changed from #3708: CLA was failing because the original PR came from a personal fork before I'd signed it. Also fixed the Apache 2.0 license header (http → https in the license URL, which was failing CI) and updated the AI disclosure section. This PR is from the MendozaLab org fork with a clean single commit. The mathematical content in 114.lean is identical.

[bengoechea]

Build was failing because hausdorffMeasure 1 left the 1 at universe level (Type) instead of ℝ. Fixed in 53d30b1 — annotated as hausdorffMeasure (1 : ℝ). CI should be green on the next run.

[bengoechea]

Note on the fix commits: the initial submission compiled locally against a slightly older Mathlib snapshot. CI's pinned toolchain exposed type class resolution differences — specifically around hausdorffMeasure elaboration. We've since set up a local environment mirroring upstream's exact lake-manifest.json so future submissions compile clean against CI on the first push.