feat(ErdosProblems): Add 114.lean with small-n EHP result (n=3–14)

FormalConjectures/ErdosProblems/114.lean

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+/-
+Copyright 2026 The Formal Conjectures Authors.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+-/
+
+import FormalConjectures.Util.ProblemImports
+
+/-!
+# Erdős Problem 114
+
+*Reference:* [erdosproblems.com/114](https://www.erdosproblems.com/114)
+
+The **Erdős–Herzog–Piranian (EHP) conjecture** (1958): among all monic polynomials
+$p$ of degree $n$, the arc length of the lemniscate $\{z \in \mathbb{C} \mid |p(z)| = 1\}$
+is maximised uniquely by $p(z) = z^n - c$, $|c| = 1$.
+
+### Known partial results
+
+- **$n = 2$**: Eremenko–Hayman (1999) [EH99]
+- **$3 \leq n \leq 14$**: Mendoza (2026), IEEE 1788-rigorous interval arithmetic,
+  [doi:10.5281/zenodo.19480329](https://doi.org/10.5281/zenodo.19480329)
+- **Sufficiently large $n$**: Tao (2025) [Ta25]
+- **Upper bound**: $L(p) \leq 2\pi n$ for all monic degree-$n$ $p$ (Pommerenke, 1961)
+
+### References
+
+- [EHP58] Erdős, P., Herzog, F., Piranian, G. (1958). *Metric properties of polynomials*.
+  J. Analyse Math. 6, 125–148.
+- [Po61] Pommerenke, Ch. (1961). *Über die Kapazität ebener Kontinuen*.
+  Math. Ann. 144, 115–120.
+- [EH99] Eremenko, A., Hayman, W. (1999). *On the length of lemniscates*.
+  Michigan Math. J. 46, 409–415.
+- [Ta25] Tao, T. (2025). *The Erdős–Herzog–Piranian conjecture for large $n$*.
+-/
+
+namespace Erdos114
+
+/-- The level curve (lemniscate) of a polynomial `p` at level 1:
+$\{z \in \mathbb{C} \mid \|p(z)\| = 1\}$. -/
+def levelCurveUnit (p : Polynomial ℂ) : Set ℂ :=
+  {z : ℂ | ‖p.eval z‖ = 1}
+
+/-- The arc length of the lemniscate of `p`, measured as the
+1-dimensional Hausdorff measure of the level curve. -/
+noncomputable def arcLength (p : Polynomial ℂ) : ℝ≥0∞ :=
+  MeasureTheory.Measure.hausdorffMeasure 1 (levelCurveUnit p)
+
+/-- **Erdős Problem 114** (open conjecture). Among all monic polynomials of
+degree $n$, $p(z) = z^n - c$ (with $|c| = 1$) maximises the arc length of the
+lemniscate $\{z \in \mathbb{C} \mid |p(z)| = 1\}$.
+
+Originally posed by Erdős, Herzog, and Piranian [EHP58]. Solved for $n = 2$
+[EH99], $3 \leq n \leq 14$ (Mendoza, 2026), and sufficiently large $n$ [Ta25].
+Open in general. -/
+@[category research open, AMS 30]
+theorem erdos_114 (n : ℕ) (hn : 1 ≤ n)
+    (p : Polynomial ℂ) (hp : p.Monic) (hp_deg : p.natDegree = n) :
+    arcLength p ≤ arcLength (Polynomial.X ^ n - 1) := by
+  sorry
+
+/-- **Erdős Problem 114, small $n$** (solved, computationally certified).
+For $3 \leq n \leq 14$, the polynomial $z^n - c$ ($|c| = 1$) maximises the arc
+length of the lemniscate among all monic polynomials of degree $n$.
+
+Verified by IEEE 1788-rigorous interval arithmetic using branch-and-bound
+(Python/mpmath for $n \leq 12$, Rust/inari for $n \leq 14$). Each case reduces
+to a finite computation over a compact parameter space, with all bounds certified
+to floating-point rigor per IEEE 1788-2015.
+
+*Reference:* K. Mendoza (2026).
+[doi:10.5281/zenodo.19480329](https://doi.org/10.5281/zenodo.19480329) -/
+@[category research solved, AMS 30]
+theorem erdos_114_small_n (n : ℕ) (hn : 3 ≤ n) (hn14 : n ≤ 14)
+    (p : Polynomial ℂ) (hp : p.Monic) (hp_deg : p.natDegree = n) :
+    arcLength p ≤ arcLength (Polynomial.X ^ n - 1) := by
+  sorry
+
+end Erdos114