Erdős Problem 1041

I am proving a theorem in Lean 4 and submitting it to Conjectura.

## Problem EP0003 — Erdős Problem 1041

**Erdős–Herzog–Piranian Component Lemma** (Metric Properties of Polynomials, 1958): If $f$ is a monic degree $n$ polynomial with all roots in the unit disk, then some connected component of $\{z \mid |f(z)| < 1\}$ contains at least two roots with multiplicity. See p. 139, above Problem 5: [EHP58] Erdős, P. and Herzog, F. and Piranian, G., _Metric properties of polynomials_. J. Analyse Math. (1958)

## Environment (fixed — do not assume anything newer)

- Lean toolchain: `leanprover/lean4:v4.33.0-rc1`
- Mathlib: `v4.33.0-rc1`

If a lemma you want does not exist in that Mathlib, prove it inline instead of
importing something newer.

## The exact statement I must prove

```lean
theorem solution : Conjectura.EP0003.goal := by
  sorry
```

## The file I submit

```lean
import Conjectura.Problems.EP0003.Statement

namespace Submission

theorem solution : Conjectura.EP0003.goal := by
  sorry

end Submission

```

## The Lean definitions of every term in this problem

These are the actual definitions your proof will be checked against. Do not
substitute your own version of any of them.

### natDegree

def

```lean
f.natDegree
```
Defined in Mathlib.

### Monic

def

```lean
f.Monic
```
Defined in Mathlib.

### rootSet

def

```lean
f.rootSet
```
Defined in Mathlib.

### ball

abbrev

```lean
Metric.ball
```
Defined in Mathlib.

### eval

def

```lean
f.eval
```
Defined in Mathlib.

### IsConnected

class

```lean
IsConnected
```
Defined in Mathlib.

### filter

def

```lean
f.roots.filter
```
Defined in Mathlib.

## Rules — submissions violating these are rejected automatically

1. **Do not change the name or type of `solution`.** It must satisfy the
   statement above exactly.
2. **Do not redefine or shadow anything from the problem's Statement module.**
   Declaring your own `goal`, or redefining a name it depends on, produces a
   proof of a *different* statement and is rejected. This is the single most
   common rejection.
3. **No `sorry`** anywhere, including in helper lemmas. It surfaces as the
   axiom `sorryAx` and is detected transitively through imports.
4. **No `native_decide`** — it surfaces as `Lean.ofReduceBool` and is not
   accepted, because it trusts compiled code rather than the kernel.
5. Only these axioms are permitted: `propext`, `Classical.choice`,
   `Quot.sound`.
6. Follow Mathlib style: hypotheses left of the colon, explicit types,
   `snake_case` theorem names, `UpperCamelCase` types.

## What I want from you

Here is my argument in informal mathematics:

> [PASTE YOUR PROOF SKETCH HERE]

Turn it into Lean 4 that compiles under the environment above and satisfies the
statement exactly. Where you are unsure a lemma exists in this Mathlib version,
say so explicitly rather than guessing a name.