Erdős 243: bounded-negative-error rigidity variant

`243.lean` currently carries one declaration, the open problem. This issue
proposes one solved variant beside it.

Erdős 243 concerns a sequence of integers $1 \le a_1 < a_2 < \cdots$ with
$a_n / a_{n-1}^2 \to 1$ and $\sum 1/a_n \in \mathbb{Q}$, and asserts
$a_n = a_{n-1}^2 - a_{n-1} + 1$ for all large $n$.

Clearing denominators along such a sequence gives an integer state
$D_0 = q$, $D_{n+1} = a_n D_n$, $C_0 = p$, $C_{n+1} = a_n C_n - D_n$, where
$\sum 1/a_n = p/q$. Then $C_n = D_n \sum_{k \ge n} 1/a_k$ is a positive integer
at every index. The centred state $E_n = D_n - (a_n - 1) C_n$ vanishes
identically on Sylvester's sequence $2, 3, 7, 43, 1807, \dots$, and the problem
is the assertion that $E_n$ eventually vanishes. These are the coordinates of
Koizumi, [arXiv:2504.05933](https://arxiv.org/abs/2504.05933), Lemma 15.

Erdős and Straus, Pacific J. Math. 14 (1964), 128-137, settle the case
$E_n \ge 0$. The variant proposed here reaches the same conclusion from a
bounded negative part $-B \le E_n$, with no periodicity assumption on the sign
pattern of $E$, under a normalised vanishing hypothesis $K|E_n| < C_n$ for every
$K$ and all large $n$. It leaves `erdos_243` open: the remaining obstruction is
a centred state with cofinally unbounded negative excursions.

The proof is Lean, external, and sorry-free, and will be linked with
`formal_proof using lean4 at`. The three definitions are one-off and stay inside
`243.lean`.

Start by reading 243.lean and its existing Erdős 243 declaration, then inspect the externally supplied proof referenced by `formal_proof using lean4 at`. Done means adding the three one-off definitions and a sorry-free formal proof of the bounded-negative-error variant while leaving `erdos_243` open.

Written by the indexing model from the issue text.