Balister–Győri–Schelp conjecture on difference partitions in Z/2^s

### What is the conjecture
Let $s \ge 2$ and $n = 2^{s-1}$. Let $M = [v_0, \dots, v_{n-1}]$ be an $s \times n$ matrix over $\mathbb{F}_2$ with nonzero columns satisfying

$$
\sum_{i=0}^{n-1} v_i = 0.
$$

Then $M$ has a *Hadamard pair solution*, i.e. the vector space $\mathbb{F}_2^s$ can be partitioned into $n$ disjoint pairs $\{x_i, y_i\}$ such that

$$
x_i + y_i = v_i \quad \text{for every } i \in \{0, \dots, n-1\}.
$$

* **Source Paper:** *Perfect Matchings with Prescribed Differences Beyond Hall: The Two-Hole Problem* ([arXiv:2607.08630](https://arxiv.org/abs/2607.08630), [HTML](https://arxiv.org/html/2607.08630), [PDF](https://arxiv.org/pdf/2607.08630))
* **Authors:** Aryeh Lev Zabokritskiy
* **Original Reference:** P. Balister, E. Győri, R. H. Schelp (2011), *Coloring vertices and edges of a graph by integers*, European Journal of Combinatorics 32(4), 533–537.
* **In-Paper Location:** [Conjecture 1 (Page 6)](https://arxiv.org/pdf/2607.08630#page=6) in Section 2 "Preliminaries and the Hyperplane Viewpoint"
* **OpenConjecture ID:** 4415 ([OpenConjecture](https://github.com/davisrbr/conjectures-arxiv))

### Prerequisites needed
None. Standard finite combinatorics and finite vector spaces over $\mathbb{F}_2 = \mathrm{ZMod } 2$ are available in Mathlib4.

A Lean 4 statement sketch:
```lean
import Mathlib

/-- The Balister–Győri–Schelp (BGS) Conjecture (2011). -/
theorem bgs_conjecture (s : ℕ) (hs : 2 ≤ s)
(v : Fin (2^(s - 1)) → (Fin s → ZMod 2))
(hv_nonzero : ∀ i, v i ≠ 0)
(hv_sum : ∑ i, v i = 0) :
∃ (p : Fin (2^(s - 1)) → (Fin s → ZMod 2) × (Fin s → ZMod 2)),
(∀ i, (p i).1 + (p i).2 = v i) ∧
(Function.Injective (fun (b : Bool, i : Fin (2^(s - 1))) => if b then (p i).1 else (p i).2)) := by
sorry
```

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This issue was generated by an AI agent and reviewed by me.