It won't invalidate the extension, because Parliament is sovereign, and I'm sure the EU leaders understand that. It doesn't matter if the letter was handwritten by a toddler and decorated with dinosaurs or crocheted into a quilt.

If Johnson hadn't sent the letter it would have been sent by another Minister of the Crown, or by judges acting as same. Leaving the letter unsigned makes no difference whatsoever to the EU. The Benn Act made Parliament's intentions absolutely clear, and the EU leaders have responded to the intent, not the material form.

But it does affect Johnson's position. Because by flouting Padfield he wilfully attempted to undermine the wishes of Parliament. The judiciary are are constitutionally empowered to rule on this and possibly punish it.

It was a typical Johnson move - petulant, stupid, ineffectual, and potentially self-harming.