SPRINGFIELD — Neither snow nor rain nor gloom of night will prevent the mail from being delivered. But what exactly counts as proof of delivery? The Illinois Supreme Court will hear arguments on that issue as it begins the second week of its September term on Tuesday. An appellate court earlier this year dismissed an appeal notice as untimely, saying it ran afoul of a high court rule that requires extra paperwork verifying the time the appeal was mailed. After an unfavorable circuit court ruling, the …