We've been discussing a lot and in order to avoid a spaghetti discussion, let's recap and agree on what we're exactly arguing for now and what we've already covered...Is there really any chance that a child will choose School and Homework with no external requirement?
The logical merits of single predestination was argued over first which led to a fork over whether God had provided a valid path of salvation to the non-elect outside of predestined salvation. I replied in the affirmative and argued that God not only desired the non-elect to repent and live but also showed conditional mercy in giving them a new heart unto repentance with knowledge of the truth in which if they do self-determine to endure in faith to the end, they will be saved (Scriptural references provided). Both of us agree that they themselves certainly self-determine to fall away.
You count this certainty as grounds for predestination but I count it simply as man's self-determinism - which occurs by definition only after predestination and not before any man's good or evil. I'd asked how you hold God to have factored in man's certain destruction, on account of his evil innate nature, without factoring in Adam's future evil that would stand in violation of Rom 9:11 and that's been left open-ended.
On my part, I'd raised the question on how God's nature can be consistent if He desires against what He Himself earlier sovereignly decreed/counselled. This is being discussed now. And what of the Hebrews falling away passages which reveal an initial conditional supernatural work done by God in the non-elect towards salvation. Yet to discuss this here.
You'd asked for single predestination's interpretation of Rom 9 and I listed out related beliefs in a recent post. Hardening is not from predestination but only after the non-elect fill up their measure of iniquity.
So your above question is more about the child self-determining its destiny. And this self-determined choice by definition is not predestined. Which brings us back to the open-ended question earlier - If the certainty is derived only on account of man's own self-determinism, then that's not predestined since such a choice by definition cannot be before any man's good or evil. However, if you can show that the certainty was derived from God's own sovereign decrees to not provide Christ or regeneration as means of salvation to the non-elect, independent of man's own good or evil, then I'd argue over different things but at least we'd have made some progress.