"Technically" might better be expressed as "Speculatively." This is the realm of pure theory. We have an artificial child, which we have reduced to a single descriptive essence. I hear you saying, "I want to know concerning this representative child the way God knows."Thanks, brother. I’ll have to go through this when I have time.
For sake of the discussion, let’s say it is a month old, unregenerate (from God’s perspective) baby.
Technically, are they in the CoW substantially, and in the CoG administration?
Perhaps (speculating; we have no supervision of the independent moves of the Spirit, Jn.3:8) God has already begun a lengthy process of spiritual transformation in the one-month old child, with eternal blessedness in view. If God's elect, it will begin sometime by the due use of ordinary means. In baptism, as Presbyterians we know one thing for certain: God makes a visible claim on this infant person. He makes that claim in connection with the outward profession of the parent, a profession that states in part: "I belong to God."
Because God claims the parent in toto, everything that belongs to him belongs to God; and in biblical terms children belong to their parents. God does not ordain baptism for one's house, car, or monogrammed luggage; but ever since Abraham he has placed a special claim on image-bearing possessions (typically children) of his covenanted image bearers. He wants his Name on them also.
What is the point? That the baptism men perform or submit to isn't the spiritual reality to which it points, but is the sign designed to so point. Water baptism is the claim. A baptized person--of any age--has a claim placed on them, designating them as individuals who ought to be regarded as God's covenant possession. So, the covenant of grace is claiming that one-month old child, and his regenerate STATUS (if it could be known) is irrelevant. This is how the Presbyterian sees the matter.
All we know is the administration. Even in regard to adults, for baptism and profession, we simply take in some additional evidence being offered suitable to the person. We act not on the secret things but on what is revealed. We don't ask: "Has this person truly passed from death to life from God's perspective." We don't get to know things the way God knows. Someone who has nothing more than a said-faith is still in his sin, still bound to Adam and the claims of the covenant of works. But we still baptize the man who makes the profession, because we can't see the heart or future evidence beyond today. It is not for us to harbor doubts, supposing he may be substantially bound to works--rather like hopelessly attending a wedding when we're aware many marriages fail.
God's claim is announced in the man's baptism. He actually speaks through his church; not effecting spiritual reality by means of its work, but making meaningful even powerful statements. The administration, though visible, is not without weight or significance precisely because it is divine in origin, and has a God ordained connection with the spiritual realm. Thus it is highly offensive and wicked to despise his gracious covenant claim, exactly what Heb.6:4ff warns against. It isn't "just" administrative covenant connection that's in view; but as God's administration, it is a most serious claim.
Getting back to the one-month old child, who may be wholly unregenerate or may have already the beginning of his salvation conveyed to him on the earliest whispering breezes of the Spirit's mighty gusting to come. God administratively claims him, we baptize him, the baptism stands for the claim and "is" the claim of the covenant of grace. We are duty bound to go to battle with any other claims, be they substantive or administrative. In due season, through the Spirit's personal ministry and his church instruments, we desire to see such a child soon singing Christ's praises, understanding his need of such a Savior, praying to so august a Person who has stooped to claim him, and believing in the Lord Jesus for his entire and only hope of redemption. May he never willfully challenge that covenant claim so early staked to him.