We are its living word, and not/ a book it wrote
The statement "we are its living word" has religious connotations. "living word" usually refers to the 'Word of God' so the implication is that our beings are bound up with the very fate that determines existence. Paterson makes clear we are not entirely powerless. Fate may have written “a book” for us but we are “living”. This suggests we can influence what happens to us.
The "book it wrote and then forgot" could denote any spiritual text, while "its fourteen-billion-year-old song" refers to the age of our universe. "Song" makes us think of a melody or patterning that is passed from generation to generation like poetry itself. We might conclude from this that whether we realise it or not there is some force at work in our actions, in our "right and wrong" and it might even be ourselves.
At this point, Paterson returns to the personal anecdote that begins the poem: his son's drawing. There is a symmetry to the structure as the reference in the penultimate stanza to the "spoiled work and useless kit" takes us back to the second stanza's screwed up picture. The child's fist coming "down like a stone" has a sense of finality about it, as if he simply wants to give it all up. There is a perhaps a biblical link here to the stone that sealed Christ's tomb. In this sense, we might read the final stanza as a sort of resurrection when the stone of despair is rolled away to reveal something miraculous.
look at the little avatar/ of your muddy water-jar
Whether it is down to science, or fate or some kind of god, perfection is now to be found in the "muddy water-jar" – in the humblest thing. The rhyme between "avatar" and "water-jar" confirms this. Avatar returns us to the image of Krishna. The "muddy" water suggests the mud from which the God of the Old Testament moulded Adam and Eve.
Paterson directs his son and the reader to look to the jar and to find "the perfect ring" which has been there all along. Here is the perfection, which has been created without even trying. Again he refers to "singing" as if this "living" voice is always there, if we just listen to it. The present participle also suggests its perpetuity. Instead of the heavens and space above us, this "ring" now resounds "under everything", which implies it is hidden and that it will accompany us whether we have high aspirations or not. But the final point to make is that, despite its perfection, the ring exists in muddied water and therefore is imperfect. We must accept therefore that perfection is found in imperfection - after all that is what it is to be human.