Your question is a deceptively simple/basic one, but its not that simple to answer, which is probably why no one has so far answered it.
I'll have a go, but you might get a more definitive response from the guru's that minister to the "Science of Lubricants and Additives" forum.
If I understand it correctly, you're asking what the relationship between TBN and sludge formation is.
"Relationship" here could mean "statistical/temporal association" describing (positive or negative) correlation between the two phenomena, or it could mean causal relationship, as in, one thing (or its converse) causes another.
The first meaning is fairly straightforward. TBN and sludge formation are negatively correlated, so sludge formation is associated/corresponds/tends to coincide with low/decreasing TBN numbers.
Establishing a causal relationship is trickier, since there is a lot going on as oil ages. As TBN goes down, acid number (AN) tends to go up, but I think it'd be a stretch to say that AN causes sludge formation.
Sludge formation is mostly due to oxidative polymerisation, which will also generate organic acids, so, (at least to an extent) a rise in AN is an effect, rather than a cause, of sludge (or sludge precursor) formation.
Since sludge formation is mostly due to oxidative polymerisation, a decline in antioxidant additives (which tends to coincide with decline in TBN) is likely to be causally related to it. Whether this is directly related to TBN might depend on the extent to which the basic/buffering additives are likely to be also antioxidants, which I don't know, but I THINK the TBN contributors are mostly detergent/dispersents like sulphonates and generally not also significant antioxidants (?).
It might also depend on the extent to which oxidation is directly influenced by (psuedo) pH, which I don't know either.
This article dances around this question but doesn't directly answer it. Here sludge formation is related to decline in ZDDP, which is normally thought of as an anti-wear agent but is also an anti-oxidant. Others include aromatic amines and phenols.
http://www.machinerylubrication.com/Read/1052/acid-number-test
Short version (my best guess): TBN can be used to predict sludge formation but it (or its Acid Number converse) don't cause it.