I found a really cool article that explains in depth what oxidation,varnish, and sludge and their consequences are and also details the testing methods to find these. It makes for a worthwhile read:
Sludge, Varnish and Oxidation
Here are a few salient points I would like to recapitulate:
1. "When oil oxidizes numerous decomposition products are formed, including acids. Heat and the presence of metals such as iron or copper particles accelerate the process. So too, highly aerated oils are far more susceptible to oxidation. Primary oxidation products, known as aldehydes and ketones, develop (grow in size) through a series of steps (free-radical chain-reaction) to form polymers and other high molecular weight condensations. Eventually, the oil's viscosity begins to increase and the dense oxide suspensions can no longer be held in a stable oil-dissolved state. It is at this point when the oil is said to "throw sludge" leading to the formation of deposits and varnish."
2. "When passing a small amount of solvent-diluted oil through a one-micron membrane you will often see sludge and amber colored polymers present on the membrane surface"
3. "Additionally, some very dense depth-type filter media, often used as off-line filters, are said to have a strong affinity for carbon suspensions."
Toilet paper bypass filters are advertised to be depth-type filters and remove particles below 1 micron in size. From these excerpts, it would seem that any of the TP Bypass Filters can remove most of the decomposition by-products of oxidation and sludge precursors: acids, and polymers formed by oxidation of the oil. The article explained to me the reason why my used TP roll would excrete some very thick, black oil when squeezed.