One of the quckest ways to start a food fight in a room full of pool guys is to start talking water chemistry. In a recent Pentair class, talk got around to whether CYA needed to be factored into (or rather, out of) the total alkalinity reading when testing for the LSI. Some say yes, others no since it is itself part of the total alkalinity. Does the LSI look for TOTAL alkalinity or is it better to separate out just the carbonate alkalinity? As is usually the case, most guys just echo what they heard in a water chemistry class taught in the industry, most of us did not study water chemistry at the college level.
The saturation index most certainly requires the Carbonate Alkalinity in its calculation, not the Total Alkalinity (TA), so the TA needs to be adjusted to remove the effects of Cyanuric Acid (CYA) and, if present, Borates.
The saturation index is nothing more than a measure of the saturation of calcium carbonate in the water so the relevant parameters are calcium and carbonate concentrations (technically, activities, which are effective concentrations as influenced by ionic strength). Calcium you can mostly measure directly as Calcium Hardness (CH), but carbonate is not measured directly. Instead, the Total Alkalinity (TA) test measures all forms of alkalinity so if you remove the contributions from CYA (and borates, if present), then from the resulting carbonate alkalinity and from the pH one can calculate the carbonate amount. That is what the saturation index formula does. The pH is needed because that determines how much of the carbonates are in the form of carbonate ion (which is what we want for the saturation calculation), how much is in the form of bicarbonate ion and how much is in the form of carbonic acid and aqueous carbon dioxide.
The other factors to the saturation index formula are less direct. The temperature affects the equilibrium constants, so affects the product of calcium and carbonate concentrations that result in the saturation of the water (at the edge of precipitating/scaling vs. dissolving calcium carbonate). The Total Dissolved Solids (TDS) is a proxy for the ionic strength in the water where having more ions (and higher charges) in the water partially shields the calcium and carbonate ions lowering their effective concentration.
Now in practice some of these adjustments aren't very large. At a pH of 7.5 and TA of 80 ppm, the difference in the saturation index between 30 ppm CYA and no CYA is only 0.05. However, the difference between 80 ppm CYA and no CYA is 0.16 (the higher CYA level has a lower saturation index, if the TA is the same in both cases). As for TDS, the difference between 3000 ppm salt and a fresh fill with 500 ppm TDS (around 350 ppm salt) is 0.2 in the saturation index (the higher salt level having a lower saturation index).