In other words, I am not suggesting (nor have I ever suggested) that one can simply use a1 (ignoring a2, a3, etc.) that has been obtained from a higher-order fit.
I'll concede that for A.J.'s measurement of Roasted Barley, the difference in mEq/kg is only 5. So in that specific case, it's not a good example. Also, we'd have to agree that 300L roasted barley was a "dark" roasted malt.
In other cases, however, I guess I wonder how this differs. If you measure a1 for a specific grain (like you did in your paper, which I think is great BTW) and use it in:
dQMalt = a1 * (pHz - pH DI)
how is that any different than having measured a2 and a3 as well and just disregarded them? Whether you measure a2 and a3 or not, the malt still has those values. Isn't approximating dQMalt for a specific malt with only a linear approximation of it's full acidity going to introduce error in more acidic malts?
For instance, If I take the data from A.J.'s measurement of Crisp Chocolate:
pH DI = 4.70, a1 = -76.43, a2 = -0.40, a3 = -3.84
and plug and chug for pHz = 5.40:
Linear dQMalt = ~ -155 mEq/kg
Poly dQMalt = ~ -27 mEq/kg
That's certainly not a small difference.
As far a Sauermalz goes, I believe it is best to simply treat it as a source of lactic acid (which it is), rather than as a malt.
Here is where we get into more trouble though. Sauermalz is a malt treated with biological acid, i.e. Sauergut, and really can't be treated with any degree of accuracy as a straight source of Lactic Acid by weight. We can't even treat Sauergut as an equivalent to Lactic Acid, so how can we reliably do it for Sauermalz?
I use Sauermalz eclusively as my acidifying agent ("Agent of Acidfication" would be an awesome brewing superhero!) and for batches using Brun Water, My old sheet, and MpH, I have never had a time where I didn't have to alter the acid % or strength modifier in Brun Water significantly to match values in the field. Strictly speaking, I have never used the nominal percentages quoted by some for Sauermalz.
I believe the chart I have seen from A.J. with the Lactic and Sauermalz overlayed, only shows coincidence at one specific pH.
And again, we have valuable titration data measured for Sauermalz, so why wouldn't we use it? Certainly it's better than estimating it's acidity based on something that it isn't, no? Granted, it may not be perfect, but it seems a step in a better direction.
I just want to state that I am not trying to antagonize you at all. I am a curious about all these things and actively looking into figuring them out. I appreciate your papers and work.
I am disagreeing with you in the best and most productive terms because i want to know these things and how they work.
Cheers!