Some of the sugar is still inside the grain until you raise the temperature it cannot easily be rinsed out... just one reference to start off with:
http://www.howtobrew.com/section3/chapter17.html
Edit: If the sugar was already all in solution, fly sparging would have no benefit.
So, I went back and reread the section of How to Brew that you linked. Couldn't find anything in there that contradicted what I said. Can you point out the part that you think does contradict what I said? The grain bed is kind of like a sponge. If you dip a dry sponge in water, and then remove it, some of the water drains off and some remains in the sponge. If you dipped the sponge in sugar water, then some of the sugar would be retained in the sponge along with the water, but this doesn't mean that the sugar has entered the cellulose of the sponge, it's just trapped in the pores and still dissolved in the water. Same with sugar in wort retained in the grain bed. It's dissolved in the water.
As to your Edit question, let's assume we are making a 5 gal batch with 12 lbs of grain, and that the weighted average of the grain potentials is 1.036. We then have 12*36 = 432 potential gravity points, So, if we could actually get 5 gal of wort with all of the available sugar, we would have an SG of
1 + 432 / (5 * 1000) = 1.0864.
Now an SG of 1.0864 converts to Plato as 20.51. Thus our 5 gals of wort would have:
0.2051 * 1.0864 *8.329 *5 = 9.2634 lb of sugar
If we used 1.5 qt/lb of strike water we would have
1.5 * 12 = 18 qts or 4.5 gal of strike water
If we got 100% conversion efficiency, then the wort in the mash would have:
4.5 * 8.329 = 37.48 lb of water
and
9.2634 lb of sugar
The wort would be
9.2634 / (9.2634 + 37.48) = 0.198176 => 19.8176% sugar or 19.8176 Plato
19.8176 Plato converts to an SG of 1.083
The total volume of wort in the mash is
(9.2634 + 37.48) lb / (1.083 * 8.329 lb/gal) = 5.18 gal
If we have a typical apparent grain absorption of 0.125 gal/lb, then we would have
12 * 0.125 = 1.5 gal apparent absortion
So, if we ran off all of the wort in the MLT without fly sparging, we would collect
4.5 - 1.5 = 3.0 gal
But the actual volume of wort in the mash was 5.2 gal so the real grain absorption was:
5.18 - 3.0 = 2.18 gal of 19.8176 Plato wort
This 2.18 gal of retained wort is 19.8176% sugar by weight (19.8176 Plato) so the amount of sugar retained in the wort is
0.198176 * 2.18 * 1.083 * 8.329 = 3.897 lb
The sugar collected in the BK would be
0.198176 * 3.0 * 1.084 * 8.329 = 5.363 lb
Without sparging we would be leaving
3.897 / (3.897 + 5.363) = 0.4208 => 42.08%
of the sugar we created in the mash as wort held in the grain bed, and our efficiency would only be 57.92%. We need to sparge to get more of this sugar out of the grain bed and into the BK. Either fly sparging or batch sparging would dilute the wort in the grain bed, and the sparge run off would contain most of the remaining sugar in the wort retained in the grain bed. Since we have collected more of the sugar by sparging, our efficiency went up.
Brew on
