Shielding effect stays constant across a period because number of inner electrons stays the same.
Well, that's not true.
It only works for the period =1,2,3 but in period 4, the number of inner electrons of $K$ to $Cu$ is 18,while the number of inner electrons from $Zn$ to $Kr$ is 28. So the number of inner electrons is not the same, so the shielding effect isn't constant.
Now there is another definition i saw at more than 3 sites which i found it more sensible and works well, is that :
The shielding effect stays constant across a period because number of inner shells stays constant.
And that is true because at period 4, number of inner shells is 3 for all elements in period 4, and this works for all periods.
So what is the right definition?