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Which is better and faster?
- recognising that you have PLL + parity, and doing the parity from any angle, not regarding what PLL you get afterwards
OR
- Recognising that you have PLL + parity, finding and angle to do a suitable PLL from, and doing said PLL so you end up with opposite swap PLL Parity
Third option: solve PLL parity in a single alg.

Around half of the adjacent-CP cases have algs that are (probably) better than doing normal PLL + normal PLL parity in either order. E.g. R U' L U2 R' U 3r' r2 U2 r2 u2 2R2 u2 is much shorter than normal parity + J perm and is mostly RU-ish.

i hate doing slice moves

which is faster parity alg ?

Rw2 R2 U2 Rw2 R2 Uw2 Rw2 R2 Uw2

or

Rw2 F2 U2 Rw2 R2 U2 F2 Rw2 ???
Third option again: u2 r2 U2 2R2 U2 r2 u2. Pretty much a toss-up between this and r2 F2 U2 2R2 U2 F2 r2 if you don't want (many) slice moves.
 
my 4x4 PB is around 40.xx

around 2 or 3 days ago i almost get PB. my time is 41.37 with PLL parity

i know nobody care if i get parity or not, but i just want to tell you if i get no PLL parity, maybe around 39.xx
I agree. I broke my pb and got my first sub-1 minute solve on 4x4 a couple of weeks ago, it was a 58.31 with pll parity. And if I hadn't gotten pll parity, I believe my solve could have been a 56 or 57 second solve (OLL Parity doesn't faze me, and also I am not very fast at pll parity. Lol).
 
I agree. I broke my pb and got my first sub-1 minute solve on 4x4 a couple of weeks ago, it was a 58.31 with pll parity. And if I hadn't gotten pll parity, I believe my solve could have been a 56 or 57 second solve (OLL Parity doesn't faze me, and also I am not very fast at pll parity. Lol).
PLL Parity is a 1.5 second alg. OLL parity on the other hand, for me, is 5 seconds
 
No, I said it could have been. I don't count it. If all my pb are 7.00 seconds, I say they could have been better if the alg was better and easier, but I still count it as a 7 second pb. It is a thought of "could have been " not of "what I count it as ".
 
What's the notation for the inner L or R slice on the 4x4?
In SiGN, the notation most people should be using unless they have an extremely good reason not to:
2L, 2R

In Singmaster notation, the old notation a lot of people still use, but is kind of useless outside of blindsolving:
l, r
(note that lowercase letters mean wide moves in SiGN)

In WCA notation, which does not allow notating slice moves directly:
Lw L', Rw R'
 
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