Two flipped edges, nothing weird about that -.-

My confusion stemmed from the fact that the rest of the cube is solved; those are the only two edges remaining, and they need to be swapped. However, it is impossible to perform only a single swap, so how is this case possible?

These are flipped edges, or mis-oriented edges. So, this is not an impossible case, you did Step 4a) wrong in Roux, when you didn't oriented all the edges correctly (or mixed up the orientation in a later step).

"since the laws of cubing dictate that there can only be an even number of swaps" - this is true, although, your case is not a swap of two pieces. It's a flip of two pieces.

Thanks for clearing up the confusion.
 
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Could someone help me with my solves to see what I'm doing wrong?
Scramble: U B2 F2 U L' D F U L2 R2 B' R2 B' R D F U2 L2 B' R L F' U' B' U2
Inspection: x
First Block: L2 D' F' U' M2' B
Second Block: U R2 U M2' U R2' U2 r' U' r R U R' U2 R U2 R' U2 R U' R'
CMLL: R U R' U R U2 R' U R U R' U' R' F R2 U' R' U' R U R' F'
LSE: M2 U2 M' U2 M U' M U M2' U' M' U2 M2' U2 M
That one took forever to do XD I'll edit another one on here tomorrow.
 
Could someone help me with my solves to see what I'm doing wrong?
Scramble: U B2 F2 U L' D F U L2 R2 B' R2 B' R D F U2 L2 B' R L F' U' B' U2
Inspection: x
First Block: L2 D' F' U' M2' B
Second Block: U R2 U M2' U R2' U2 r' U' r R U R' U2 R U2 R' U2 R U' R'
CMLL: R U R' U R U2 R' U R U R' U' R' F R2 U' R' U' R U R' F'
LSE: M2 U2 M' U2 M U' M U M2' U' M' U2 M2' U2 M
That one took forever to do XD I'll edit another one on here tomorrow.

Second block is quite long.
U M2 U r U' R could be used for one half of the SB

I'm no rouxer, so don't take my word.
 
Could someone help me with my solves to see what I'm doing wrong?

Your 2nd block has too much CFOP influence, and unfortunately there are a lot of moves, even for CFOP. But this is Roux!

The initial part of 2nd block is fine, but there are still more moves than needed.
You did:

Second Block: U R2 U M2' U R2' U2 r' U' r

Which is:

U R2 // pair
U M2' U R2' // insert edge
U2 r' U' r // insert pair
View at alg.garron.us

Instead, try this:

U M2' U r U' r // RB block

If it helps, look at it like this:

U R2 // pair
r2 U // place edge
r U' r // finish block
View at alg.garron.us

-*-*-

You solved the last edge like this:

R U R' U2 R U2 R' U2 R U' R'

Which is way too many moves.

Try this:

U r U' M U R'
=
U (r U' r') (R U R') // RF pair

Even preserving M slice there is:

U F' R U R' U' R' F R // RF pair
or
R' U' R' U' R U R U R // RF pair
 
Could someone help me with my solves to see what I'm doing wrong?
Scramble: U B2 F2 U L' D F U L2 R2 B' R2 B' R D F U2 L2 B' R L F' U' B' U2
Inspection: x
First Block: L2 D' F' U' M2' B

SB:
U M2 U r U' // 1x2x2
R U' r U' r' U2 r U r' // Insert last pair

Or:
U2 r2 U r U2 R' U' R U // 1x2x2
R U M U r U2 r' U' r // Insert last pair
 
I've heard from a variety of sources that experienced Roux solvers complete step 4a (EO) and 4b (UL & UR pieces) as a single step. I seem to have trouble transitioning fluently, leading me to believe this is a heavy contributor to increased times.

Any tips to look out for in these steps? Perhaps I need a new perspective of this.
 
I've heard from a variety of sources that experienced Roux solvers complete step 4a (EO) and 4b (UL & UR pieces) as a single step. I seem to have trouble transitioning fluently, leading me to believe this is a heavy contributor to increased times.

Any tips to look out for in these steps? Perhaps I need a new perspective of this.

Just practice tracking your UL and UR edges. This is the only way until you become more advanced and try to do them simultaneously.
 
I've heard from a variety of sources that experienced Roux solvers complete step 4a (EO) and 4b (UL & UR pieces) as a single step. I seem to have trouble transitioning fluently, leading me to believe this is a heavy contributor to increased times.

Any tips to look out for in these steps? Perhaps I need a new perspective of this.

See for example the following scrambling:
D B F' R' D2 U2 L' B' F U x2

The EO is simple here, just a regular U' M' U M. However, instead of that, if we choose a different solution, we can influence the UL/UR edges to a better position:
U' M' U' M
the remaining part is fast and easy:
U M2 U E2 M' E2 M'

Another example:
B L' B D' R' L B U' L U R2 L2 D2 x

Instead of M' or M, we start with U M, which lead us to an easy EO, where we can put UL/UR in its place again directly: U M' U' M' U', and finally, E2 M' E2 M'.
(whole solution: U M U M' U' M' U' E2 M' E2 M')

This unfortunetly not always work (at least for me...), but in 1 out of 3 or 4 cases EO can be merged with UL/UR quite easily. Just keep practicing of tracking elements.
 
Could someone help me with my solves to see what I'm doing wrong?
Scramble: U B2 F2 U L' D F U L2 R2 B' R2 B' R D F U2 L2 B' R L F' U' B' U2
Inspection: x
First Block: L2 D' F' U' M2' B
Second Block: U R2 U M2' U R2' U2 r' U' r R U R' U2 R U2 R' U2 R U' R'
CMLL: R U R' U R U2 R' U R U R' U' R' F R2 U' R' U' R U R' F'
LSE: M2 U2 M' U2 M U' M U M2' U' M' U2 M2' U2 M
That one took forever to do XD I'll edit another one on here tomorrow.
Another 2

Scramble: U2 F U L2 R D2 L D F B U' D B' L B2 L2 F D L2 U' B' D2 B2 F' R
z2
First Block: B D M' U x2 L' D' U' R' U R L U L' U r B'
Second Block: R U2 R' U' R2 U R M U R U2 R2 F R F'
CMLL: U2 R' U' R U' R' U2 R2 U R' U' R' F R2 U' R' U' R U' R' F'
LSE: U2 M' U M' U M2' U M2' U' M U2 M

Scramble: D U' B F' L' D' U R2 D B' U R2 B' R2 F2 D' L' F' D2 L' R F' D' F2 B
y'
First Block: U2 B2 M U L U L2 U' M U' M2' U2 F'
Second Block: U2 M2' R U2 r U r' M' R' U' R U r' U' R
CMLL: U2 R' F R U R' U' R' F' R U R F R U' R' U' R U R' F' R U R' U' R' F R F'
LSE: U2 M U M U M2' U M2' U' M' U2 M U2 M2'
 
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Another 2

Scramble: U2 F U L2 R D2 L D F B U' D B' L B2 L2 F D L2 U' B' D2 B2 F' R
z2
First Block: B D M' U x2 D L' D' U' R' U R L U L' U r B'
Second Block: R U2 R' U' R2 U R M U R U2 R2 F R F'
CMLL: U2 R' U' R U' R' U2 R2 U R' U' R' F R2 U' R' U' R U' R' F'
LSE: U2 M' U M' U M2' U M2' U' M U2 M

What is that first block?
 
Solution 2

Scramble: D U' B F' L' D' U R2 D B' U R2 B' R2 F2 D' L' F' D2 L' R F' D' F2 B
y'
First Block: U2 B2 M U L U L2 U' M U' M2' U2 F'
Second Block: U2 M2' R U2 r U r' M' R' U' R U r' U' R
CMLL: U2 R' F R U R' U' R' F' R U R F R U' R' U' R U R' F' R U R' U' R' F R F'
LSE: U2 M U M U M2' U M2' U' M' U2 M U2 M2'



To finish the first block you did a pair as (U') M U' M2' U2 F'. You don't need to build the pair on the top layer, try something like (U') M U' R2 F.
 
Solution 1

Your solution does not work, so I have to guess which block colors do you prefer.

Scramble:
U2 F U L2 R D2 L D F B U' D B' L B2 L2 F D L2 U' B' D2 B2 F' R

z2 y'
M' U2 M R U x' // 1x2x2
B M2 r B' // FB
U M2 U M' U R' U' R2 // 1x2x2
U' M U M' r U R' // SB
U R' U' R U' R' U R U' R' U R U' R' U2 R // CMLL(preserve EO)
U' M' U2 M' U' M2 U // ULUR
M' U2 M2 U2 M // Mperm



Solution 2

Scramble:
D U' B F' L' D' U R2 D B' U R2 B' R2 F2 D' L' F' D2 L' R F' D' F2 B

y' x'
r U r R D' // 1x2x2
R2 U' r' M U x' // SB
U2 R U' M2 U' // 1x2x2
R U R // SB
U R' U' R U' R' U2 R2 U R' U R U2 R' // CMLL
M U' M' U2 M' U M' // EO
U M' U2 M U // ULUR
M2 U2 M U2 M // Mperm
 
Solution 1

Your solution does not work, so I have to guess which block colors do you prefer.

Scramble:
U2 F U L2 R D2 L D F B U' D B' L B2 L2 F D L2 U' B' D2 B2 F' R

z2 y'
M' U2 M R U x' // 1x2x2
B M2 r B' // FB
U M2 U M' U R' U' R2 // 1x2x2
U' M U M' r U R' // SB
U R' U' R U' R' U R U' R' U R U' R' U2 R // CMLL(preserve EO)
U' M' U2 M' U' M2 U // ULUR
M' U2 M2 U2 M // Mperm



Solution 2

Scramble:
D U' B F' L' D' U R2 D B' U R2 B' R2 F2 D' L' F' D2 L' R F' D' F2 B

y' x'
r U r R D' // 1x2x2
R2 U' r' M U x' // SB
U2 R U' M2 U' // 1x2x2
R U R // SB
U R' U' R U' R' U2 R2 U R' U R U2 R' // CMLL
M U' M' U2 M' U M' // EO
U M' U2 M U // ULUR
M2 U2 M U2 M // Mperm
Fixed my first solution (I think). I'm white/yellow neutral.
 
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Another 2

Scramble: U2 F U L2 R D2 L D F B U' D B' L B2 L2 F D L2 U' B' D2 B2 F' R
z2
First Block: B D M' U x2 L' D' U' R' U R L U L' U r B'
Second Block: R U2 R' U' R2 U R M U R U2 R2 F R F'
CMLL: U2 R' U' R U' R' U2 R2 U R' U' R' F R2 U' R' U' R U' R' F'
LSE: U2 M' U M' U M2' U M2' U' M U2 M

Scramble: D U' B F' L' D' U R2 D B' U R2 B' R2 F2 D' L' F' D2 L' R F' D' F2 B
y'
First Block: U2 B2 M U L U L2 U' M U' M2' U2 F'
Second Block: U2 M2' R U2 r U r' M' R' U' R U r' U' R
CMLL: U2 R' F R U R' U' R' F' R U R F R U' R' U' R U R' F' R U R' U' R' F R F'
LSE: U2 M U M U M2' U M2' U' M' U2 M U2 M2'
First Block should be finished in less than 10 moves or so.
I recommend you make block many times with same scramble and go over some top rouxers' reconstructions.
 
Scramble: U2 F U L2 R D2 L D F B U' D B' L B2 L2 F D L2 U' B' D2 B2 F' R
y
first block: r U' R2 U B2 F R r2 F
second block: R' M2 U2 r U R' U' M U r'
cmll: F R U R' U' F' L' U R U' L U R'
LSE: M2 U M U M U M' U2 M'

Scramble: D U' B F' L' D' U R2 D B' U R2 B' R2 F2 D' L' F' D2 L' R F' D' F2 B
x2 y
first block: r' U' R U M' U2 B r' U2 M F
second block: r U R2 U2 R' U' R2 U R' U' R U R'
cmll: L U L' U L U2 l'
LSE: M' U' M' U2 M U' M U' M' U' M' U' M2 U2 M2 U M2
 
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