A3

A3 is a modular speedsolving system for 3x3x3. It consists of techniques involving the combination of freestyle matching and pseudo blockbuilding with algorithms used to solve cases while positioning any pseudo blocks or pieces.
Steps
The system consists of three main, interconnected, techniques:
1. Passive Blockbuilding  This is when a user freely builds blocks or pieces and places them without regard to orientation or permutation. When placed, the pieces can be correctly built but not permuted, not oriented, or they can have simply been placed correctly. Blockbuilding can consist of 1x2x3s, 1x2x2s, 1x1x2s, or even lone pieces.
2. Resolution  Once the user reaches a point where they recognize a case for which they know an algorithm, the algorithm is applied. This algorithm simultaneously positions pseudo blocks or pieces that were built in the previous step. If there are other steps after this resolution algorithm is applied, the user can continue solving what is left. Resolution can be applied during any algorithm set that is a part of the solve. In ZZ, resolution could be used in PLL, NMLL, EPLL, ZBLL, or any other variant. In Roux, resolution would be CMLL.
3. Progression  With passive blockbuilding, the user can start out simple. This means going from matching blocks to building nonmatching blocks, such as blocks that are an R or R' turn away in the Roux or ZZ method. The user can then progress to building blocks with pairs that aren't connected to the correct edge. The associated algorithms and recognition for the current case and correcting these blocks would be learned. From there, users can move to solving other types such as misoriented blocks or pieces. The further the solver goes, and the more algorithms learned for those cases, the more potential there is for saving moves. This system of progression means freedom to build however is desired and to continue to add to the solver's abilities.
As a system, A3 is the application of the three techniques above to another method. In this way it wouldn't be defined as a traditional method. A3 can be used as a framework to improve existing methods. Because of the free nature of this system, there are many addons that users can create. These addons are the combination of new types of pseudo blocks and the associated algorithms that later solve a case and adjust the pseudo blocks. Additional techniques, such as Transformation, can be used to further reduce the move count and reduce the number of cases. To be used as a traditional method of its own, A3 solvers would make use of freestyle, passive blockbuilding to reach a point where they apply a resolution algorithm. At this point in time speedsolvers don't use freestyle solving so A3 is best used when applied to other methods. This system can also be applied to big cubes and other puzzles. In direct solving methods, for example, the inner layers can be pseudo built in the same way that the outer layers are when A3 is applied to the 3x3.
The Resolution step works best in steps with few cases. This is because as the solver adds to the types of blocks they build, a new algorithm set is learned. In the ZZ method, ZBLL is a good fit for the nonmatching blocks that are an R/R'/R2 away because solvers don't have to learn new algorithms. However, it probably wouldn't be a good choice for anything further because of ZBLL's high case count. EPLL, NMLL, or PLL would be easiest in ZZ.
Progression
The point isn't to build whatever is seen first and place things without thinking. It is to be done progressively and it is likely best for solvers to start out with simple nonmatching blocks. These are easiest because they don't change recognition much for the resolution step. Then solvers move on to learning another type that is easy to recognize and build. It probably isn't worth it to go to the extreme depths. However, what is considered extreme is method relative. A misoriented edge or pair would be extreme in ZZ, but not so much in an F2L method that doesn't orient edges early. The obvious benefits are the ability to plan more during inspection and reduced move count throughout the solve. Below is an example of the various ways pseudo can be applied to a 1x2x3 block. This chart doesn't contain the many mirrored, inverse, and other similar versions. Below each type are the setup moves to reach its state.
Examples
ZZ
Example 1:
 Scramble: F' D2 F2 D2 R F2 L D2 F2 D2 B2 L R' D' L' F' R F' L B D'
 EOLine: L B' D2 L' R' F L D
 1x2x2: L U2 R U' R U2
 1x2x2: L U2 R' U2
 Pair: R' U R' U' R U2 R'
 Pair: U2 L U L' U' L U2 L'
 ZBLL
 L U' R U L' U R' U2 L U' R U R' L2
 NMLL
 Separation: U2 R' U' R U' R' U2 R
 Permutation: U2 F R2 U' L' U R2 U' L U F' (U r')
 COLL/EPLL
 COLL: U2 R2 D R' U2 R D' R' U2 R'
 EPLL: M2 U' M U2 M' U' M2 U' r'
 OCLL/PLL
 OCLL: R2 D' R U2 R' D R U2 R
 PLL: U2 F R U' R' U' R U R' F' R U R' U' R' F R F' r'
Example 2:
 Scramble: R2 F' R2 D F2 U F2 D2 B2 U R2 U2 B L F2 D' U' L2 R' D2
 EOLine: F U2 L R F L' D'
 1x2x3: L' R' U' L' U' L R' U2 R U L
 1x2x3: R' U R' U R'
 ZBLL
 U R U2 R' U' R U' R' U' L' U' L U R U R' U' R2
 NMLL
 Separation: U2 R U2 R2 U' R2 U' R2 U2 R
 Permutation: L' U2 L R U2 R' U2 R2
Roux
 Scramble: U B' L F2 D L F D R' D2 R U2 L' B2 L D2 R B2 D'
 FB: y2 U L' U' l2 B'
 SB: R' M2 U2 r U R' U' r' U R
 CMLL: U2 l U' R' F r U' B L' B'
 LSE: M2 U' M U M' U2 M2 U M' U2 M U2 r'
CFOP
 Scramble: L F' B' U2 L B' R2 D' L2 D F2 R2 D' L2 D2 L2 D' R2 L'
 XCross: y' x' L D' R L2 u2 F2 u
 Pair 2: R U R'
 Pair 3: y R U2 R' U' R U R'
 Pair 4: L U L'
 OLL: F' r U R' U' r' F R
 PLL: U R U' D R U R' D R D' R U' R' D' R' U' L2