Sequence Theory

Sequence Theory, written by Scaffold. If you have questions, reach out to me through my discord scaffoldhax.

Credits to Crescendo (https://www.youtube.com/@crescebdo and https://space.bilibili.com/8252252) and Nastra-OH (https://space.bilibili.com/1709547367)

Instead of overloading yourself with info, I would recommend you understand the basic concepts and use this as a reference page for the more complicated and intricate pieces of info.


Link to basic version: https://pastebin.com/2XVr6hcU
^ I would recommend reading this first if you have little to no prior knowledge




a) Definitions-


SE- Survival Endless
DE- Day Endless
NE- Night Endless
PE- Pool Endless
FE- Fog Endless
RE- Roof Endless
ME- Moon Endless (Night Roof Endless, I would advise against calling it NRE)
MGE- Moon Garden Endless (Requires mods to access)
AQE- Aquarium Endless (Require mods to access)


Instant- A plant that instantly kills zombies when activated or automatically activates when planted. The instants are ice-shrooms, doom-shrooms, jalapeños, cherry bombs, squashes, hypno-shrooms, chompers, potato mines, cob cannons and blovers.

Operation/Action- Anything that you can do inside of the game.

Symbol- A representation of an operation using letters. (P, c, N, etc.)

Sequence- The arrangement of operations.

Acceleration Plants- Instants that are used to speed up wave spawns. (Also called additive plants.)

Deceleration Plants- Instants that are used to slow down wave spawns. (Also called subtractive plants. This includes only ice-shroom.)

Accelerated Wave- A wave that is sped up by an acceleration plant. (Also called additive waves.)

Decelerated Wave- A wave that is slowed down by a deceleration plant. (Also called subtractive waves.)
This is usually called an ice wave which is misleading because not all waves that use ices are ice waves but I will keep calling it that for the rest of the pastebin because it's easier to understand for everyone.
Honestly, the definition of an ice wave is kinda debatable but usually waves that pre-ice and are 10s+ are 100% considered ice waves while waves that are 8s-9s are kinda debatable. It's debatable because in order to have an ice wave of that length, you have to ice extremely late to the point where all of the zombies have already walked in range which kinda ruins the point of an ice wave.

Pao- Chinese word for cob (炮).

Tick- One in-game tick is 1/100th of a second aka one centisecond (cs) and it is the shortest time unit the game uses.

Intro- The first couple waves of a setup before the actual sequence/cycle starts.

Imp Intersection- The action of killing imps with an instant while they're in the air or while they're landing in the rear of the setup. This technique is 0.19 seconds precise in most cases when unslowed (1.07s-1.26s after the garg/giga is hit if you kill the imp in the front) and roughly 0.3s precise when slowed. The numbers change depending on the scenario but these are usually what they are. This is also called 'imp-shot.'

Air Bomb- The action of killing imps in the front of the setup (direct Chinese translation of 空拦)

Tail Bomb- The action of killing imps in the back of the setup (direct Chinese translation of 尾炸)

Roof Wind- A glitch in the original version of PvZ in which cobs shot from columns 1-4 won't shoot correctly in the roof or moon venue. The cobs will look like they land in the row above where they're shot, the column 1-2 cobs will only kill 2 rows of zombies sometimes and they can kill 3 rows of pogos and imps. This is called 'wind' because the Chinese players apparently thought the cob looked like it was being pushed by wind. Example: https://youtu.be/U9x8KZb38Hg (the imp intersection cobs)

Rigorous- Completely harmless and stable (usually when using extreme spawning).


Multi-Cob- A setup with 4 or more cobs roughly.

Low-Cob- A setup with 8 or less cobs roughly (A setup being low-cobbed or not is often based off the rhythm. Setups with slow rhythms like C4u/C5u are usually considered low-cobbed even if it has a lot more than 8 cobs).

Non-Cobbed- A setup that relies on io to accelerate the sequence. This is not the same as cobless (not confusing at all). Examples; Traditional 4 cob, classic 2 cob, 1 cob suppression, and any cobless setup.

Cobless- A setup without cobs.


Wavelength- The length of a wave.

Hard Wavelength- Wavelengths in which a certain zombie has 0 threat at all. (e.g. a 7.47s wave is the shortest wave a catapult can shoot a basketball so a wavelength of 7.46s is a hard wavelength.)

Soft Wavelength- Wavelengths that control the threat of certain zombies within a certain range. (e.g. in the example above, a wavelength of 7.50s would be a soft wavelength. We usually use soft wavelengths for threats that aren't fatal, like basketballs, while we use hard wavelengths for threats that are, like diggers)


Cob Reuse- Repeating the use of cobs in a sequence after the cobs have recharged.

Linear Cob Reuse- The type of cob reuse used in almost all rhythms besides I-series rhythms (Ix, Ixi, Ixu and Ixs). When cobs are reused with this type of cob reuse, all of the cobs are reused to their respective waves.

Non-Linear Cob Reuse- The type of cob reuse used in all I-series rhythms (Ix, Ixi, Ixu and Ixs). When cobs are reused with this type of cob reuse, not all of the cobs are reused to their respective waves. This is sometimes used in outlier sequences like ME20 I8u or sideless 15 FE I9 (https://youtu.be/zBQmV8z-zws and https://www.bilibili.com/video/BV1x4UsYGE4G)

The above two definitions are trash and I can't figure out better definitions so you kinda have to figure out what they mean by exposure. You can ask me if you have questions about that.


Rhythm- The way in which operations are arranged.

Adjacent Rhythms: Rhythms in which the ice waves are back to back. Also called "ortho" rhythms.
Ortho is referring to benzene rings.

Opposite Rhythms: Rhythms in which the ice waves are separated by an equal number of waves. Also called "para" rhythms.
Para is referring to benzene rings.

Inter Rhythms: Rhythms in which the ice waves are separated by an unequal number of waves. Also called "meta" rhythms
Inter means "between" implying that rhythms of this category are in between adjacent and opposite.
Meta is referring to benzene rings.

Wave-by-Wave Rhythms: Rhythms that use special operations for each wave instead of repeating a cycle.


Early Jack: Jack that blows up within 4.4s-7.4s after they spawn. Early jacks are only 5% of all jack spawns.

Late Jack: Jack that blows up within 13.22s-22.68s after they spawn. Late jacks are 95% of all jack spawns.

Grace Period: The time window in which jacks are incapable of blowing up.


Limit Setup: A setup that cannot play endlessly.

Endless Setup: A setup that can play endlessly.



b) Symbols that are commonly used to denote actions.


1) Actions involving cob cannons.

P  : Using a cob. (P cobs are usually the first cob of a wave/operation. P stands for pao)
S  : Using a cob syncing with another cob operation or instant. (Always follows an operation of that wave. S stands for sync)
B  : Using a cob to kill a select group of zombies without spawning the next wave. Often used before P/N/A.
D  : Using a cob to attack the front directly after a cob operation or instant. (D stands for delayed cob)
d  : Using a cob to attack the back directly after a cob operation or instant.
b  : Using a cob to attack the back not after a cob operation or instant.
P' : Shoveling a cob to restart the cooldown in order to reuse the cob early. (Add ' after any cob operation and it means shovel)
G  : Modern glitch where you use a cob on row 7 by either using script or shooting the cob outside of the game's resolution.
I2 : Used when a cob or an instant lands at the same time as an ice activates. Stands for ice2. (Instead of writing PPI, sometimes people write PPI2 although I personally never write this)
I3 : Used when an ice activates on the 105th tick of a garg's imp throwing animation (211th tick if slowed) which causes the animation to restart due to an oversight in coding. Stands for ice3. Example: http://b23.tv/kr2dDzy
Ice1 is extremely obscure but it's ice2 except the cob/instants are delayed enough to kill the pole vaulters. Ice4 is when you ice on the last tick of a garg's plant smashing animation which causes the plant to be unharmed (http://b23.tv/hNuTB39). Neither of these are ever used when writing sequences.
These "icex" techniques are named the way they are just because they were named in order of being discovered.

2) Actions involving cards.

I  : Using ice-shroom. (I stands for ice)
I' : Using imitater ice-shroom. (This is only used in cases where the imitater matters. It's best to call it I when it doesn't matter which ice you use.)
N  : Using doom-shroom. (N stands for nuclear)
n  : Using doom-shroom to attack the back after a cob operation or instant.
A  : Using cherry bomb.
A' : Using imitater cherry bomb. (A' used to be notated as jalapeño)
J  : Using jalapeño.
a  : Using squash.
c  : Using cannon fodder. (c stands for cannon fodder)
m  : Using potato mine. (m stands for mine)
C  : Using spikeweed. (I've seen some English-speaking players use "s" to signify spikeweed. This is unofficial though)
i  : Using cherry/jala/squash/spikeweed to kill zombonies. (i stands for instant, unofficial)
T  : Using tall-nut to stop pole vaulters or pogos. (T stands for tall-nut)
You may add an apostrophe ' after any of these operations to denote using its imitater form like shown with A' or I'.

3) The damage from non-instant plants.

io or IO : The damage that comes from non-instant plants. IO stands for implicit operations.

This can be used in an "empty" wave where you only depend on io and do nothing else but otherwise this isn't really used when writing sequences.




Symbols that are commonly used to arrange our actions.

1) Splitting symbols.

| : To distinguish the actions to be done in different waves.
- : There is a time between two actions.
/ : To distinguish the actions on the upper half and the lower half at the same time.
( ) : The action within the parentheses replaces another action sometimes. This symbol is also used to indicate when an operation is optional or to indicate when to shovel a cob.
Note: If a Chinese player indicates when to shovel a cob, they'll say (铲种) literally meaning "shovel".



c) Rules to writing sequences.

Note: The rules to writing sequences change slightly depending on whether you're in the Chinese community or the English speaking community. These rules are meant to be as accurate as possible but there might be some differences depending on who's writing the sequence.
Also, don't even think about memorizing all 8 of these rules. These are rules that are learned through exposure but I think it's a good idea to write them out anyways.

1. Sequence writing splits the setup between the top half (rows 1-3) and the bottom half (rows 4-6 in a 6 lane venue, 3-5 in a 5 lane venue).
If the top half or bottom half is different, you must specify using the / symbol. One exception is simple doom waves which are sometimes just written without splitting the sides. However, doom waves that require more operations are usually written normally.
Examples: W1 is written PD/PSD instead of PDPSD or PPSDD or whatever other combination of symbols: https://youtu.be/ER_K_Mb7lA8
I wrote the doom wave on w5 NDA instead of NA/ND which is an exception: https://youtu.be/UD_84-3LXUc

2. If one of the sides are different, it's standard to write the top half operation first before writing the bottom half operation. This isn't necessarily 100% required but it's easier to read.
Example: PSD/PD instead of PD/PSD in this setup because top half operations go first; https://youtu.be/5NfO2u23bLU

3. If both of the sides have the same exact operations, you never want to split the sides obviously. Instead, you want to write each individual operation twice. Sometimes, if you want to really compress your sequence writing, you would write each individual operation once. This is theoretically a better way of writing it but this isn't what's standard but I did do this below when writing rhythm example sequences because I wanted them to be compressed.
Example: PPSSDD or PSD compressed instead of anything else in this setup: https://youtu.be/HPcT-7OGIY8

4. The exact standard format for writing sequences in the English speaking community is as follows (x=operation)-
Sequence: P6, x | x | x | x | x | x ( 6 | 6 | 6 | 6 | 6 | 6 )
The numbers to the right are the wavelengths and P6 is the rhythm.
In the Chinese community, often people compress their sequences and write it something like this-
P5u, x|x|x|x|x (6,9)
The operations like PPSSDD are compressed to PSD, the rhythm is sometimes not named, and each wavelength is only written once separated by a comma. Much uglier, much more difficult to read but the info is very dense and compressed. This is never used in the English speaking community though.

5. After an ice is used in an ice wave, the - symbol must be used to denote a break in the operations. If there are pre-cobs killing the diggers + zombonies, then the - is placed after the cobs.
Example: IPP-PPDD on the ice wave instead of I-PP-PPDD or anything else: https://youtu.be/HPcT-7OGIY8

6. If an ice is used at the same time as an instant or cob cannon (ice2), the ice is denoted after the instant to avoid confusion.
Example: PPI or PPI2 in this setup instead of IPP: https://youtu.be/yS9FC2tj6mU
ANI or NAI in the doom wave instead of any other combination: https://youtu.be/bviF1XRZqBM

7. This is stated above when explaining the different types of cob symbols but I'll state it again. Some cob operations absolutely require another operation of that wave (usually a normal cob) in order to make sense. The operations that do this are S cobs, D cobs, and d cobs. This is because these cob operations are dependent and don't make sense on their own by definition.
Example: In this setup, the sequence is written IPP-PPDD | PPDDcccc instead of IPP-PPDD | DDDDcccc because you can't start a wave out with D cobs since there's nothing before it to be delayed from: https://youtu.be/FOigeuQKZaQ
This also means that there's no way of indicating whether you're catching imps with the P cobs or not but this is easily identifiable based off context.

8. Pre-cobs on ice waves used to kill diggers/zombonies are denoted as P cobs. Technically by definition, these should be notated as B cobs because they don't spawn the next wave but this is just what's standard.
Example: IPP-PPDD instead of IBB-PPDD: https://youtu.be/HPcT-7OGIY8



d) Hard/Soft Wavelengths

As stated above, hard wavelengths are wavelengths that 100% eliminate the threat of a certain zombie while soft wavelengths don't 100% eliminate the threat. It's possible to know 100% whether a wavelength works against jacks or not although I'm gonna have to go over a lot of theory first.
1. Early jacks blow up between 4.4s-7.4s while late jacks blow up between 13.22s-22.68s.
2. A jack's timer doesn't slow down if the zombie is slowed. However, it does pause if the jack has been iced.
3. If you ice, all zombies including jacks will be frozen for minimum 4 seconds.
4. The jack explosion animation takes 1.1 seconds so the jack doesn't actually explode until the animation ends. The animation is always 1.1 seconds even if the jack is slowed.
5. Waves take 2 seconds to spawn in after all necessary damage has been dealt.

Using this info, you can use math to figure out the hard wavelengths for all types of waves.
1. If you want to completely avoid early jacks on accelerated waves, you take the earliest jack explosion (4.4), add the animation time (1.1) and add the 2 second wave buffer. Add this up and you learn that the longest accelerated wave that can avoid early jacks is 7.5 seconds. This exact wavelength is used in pure 20 cob P5u or most RE 8 cob setups.
2. If you want to completely avoid early jacks on ice waves, you take the earliest jack explosion (4.4), add the 4 second ice buffer, add the animation buffer (1.1) and the 2 second wave buffer. Add this up and you learn that the longest ice wave that can avoid early jacks is 11.5 seconds. This exact wavelength is used in precise ch6.
3. If you want to completely avoid late jacks on accelerated waves, you take the earliest late jack explosion (13.22), add the animation time (1.1) and add the 2 second wave buffer. Add this up and you get 16.32. This is completely unused but I thought I might as well show it.
4. If you want to completely avoid late jacks on ice waves, you take the earliest late jack explosion (13.22), add the 4 second ice buffer, add the animation buffer (1.1) and the 2 second wave buffer. Add this up and you get 20.32 seconds.

This info might seem somewhat useless but it's utilized in every single cobbed setup without you even realizing. It's useful to understand how this works so you can hopefully avoid jacks better in your gameplay and/or your setups and solutions.



e) Rhythms

Here I summarize the characteristics/properties of each rhythm and list out some solutions that are usually used in each rhythm.
Note: t stands for the amount of seconds in a cycle with s standing for seconds. (e.g. Cxu-ts becomes C8u-50s)


Non-Cobbed Rhythms-


1. Cx-ts
characteristics:
(1) x is the number of waves in a cycle.
Short for Card x.
This rhythm often has -io attached to the end of it to indicate that the it relies on io plants to accelerate the tempo.


Cobbed Rhythms-


1. Px-ts
characteristics:
(1) x = the number of waves in a cycle.
(2) The wavelength of each wave must be the same.
(3) No ice waves are used.
Short for Pao x (This can be translated to Cob x like I've seen Ming do)
This is the basic fundamental cobbed rhythm that is used in the easiest cobbed setups like classic 10 cob or classic 8 cob.

2. Pxu-ts
characteristics:
(1) x = the number of waves in a cycle.
(2) A cycle must have 2 or more distinct wavelengths.
(3) No ice waves are used.
This is Px except the waves are different lengths. Very rarely required.

3. chx-ts
characteristics:
(1) x = number of ice waves + number of waves in a cycle.
(2) The wavelength of each ice wave must be the same if x≤6.
(3) Exactly two ice waves are used in a cycle.
(4) If t>36, it is referred to as chx* with an asterisk. Cycles that are 50s or close to 50s are classified as a Cxu rhythm.
Short for change x.
(5) t≥52.13s for upper chx rhythms.
This is the daytime double ice rhythm and it is very important to learn early on

4. chxu-ts
characteristics:
(1) x = number of ice waves + number of waves in a cycle.
(2) Exactly two ice waves are used in a cycle.
(3) The wavelength of each ice wave must be different.
(4) x≤6.
(5) If t>36, it is referred to as chxu* with an asterisk.
u stands for uneven wavelength.
Chx except the ice waves have different wavelengths

5. Sx-ts
characteristics:
(1) x = number of ice waves + number of waves in a cycle.
(2) Exactly one ice wave is used in a cycle.
(3) t≥34.75s.
(4) x≤6.
(5) If t>36, it is referred to as Sx* with an asterisk.
S stands for single ice wave

6. Cxu-ts
characteristics:
(1) x = number of ice waves + number of waves in a cycle.
(2) Exactly two ice waves are used in a cycle.
(3) t≥50s in nighttime.
(4) Often requires linear cob reuse.
This is the nighttime double ice rhythm. This isn't very useful to know until you start playing complicated cobbed setups in the nighttime venues.
The wavelength can be a bit less than 50s in daytime because you don't have to worry about ice reuse but this is very rarely used.

7. Cxn-ts
characteristics:
(1) x=number of ice waves + number of waves in a cycle.
(2) Exactly one ice wave is used in a cycle.
(3) t≥50s in nighttime.
(4) Doom is used when an ice would be used in its Cxu equivalent.
This is pretty much forced to play double doom.

8. Cxi-ts
characteristics:
(1) x = number of ice waves + number of waves in a cycle.
(2) Exactly three ice waves are used in a cycle.
(3) If x=6 and t>36s, or x>6 and t>54s, it is referred to as Cxi* with an asterisk.
There's actually only one valid 36s rhythm, C6i. C7i-C9i can exist with 50s+ cycle length though.

9. Cxx-ts (example: C7x)
characteristics:
(1) x = number of ice waves + number of waves in a cycle.
(2) Exactly three ice waves are used in a cycle.
(3) t>50s.
(4) There are at least 2 distinct ice wavelengths.
This is an extremely obscure rhythm with very limited use in real gameplay.

9. Ix-ts
characteristics:
(1) x = number of ice waves + number of waves in a cycle.
(2) Exactly two ice waves are used in a cycle.
(3) The wavelength of each ice wave must be the same.
(4) There are accelerated waves between ice waves.
(5) Requires non-linear cob reuse in the first 34.75s of the cycle.
Cycles that require more cobs than is written in the sequence can be notated Ix-~ts and cycles that require less cobs than is written in the sequence can be notated Ix+~ts. This notation is applicable to Ixi, Ixu, Ixs, Px and Pxu but it's only used for Px/Pxu when there's non-linear reuse.
Short for Ice x.

10. Ixi-ts
characteristics:
(1) x = number of ice waves + number of waves in a cycle.
(2) It cannot have accelerated waves between ice waves unless there are 3 or more ice waves.
(3) Requires non-linear cob reuse in the first 34.75s of the cycle.

11. Ixu-ts
characteristics:
(1) x = number of ice waves + number of waves in a cycle.
(2) Exactly two ice waves are used in a cycle.
(3) The wavelength of each ice wave must be different.
(4) There are accelerated waves between ice waves.
(5) Requires non-linear cob reuse in the first 34.75s of the cycle.

12. Ixs-ts
characteristics:
(1) x = number of ice waves + number of waves in a cycle.
(2) Exactly one ice wave is used in a cycle.
(3) Requires non-linear cob reuse in the first 34.75s of the cycle.
Non-linear cob reuse that causes the cycle to become shorter can be notated Ixs+~ts and reuse that causes the cycle to become longer can be notated Ixs-~ts.
S stands for single ice wave

Here are some common solutions for certain rhythms-

Note: If x>6, the rhythm is usually used in the nighttime endless modes only because you can't store ices in night (with exceptions of course).
Also, typically the ice waves are placed first when writing the sequence just for courtesy but this is optional. I don't do this when I write sequences in my descriptions.

<Cx-ts>
C2-50s: I-N|I-AJa (25|25)
C3-50s: I-N'|AJa|N (17|17|17) or (18|16|16)
C4-io: N|a-I|J-I|A (for io-dependent rhythms like this one, wavelengths are imprecise)
C5-io: I|J-I|A|N|A'
C6-io: I|I|I|I|N|AJ (radio clock)
<Px-ts>
P3-36s: PP|PP|PP (12|12|12)
P4-36s: PP|PP|PP|PP (9|9|9|9)
P5-35s: PP|PP|PP|PP|PP (7|7|7|7|7)
P6-36s: PP|PP|PP|PP|PP|PP (6|6|6|6|6|6)
Due to the fact that the fastest wave can only be 6.01s, P6 is the natural upper boundary for the Px rhythm.
P2 would be 18|18 but it's unusable
<Pxu-ts>
P5u-36s: PP|PP|PP|PP|PP (6|9|6|9|6)
P5u-34.75s: PPDD|PPdd|PPDD|PPdd|PPDD (6.57|7.52|6.57|7.52|6.57) (pure 20 P5u)
P6u-40s: PP|PP|PP|PP|PP|PP (6|6|6|7|7.5|7.5)
P2u-P4u are not used at all. P6u is rare and it's used in nuclear 4 cob due to the coffee bean reload (https://youtu.be/xIbTkRewI1s). P5u is used somewhat often but it can usually be replaced with P5. The strange rhythm written is the P5u solution for pure 20 cob which is almost the only practical use of the rhythm although it’s not immune to catapults or jacks.
<Sx-ts>
S3-36s: I-PP|PP (21|15)
S4-36s: I-PP|PP|PP (18|9|9)
S4-34.75s: I-PP|PP|PP (22.73|6.01|6.01)
S4-34.75s: I-PP|PP|PP (19.83|7.46|7.46)
S5-36s: I-PP|PP|PP|PP (18|6|6|6)
S5-34.75s: I-PP|PP|PP|PP (16.72|6.01|6.01|6.01)
S5-35.09s: I-Pd|Pd|PD|PD (14.65|7.3|6.57|6.57)
S5-34.75s: I-PP|PP|PP|PP (13.09|7.22|7.22|7.22)
S6-36s: I-PP|PP|PP|PP|PP (12|6|6|6|6|6)
S6-35.5s: I-PP|PP|PP|PP|PP (11.5|6|6|6|6) (jack-proof)
Sx with x>6 can't exist because S7 would need to fit 5 accelerated waves and 1 ice wave within 36s which is impossible
<Cxu-ts>
C4u-50s: I-PP|I-N (25|25)
C4u-50s: I-B-Nd|I-B-JPA (25|25) (4 cob cycle, 3 cobs required. technically this is also I4i+ but not a soul would ever call it that)
C5u-50s: I-PP|I-PP|N (20|20|10) or (21|21|8)
C6u-51s: I-PP|PP|I-PP|N (18|7|18|7) (opposite C6u)
C6u-50.02s: IP-PD|Nd|IP-PD|Pd (17.55|7.46|17.55|7.46) (technically I6 but most people wouldn't call it that)
C6u-54s: I-PP|I-PP|N|PP (18|18|9|9) (adjacent C6u)
C6u-52.5s: IP-PD|IP-PD|Nd|Pd (19.75|19.75|7.5|7.5)
C7u-50s: 16|16|6|6|6 (adjacent C7u)
C7u-50s: 16|6|16|6|6 (inter C7u)
C7u-50.01s: 15.26|16.72|6.01|6.01|6.01 (common wavelengths)
C7u-50s: 14.5|14.5|7.5|7.5|6 (alternate wavelengths)
C7u-51.5s: 14.5|7.5|14.5|7.5|7.5 (alternate wavelengths)
C7u-51s: I-PP|PP|I-PP|N|PP (13|8.75|13|7.5|8.75) (alternate wavelengths)
C8u-50s: I-PP|I-PP|AJa|PP|N|PP (13|13|6|6|6|6) (adjacent C8u)
C8u-50s: I-PP|PP|I-PP|PP|N|PP (13|6|13|6|6|6) (inter C8u)
C8u-50s: I-PP|PP|PP|I-PP|N|PP (13|6|6|13|6|6) (opposite C8u)
C8u-52s: I-PP|PP|PP|I-PP|N|PP (12.5|7.5|6|12.5|7.5|6) (alternate wavelengths)
C8u-48s: I-PP|PP|PP|I-PP|PP|PP (12|6|6|12|6|6) (daytime only)
C9u-50s: (10|6|10|6|6|6|6)
C4u and C2 refer to the same rhythm but in different ways.
C9u is pretty much the same as ch9 as shown below. C9u encompasses all 7 wave sequences with double ice wave while ch9 is a bit more specific.
C7u wavelengths (16s) usually can't work without imp intersection. If you have no imp intersection in a setup, you'll usually have to use ch7 with 18s wavelengths.
(Cxn-ts>
C6n-54s: I-P|N|P|N|P (18|9|9|9|9)
C7n-50s: I-P|N|P|P|N|P (15|7|7|7|7|7)
C8n-54s: I-P|P|N|P|P|N|P (12|6|6|6|6|6|6)
C9n-60s: I-P|P|N|P|P|N|P|P (12|6|6|6|6|6|6|6)
Rhythm made by crescendo and it's honestly not found too much practicality yet. Cxn is demonstrated here: https://www.bilibili.com/video/BV12vK7eeERR
<chx-ts>
ch4-36s: I-PP|I-PP (18|18)
ch5-36s: I-PP|I-PP|PP (15|15|6)
ch6-36s: I-PP|PP|I-PP|PP (12|6|12|6) (opposite ch6)
ch6-36s: I-PP|I-PP|PP|PP (12|12|6|6) (adjacent ch6)
ch6-35s: I-PP|PP|I-PP|PP (11.5|6|11.5|6) (jack-proof)
ch6-34.76s: I-PP|PP|I-PP|PP (10.32|7.06|10.32|7.06) (col 7 pumpkins)
ch7-35s: I-PP|I-PP|PP|PP|PP (8.5|8.5|6|6|6) (this can be considered P5u)
ch7-54s: I-PP|I-PP|N|PP|PP (18|18|6|6|6)
ch8-54s: IP-PP|IP-PP|PP|PA|N|PP (15|15|6|6|6|6)
ch9-54s: I-PP|I-PP|PP|PP|N|PP|PP (12|12|6|6|6|6|6) (adjacent ch9)
ch9-54s: I-PP|PP|I-PP|PP|N|PP|PP (12|6|12|6|6|6|6) (inter ch9)
ch9-54s: I-PP|PP|PP|I-PP|N|PP|PP (12|6|6|12|6|6|6) ('opposite' ch9 aka opposite to inter C8u)
ch9-54s: I-PP|PP|N|PP|PP|I-PP|PP (12|6|6|6|6|12|6) (staggered ch9)
Note: The difference between upper chx rhythms and Cxu rhythms is that chx is built around the length of a cob's reload (34.75s) while Cxu is built around the length of an instant's reload (50s). This means that all upper chx rhythms are 52.13s+ while Cxu is always 50.01s (unless the waves are forced to be lengthened or it's daytime). This makes them pretty similar but you can definitely feel a difference when comparing the two.
Opposite ch9 isn't truly opposite because there are 2 accelerated waves to 3 accelerated waves. This is why I called it opposite to inter C8u because it transitions the double accelerated waves in opposite C8u with the triple accelerate waves in inter C8u but idk it's a tough rhythm to name.
Staggered ch9 is just the same as normal ch9 except the doom wave is before the second ice wave. You don't have to call it staggered if you don't want to and I've seen some people call this S6-ch9 which is also correct.
<chxu-ts>
ch4u-35s: I-PP|I-PP (16|19)
ch5u-36s: I-PP|I-PP|PP (14|16|6)
ch5u-35s: I-PP|I-PP|PP (16|13|6)
ch6u-42s: I-PP|PP|I-PP|PP (12|6|16|6)
Ch6u is redundant and is only used in some DE20 and RE20 solutions meanwhile ch5u is extremely op.
The upper chxu solutions (ch7u, ch9u) aren't really used and even if they are, they are just named by their chx counterpart. Ch7u is used in the manual solution for string 14 cob: https://youtu.be/5ysANo2WyP8 But as you can see in the description, this rhythm is just called ch7.
<Cxi-ts>
C6i-36s: I-PP|I-PP|I-PP (12|12|12)
C6i*-39s: I-PPDD|I-PPdd (13|13)
C7i-36.97s: I-PP|I-PP|I-PP|PP (10.32|10.32|10.32|6.01) (col 7 pumpkins)
C7i-54s: I-PP|I-PP|I-N|PP (18|18|12|6) (ch4-N)
C8i-51s: I-PP|I-PP|I-PP|N|PP (13|13|13|6|6) (C6i-N)
C8i-51s: I-PP|I-PP|I-PP|N|PP (12|12|12|7.5|7.5) (C6i-N)
C8i*-62s: Ia-N|PP|Ia-PP|PA|I-PP (17.5|7.5|17.5|7.5|12)
C9i-54s: I-PP|PP|I-PP|PP|I-PP|PP (12|6|12|6|12|6) (ch6)
C9i-54s: I-PP|I-PP|PP|PP|I-PP|PP (12|12|6|6|12|6)
C9i-54s: I-PP|I-PP|I-PP|PP|PP|PP (12|12|12|6|6|6)
C7i can be called ch4-N and C8i can be called C6i-N in cases where there's no fixed cycle and doom is used whenever it can be to prevent you from losing too many ices. However, these names make no sense to use when there is a fixed cycle.
The first C9i case is just an extension of ch6 so we call it ch6 even if the cycle is 54s. The other two C9i cases haven’t found uses yet.
<Cxx-ts>
C6x-54.62s：I-PP|IAJ-PP|IN-PP (16.1|19.26|19.26) (https://www.bilibili.com/video/BV1PK4y1J7tN)
C7x-53.26s: I-PPd|IA-PPD|I-NJ|PPD (16.05|18.2|13|6.01)
Odd rhythm with few uses. Cxx is a subset of Cxi so the upper Cxx rhythms can be named by it's Cxi equivalent if you want.
<Ix-ts>
I5-32.5s: IP-PD|IP-PD|Pd (13.5|13.5|6) (16 cob cycle, 18 cobs required)
I6-~28.92s: I-PP|PP|I-PP|PP (8.45|6.01|8.45|6.01) (8 cob cycle, 10 cobs required, this can also be considered P4u-)
I6-~34.2s: I-PD|Pd|I-PD|Pd (11.09|6.01|11.09|6.01) (16 cob cycle, 18 cobs required)
I6+~44s: IP-PD|Pd|IP-PD|Pd (16|6|16|6) (20 cob cycle, 18 cobs required)
I6+~46s: IPP-PP|PP|IPP-PP|PP (17|6|17|6) (12 cob cycle, 10 cobs required)
I6+~47.33s: IP-PD|IP-PD|Nd|PD (17.2|17.2|6.92|6.01) (adjacent I6, 18 cob cycle, 16 cobs required)
I7-42s: (12|6|12|6|6) or (12|12|6|6|6)
I7-50.43s: 16.2|16.2|6.01|6.01|6.01 (adjacent I7, full-defensive ne18)
I7-50.43s: 16.2|6.01|16.2|6.01|6.01 (inter I7, ne20)
C8u-I8+~50s: IP-P|PD|IP-P|PD|PN|PD (13|6|13|6|6|6) (18 cob cycle, 16 cobs required)
I8-54.84s: 15.4|6.01|15.4|6.01|6.01|6.01 (me20)
I8-55.06s: I-PP|PP|PP|I-PP|PP|PP (13.09|7.22|7.22|13.09|7.22|7.22) (10 cob cycle, 8 cobs required)
Ch9-I9+~55s: IP-P|PD|IP-P|PD|PN|PD|Pcc (12|6|13|6|6|6|6) (18 cob cycle, 16 cobs required. No one would consider this Ch9-I9u)
I9-~52.23s: I-PD|Pd|I-PD|Pd|ND|Pd|PD (11.09|6.01|11.09|6.01|6.01|6.01|6.01) (18 cob cycle, 20 cobs required)
I9+~56.61s: 13.28|6.01|13.28|6.01|6.01|6.01|6.01
Anything below I5 can't exist. The +- notation isn't required by any means but it helps distinguish everything.
C8u-I8 and Ch9-I9 don't have non-linear reuse in the first 34.75s which is why they're not I8 and I9.
I didn't write some of the sequences because there's no point.
<Ixi-ts>
I4i-~29.02s: I3Pd/PD-PP|I3Pd/PD-PP (14.51|14.51) (12 cob cycle, 15 cobs required)
I4i-~31.16s: I-Pd|I-Pd (15.58|15.58) (8 cob cycle, 10 cobs required)
I4i-~32.4s: I-PD|I-PD (16.2|16.2) (8 cob cycle, 10 cobs required)
I4i-~32.44s: I3Pd-P|I3Pd-P (16.22|16.22) (12 cob cycle, 14 cobs required)
I4i-~33.08s: I3PD-P|I3PD-P (16.54|16.54) (12 cob cycle, 14 cobs required)
I4i+~46.34s: I-B-P|I-B-P (23.17|23.17) (4 cob cycle, 3 cobs required, I've seen this called I2i and you can also call it C4u)
I6i-~34.5s: I-Pd|I-Pd|I-Pd (11.5|11.5|11.5) (12 cob cycle, 14 cobs required)
I6i+~40.86s: I3PD-P|I3PD-P|I3PD-P (13.62|13.62|13.62) (18 cob cycle, 16 cobs required)
I7i-52.5s: IP-PD|IP-PD|IP-ND|Pd (15.5|15.5|15.5|6) (12 cob cycle, 14 cobs required)
I7i-52.5s: IP-PD|IP-PD|IP-PD|Pd (15.5|15.5|15.5|6) (14 cob cycle, 16 cobs required)
I4i should definitely be called I4 and it technically is I4 but it's too late to change the name now.
<Ixu-ts>
I8u+~53.98s: 13|6.01|16.94|6.01|6.01|6.01
I8u+~55.68s: 14.09|6.01|14.65|7.46|7.46|6.01
I8u+~54.94s: I-PP|IPP-PP|PP|N|PP|PP (13.58|17.32|6.01|6.01|6.01|6.01) (10 cob cycle, 8 cobs required, adjacent ice waves)
I9u: 12|6.01|6.01|16.94|6.01|6.01|6.01
I7u exists but has no practical application at the moment
<Ixs-ts>
I3s-32.96s: 25.01|7.95 (https://www.bilibili.com/video/BV1TKZSYKEwA)
I4s-31.68s: IP-PD|PD|PD (18.54|6.57|6.57) (14 cob cycle, 16 cobs required)
I5s-31.5s: IP-P|PD|PD|Pcc (13.5|6|6|6) (14 cob cycle, 16 cobs required)
I5s-30s: IP-P|P|P|P (12|6|6|6) (10 cob cycle, 12 cobs required)
I7s-42s: IP-P|P|P|P|P|P (12|6|6|6|6|6) (14 cob cycle, 12 cobs required)
All of these upper I-series rhythms can be notated as Cxu and it's not technically wrong since the I-series rhythms are subsets of Cxu. However, I'd recommend being more specific when naming your rhythms if possible because Cxu creates a lot of ambiguity.

The solutions above are just for reference. They may be changed in different cobbed setups and the solution's changes depend on your demands and the situation.