Symbols

1. <|-|>
- Reads from/writes to(binary, left to right)
2. |>-|>
- Points to without chaining(binary, left to right)
3. |>=|>
- Points to with chaining(binary, left to right)
4. =|>
- Effectively means(binary, left to right)
5. <|=|>
- Is effectively the same as(binary, left to right)

Rules

1. a |>=|> a <|=|> a |>-|> a
2. a |>=|> b <|-|> b <|=|> a |>-|> b =|> a <|-|> b
3. a |>=|> b |>-|> c =|> a |>-|> c && b |>-|> c
4. (c must be different from a) a |>=|> b |>=|> c =|> a |>=|> c && b |>=|> c
5. a |>=|> b |>=|> a =|> a |>=|> a && b |>=|> b

Theorems

1. a |>=|> b |>=|> c |>=|> b =|> a <|-|> b && b <|-|> b && c <|-|> c
Proof:
a |>=|> b |>=|> c |>=|> b =|> a |>=|> b && b |>=|> c |>=|> b (Rule 4)
=|> a |>=|> b && b |>=|> b && c |>=|> c (Rule 5)
=|> a |>=|> b && b |>-|> b && c |>-|> c (Rule 1)
=|> a |>-|> b && b |>-|> b && c |>-|> c (Rule 3)
=|> a <|-|> b && b <|-|> b && c <|-|> c (Rule 2)

Applications

1. Doubly circular linked list
From Theorem 1:
- a1 |>=|> a2 |>=|> a3 |>=|> ... |>=|> an |>=|> ai =|> a1 <|-|> ai
What a1 reads from/writes to can be controlled by controlling what an points to without changing any other pointer record, making implementing doubly circular linked lists feasible.
To be more effective, efficient and simple, always keep track of the ids of the variable being the 1st, last and currently pointed elements respectively(currently pointed means being actually read from/written to).
In the below setup, head_var_id, cur_var_id and tail_var_id keeps track of the ids of the variable being the 1st, last and currently pointed elements respectively.
a) Setting up the list
- Use this script call:

cur_var_id = head_var_id
(list_length - 1).times { |index|
  $game_variables.var_pointers[cur_var_id] = [var_id_list[index], true]
  cur_var_id = $game_variables.var_pointers[cur_var_id][0]
}
tail_var_id = cur_var_id
$game_variables.var_pointers[tail_var_id] = [head_var_id, true]

$game_variables[head_var_id] # Using $game_variables[cur_var_id] also works

$game_variables.var_pointers[tail_var_id][0] = head_var_id
i.times { cur_var_id = $game_variables.var_pointers[cur_var_id][0] }
$game_variables.var_pointers[tail_var_id][0] = cur_var_id

$game_variables.var_pointers[tail_var_id][0] = head_var_id
i.times { cur_var_id = $game_variables.var_pointers.index([cur_var_id, true]) }
$game_variables.var_pointers[tail_var_id][0] = cur_var_id

next_var_id = $game_variables.var_pointers[cur_var_id][0]
$game_variables.var_pointers[cur_var_id][0] = new_var_id
$game_variables.var_pointers[new_var_id] = [next_var_id, true]

old_var_id = $game_variables.var_pointers[cur_var_id][0]
next_var_id = $game_variables.var_pointers[old_var_id][0]
$game_variables.var_pointers[cur_var_id][0] = next_var_id
$game_variables.var_pointers[old_var_id][0] = 0

$game_variables.var_pointers[new_var_id] = [head_var_id, true]
head_var_id = new_var_id
$game_variables.var_pointers[tail_var_id][0] = head_var_id

$game_variables.var_pointers[tail_var_id] = [new_var_id, true]
tail_var_id = new_var_id
$game_variables.var_pointers[tail_var_id][0] = head_var_id

new_head_var_id = $game_variables.var_pointers[head_var_id][0]
$game_variables.var_pointers[head_var_id][0] = 0
head_var_id = new_head_var_id
$game_variables.var_pointers[tail_var_id][0] = head_var_id

new_tail_var_id = $game_variables.var_pointers.index([tail_var_id, true])
$game_variables.var_pointers[tail_var_id][0] = 0
tail_var_id = new_tail_var_id
$game_variables.var_pointers[tail_var_id][0] = head_var_id

$game_variables.var_pointers[tail_var_id][0] = head_var_id
i.times { tail_var_id = $game_variables.var_pointers[tail_var_id][0] }
head_var_id = $game_variables.var_pointers[tail_var_id][0]

$game_variables.var_pointers[tail_var_id][0] = head_var_id
i.times { tail_var_id = $game_variables.var_pointers.index([tail_var_id, true]) }
head_var_id = $game_variables.var_pointers[tail_var_id][0]