Octive

Octive is an esolang created by CreeperBomb in 2023. It has eight unique operations compacted into 72 characters, and has an accumulator which can either be a natural number (incl. 0) or infinity. The eight operations are:

Name	Description
Even	Multiplies accumulator by 2, mapping infinity to 0
Odd	Multiplies accumulator by 2 and then adds 1, mapping infinity to 1
Invert	Infinity goes to 0, 0 goes to infinity, and every other number $n$ goes through $\sum _{i=0}^{\log _{2}(n)}\left(2^{\left\lfloor \log _{2}(n)\right\rfloor -i}\left\lfloor {\frac {n}{2^{i}}}\right\rfloor -2^{\left\lfloor \log _{2}(n)\right\rfloor +1-i}\left\lfloor {\frac {n}{2^{i+1}}}\right\rfloor \right)$
Small	Divides by 2, flooring the result and mapping infinity to infinity
Beginner	Starts a loop between it and the matching Terminator
Terminator	Marks the end of a loop and goes back to the matching Beginner
Antinull	Exits the loop if the accumulator is even
Interact	If the accumulator is even, the next Even will additionally output 1 or the next Odd will additionally output 0, depending on whichever is run first. If the accumulator is odd, then the user can choose to either do nothing, run an Even, or run a Small

These are then compacted as such:

↓ First \ Second →	-	- Even	- Terminator	- Odd	- Small	- Interact	- Beginner	- Invert	- Antinull
Invert -	q	1	u	v	c	h	>	e	9
Even -	y	"	]	j	@	^	x	;	t
Beginner -	}	i	a	w	8	$	7	(	%
Antinull	[	g	.	'	~	`	f	d	,
Odd -	=	+	6	m	:	4	p	?	n
Terminator -	5	r	/	{	<	*	k	[newline]	b
Interact -	0	-	!	#	[horiz. tab]	o	z	№	)
Small -	&	[space]	3	l	2	s	ü	_	\

The accumulator starts at 0.

Example programs

Cat, where even -> 0 and small -> 1:

4%s:a*=

Truth machine, where even -> 1 and small -> 0:

4ü~-6#