The Ouroboros Operator

Two snakes, not one

The operator $\circlearrowright$ is the ouroboros operator — named after the ancient image of a snake consuming its own tail. But here there are two snakes, each eating the other's tail. What emerges from their encounter depends entirely on which one leads.

$$\{\}_- \circlearrowright \{\}_+ = \{\{\}\}_+ = +1$$ $$\{\}_+ \circlearrowright \{\}_- = \{\{\}\}_- = -1$$

The same two objects, the same operator, opposite results. Order is everything.

Two rules

The behaviour of $\circlearrowright$ follows two simple rules:

Rule A — opposite orientations meet (generative): When a positive void meets a negative void, or vice versa, the encounter promotes — it climbs to the next nesting level. The left operand determines the sign of the result. This is the operation that creates number from nothingness.

Rule B — same orientations meet (absorptive): When a void meets its own kind, nothing new is generated. The higher-ranked one simply absorbs the lower and persists unchanged. A void meeting only itself has nothing to push against — no creation, no rank increase.

The truth table

Left	Right	Result	Rule
$\{\}_-$	$\{\}_+$	$\{\{\}\}_+ = +1$	A: generative
$\{\}_+$	$\{\}_-$	$\{\{\}\}_- = -1$	A: generative
$\{\}_-$	$\{\}_-$	$\{\}_-$	B: absorptive
$\{\}_+$	$\{\}_+$	$\{\}_+$	B: absorptive

The first two rows are $0^0 = \pm 1$ — the result depends on which oriented zero came first. The last two say a void meeting only itself simply persists.

The rules extend to higher ranks

The same two rules apply at every nesting level:

Left	Right	Result
$\{\{\}\}_+$	$\{\}_-$	$\{\{\{\}\}\}_-$ (A: climbs one level, sign flips)
$\{\{\}\}_-$	$\{\{\}\}_+$	$\{\{\{\{\}\}\}\}_+$ (A: climbs one level, sign flips)
$\{\{\}\}_-$	$\{\}_-$	$\{\{\}\}_-$ (B: same sign, higher rank absorbs)

Each opposite-orientation encounter threads into the next nesting level — a dimensional step up, not a movement within a fixed space.

Order matters — and this is not a defect

$\circlearrowright$ is not commutative under Rule A, and it is not associative. Neither of these is a flaw. The order of encounter determines the sign of what is produced, and grouping changes the path through rank. This is exactly how crossings work in knot theory — the over/under assignment at each crossing determines the knot, and changing the order of crossings changes the knot.

What zero actually requires

In ordinary computing, $0$ is a value you can assign directly. In MMP, expressing zero in a circuit requires one positive void and one negative void held in balance — a tension maintained, not a value stored. It is not that zero is absent; it is that zero is the result of equilibrium between two oriented poles.

Classical bit	MMP equivalent
$0$	$\{\}_-$ and $\{\}_+$ in balance (not directly assignable)
$+1$	$\{\}_- \circlearrowright \{\}_+$
$-1$	$\{\}_+ \circlearrowright \{\}_-$
NOT	swap orientation: $\{\}_+ \leftrightarrow \{\}_-$
same-kind encounter	absorptive — Rule B
opposite-kind encounter	generative — Rule A, climbs rank

The integers are not assumed — they are produced by ordered encounters of oriented nothingness with itself.