The Ouroboros

72 transitions. The snake eats its tail. 3^2 x 2^3 = 9 x 8 = 72. Starting from any seed, applying AND/XOR/MAJ operations cycles back. Pattern 7 (111) persists at every level.

The 72-Transition Cycle

3^2 x 2^3 = 9 x 8 = 72. Starting from any of 8 patterns, applying AND, XOR, and MAJ repeatedly cycles through 72 transitions before the highest level feeds back into level 0. The snake eats its own tail.

Pattern 7 (111 -- all three bits on) is the unique fixed-point attractor. Every seed, every level, every meta-level -- all roads lead to 7. Different journeys, same destination.

Self-referenceHofstadter's strange loops -- metaphorical, philosophical72-transition deterministic cycle. Arithmetic, not metaphor.OperationsArbitrarily chosen (Conway: neighbor count)AND, XOR, MAJ: three natural 3-bit operations.States2 (alive/dead) in Conway8 = 2^3. Three bits, eight patterns.Fixed pointMany (gliders, oscillators)One: pattern 7 (111). Unique across all meta-levels.ClosureNo -- runs forever or diesYes -- 72 transitions, level N feeds level 0.

Why 72?

3^2 = 9: the number of operation triples (AND/XOR/MAJ applied to 3 inputs). 2^3 = 8: the number of distinct 3-bit patterns. 9 x 8 = 72 transitions to exhaust all operation-pattern combinations. The cycle length is not chosen -- it falls out of 2 and 3.

Ouroboros Closure
For any seed pattern p in {0,...,7}, applying the 72-transition cycle of (AND, XOR, MAJ) operations returns to p. The attractor state 7 (111) appears at every meta-level.

The compiler .ax compiling itself is this same loop made literal. wasm_from_src applied to its own source produces itself.

Source code · Public domain (CC0)

Report issue

.ax source compiled to WASM via self-hosting compiler. Zero HTML authored.