Conway's Game of Life: 2D grid, 2 states, 1 rule (neighbor count). Turing-complete but no self-closure. Axiom Ouroboros: 3 operations (sigma/D/K), 8 patterns, 72-transition recursive cycle. Self-closing.
| PROPERTY | CONWAY | AXIOM OUROBOROS |
|---|---|---|
| Operations | 1 (neighbor count threshold) | 3 (AND, XOR, MAJ) — complete basis |
| States | 2 (alive/dead) | 8 (DK = 23 patterns) |
| Self-closure | No — runs forever or dies | Yes — 72 transitions, level N feeds level 0 |
| Meta-levels | None (flat grid) | Infinite — K=3 ouroboroses gattai upward |
| Fixed point | Gliders, oscillators (many) | LIFE=7 (unique, all domains present) |
| Derivation | Chosen rules | From sigma=1, D=2, K=3 only |
Source: axiom_octave_generic.c (Session 48). 991 lines. KD x DK = 9 x 8 = 72 = the number of transitions for the ouroboros to close. After 72 transitions, the highest level's pattern becomes the seed for level 0. Different seeds, different journeys, same destination: LIFE=7.
Standard view: Self-reference is a logical curiosity. Hofstadter's strange loops are metaphorical.
Axiom view: The 72-transition cycle IS the ouroboros: starting from any operation, applying sigma/D/K repeatedly returns you to the start. .ax interpreting .ax is this loop made literal. The axiom doesn't describe self-reference — it performs it.