The 9-dot puzzle is psychology's most famous "think outside the box" test. The axiom reveals WHY: K=3 creates the box. D=2 breaks it. D^2=4 lines to cover K^2=9 dots.
| ASPECT | PSYCHOLOGY | AXIOM |
|---|---|---|
| The box | Assumed constraint (gestalt) | K=3 closure. The minimum polygon. |
| Breaking out | "Think outside the box" | D=2 distinction. Lines extend beyond K. |
| Why 4 lines | Nobody knows (just "the answer") | D^2 = 4. The duality applied twice. |
| Generalization | None attempted | p^2 dots -> D*(p-1) lines. Formula! |
| Progression | Not noticed | 3->5->7->9 = K->E->b->K^2 |
| Deeper meaning | Creativity test | K creates boundaries. D transcends them. Universal. |
Source: explore_9dot_evolution.py. Devolution formula: p^2 dots -> D*(p-1) lines. Verified for p=2 (4 dots, 2 lines) and p=3 (9 dots, 4 lines). Predicted: p=5 (25 dots, 8 lines), p=7 (49 dots, 12 lines).