Theorem 6: Minimal Stable Spatial Dimension is d = 3
Only 3D supports non‑erasable linking of dual cycles
Statement
Stable dual cycles with nontrivial linking require exactly three spatial dimensions. In 2D, links are impossible; in d ≥ 4, all links can be unlinked by ambient isotopy.
In plain English
To leave a permanent mark, two loops have to hook each other in a way you can’t undo without cutting. That can’t happen in flatland (2D), and in roomy worlds (4D+) you can always slide them apart. Only in 3D do links “stick.” So space chooses 3.
- Why inevitable: standard knot facts: no linking in 2D; unlinking in ≥4D; 3D is uniquely sticky.
- What it buys: a stable record you can’t erase for free—fuel for memory, structure, and dynamics.
Sketch
- Jordan/planarity forbids linked, disjoint 1‑cycles in 2D.
- Alexander duality/ambient isotopy unlinks in d ≥ 4.
- Only d = 3 supports a Hopf link with invariant linking number.