Minimal Embedding Dimension
Phase-preserving embeddings of cyclic processes require k ≥ 3 to remain temporally distinct.
Implements Theorem 10: k = 2 collapses meta-time; k = 3 separates it.
Challenge: When am I?
StatusDistinct
Collisions0
Spaceℝ³ (helix)
k=3 (Helix): Cycles separated by meta-time τ. Temporal distinctness preserved.
τ=0
τ=T
Collision
The Obstruction
In 2D, phase-preserving maps force cyclic trajectories onto a circle. Different cycles must overlap—collisions are geometrically unavoidable.
Why Not Spiral?
Spirals avoid collisions by encoding time in radius. But this violates bounded recurrence—the system drifts rather than cycles. Toggle to see.
Key Insight
When k < 3, systems cannot distinguish when, only what. This forces categorical representations—continuous dynamics become discrete symbols.