S4-S6: Suspension And Warning Layer
Past S^3, the spheres get impossible to picture — but the bookkeeping does not get harder. One rule, χ → 2 − χ, carries all the way up. This short unit shows how a single finite lemma covers S^4, S^5, and S^6 at once.
Unit Goal
Understand how finite suspension counts and the Euler-parity rhythm (2, 0, 2, …) get reused across several higher-dimensional scaffolds without re-proving anything.
The Idea
Each suspension flips the Euler characteristic between 2 and 0, so even spheres read 2 and odd spheres read 0. Because it is one rule applied repeatedly, a single finite suspension fact certifies the whole S^4–S^6 stretch. Richer quaternionic, projective, and octonionic-shadow language for these dimensions stays roadmap material unless a theorem card says otherwise.
Lesson Path
- Suspension Euler Parity — read the
2, 0, 2rhythm off the finite cell counts.
The theorem card is the status source: a finite suspension-count fact is not a complete continuous theory of S^4, S^5, or S^6, which is future work.
This is a scaffold unit: the theorem cards are the status source, and the widgets and diagrams are explanations, not proofs. See What “Proved” Means Here.
Checkpoint
What is the difference between proving a finite suspension-count fact and proving a complete theory of S^4, S^5, or S^6?