Phase II: Infrastructure Seeds
Phase II is the toolbox unit. Once you have circles and spheres, you start needing machinery — ways to repeat, to attach data to points, to track orientation, and to badge a diagram with the proof behind it. Each tool here starts at its simplest, honest form.
Unit Goal
Meet five pieces of infrastructure — stable suspension, a period-8 clock, trivial bundles, directed boundaries, and proof-carrying glyphs — as small, checked seeds that later theory can build on.
The Idea
Every page in this unit is deliberately the trivial case, taught confidently as exactly that. A double suspension is Euler-neutral. A period-8 clock is C_8 reused for dimension counting. A trivial bundle is an ordered pair (base, fiber). A directed boundary is a signed endpoint difference. A glyph is a metadata record that points at a real theorem. None of these are the full theory they seed — and the theorem cards and target indexes are what say which parts are proved, planned, exploratory, or future work.
Lesson Path
- Stable Spheres — suspend twice, return to the same Euler characteristic.
- Bundles — a base coordinate and a fiber coordinate, kept separate.
- Bott And Clifford — the period-8 clock that anchors the deeper periodicity.
- Boundary Cobordism — directed intervals and signed boundaries.
- Proof-Carrying Glyphs — badges that carry their proof’s identifiers.
This is a scaffold unit: theorem cards and generated target indexes determine what is proved, planned, exploratory, or future work; the widgets and diagrams are explanations, not proofs. See What “Proved” Means Here.
Checkpoint
Why should a proof-carrying glyph point to a theorem manifest instead of carrying its own unverified proof-status claim?