S15: Active Hopf Horizon
S^15 is the horizon of the project — the next octonionic Hopf step, where the book has planted finite seeds and guardrails but not a finished theory. This unit is honest about being a frontier: it shows you the checked footholds and, just as importantly, a counterexample that stops a tempting wrong claim.
Unit Goal
See what is actually nailed down at the horizon — finite suspension/Euler facts, bounded coordinate landings, and safe right-phase facts — and understand why one counterexample keeps a naive invariance claim from being promoted.
The Idea
A horizon chapter records progress without overstating it. The finite eight-suspension fact gives χ(S^15) = 0 by the usual parity rhythm. The bounded octonionic Hopf landing facts hold under explicit hypotheses. And the counterexample is the hero of the unit: it proves that naive shared right-multiplication does not preserve the full bounded Hopf coordinate, so the honest claim has to stay narrow. This is roadmap material; the Living Book does not claim a complete implemented S^15 theory.
Lesson Path
- Future Hopf Horizon — the finite seeds, the safe right-phase facts, and the guardrail counterexample.
This is a horizon scaffold: it stays future/roadmap-facing, the theorem cards are the status source, and the widgets and diagrams are explanations, not proofs. See What “Proved” Means Here.
Checkpoint
Which theorem card records the counterexample showing that naive shared right unit-octonion multiplication does not preserve the full bounded Hopf coordinate? (Look for S15-T0008.)