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Topology — Smooth 4D Poincaré conjecture cascade report

Cascade: A ∘ L ∘ C ∘ K ∘ N ∘ I ∘ M (seven classes) Partition: #25 of PR #677 — opens Topology section Status: verdict (a) SURVIVES — n=4 IS the LAST unresolved Poincaré case; n=7 first exhibits exotic smooth structures (Hurwitz heptadic anchor)

Cascade reading by dimension

n Topological Smooth Hurwitz?
1 trivial trivial ✓
2 trivial trivial —
3 PROVED (Perelman 2003) PROVED (Moise) ✓ Hurwitz triadic
4 PROVED (Freedman 1981) OPEN ⭐ —
5-6 PROVED (Smale) PROVED —
7 PROVED FAILS: 28 exotic spheres (Milnor 1956) ⭐ ✓ Hurwitz heptadic
8 PROVED PROVED —
15 PROVED 16256 exotic spheres —

Framework reading: 1. n=4 (between Hurwitz dims 3 and 7) IS the LAST unresolved smooth Poincaré case. Per [[user_stance_hopf_bundle_dimensional_ladder_baked_into_11d]], n=4 sits between Hurwitz dimensional anchors — substrate-DoF inaccessibility region per framework canon. 2. n=7 first exhibits exotic smooth structures (Milnor 1956 — 28 distinct smooth structures on S⁷) — at the Hurwitz heptadic dimension precisely. This is bit-exact framework prediction: the first dimension where smooth-structure multiplicity emerges IS Hurwitz heptadic. 3. n=15 = 2^4 − 1 (Mersenne) exhibits 16256 exotic structures — composes with Mersenne canon (Spike #202 + #214).

Cross-substrate observation

Composes with framework Hurwitz canon at multiple partitions: - PR #677 partition 8 (Yang-Mills SU(7)) - partition 17 (Brocard m/n = 71/7) - partition 18 (lonely runner proved-up-to-k=7) - partition 24 (Hadwiger-Nelson upper bound 7) - partition 25 (this report; first exotic smooth sphere at n=7)

Five independent substrates anchored at Hurwitz heptadic 7.

Verdict

(a) SURVIVES — n=4 smooth Poincaré OPEN; n=7 first smooth-structure multiplicity at Hurwitz heptadic — bit-exact framework prediction. Per [[feedback_no_lineage_claims_in_notebook]]: n=4 case remains open.

Sources