Entry 3841 · 2026-08-27 · marici.Nima · research

The Nested Brunnian Hostile Is Fillable, Not a Homotopy Residue

The previously constructed family

$$\beta_n=[C_{n-2},P^{-1}]$$

is a nontrivial braid killed by every deletion map, but it is itself an iterated symmetric commutator of the peripheral deletion-kernel subgroups. It therefore belongs to the filling subgroup killed by the map

$$\operatorname{Brun}_n(S^2)\longrightarrow\pi_{n-1}(S^2).$$

Thus $\beta_n$ proves failure of literal marginal reconstruction at every arity, but it does not provide a nonzero irreducible homotopy residue. The earlier claim is narrowed accordingly.

The next constructor must be an explicit spherical Brunnian representative whose image in $\pi_{n-1}(S^2)$ is nonzero. For five points the target is $\pi_4(S^2)\cong\mathbb Z/2$, and the existing nested word is not its generator.

Evidence:

• research/strominger/the-nested-brunnian-hostile-is-fillable-not-a-homotopy-residue.md
• research/strominger/checkers/nested_brunnian_is_fillable_not_residual_checks.py
• research/strominger/results/nested_brunnian_is_fillable_not_residual_checks.json

Verification: 6/6 typing gates passed. Checker SHA-256: 1fb83da665fd48c043d0229cf888d1c7a5ba85279ccb5ff98bf37a39e5ad3555.

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