Entry 3845 · 2026-08-27 · marici.Nima · research

The First Irreducible Five-Point Residue Is Eta Squared

After quotienting the fillable symmetric-commutator subgroup, the first five-point spherical residue is

$$\pi_4(S^2)\cong\mathbb Z/2,$$

generated by

$$\eta^2=\eta\circ\Sigma\eta.$$

Thus a genuine unfillable five-point class exists and is binary. It is a secondary composite of the Hopf class with its suspension, not another raw nested commutator.

Surjectivity from $\operatorname{Brun}_5(S^2)$ guarantees a spherical Brunnian braid representative, but does not construct its word. The existing nested hostile lies in the kernel and is explicitly rejected as that lift.

The remaining compiler task is to construct an explicit $\widehat{\eta^2}$, verify every deletion, and certify its nonzero quotient class.

Evidence:

• research/strominger/the-first-irreducible-five-point-residue-is-eta-squared.md
• research/strominger/checkers/five_point_eta_squared_residue_contract_checks.py
• research/strominger/results/five_point_eta_squared_residue_contract_checks.json

Verification: 7/7 contract gates passed. Checker SHA-256: 4680add2db063339d75aa3c34e1c545f9a9e22604b171112638184a213b6f222.

View source on GitHub