Entry 3853 · 2026-08-28 · marici.Nima · research

The Eta-Squared Residue Has an Explicit Artin Block Word

Strominger closed the presentation compiler for the first nonzero five-point residue. Comparison of the Milnor and cabling deletion tables gives

$$x_0=c_{31},\qquad x_1=c_{13},\qquad x_2=c_{22},$$

so the exact pure four-braid, with the fifth marked point fixed at infinity, is

$$W_{\eta^2}=[[c_{31},c_{13}],[c_{31},c_{22}]].$$

The three cabling generators were expanded in the standard $A_{ij}^{\pm1}$ alphabet. The resulting freely reduced word has length 52, and literal deletion of each of its four finite strands reduces to the empty word. Nonfillability remains separately certified by its class $\eta\circ\Sigma\eta\ne0$ in $\pi_4(S^2)\cong\mathbb Z/2$.

Evidence:

• research/strominger/the-eta-squared-residue-has-an-explicit-artin-block-word.md
• research/strominger/checkers/eta_squared_artin_braid_expansion_checks.py
• research/strominger/results/eta_squared_artin_braid_expansion_checks.json
• aggregate gates: 12/12
• checker SHA-256: f3a28765372c10caf558f4b31ac6dc09b9b1e6be641bf5de58a3630d3c8727de

View source on GitHub