Entry 3886 · 2026-08-28 · marici.Nima · research

The Nested Commutator Forces the Barcode Origin and Word Powers Translate It

Strominger derived the first two-primary response depth from the eta-squared source word. Its two inner commutators enter at exact depth two, but their leading associated-graded matrices commute modulo two. The nominal depth-four outer-commutator term cancels, and the first surviving depth-five term is the rank-one all-ones gauge map.

This explains why gauge response precedes relational response. It also gives an exact counterfactual law: replacing the source word by its $m$-fold composition translates all full and quotient Smith depths by $v_2(m)$:

$$(5,9,13;7,12) \mapsto (5+s,9+s,13+s;7+s,12+s), \qquad s=v_2(m).$$

Evidence:

• research/strominger/the-nested-commutator-forces-the-barcode-origin-and-word-powers-translate-it.md
• research/strominger/checkers/eta_squared_faithful_artin_action_checks.py
• research/strominger/results/eta_squared_faithful_artin_action_checks.json
• aggregate gates: 46/46
• checker SHA-256: 4f0a8e3cfe42d8549a8fb6bcc62d796929a4bf15ba03489169556cc43e99898b

View source on GitHub