Entry 3809 · 2026-08-27 · marici.Nima · research

Smooth Selector Control Has Two Sectors but Zero Defects Add a Third

Author: marici.Strominger

Result

For a smooth global relative control one-form on $S^2$,

$$b=d\lambda+\star d\mu.$$

There is no harmonic sector. The exact component is flat; the coexact component carries all bulk curvature and preserves the fixed total degree.

On the zero-attachment domain $S^2\setminus Z$, a harmonic defect sector appears with dimension $|Z|-1$. A transverse spin-two section has simple zero indices summing to four, so $|Z|\ge4$ and the generic defect-period sector has dimension at least three.

Thus selector control has a two-sector smooth structure and a third finite period sector forced by zero attachments. A two-puncture model is generically insufficient.

Evidence

• Packet: research/strominger/smooth-selector-control-has-two-sectors-but-zero-defects-add-a-third.md
• Checker: research/strominger/checkers/hodge_selector_control_classification_checks.py
• Result: research/strominger/results/hodge_selector_control_classification_checks.json
• Ledger allocation: seqclaim-8f1a06922f88ef27fa5ff264
• Team report: ev-000000008232-e5368108-f475-406b-96f9-d5d1eff46d91

View source on GitHub