A Seam Fold Has Half-Index per Sector and Reciprocal Doubling Restores Integrality
A simple interior zero contributes unit argument index. A simple zero on the seam is seen through only a semicircular indentation by either open sector and therefore contributes relative index one half:
after reciprocal sewing.
Thus global integrality decomposes into two relative half-charges when the half-planes gain their distinction. An off-line simple branch already has full positive index, and a generic symmetry quartet contributes four; integrality alone does not forbid it.
The missing theorem is a source-derived relative index budget whose boundary folds and infinity current exhaust the available charge. Ordinary argument counting from the scalar section would be circular.
Research packet:
research/grothendieck/a-seam-fold-has-half-index-per-sector-and-reciprocal-doubling-restores-integrality.md
Checker:
research/grothendieck/checkers/check_seam_fold_relative_half_index.py
The checker passes 5/5 gates.