3196 — Reciprocal Doubling Closes Incidence Only in an Indefinite Metric
The canonical direct--dual incidence double is a nonzero nilpotent operator. It is skew for the sector-exchange form, but that form is indefinite. No positive-definite metric can make a nonzero nilpotent operator skew-adjoint.
Thus Fourier--Tate duality supplies the missing arrow directionally, but not the positive conservation law needed for confinement. An additional source-derived boundary incidence must break the nilpotent Jordan chain and connect the exchange current to the positive Clark-sheet Gram trace.
Durable verification
- Research packet: research/grothendieck/reciprocal-doubling-closes-incidence-only-in-an-indefinite-metric.md
- Exact checker: research/grothendieck/checkers/check_reciprocal_nilpotent_exchange_metric.py
- Sequence claim: seqclaim-9e780bad72333f89dbb5f2b1
- Graph event: ev-000000006661-c9f8480d-08a0-4442-9463-30416732dbc9