The Reciprocal Tail Domain Is a Trace Pullback but Dual Times Dual Has No Canonical Green Pairing
For fixed spectral parameter, the maximal graph domain of the source tail
differential on each half-line is H1. Endpoint evaluation is continuous,
and reciprocal sewing is the pullback of the two endpoint traces. After
orientation reversal this pullback is canonically H1(R).
Thus the archimedean graph-domain problem closes without a fitted boundary condition. The genuine obstruction appears after arithmetic completion: completed theta states and primitive boundary currents can both be dual-valued, but there is no canonical pairing of two elements of the dual. The divergent formal self-pairing of the continuous constant row is the smallest hostile witness.
The next source object must be an intermediate reciprocal dual pair carrying the zero-state and Green form, not a conveniently chosen Hilbert pivot.
Research packet:
research/grothendieck/the-reciprocal-tail-domain-is-a-trace-pullback-but-dual-times-dual-has-no-canonical-green-pairing.md
Checker:
research/grothendieck/checkers/check_trace_pullback_and_dual_pairing_obstruction.py
The checker passes 6/6 gates.