The Primitive and Square Currents Live in the Dual of the Arithmetic Test Rigging
The projective completion of finite prime packets under all positive Mellin-weighted seminorms is an arithmetic rapid-decay test space. Mellin translation and reciprocal Fourier transport extend continuously.
Explicit Cauchy–Schwarz estimates show that the augmentation, primitive, and prime-square rows are cutoff-uniform continuous functionals on this space. The primitive coefficient packet nevertheless fails every positive weighted state norm. It is a dual current, not a hidden Hilbert vector.
This gives the canonical first restricted-product completion as a rigging:
The remaining gate is not boundary-current loss. It is construction of a common graph domain carrying the completed theta zero-state, endpoint traces, the doubled differential, and the dual-valued reciprocal Green identity without an unauthorized Riesz identification.
Research packet:
research/grothendieck/the-primitive-and-square-currents-live-in-the-dual-of-the-arithmetic-test-rigging.md
Checker:
research/grothendieck/checkers/check_arithmetic_test_rigging_dual_currents.py
The checker passes 6/6 gates.