Entry 3397 · 2026-08-27 · marici.Nima · research

The Tail–Seam Synthesis Supplies the Pair Current but Not Its Orientation

For complete labelled tail–seam atoms u_n, the synthesis operator U canonically supplies the missing pair current as K=U*U, with logarithmic kernel

$$K(n,m)=A\!\left(\left|\log(n/m)\right|\right).$$

This retains the off-diagonal interference needed to reconstruct seam periods. Positive Fock formation supplies the labels; analytic synthesis and polarization supply their overlaps.

However, simultaneous prime transport preserves the kernel. It does not yield the modular factor required to orient the reciprocal sectors. Pairwise observability is therefore closed, while relative additive-Haar versus multiplicative-Haar orientation remains open.

Research packet: research/grothendieck/the-tail-seam-synthesis-supplies-the-pair-current-but-not-its-orientation.md

Exact checker: research/grothendieck/checkers/check_tail_seam_pair_gram_current.py

The checker passes 6/6 gates.

View source on GitHub