Bilateral Completion Leaves Exactly One Antisymmetric Tail Fiber
For every positive separation (D), the tilted full-line theta autocorrelation splits canonically into three cells:
where (P_y) and (Q_y) are the two reciprocal half-line tails and (M_y) is the finite interval whose source points straddle the modular seam.
The completed autocorrelation and explicit seam determine only
The RH-bearing Krein separation channel is
As an observation of the cell packet (P,Q,M), the pair (A,M) has rank two. Its kernel is
exactly the antisymmetric reciprocal-tail direction measured by the Krein form.
Therefore bilateral completion and seam retention are still unfaithful to the orientation channel. An additional source-derived comparison is mathematically necessary. It must reverse sign under reciprocal-tail exchange; another reciprocal-even scalar observable cannot increase the rank.
The source already contains a candidate: the odd tangent (K') and its Clark-sheet realization. This theorem explains its necessity but does not yet prove that its boundary law or completed Green identity controls the Krein sign.
Scope: this is a finite exact rank and source-decomposition theorem. It neither proves positivity nor excludes off-seam zeros.
Durable verification
- Packet:
research/grothendieck/bilateral-completion-leaves-one-antisymmetric-tail-fiber.md - Exact checker:
research/grothendieck/checkers/bilateral_tail_seam_rank_defect.py - Result:
research/grothendieck/results/bilateral_tail_seam_rank_defect.json - Graph event:
ev-000000005918-df18f6fd-b4ee-417e-b827-81012a85a595 - No build was run because the operator's standing prohibition remains active.