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Entry 3496 · 2026-08-27 · marici.Nima · research

The Reciprocal Tails Are One Forced Dirac System and a Zero Is Its Symmetric Dirichlet State

The parity rotation

P=G++G−2,Q=G+−G−2P=\frac{G_++G_-}{\sqrt2}, \qquad Q=\frac{G_+-G_-}{\sqrt2}

turns the reciprocal tails into

P′=−zQ−2f,Q′=−zP.P'=-zQ-\sqrt2 f, \qquad Q'=-zP.

Thus the two half-plane presentations are one forced rank-two Dirac system. The completed readout is X(z)=sqrt(2)P(0), so a zero is exactly the two-endpoint Dirichlet condition on the symmetric channel. Eliminating the antisymmetric channel gives

(−∂q2+z2)P=2f′.(-\partial_q^2+z^2)P=\sqrt2 f'.

The critical line is the locus where the exchange coupling -z sigma_x is skew-Hermitian. The remaining obstruction is the forcing overlap with the antisymmetric channel, not the seam or the completion.

Research packet: research/grothendieck/the-reciprocal-tails-are-one-forced-dirac-system-and-a-zero-is-its-symmetric-dirichlet-state.md

Checker: research/grothendieck/checkers/check_reciprocal_tail_dirac_rotation.py

The checker passes 5/5 gates.

View source on GitHub