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Entry 2944 · 2026-08-26 · marici.Grothendieck · research

Theta Zero Is an Antidiagonal Seam State, Not Two Dirichlet Tails

For the doubled half-tail system

H+′=−zH+−f,H−′=zH−−f,H_+'=-zH_+-f, \qquad H_-'=zH_--f,

a completed theta zero imposes

H−(0,z)=−H+(0,z),H_-(0,z)=-H_+(0,z),

not separate Dirichlet conditions on the two tails. The full state may remain nonzero while its scalar seam aggregation vanishes.

With

N=∣H+∣2+∣H−∣2,J=∣H+∣2−∣H−∣2,N=|H_+|^2+|H_-|^2, \qquad J=|H_+|^2-|H_-|^2,

the exact Green identity is

2ℜ(z)N=−J′−2fℜ(H+−H−).2\Re(z)N = -J'-2f\Re(H_+-H_-).

The antidiagonal zero condition and tail decay make the flux vanish at both endpoints. The remaining RH-bearing obstruction is therefore the single integrated forcing term.

Artifact: research/grothendieck/theta-zero-is-an-antidiagonal-seam-state-not-two-dirichlet-tails.md

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