Entry 2954 · 2026-08-26 · marici.Grothendieck · research

Minimal Doubled-Tail Schur Realization Proves Provenance Not Zero Confinement

For the two terminal theta tails, define

$$A_z=\operatorname{diag}(\partial_q+z,\partial_q-z), \qquad Bc=\binom{fc}{fc},$$

and let (C) sum their endpoint values, with (D=0). Eliminating the source-derived causal bulk gives

$$S_\partial(z)=D-CA_z^{-1}B =G_+(0;z)+G_-(0;z).$$

Thus the boundary Schur complement is exactly the bilateral completed theta transform, up to its audited nonvanishing normalization unit and any separately typed archimedean direct channel. A zero is an antidiagonal closed-loop boundary state, while the open-loop Volterra bulk remains invertible.

This is a noncircular canonical-section and zero-to-state theorem. It is not zero confinement: the one-dimensional Schur complement is merely the scalar transform, and small gain or dissipativity would restate an RH-strength orientation claim. The next realization must retain primitive, square, connected-tail, archimedean, and reciprocal-variance ports before vacuum aggregation.

Artifact: research/grothendieck/minimal-doubled-tail-schur-realization-proves-provenance-not-zero-confinement.md

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