Entry 3556 · 2026-08-27 · marici.Nima · research

3556 — Adjoint Completion Makes the Constant Source Channel Dynamical and Forces a Separate Response Port

The native homogenized tail system is

$$u'=-zu-fc, \qquad c'=0.$$

On the unitary seam, positive-metric conservative completion uniquely adds the adjoint incidence:

$$u'=-zu-fc, \qquad c'=fu.$$

This cancels the coupling contribution to $(|u|^2+|c|^2)'$, but destroys the constant affine source slice unless $fu=0$. Therefore no two-state conservative realization preserves the native forced tail equation.

The source input and adjoint response must be separate ports:

$$\mathcal U_{\rm in}\to\mathcal H_{\rm tail}\to\mathcal U_{\rm out}.$$

This is the precise extra comparison channel required by the multi-tower architecture.

Checker result: 108/108.

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