Structured Leakage Circulation Generates Nontrivial Phase-Lag Return

Summary

Theorem 111 establishes that delayed return is a structural consequence of the leakage geometry and not an independent dynamical assumption. The theorem replaces earlier attempts to obtain an additional reduced operator direction and instead identifies the source of phase-lag transport within the internal structure of the leakage circulation generator.

Using the explicit leakage adjacency operator from T103,

\[
\mathcal{A}_L
=
\frac{1}{\sqrt{80}}
\begin{pmatrix}
0 & e^{i\pi/4} & 0 \\
e^{-i\pi/4} & 0 & e^{i\pi/4} \\
0 & e^{-i\pi/4} & 0
\end{pmatrix},
\]

The leakage sector is shown to form the mediated transport chain

\[
e_3 \leftrightarrow e_4 \leftrightarrow e_5.
\]

The theorem proves that \(\mathcal{A}_L\) is not proportional to the identity and therefore possesses genuine internal transport structure. Direct coupling between the observable boundary modes is absent,

\[
(\mathcal{A}_L)_{35}
=
(\mathcal{A}_L)_{53}
=
0,
\]

forcing transport between \(e_3\) and \(e_5\) to proceed through the intermediate leakage mode \(e_4\).

This mediated structure provides the geometric explanation for the delayed-return computation of T104. The first-order return channel vanishes,

\[
L^\dagger B L = 0,
\]

while the second-order return channel survives,

\[
L^\dagger B^2 L \neq 0.
\]

The theorem, therefore interprets the intermediate leakage mode \(e_4\) as a temporary storage channel through which amplitude must circulate before reappearing at the observable boundary.

The principal conclusion is that phase-lag transport arises naturally from mediated leakage circulation rather than from an additional operator degree of freedom. Delayed return is therefore geometrically forced by the leakage architecture itself.

The theorem closes Arc 13 by completing the progression

\[
\text{structured leakage}
\rightarrow
\text{delayed return}
\rightarrow
\text{phase lag}
\rightarrow
\text{observable bias},
\]

and provides the structural foundation for the phase-lag development of Arc 14.

One-Line Summary

\[
\boxed{
\text{Phase-lag transport exists because leakage circulation is mediated rather than direct.}
}
\]

Epistemic Status

Solid: \(\mathcal{A}_L \not\propto I\); mediated chain structure \(e_3 \leftrightarrow e_4 \leftrightarrow e_5\); absence of direct \(e_3 \leftrightarrow e_5\) coupling; \(L^\dagger B L = 0\); \(L^\dagger B^2 L \neq 0\); delayed return as a structural consequence of leakage geometry.

Conditional: interpretation of delayed return as the source of later observable asymmetries in the T112-T119 development.