Phase-Lag Transport as the Third Boundary Channel

T107 proved that the reduced boundary operator space has rank two,

\[
\dim(\mathcal B_Y^{\mathrm{red}})=2,
\]

so the third boundary contribution cannot appear as an independent reduced operator direction. T108 shows that the missing structure survives instead as delayed transport, namely, phase lag.

Let \(\Pi_Y\) denote the admissibility projector, \(K_\perp = I-\Pi_Y\) the leakage-complement projector, \(L\) the leakage injection map, and \(B=i\mathcal A_L\) the leakage circulation generator. The reduced return family is

\[
R(t)=L^\dagger e^{tB}L.
\]

The key observation comes from T104. The first-order return vanishes:

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

Because a single application of the leakage generator transfers amplitude into the gate-null middle leakage mode, producing no observable boundary-to-boundary return.

However, the second-order return survives:

\[
C_2=L^\dagger B^2L
=
-\frac{1}{80}(I-i\sigma_y)
\neq 0.
\]

Thus, complement-sector leakage is not destroyed. It is delayed.

T107 further established that

\[
C_2\in\operatorname{span}\{\Pi_Y,K_\perp\},
\]

So the delayed return contributes no additional reduced operator direction. The reduced boundary algebra therefore remains two-dimensional even though a nontrivial return process exists.

The central conclusion of T108 is that the third boundary transport class is not encoded as operator rank. It is encoded as transport history. The return channel survives through higher-order leakage circulation and reappears as a delayed observable contribution.

Accordingly, the three transport classes established in T106 are:

\[
T_{\mathrm{admit}},
\qquad
T_{\mathrm{return}},
\qquad
T_{\mathrm{residual}}.
\]

The admitted and leakage-complement sectors possess immediate reduced operator representations, while the return channel survives through phase-lag transport.

This theorem provides the conceptual bridge between Arc 13 and Arc 14. The mixing-operator program culminates in the recognition that delayed return survives despite the rank-two reduction, and the next stage of the framework becomes the study of leakage holonomy, projection bias, and observable asymmetry generated by this phase lag.

Dependencies:
T96 (Gate Projector and Leakage Complement),
T103 (Surface Circulation Generator),
T104 (Explicit Interference Correction),
T106 (Boundary Transport Multiplicity),
T107 (Reduced Boundary Rank Is Two).

Epistemic Classification

Solid:

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

\[
C_2=L^\dagger B^2L\neq0,
\]

\[
\dim(\mathcal B_Y^{\mathrm{red}})=2.
\]

Solid within the reduced framework:

\[
C_2\in\operatorname{span}\{\Pi_Y,K_\perp\}.
\]

Conditional:

Identification of the delayed return with the transport class

\[
T_{\mathrm{return}},
\]

since this relies on the T106 transport classification and the later residual-source architecture developed in T129-T136.

One-Line Summary

\[
\boxed{
\text{The third boundary transport channel survives as phase-lag transport rather than as a third reduced operator direction.}
}
\]