Mixing Operator from Leakage Return

T102 derives the first-principles origin of the conjugate-half mixing operator introduced abstractly in T100–T101. While the earlier theorems established that weak mixing between the observable sector and its conjugate counterpart produces first-order corrections to Born probabilities, the mixing map itself remained an assumed object. T102 removes that assumption by deriving the operator directly from the leakage-return dynamics of the extended Q5 transport system.

The starting point is the leakage architecture developed in T93–T96. The extended state space is

\[
\mathcal{H}_{\mathrm{ext}}
=
\mathbb{C}^{2}
\oplus
\mathcal{H}_{L},
\]

with a coupled generator

\[
\mathcal{G}
=
\begin{pmatrix}
A & -L^{\dagger}\\
L & B
\end{pmatrix},
\]

where

\[
K_\perp = I-\Pi_Y
\]

is the leakage-complement projector introduced in T96, while

\[
L:\mathbb{C}^{2}\rightarrow\mathcal{H}_{L}
\]

is the leakage injection map into the leakage sector. The operator \(B\) governs circulation within the leakage sector.

From T94, leakage-return dynamics are described by

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

Expanding the leakage evolution operator,

\[
e^{tB}
=
I+tB+O(t^2),
\]

gives

\[
R(t)
=
L^{\dagger}L\psi
+
t\,L^{\dagger}BL\psi
+
O(t^2).
\]

The theorem identifies the first nontrivial mixing contribution as the circulation-induced term

\[
t\,L^{\dagger}BL\psi.
\]

Introducing the coupling scale

\[
\eta\sim t,
\]

The effective observable state becomes

\[
\psi_{\mathrm{eff}}
=
\psi
+
\eta\,C\psi
+
O(\eta^2),
\]

with the derived mixing operator

\[
\boxed{
C
=
L^{\dagger}BL.
}
\]

The theorem, therefore, demonstrates that conjugate-half mixing is not an independent postulate. It emerges naturally from the sequence:

\[
\text{leakage}
\;\xrightarrow{L}\;
\text{surface circulation}
\;\xrightarrow{B}\;
\text{return}
\;\xrightarrow{L^{\dagger}}\;
\text{observable correction}.
\]

A key structural property follows immediately. Since

\[
B^{\dagger}
=
-B,
\]

The mixing operator satisfies

\[
C^{\dagger}
=
-C.
\]

The operator is therefore anti-Hermitian, implying

\[
\Re\langle\psi,C\psi\rangle
=
0
\]

and ensuring preservation of normalization to first order:

\[
\|\psi+\eta C\psi\|^2
=
\|\psi\|^2
+
O(\eta^2).
\]

The theorem next establishes compatibility with the Born-correction framework of T100. Substituting

\[
\psi_{\mathrm{eff}}
=
\psi
+
\eta C\psi
+
O(\eta^2)
\]

into the observable readout yields

\[
\Delta_i
=
2\Re
\langle
\Pi_i\psi,
\Pi_iC\psi
\rangle,
\]

which reproduces the first-order correction structure previously introduced in the conjugate-half mixing program. The abstract operator \(C\) of T100–T101 is therefore identified explicitly with the leakage-return operator

\[
L^{\dagger}BL.
\]

An important stability result also follows. If a state is perfectly gate-aligned,

\[
\Pi_Y\psi
=
\psi,
\]

then

\[
K_\perp\psi
=
0.
\]

Since \(L\) injects only the leakage component,

\[
K_\perp\psi
=
0
\quad\Rightarrow\quad
L\psi
=
0.
\]

Therefore

\[
C\psi
=
L^{\dagger}BL\psi
=
0.
\]

Perfectly admitted states receive no conjugate-half mixing correction. Mixing arises only from leakage away from the gate-admissible mode.

Several conceptual consequences follow. The mixing operator is revealed as a composite object consisting of three distinct stages: leakage injection (\(L\)), leakage-sector circulation (\(B\)), and return transport (\(L^{\dagger}\)). No new free parameter is introduced at this stage. All structures arise from previously established operators within the leakage architecture.

T102 serves as the derivational bridge between the leakage program and the Born-correction program. T94 introduced residual return dynamics. T100–T101 introduced observable mixing corrections. T102 demonstrates that both descriptions arise from the same underlying mechanism, establishing

\[
\boxed{
C
=
L^{\dagger}BL
}
\]

as the first-principles origin of conjugate-half mixing within the extended Q5 framework.