Explicit Interference Correction from Q5 Leakage Return

T104 computes the explicit observable interference correction generated by Q5 leakage-return transport. T101 established the admissible structure of a hidden-half correction term but left the correction function unspecified. T102 derived the leakage-return operator \(C=L^\dagger BL\), and T103 identified the leakage generator \(B=i\mathcal{A}_L\) from the explicit Q5 leakage adjacency. T104 completes the chain by directly computing the resulting interference correction.

The central structural result is that first-order mixing vanishes identically:

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

This occurs because the leakage generator transports amplitude through the helical chain

\[
e_3 \leftrightarrow e_4 \leftrightarrow e_5,
\]

while the observable return map extracts only the boundary modes \(e_3\) and \(e_5\). A single application of \(B\) moves amplitude into the middle gate-null mode \(e_4\), preventing any first-order return contribution. Observable mixing therefore begins only at second order.

The first nontrivial return operator is

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

which is Hermitian and generates the leading observable correction. Applying the T100 renormalization procedure yields normalized probability shifts

\[
\Delta_+
=
-\frac{\sin\phi\,\sin(4\theta)}{160},
\qquad
\Delta_-
=
+\frac{\sin\phi\,\sin(4\theta)}{160},
\]

with

\[
\Delta_+ + \Delta_- = 0.
\]

The resulting observable probabilities are

\[
P_+^{\mathrm{obs}}
=
\cos^2\theta
-
\frac{\eta^2}{2}
\frac{\sin\phi\,\sin(4\theta)}{160}
+
O(\eta^3),
\]

\[
P_-^{\mathrm{obs}}
=
\sin^2\theta
+
\frac{\eta^2}{2}
\frac{\sin\phi\,\sin(4\theta)}{160}
+
O(\eta^3).
\]

The correction is therefore second order in the leakage coupling and possesses a characteristic angular envelope \(\sin(4\theta)\). This replaces the earlier minimal placeholder structure of T101 and introduces a richer node pattern, with nulls at

\[
\theta = 0,\quad \frac{\pi}{4},\quad \frac{\pi}{2},
\]

together with maxima at

\[
\theta = \frac{\pi}{8},\quad \frac{3\pi}{8}.
\]

The phase dependence is proportional to \(\sin\phi\), arising from the \(e^{\pm i\pi/4}\) phases contained in the leakage generator \(\mathcal{A}_L\). This shifts the correction into quadrature relative to the leading Born interference fringe.

T104 establishes that the Q5 leakage-return mechanism produces a fully explicit interference correction that is antisymmetric between output channels, conserves total probability, vanishes at the required endpoints, and emerges naturally from the two-step transport structure of the leakage chain. The theorem replaces the phenomenological correction function of T101 with a direct Q5-derived prediction and identifies \(\sin\phi\,\sin(4\theta)\) as the characteristic observable signature of leakage-return transport.