Core Barrier Transport Composition

T97 unifies the previously developed Q5 crossing, leakage, and gate-admission frameworks into a single operator composition describing coherent barrier transport. The theorem establishes that barrier crossing is not free propagation but a constrained sequence of transformations acting on the full incoming fibre state.

Starting from an incoming packet

\[
\psi_{\mathrm{full}}
\in
\mathcal{H}_{5\times5},
\]

The theorem identifies five sequential stages of transport:

\[
M_N
\rightarrow
D_N
\rightarrow
\mathcal{R}
\rightarrow
\Pi_Y
\rightarrow
D_QM_Q.
\]

These correspond respectively to native-side kernel interaction, native-side transport, dimensional reduction into the reduced crossing sector, gate-admissibility selection, and dual-side propagation.

The resulting transport law is

\[
\psi_{\mathrm{out}}
=
D_QM_Q\,\Pi_Y\,\mathcal{R}\,D_NM_N\,
\psi_{\mathrm{full}}.
\]

This expression provides the first complete operator-level description of barrier transport within the Q5 framework.

The theorem begins by introducing the transport operators. The native-side operators \(M_N\) and \(D_N\) act on the full \(5\times5\) fibre geometry. The reduction map

\[
\mathcal{R}
:
\mathcal{H}_{5\times5}
\rightarrow
\mathbb{C}^{2}
\]

implements the dimensional reduction that extracts the surviving \(P/Q\) crossing sector. The gate projector

\[
\Pi_Y
=
|Y\rangle\langle Y|
\]

Introduced in T96 selects the admissible quarter-turn crossing component. Finally, the dual-side operators \(M_Q\) and \(D_Q\) propagate the admitted state after barrier crossing.

The theorem establishes that the reduced crossing state is

\[
\psi_{PQ}
=
\mathcal{R}\,D_NM_N\,
\psi_{\mathrm{full}},
\]

and that the admissible crossing component is

\[
\psi_Y
=
\Pi_Y\psi_{PQ}.
\]

Only this projected component is eligible for coherent transport. The complementary component

\[
K\psi_{PQ}
=
(I-\Pi_Y)\psi_{PQ}
\]

enters the leakage sector described in T93–T96.

The outgoing transported state is therefore obtained by applying the dual-side operators to the admitted component:

\[
\psi_{\mathrm{out}}
=
D_QM_Q\psi_Y.
\]

Substitution yields the full transport composition.

Several structural consequences follow. A state crosses coherently if and only if

\[
\Pi_Y\,\mathcal{R}\,D_NM_N\,
\psi_{\mathrm{full}}
\neq
0.
\]

Thus, coherent transport requires a nonzero overlap between the reduced crossing state and the admissible \(Y\)-mode. States with zero projection onto the \(Y\)-mode fail the crossing condition and enter the leakage branch.

The theorem further demonstrates that the ordering of operators is not arbitrary. Reduction must occur after native-side processing, while gate selection must occur before dual-side propagation. The gate projector, therefore, occupies the central position within the transport architecture, separating state preparation from post-crossing propagation.

A direct connection is established with T96. Since

\[
\Pi_Y
=
|Y\rangle\langle Y|,
\]

The crossing condition becomes

\[
\psi_{PQ}
\not\perp
|Y\rangle.
\]

The gate-admission strength is therefore

\[
C_Y(\psi_{PQ})
=
\frac12
|P-iQ|^2,
\]

providing a computable measure of crossing eligibility in terms of the reduced crossing amplitudes.

T97 serves as the foundational transport-composition theorem of the Q5 framework. It links the reduced crossing sector, the leakage architecture, and the gate projector into a single operator pipeline, establishing the canonical ordering of processes that govern coherent barrier transport.