T94 extends the leakage-state framework of T93 by deriving the leading-order return dynamics of phase amplitude that leaves the \(Y\)-admissible crossing sector. Rather than treating leakage as a terminal loss mechanism, the theorem models leakage as a redistribution process in which rejected amplitude enters a secondary sector, circulates internally, and subsequently returns to the observable crossing sector. The extended state space is \[\mathcal{H}_{\mathrm{ext}}=\mathbb{C}^{2}\oplus\mathcal{H}_{L},\] with dynamics governed by the coupled generator \[\mathcal{G}=\begin{pmatrix}A & -K^{\dagger} \\K & B\end{pmatrix},\] where \(K\) transfers amplitude into the leakage sector, \(K^{\dagger}\) returns amplitude from the leakage sector, and \(B\) governs internal circulation within \(\mathcal{H}_{L}\). The theorem establishes that leakage is redistribution rather than destruction. For an initially coherent crossing state \(\psi_{PQ}(0)\), amplitude enters the leakage sector according to \[\lambda(t)=tK\psi_{PQ}(0)+O(t^{2}),\] and subsequently evolves through the internal circulation operator \[e^{tB}.\] The resulting leading-order return contribution to the crossing sector is \[R(t)=K^{\dagger}e^{tB}K\psi_{PQ}(0).\] This return term is identified as the residual surface return of the extended Q5 architecture. A central result of the theorem is that residual return is not automatically restorative. The effect of the returned phase is determined entirely by its alignment with the gate-admissible crossing component \[\psi_Y(t)=\Pi_Y \Psi(t).\] The coherence response is governed by the real overlap \[\Re\langle \psi_Y(t),R(t)\rangle.\] Three regimes emerge: \[\Re\langle \psi_Y,R\rangle > 0\] produces reinforcement of coherent crossing, \[\Re\langle \psi_Y,R\rangle < 0\] produces dephasing, and \[\Re\langle \psi_Y,R\rangle = 0\] produces a neutral contribution. The theorem, therefore, refines the decoherence picture introduced in T93. Decoherence is not interpreted as the disappearance of phase information but as the return of redistributed phase in a state that is no longer aligned with the admissible crossing mode. Misaligned return decreases gate-admission weight and reduces interference visibility even when the total norm remains conserved. Several corollaries follow. Generic residual return does not remain phase-locked to the crossing mode and therefore fails to restore coherence automatically. Exact recovery requires reconstruction of the full leakage-sector evolution \(e^{tB}\), making reversibility information-demanding even when the underlying dynamics remain unitary. Effective irreversibility is thus attributed to the information cost of reconstruction rather than to fundamental non-unitarity. T94 serves as the first formal return-transport theorem in the Q5 sequence. Together with T93, it provides a complete first-order picture of leakage dynamics: \[\text{crossing sector}\;\xrightarrow{K}\;\text{leakage sector}\;\xrightarrow{e^{tB}}\;\text{surface circulation}\;\xrightarrow{K^{\dagger}}\;\text{return transport}.\] The theorem establishes that the coherence consequences of leakage are determined not by the existence of returned phase, but by the alignment of that returned phase with the unique \(Y\)-admissible crossing mode.
T94 extends the leakage-state framework of T93 by deriving the leading-order return dynamics of phase amplitude that leaves the (Y)-admissible crossing sector. Rather than treating leakage as a terminal loss mechanism, the theorem models leakage as a redistribution process in which rejected amplitude enters a secondary sector, circulates internally, and subsequently returns to the observable crossing sector. The extended state space is [\mathcal{H}{\mathrm{ext}}=\mathbb{C}^{2}\oplus\mathcal{H}{L},] with dynamics governed by the coupled generator [\mathcal{G}=\begin{pmatrix}A & -K^{\dagger} \K & B\end{pmatrix},] where (K) transfers amplitude into the leakage sector, (K^{\dagger}) returns amplitude from the leakage sector, and (B) governs internal circulation within (\mathcal{H}{L}). The theorem establishes that leakage is redistribution rather than destruction. For an initially coherent crossing state (\psi{PQ}(0)), amplitude enters the leakage sector according to [\lambda(t)=tK\psi_{PQ}(0)+O(t^{2}),] and subsequently evolves through the internal circulation operator [e^{tB}.] The resulting leading-order return contribution to the crossing sector is [R(t)=K^{\dagger}e^{tB}K\psi_{PQ}(0).] This return term is identified as the residual surface return of the extended Q5 architecture. A central result of the theorem is that residual return is not automatically restorative. The effect of the returned phase is determined entirely by its alignment with the gate-admissible crossing component [\psi_Y(t)=\Pi_Y \Psi(t).] The coherence response is governed by the real overlap [\Re\langle \psi_Y(t),R(t)\rangle.] Three regimes emerge: [\Re\langle \psi_Y,R\rangle > 0] produces reinforcement of coherent crossing, [\Re\langle \psi_Y,R\rangle < 0] produces dephasing, and [\Re\langle \psi_Y,R\rangle = 0] produces a neutral contribution. The theorem, therefore, refines the decoherence picture introduced in T93. Decoherence is not interpreted as the disappearance of phase information but as the return of redistributed phase in a state that is no longer aligned with the admissible crossing mode. Misaligned return decreases gate-admission weight and reduces interference visibility even when the total norm remains conserved. Several corollaries follow. Generic residual return does not remain phase-locked to the crossing mode and therefore fails to restore coherence automatically. Exact recovery requires reconstruction of the full leakage-sector evolution (e^{tB}), making reversibility information-demanding even when the underlying dynamics remain unitary. Effective irreversibility is thus attributed to the information cost of reconstruction rather than to fundamental non-unitarity. T94 serves as the first formal return-transport theorem in the Q5 sequence. Together with T93, it provides a complete first-order picture of leakage dynamics: [\text{crossing sector};\xrightarrow{K};\text{leakage sector};\xrightarrow{e^{tB}};\text{surface circulation};\xrightarrow{K^{\dagger}};\text{return transport}.] The theorem establishes that the coherence consequences of leakage are determined not by the existence of returned phase, but by the alignment of that returned phase with the unique (Y)-admissible crossing mode.
T95 derives the first quantitative prediction of the extended Q5 leakage-state framework by determining the short-time decay of interference visibility for a gate-aligned crossing state coupled to leakage modes. Building on the leakage architecture of T93 and the return-transport mechanism of T94, the theorem establishes that the initial loss of coherence is governed entirely by the leakage coupling strength and is independent of the internal leakage dynamics at leading order. The extended state space is Hext=C2⊕HL, with generator \[\mathcal{G}=\begin{pmatrix}A & -K^{\dagger} \\K & B\end{pmatrix},\qquadA=i\sigma_y.\] An initially gate-aligned state Ψ(0)=(ψ00),ΠYψ0=ψ0,∥ψ0∥=1, begins entirely within the admissible crossing mode, with no initial leakage-sector occupation. The theorem first establishes that leakage amplitude enters the auxiliary sector at first order in time: λ(t)=tKψ0+O(t2). Consequently, the associated leakage weight is second order: ∥λ(t)∥2=t2∥Kψ0∥2+O(t3). Because the full evolution is norm preserving, the visible crossing-sector weight must decrease by the same amount. This leads directly to the short-time gate-admission weight ∥ΠYΨ(t)∥2=1−t2∥Kψ0∥2+O(t3). Taking the square root yields the principal result: V(t)=1−21∥Kψ0∥2t2+O(t3), where V(t)=∥ΠYΨ(t)∥ is the interference visibility introduced in T93. Using the gate decoherence measure DY(t)=1−V(t)2, The theorem obtains DY(t)=∥Kψ0∥2t2+O(t3). Thus, decoherence begins quadratically in time rather than linearly. Leakage amplitude appears immediately, but observable coherence loss emerges only after that displaced amplitude acquires measurable weight. A central consequence is that the leading decoherence coefficient depends solely on ∥Kψ0∥2, the leakage strength of the initial state. The internal leakage-sector circulation operator B does not contribute to the leading-order visibility loss and enters only at higher orders through the return dynamics derived in T94. The theorem, therefore, separates the immediate coherence-loss mechanism from the subsequent circulation and return processes. Several corollaries follow. States satisfying Kψ0=0 experience no short-time decoherence from leakage. Stronger leakage coupling produces faster visibility decay, and the coefficient ∥Kψ0∥2 may in principle be inferred from the curvature of an interference fringe at short times. Combining the result with the reduced interference law of T91 shows that leakage suppresses the interference term through a quadratic visibility factor. The theorem provides the first explicit dynamical prediction of the extended leakage architecture. It demonstrates that phase information is not destroyed when coherence is lost. Rather, phase leaves the Y-admissible crossing mode at a rate determined by the leakage coupling, producing a universal quadratic onset of decoherence in the short-time regime.
T93 extends the reduced Q5 interference framework of T91–T92 to an enlarged leakage-state space Hext=C2⊕HL, where C2 is the coherent crossing sector and HL collects non-crossing leakage modes. The theorem distinguishes between visible coherent weight C(t)=∥ΠPQΨ(t)∥2, leakage weight D(t)=∥(I−ΠPQ)Ψ(t)∥2, and gate-admission weight G(t)=∥ΠYΨ(t)∥2, where ΠY projects onto the Y-admissible crossing mode. A key result is that visible coherence and gate alignment are distinct quantities. A state may retain nonzero amplitude in the observable P/Q sector while failing admission into the Y-crossing mode. This leads to a geometric definition of decoherence: DY(t)=1−∥ΠYΨ(t)∥2. Decoherence is therefore interpreted as loss of overlap with the admissible crossing channel rather than destruction of phase information. The theorem further establishes that interference visibility is governed by the gate-admitted component of the state: V(t)=∥ΠYΨ(t)∥, yielding the extended interference law Pr(P)=21(1+V(t)sin(2t)cosϕ). Perfect visibility corresponds to complete gate alignment, while reduced visibility reflects either leakage into non-crossing modes or misalignment with the crossing sector. An extended generator G=(AK−K†B) is introduced to couple the coherent crossing sector to the leakage sector. Under the skew-adjoint conditions \[A^{\dagger}=-A,\qquadB^{\dagger}=-B,\] the evolution operator etG preserves total norm, implying C(t)+D(t)=1. Leakage therefore represents redistribution of phase weight rather than irreversible loss. The theorem provides a geometric interpretation of decoherence within the Q5 framework: phase information may remain globally conserved while becoming inaccessible to coherent crossing dynamics. In this picture, coherence is determined not merely by the presence of phase amplitude, but by its alignment with the unique Y-admissible transport mode.
T98 extends the coherent-crossing transport pipeline of T97 into a complete two-branch barrier architecture. Rather than tracking only the gate-admitted component, the theorem follows both the admitted and rejected portions of the reduced crossing state, yielding the minimal extended transport model for barrier interaction. Beginning with an incoming fibre state ψfull∈H5×5, The native-side transport and reduction operators produce the reduced crossing state ψPQ=RDNMNψfull∈C2. Using the gate-admission projector ΠY and leakage complement K=I−ΠY, Introduced in T96, the reduced state decomposes orthogonally into ψPQ=ΠYψPQ+KψPQ. The first component satisfies the Y-gate condition and is eligible for coherent crossing. The second component fails gate admission and enters the leakage branch. The theorem derives the complete extended transport operator Text=(DQMQΠYRDNMNKRDNMN), which simultaneously produces both outputs of barrier interaction. Applied to an incoming fibre state, the result is Textψfull=(DQMQΠYRDNMNψfullKRDNMNψfull). The upper component is the coherent crossing branch propagating through the barrier into the dual-side sector. The lower component is the leakage branch consisting of phase content that reaches the reduced crossing sector but fails the Y-gate condition. A key consequence is that generic barrier interaction is neither complete transmission nor complete rejection. Most states decompose into both branches simultaneously: ψPQ=ΠYψPQ+KψPQ. The barrier, therefore, acts as a filter rather than a binary gate. Coherent crossing and leakage are complementary outcomes of the same reduced crossing state. The theorem further establishes the complete-crossing and complete-leakage limits. A state crosses cleanly if and only if KRDNMNψfull=0, equivalently, RDNMNψfull∈span{∣Y⟩}. Conversely, a state exhibits complete leakage if and only if ΠYRDNMNψfull=0, so that no coherent crossing component survives. Using the orthogonal decomposition established in T96, the theorem preserves the norm split CY+DY=1, where CY=∥ΠYψPQ∥2 is the gate-admission weight and DY=∥KψPQ∥2 is the leakage weight. A major consequence of the theorem is that the leakage branch becomes the explicit source of the residual return dynamics developed in T94. Substituting the reduced crossing state into the return formula yields R(t)=K†etBKRDNMNψfull, providing a complete operator pathway from incoming fibre state to leakage circulation and subsequent return. T98 therefore completes the transport architecture initiated in T97. The barrier is revealed as a two-output filter that separates an incoming reduced state into an admitted crossing component and a rejected leakage component. Together with T93–T97, the theorem provides the first complete operator-level description of barrier transport, leakage generation, and residual return within the Q5 framework.
T103 identifies the explicit local leakage-sector circulation generator required by the leakage-return architecture developed in T93–T102. T102 established that observable conjugate-half mixing arises through the factorization C=L†BL, where L injects non-admitted crossing amplitude into the leakage sector, B governs leakage-sector circulation, and L† returns circulated amplitude to the observable sector. However, T102 left the circulation generator B unspecified. T103 supplies its local Q5 form. The theorem draws upon the leakage-generator development established later in the theorem chain. The local leakage sector is modelled by a three-mode basis (e3,e4,e5), where e3 and e5 are return-capable boundary modes and e4 is a balancing intermediary mode. The resulting local leakage adjacency has chain structure e3↔e4↔e5, with no direct coupling between e3 and e5. The corresponding local leakage adjacency operator is AL=8010e−iπ/40eiπ/40e−iπ/40eiπ/40. The theorem verifies that AL is Hermitian: AL†=AL. The leakage-sector circulation generator is then defined by B=iAL. Since AL is Hermitian, the circulation generator satisfies B†=−B, and is therefore anti-Hermitian. A direct consequence is that leakage evolution etB is unitary: (etB)†etB=I, showing that leakage circulation preserves norm. Leakage is therefore interpreted as conservative circulation rather than dissipative loss. The theorem next examines the transport structure encoded by AL. Because the matrix entries (AL)13=(AL)31=0, there is no direct coupling between the two boundary modes e3 and e5. All transport between them must pass through the middle mode e4. The middle mode, therefore, acts as a balancing mediator within the leakage sector. This structural restriction has dynamical consequences. The first-order leakage evolution etB=I+tB+O(t2) contains no direct boundary-to-boundary transfer because B13=B31=0. Consequently, transfer between e3 and e5 first appears through the second-order pathway \[e_3\rightarrowe_4\rightarrowe_5.\] Boundary return is therefore intrinsically mediated and delayed relative to local circulation. Substituting the derived generator into the T102 mixing formula yields C=L†(iAL)L. This provides the first explicit local realization of the conjugate-half mixing operator. The mixing process is no longer governed by an unspecified circulation generator but by a concrete leakage adjacency derived from the Q5 leakage architecture. Several conceptual consequences follow. Leakage transport is mediated rather than direct. Circulation preserves norm and therefore represents redistribution rather than loss. The middle leakage mode plays an essential balancing role. Most importantly, the observable mixing operator inherits its structure from a specific leakage-sector geometry rather than from an arbitrary correction term. T103 serves as the bridge between the abstract leakage-return factorization of T102 and the explicit interference calculations that follow. T102 established the form C=L†BL. T103 identifies the local generator B=iAL. Together, these results provide the explicit operator C=L†(iAL)L, which forms the starting point for the observable interference-correction analysis developed in T104. Epistemic status: Solid: The Hermiticity of AL, anti-Hermiticity of B=iAL, norm preservation of etB, absence of direct e3↔e5 coupling, and second-order nature of boundary-to-boundary transfer follow directly from the displayed local leakage matrix. Conditional: The normalization factor 1/80, the identification of (e3,e4,e5) with the physical leakage sector, and the embedding of the local leakage block into the full Q5 architecture depend upon the later global embedding results and remain conditional pending completion of the full leakage-sector construction.
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 Hext=C2⊕HL, with a coupled generator G=(AL−L†B), where K⊥=I−ΠY is the leakage-complement projector introduced in T96, while L:C2→HL 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†etBLψ. Expanding the leakage evolution operator, etB=I+tB+O(t2), gives R(t)=L†Lψ+tL†BLψ+O(t2). The theorem identifies the first nontrivial mixing contribution as the circulation-induced term tL†BLψ. Introducing the coupling scale η∼t, The effective observable state becomes ψeff=ψ+ηCψ+O(η2), with the derived mixing operator C=L†BL. The theorem, therefore, demonstrates that conjugate-half mixing is not an independent postulate. It emerges naturally from the sequence: leakageLsurface circulationBreturnL†observable correction. A key structural property follows immediately. Since B†=−B, The mixing operator satisfies C†=−C. The operator is therefore anti-Hermitian, implying ℜ⟨ψ,Cψ⟩=0 and ensuring preservation of normalization to first order: ∥ψ+ηCψ∥2=∥ψ∥2+O(η2). The theorem next establishes compatibility with the Born-correction framework of T100. Substituting ψeff=ψ+ηCψ+O(η2) into the observable readout yields Δi=2ℜ⟨Πiψ,ΠiCψ⟩, 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†BL. An important stability result also follows. If a state is perfectly gate-aligned, ΠYψ=ψ, then K⊥ψ=0. Since L injects only the leakage component, \[K_\perp\psi=0\quad\Rightarrow\quadL\psi=0.\] Therefore Cψ=L†BLψ=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†). 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 C=L†BL as the first-principles origin of conjugate-half mixing within the extended Q5 framework.
TA38 derives the reinjection block CB:HR→HM of the barrier-scattering matrix from the Q5 admissibility architecture. Given the residual state circulated by TA36 and TA37, only the component of residual return lying in the Y-admissible gate subspace may re-enter the coherent carrier sector. This forces the reinjection block to take the formCB=ΠYUreturnHR.The theorem also gives the exact nonvanishing condition: reinjection occurs precisely when the residual return transport has nonzero overlap with im(ΠY). The resulting carrier output consists of direct carrier transport plus a lagged residual reinforcement term,ψMout(t)=ABψM+CBetEBDBψM.Together with TA36 and TA37, this completes the forced off-diagonal block trilogy: extraction, mediated circulation, and admissible reinjection. The result shows that the residual sector of the barrier-scattering matrix is geometrically constrained by the Q5 architecture rather than introduced as a phenomenological model. Status: solid for the forced structure of CB; conditional for nontrivial reinjection pending explicit computation of the leakage generator AL.