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: \[T_{\mathrm{admit}},\qquadT_{\mathrm{return}},\qquadT_{\mathrm{residual}}.\] 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 \[\boxed{\text{The third boundary transport channel survives as phase-lag transport rather than as a third reduced operator direction.}}\]
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}},\qquadT_{\mathrm{return}},\qquadT_{\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.}}]
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†BL, and T103 identified the leakage generator B=iAL 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†BL=0. This occurs because the leakage generator transports amplitude through the helical chain e3↔e4↔e5, while the observable return map extracts only the boundary modes e3 and e5. A single application of B moves amplitude into the middle gate-null mode e4, preventing any first-order return contribution. Observable mixing therefore begins only at second order. The first nontrivial return operator is C2=L†B2L=−801(I−iσy), which is Hermitian and generates the leading observable correction. Applying the T100 renormalization procedure yields normalized probability shifts Δ+=−160sinϕsin(4θ),Δ−=+160sinϕsin(4θ), with Δ++Δ−=0. The resulting observable probabilities are P+obs=cos2θ−2η2160sinϕsin(4θ)+O(η3),P−obs=sin2θ+2η2160sinϕsin(4θ)+O(η3). The correction is therefore second order in the leakage coupling and possesses a characteristic angular envelope sin(4θ). This replaces the earlier minimal placeholder structure of T101 and introduces a richer node pattern, with nulls at θ=0,4π,2π, together with maxima at θ=8π,83π. The phase dependence is proportional to sinϕ, arising from the e±iπ/4 phases contained in the leakage generator AL. 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 sinϕsin(4θ) as the characteristic observable signature of leakage-return transport.
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.
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 Hext=C2⊕HL, with dynamics governed by the coupled generator G=(AK−K†B), where K transfers amplitude into the leakage sector, K† returns amplitude from the leakage sector, and B governs internal circulation within HL. The theorem establishes that leakage is redistribution rather than destruction. For an initially coherent crossing state ψPQ(0), amplitude enters the leakage sector according to λ(t)=tKψPQ(0)+O(t2), and subsequently evolves through the internal circulation operator etB. The resulting leading-order return contribution to the crossing sector is R(t)=K†etBKψPQ(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 ψY(t)=ΠYΨ(t). The coherence response is governed by the real overlap ℜ⟨ψY(t),R(t)⟩. Three regimes emerge: ℜ⟨ψY,R⟩>0 produces reinforcement of coherent crossing, ℜ⟨ψY,R⟩<0 produces dephasing, and ℜ⟨ψY,R⟩=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 etB, 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: crossing sectorKleakage sectoretBsurface circulationK†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.
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.
T96 provides the explicit matrix realization of the gate-admission and leakage operators that were introduced abstractly in T93–T95. The theorem establishes a canonical orthogonal decomposition of the reduced Q5 crossing sector HPQ=C2 into a Y-admissible crossing mode and its leakage complement. This converts the leakage architecture from an abstract projector framework into an explicitly computable model. The admissible crossing mode is defined as ∣Y⟩=21(1i), representing the quarter-turn phase relation associated with the Y-gate. The corresponding gate projector is ΠY=∣Y⟩⟨Y∣=21(1i−i1), and the leakage complement is K=I−ΠY=21(1−ii1). The theorem proves that both operators are orthogonal projectors satisfying \[\Pi_Y^2=\Pi_Y,\qquadK^2=K,\qquad\Pi_YK=K\Pi_Y=0,\] yielding the canonical decomposition HPQ=span{∣Y⟩}⊕ker(ΠY). Every reduced crossing state ψ=(PQ) admits the unique decomposition ψ=ΠYψ+Kψ, with norm identity ∥ψ∥2=∥ΠYψ∥2+∥Kψ∥2. The theorem further derives explicit formulas for gate admission and leakage strength. The crossing-mode amplitude is ⟨Y∣ψ⟩=21(P−iQ), and therefore the gate-admission strength is CY(ψ)=∥ΠYψ∥2=21∣P−iQ∣2. For normalized states, DY(ψ)=∥Kψ∥2=1−21∣P−iQ∣2, so that CY(ψ)+DY(ψ)=1. Gate coherence and leakage are therefore complementary quantities. Several structural consequences follow. A state crosses cleanly if and only if Kψ=0, equivalently ψ=c(1i) for some complex scalar c. Any deviation from this quarter-turn phase relation produces a nonzero leakage component. The theorem also provides the explicit realization of the abstract leakage operator appearing in T93–T95. Quantities such as the leakage norm ∥Kψ0∥2 and the short-time visibility decay derived in T95 become directly computable from the crossing amplitudes P and Q. The projector formalism, therefore, acquires a concrete matrix representation suitable for explicit calculations. T96 serves as the foundational algebraic theorem of the leakage program. It identifies the admissible crossing mode, constructs the associated projector and leakage complement, establishes the orthogonal decomposition of the reduced crossing sector, and converts the abstract coherence and leakage measures of T93–T95 into explicit computable formulas.