T93 extends the reduced Q5 interference framework of T91–T92 to an enlarged leakage-state space \[\mathcal{H}_{\mathrm{ext}}=\mathbb{C}^{2}\oplus\mathcal{H}_{L},\] where \(\mathbb{C}^{2}\) is the coherent crossing sector and \(\mathcal{H}_{L}\) collects non-crossing leakage modes. The theorem distinguishes between visible coherent weight \[C(t)=\|\Pi_{PQ}\Psi(t)\|^{2},\] leakage weight \[D(t)=\|(I-\Pi_{PQ})\Psi(t)\|^{2},\] and gate-admission weight \[G(t)=\|\Pi_{Y}\Psi(t)\|^{2},\] where \(\Pi_Y\) 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: \[\mathcal{D}_{Y}(t)=1-\|\Pi_{Y}\Psi(t)\|^{2}.\] 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: \[V(t)=\|\Pi_{Y}\Psi(t)\|,\] yielding the extended interference law \[\Pr(P)=\frac12\left(1+V(t)\sin(2t)\cos\phi\right).\] 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 \[\mathcal{G}=\begin{pmatrix}A & -K^{\dagger}\\K & B\end{pmatrix}\] 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 \(e^{t\mathcal{G}}\) 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.
T93 extends the reduced Q5 interference framework of T91–T92 to an enlarged leakage-state space [\mathcal{H}{\mathrm{ext}}=\mathbb{C}^{2}\oplus\mathcal{H}{L},] where (\mathbb{C}^{2}) is the coherent crossing sector and (\mathcal{H}{L}) collects non-crossing leakage modes. The theorem distinguishes between visible coherent weight [C(t)=|\Pi{PQ}\Psi(t)|^{2},] leakage weight [D(t)=|(I-\Pi_{PQ})\Psi(t)|^{2},] and gate-admission weight [G(t)=|\Pi_{Y}\Psi(t)|^{2},] where (\Pi_Y) 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: [\mathcal{D}{Y}(t)=1-|\Pi{Y}\Psi(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)=|\Pi_{Y}\Psi(t)|,] yielding the extended interference law [\Pr(P)=\frac12\left(1+V(t)\sin(2t)\cos\phi\right).] 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 [\mathcal{G}=\begin{pmatrix}A & -K^{\dagger}\K & B\end{pmatrix}] 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 (e^{t\mathcal{G}}) 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.
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.
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.
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.
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 ψfull∈H5×5, The theorem identifies five sequential stages of transport: \[M_N\rightarrowD_N\rightarrow\mathcal{R}\rightarrow\Pi_Y\rightarrowD_QM_Q.\] 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 ψout=DQMQΠYRDNMNψfull. 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 MN and DN act on the full 5×5 fibre geometry. The reduction map R:H5×5→C2 implements the dimensional reduction that extracts the surviving P/Q crossing sector. The gate projector ΠY=∣Y⟩⟨Y∣ Introduced in T96 selects the admissible quarter-turn crossing component. Finally, the dual-side operators MQ and DQ propagate the admitted state after barrier crossing. The theorem establishes that the reduced crossing state is ψPQ=RDNMNψfull, and that the admissible crossing component is ψY=ΠYψPQ. Only this projected component is eligible for coherent transport. The complementary component KψPQ=(I−ΠY)ψPQ 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: ψout=DQMQψY. Substitution yields the full transport composition. Several structural consequences follow. A state crosses coherently if and only if ΠYRDNMNψfull=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 ΠY=∣Y⟩⟨Y∣, The crossing condition becomes ψPQ⊥∣Y⟩. The gate-admission strength is therefore CY(ψPQ)=21∣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.
T91 establishes the reduced Q5 two-channel interference law after the 5→3→2 reduction. The surviving reduced state space is C2, with crossing channels P and Q, and transport generated by A=iσy, satisfying A2=−I. Continuous reduced evolution is given byψ(t)=etAψ(0),withetA=(cost−sintsintcost). For the initial relative-phase stateψ(0)=21(1eiϕ),The evolved channel probabilities arePr(P)=21(1+sin(2t)cosϕ),andPr(Q)=21(1−sin(2t)cosϕ).The observable channel imbalance is thereforePr(P)−Pr(Q)=sin(2t)cosϕ. The theorem shows that reduced Q5 transport naturally produces a two-channel interference structure on the reduced crossing plane. The transport amplitude is controlled by accumulated phase t=kε, while the coherence factor is controlled by the relative phase ϕ. Phase averaging removes the interference contribution:⟨cosϕ⟩=0⟹⟨Pr(P)⟩=⟨Pr(Q)⟩=21.Loss of coherent phase relation, therefore restores equal channel probabilities. T91 converts the earlier transport-generator chain into an explicit observable probability law. Earlier theorems established\[A^2=-I,\qquadT^k\to e^{tA},\qquad\mu_k=\sin(k\varepsilon),\]while T91 derives the resulting two-channel interference structure generated by the same reduced transport algebra. The discrete form follows fromt=kε,ε=3201,givingPr(P)=21(1+sin(3202k)cosϕ).The fringe contrast oscillates with transport step number k, with the first maximal contrast neark≈80π≈251,matching the near-integer phase-maximization structure identified previously in T84. Classification:Solid: matrix exponential, probability law, normalization, and phase averaging.Conditional: identification of the reduced two-channel Q5 state space and the generator chain A=iσy from T37 and T80-T89.Not claimed: direct experimental observability.