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.
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.
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.
T101 applies the conjugate-half mixing framework of T100 to a two-output interference experiment and derives the resulting structure of observable probability corrections. While T100 established that weak mixing between the observable sector and its conjugate counterpart produces first-order corrections constrained by total-probability conservation, T101 specializes those corrections to interference measurements and determines the form that any physically admissible deviation must take. The starting point is the standard two-output Born readout \[P_+^{(0)}=\cos^2\theta,\qquadP_-^{(0)}=\sin^2\theta,\] where θ parameterizes the interferometer configuration and ϕ denotes the relative phase. From T100, observable probabilities in the presence of weak conjugate-half mixing satisfy P±obs=P±(0)+ηΔ±+O(η2), together with the conservation law Δ++Δ−=0. The theorem shows that, in a two-output experiment, conservation immediately forces an antisymmetric correction structure: Δ+=−Δ−=f(θ,ϕ), yielding P+obs=cos2θ+ηf(θ,ϕ)+O(η2),P−obs=sin2θ−ηf(θ,ϕ)+O(η2). Thus, conjugate-half mixing cannot appear as an unconstrained additive deviation. Any increase in one output channel must be accompanied by an equal decrease in the other. The theorem next derives positivity constraints on the correction function. Since probabilities must remain between zero and one, the correction cannot remain finite at the interference endpoints θ=0,θ=2π, where one channel has unit probability and the other has zero probability. Consequently, \[f(0,\phi)=0,\qquadf\!\left(\frac{\pi}{2},\phi\right)=0,\] or must be suppressed sufficiently rapidly near those points. A minimal admissible envelope satisfying these constraints is α(θ)=sin(2θ), which vanishes at both endpoints while remaining positive throughout the interior of the interference interval. Incorporating the leading phase dependence yields the simplest structured correction f(θ,ϕ)=csin(2θ)cosϕ, where c is a coupling parameter describing the strength of the projected conjugate-half contribution. Substitution gives the minimal interference-correction model P+obs=cos2θ+ηcsin(2θ)cosϕ+O(η2),P−obs=sin2θ−ηcsin(2θ)cosϕ+O(η2). The theorem therefore predicts that any observable signature of conjugate-half mixing must appear as antisymmetric redistribution between interference outputs rather than as excess total probability. The characteristic experimental signal is δP+=−δP−, with the correction vanishing near the interference endpoints and reaching maximal magnitude in the interior of the interference pattern. T101 serves as the interference-level realization of the conjugate-half mixing framework. T100 established the general conservation law governing first-order Born corrections. T101 translates that abstract constraint into a concrete interference prediction, showing that weak conjugate-half mixing manifests as differential redistribution of probability across interference channels while preserving total probability to first order.
T107 proved that the reduced boundary operator space has rank two, dim(BYred)=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 ΠY denote the admissibility projector, K⊥=I−ΠY the leakage-complement projector, L the leakage injection map, and B=iAL the leakage circulation generator. The reduced return family is R(t)=L†etBL. The key observation comes from T104. The first-order return vanishes: L†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: C2=L†B2L=−801(I−iσy)=0. Thus, complement-sector leakage is not destroyed. It is delayed. T107 further established that C2∈span{ΠY,K⊥}, 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†BL=0,C2=L†B2L=0,dim(BYred)=2. Solid within the reduced framework: C2∈span{ΠY,K⊥}. Conditional: Identification of the delayed return with the transport class Treturn, since this relies on the T106 transport classification and the later residual-source architecture developed in T129-T136. One-Line Summary The third boundary transport channel survives as phase-lag transport rather than as a third reduced operator direction.
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.
This record presents TA19 (Observable Deviation from Leakage Spectrum), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework. TA19 closes the depth architecture arc TA14 through TA18 by connecting the leakage spectrum to observable probability deviations, and feeds back into the T71 through T79 experimental prediction arc. The depth program is no longer self-contained: it now connects the internal transport hierarchy to measurable interference structure. The full mechanism chain established across TA14 through TA19 is: depth mode n produces roughness scaling as n to the fourth power, which produces leakage amplitude scaling as n squared in the low-mode regime, which drives leakage return through the complement sector, which produces observable probability deviation. Each stage has its own object, its own scaling law, and its own epistemic status. The central result is that in the perturbative low-mode regime where c times n squared is much less than one, the depth-mode observable deviation scales as approximately eta times c times n squared. The full implied structure gives a deviation proportional to eta times cn^2 divided by (1 + cn^2), which saturates to a global bound of eta as mode number increases. Observable anomaly; therefore, initially grows with depth-mode frequency before saturating; it does not grow indefinitely. The observable deviation is connected to the T71 quadrature deviation structure. The T71 interference deviation has the form involving coefficients A and B multiplying the cosine and sine of the phase angle. Under TA19, the depth-mode-dependent deviation coefficients scale as eta times n squared at low mode number, giving a depth-mode-refined interference prediction where the deviation amplitude carries an explicit n squared factor. Exact coefficients and phase prefactors are not computed here. Three lemmas establish the result: leakage return generates observable correction at leading order in the leakage coupling parameter eta (Lemma 1); the depth-mode deviation scaling follows as eta times n squared in the low-mode regime and eta times the square root of the roughness functional equivalently (Lemma 2); and the T71 quadrature deviation coefficients inherit the n squared scaling (Lemma 3). A critical epistemic note is maintained throughout: the theorem separates the robust qualitative claim from the model-dependent quantitative claim. The robust claim is that observable deviation increases monotonically with depth-mode frequency before saturating. The model-dependent claim is the specific n squared scaling, which requires the continuum depth approximation from TA15, the low-pass gate model from TA18, and the leading-order perturbative coupling assumed here. These must not be conflated in experimental interpretation. The leakage-deviation coupling is explicitly identified as a first-order perturbative ansatz rather than a structural inevitability, since probabilities are nonlinear objects and the full leakage return operator can contribute phase-sensitive cross terms at higher order in eta. Together, TA14 through TA19 form a complete emergence ladder: discrete adjacency transport, continuum evolution, smoothness selection, quartic roughness spectrum, leakage from admissibility misalignment, and perturbative observable consequences. This is a coherent multistage transport framework connecting discrete Q5 geometry to measurable interference deviations. The theorem chain progressively derives the structure of the effective transport generator Geff=PiYGPiY+K†BK, from which observable phase, leakage, decoherence, and residual correction emerge as structural consequences of projected transport closure on Q5.