T75 develops a model-dependent prediction for the observable Q5 fringe-bias scale arising from leakage into hidden quarter-phase transport classes. Using the weight parameterization\[w_0=1-\epsilon,\qquadw_1=\epsilon_1,\qquadw_2=\epsilon_2,\qquadw_3=\epsilon_3,\]with\[\epsilon_1+\epsilon_2+\epsilon_3=\epsilon\ll1,\]The theorem derives the exact bias relation\[\eta^2=(\epsilon+\epsilon_2)^2+(\epsilon_3-\epsilon_1)^2.\]The resulting fringe bias is therefore controlled not only by total leakage magnitude but also by the structure of the leakage distribution across the quarter-phase classes. The cosine-sector deviation arises from the overall leakage displacement term \((\epsilon+\epsilon_2)\), while the sine-sector deviation is governed by the antisymmetric imbalance mode \((\epsilon_3-\epsilon_1)\). T75 is structurally important because it converts the earlier qualitative leakage architecture into a quantitative interferometric prediction. Under the Q5 suppression estimates inherited from T57, the theorem predicts a characteristic fringe-bias scale\[\eta_{\mathrm{Q5}}\sim10^{-4}\]for single-bridge leakage, and\[\eta_{\mathrm{Q5}}\sim10^{-7}\]under an additional return-mediated suppression layer. The theorem, therefore, supplies the missing magnitude estimate needed to complete the experimental prediction arc developed through T72-T74: signal form, structural discriminator, detection threshold, and expected signal scale. T75 does not claim a universal numerical prediction or guaranteed observability across interferometer designs. The predicted magnitude depends on the leakage-path architecture, the symmetry properties of the leakage distribution, and the mapping between the structural suppression parameter \(\epsilon\) and experimentally accessible optical leakage modes. The resulting prediction is therefore conditional rather than uniquely derived. Status: solid for the exact leakage-weight decomposition\[\eta^2=(\epsilon+\epsilon_2)^2+(\epsilon_3-\epsilon_1)^2\]and for the associated order-of-magnitude scaling analysis under the stated assumptions; conditional on the leakage-path architecture, symmetry structure, and suppression mapping inherited from T57; speculative for the existence or observability of any physical Q5 fringe residual prior to experimental detection.
T75 develops a model-dependent prediction for the observable Q5 fringe-bias scale arising from leakage into hidden quarter-phase transport classes. Using the weight parameterization[w_0=1-\epsilon,\qquadw_1=\epsilon_1,\qquadw_2=\epsilon_2,\qquadw_3=\epsilon_3,]with[\epsilon_1+\epsilon_2+\epsilon_3=\epsilon\ll1,]The theorem derives the exact bias relation[\eta^2=(\epsilon+\epsilon_2)^2+(\epsilon_3-\epsilon_1)^2.]The resulting fringe bias is therefore controlled not only by total leakage magnitude but also by the structure of the leakage distribution across the quarter-phase classes. The cosine-sector deviation arises from the overall leakage displacement term ((\epsilon+\epsilon_2)), while the sine-sector deviation is governed by the antisymmetric imbalance mode ((\epsilon_3-\epsilon_1)). T75 is structurally important because it converts the earlier qualitative leakage architecture into a quantitative interferometric prediction. Under the Q5 suppression estimates inherited from T57, the theorem predicts a characteristic fringe-bias scale[\eta_{\mathrm{Q5}}\sim10^{-4}]for single-bridge leakage, and[\eta_{\mathrm{Q5}}\sim10^{-7}]under an additional return-mediated suppression layer. The theorem, therefore, supplies the missing magnitude estimate needed to complete the experimental prediction arc developed through T72-T74: signal form, structural discriminator, detection threshold, and expected signal scale. T75 does not claim a universal numerical prediction or guaranteed observability across interferometer designs. The predicted magnitude depends on the leakage-path architecture, the symmetry properties of the leakage distribution, and the mapping between the structural suppression parameter (\epsilon) and experimentally accessible optical leakage modes. The resulting prediction is therefore conditional rather than uniquely derived. Status: solid for the exact leakage-weight decomposition[\eta^2=(\epsilon+\epsilon_2)^2+(\epsilon_3-\epsilon_1)^2]and for the associated order-of-magnitude scaling analysis under the stated assumptions; conditional on the leakage-path architecture, symmetry structure, and suppression mapping inherited from T57; speculative for the existence or observability of any physical Q5 fringe residual prior to experimental detection.
T74 establishes the statistical detection threshold for the Q5 phase-cycling signal developed in T72 and T73. Starting from the single-scan fringe residualϵ(ϕ)=2P1P2[(A−1)cosϕ+Bsinϕ],The theorem defines the intrinsic bias magnitudeη=(A−1)2+B2,and proves that the maximum balanced-path residual amplitude satisfiesϵmax=η.A key refinement is introduced: measurement of a first-harmonic bias alone is not operationally sufficient for Q5 detection, because such a bias can be absorbed into standard interferometric fitting parameters. The correct discriminator is therefore the phase-cycling invariantIQ5=∑k=03∣vk+1−Jvk∣2,which tests whether the measured quadrature evolution obeys the expected Z4 cycling structure established in T73. T74 proves that Q5 detectability requires two simultaneous conditions. First, the intrinsic signal scale must exceed the statistical shot-noise floor,η≳NNσ,where N is the photon count and Nσ is the required detection significance. Second, the observed phase evolution must remain consistent with the Q5 phase-cycling law through sufficiently small IQ5, distinguishing genuine Q5-compatible structure from ordinary calibration artifacts. Under shot-noise scaling, the minimum detectable bias obeysηmin∼NNσ. T74 is structurally important because it completes the minimal experimental prediction arc initiated in T72 and T73 by separating signal shape, structural discriminator, and statistical detectability. The theorem converts the earlier transport-cycling framework into a falsifiable interferometric prediction program with explicitly identified observables, structural consistency conditions, and detection scaling laws. Status: solid for the fringe-residual amplitude relations and shot-noise detection-threshold scaling under the stated assumptions; solid for the necessity of combining signal magnitude with structural phase-cycling discrimination; conditional on the T73 Z4 cycling framework and practical extraction of quadrature vectors; speculative for the existence or magnitude of any physical Q5 signal prior to empirical observation.
T79 completes the Q5 interferometric falsification framework by introducing a structured null-control protocol for quadrature-cycling residuals. Earlier theorems established the residual form, the Z4 quadrature-cycling structure, statistical thresholds, magnitude estimates, and systematics discrimination criteria. T79 addresses the final degeneracy problem: the cycling defectIQ5=∑k=03∣vk+1−Jvk∣2can vanish trivially when all quadrature vectors are zero. The theorem, therefore, establishes the joint operational requirementIQ5≈0and∣vk∣≈0,ensuring that the detected structure corresponds to nontrivial quadrature transport rather than the simple absence of signal. The theorem then defines four explicit null-control experiments designed to isolate ordinary instrumental artifacts from candidate Q5-consistent cycling structure. These include blocked-path operation, detector-only measurements, intentional laser-power modulation, and a classical phase-reference channel. Each control specifies a concrete pass condition under which stable, nonzero quarter-cycle transport behaviour must disappear or fail to persist. In particular, blocked-path operation forces P1P2=0, eliminating the interference residual derived in T72, while detector-only and reference-channel measurements test whether the observed cycling behaviour originates from instrumentation rather than coherent transport structure. T79 reframes the experimental objective as layered falsification rather than direct proof. A candidate signal survives only if coherent two-path interference exhibits nonzero stable quadrature vectors satisfying the quarter-cycle rotation law, while all null controls fail to reproduce the same structured behaviour. The theorem, therefore, converts the earlier invariant framework into a hardened experimental protocol capable of rejecting broad classes of ordinary systematic effects without claiming definitive proof of Q5 transport itself. Passing the protocol establishes only that the observed cycling behaviour survives the identified null controls and remains structurally consistent with the Q5 quadrature-transport prediction. Status: solid for the joint nontriviality condition IQ5≈0 together with ∣vk∣≈0; solid for the blocked-path null implied by the T72 interference structure; solid that the four null controls define experimentally testable falsification operations; conditional on whether the listed controls exhaust all relevant systematic classes for a given experimental platform; not a claim that passing the full null-control protocol would uniquely establish Q5 transport physics.
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.
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.
T73 develops a phase-cycling discriminator for Q5-consistent deviations that cannot be identified from a single interference scan. Building on T72, which shows that any first-harmonic residual can be absorbed by refitting the ordinary fringe parameters, T73 demonstrates that the lost information can survive in the constrained transformation law across multiple controlled phase settings. The theorem considers four scans separated by externally controlled π/2 phase offsets and replaces independent per-scan refitting with a single jointly constrained fit across all four settings. For these offsets, the ordinary fringe model occupies one two-dimensional phase-cycling chirality subspace, while the opposite chirality subspace is exactly orthogonal to it. Consequently, the joint fit removes the ordinary J−1-chirality component while preserving the orthogonal J-chirality component unattenuated. This yields the phase-cycling invariant IZ4=∑k=03∣vk+1−Jvk∣2, where vk is the residual quadrature vector extracted after the joint fit. A nontrivial signal detectable by this construction must exhibit the surviving quarter-turn transformation law vk+1=Jvk, giving IZ4≈0 with nonzero residual amplitude. The result is a single-chirality test rather than a choice between two equivalent rotation signs, because the opposite chirality is structurally absorbed by the joint null model. T73 does not establish that Q5's physical residual necessarily carries the surviving chirality. That identification requires a separate residual-to-observable bridge connecting the later Q5 residual structure to the experimental phase-cycling coordinates. Nor does the theorem establish immunity to correlated experimental systematics; those require separate noise and control analysis. T73 establishes the mathematical distinguishability mechanism and operational phase-cycling protocol under the stated common-reference and joint-fit assumptions. Status: solid for the exact chirality separation, joint-fit discriminator, and Z4 phase-cycling invariant under the stated four-offset protocol; conditional on a common external phase reference and jointly constrained null model; unresolved as to whether the physical Q5 residual occupies the surviving chirality; no claim of experimental detection or uniqueness against all correlated systematics.
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.