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 \(\mathbb Z_4\) quadrature-cycling structure, statistical thresholds, magnitude estimates, and systematics discrimination criteria. T79 addresses the final degeneracy problem: the cycling defect\[\mathcal I_{\mathrm{Q5}}=\sum_{k=0}^{3}|v_{k+1}-Jv_k|^2\]can vanish trivially when all quadrature vectors are zero. The theorem, therefore, establishes the joint operational requirement\[\mathcal I_{\mathrm{Q5}}\approx0\quad\text{and}\quad|v_k|\not\approx0,\]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 \(P_1P_2=0\), 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 \(\mathcal I_{\mathrm{Q5}}\approx0\) together with \(|v_k|\not\approx0\); 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.
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 (\mathbb Z_4) quadrature-cycling structure, statistical thresholds, magnitude estimates, and systematics discrimination criteria. T79 addresses the final degeneracy problem: the cycling defect[\mathcal I_{\mathrm{Q5}}=\sum_{k=0}^{3}|v_{k+1}-Jv_k|^2]can vanish trivially when all quadrature vectors are zero. The theorem, therefore, establishes the joint operational requirement[\mathcal I_{\mathrm{Q5}}\approx0\quad\text{and}\quad|v_k|\not\approx0,]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 (P_1P_2=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 (\mathcal I_{\mathrm{Q5}}\approx0) together with (|v_k|\not\approx0); 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.
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.
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.
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\[w_0=1-\epsilon,\qquadw_1=\epsilon_1,\qquadw_2=\epsilon_2,\qquadw_3=\epsilon_3,\]withϵ1+ϵ2+ϵ3=ϵ≪1,The theorem derives the exact bias relationη2=(ϵ+ϵ2)2+(ϵ3−ϵ1)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 (ϵ+ϵ2), while the sine-sector deviation is governed by the antisymmetric imbalance mode (ϵ3−ϵ1). 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 scaleηQ5∼10−4for single-bridge leakage, andηQ5∼10−7under 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 ϵ and experimentally accessible optical leakage modes. The resulting prediction is therefore conditional rather than uniquely derived. Status: solid for the exact leakage-weight decompositionη2=(ϵ+ϵ2)2+(ϵ3−ϵ1)2and 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.
This record presents TA21 (Experimental Quadrature Extraction Protocol), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework. TA20 identified the three-part diagnostic signature for Q5 depth-mode leakage: monotone mode scaling, smooth-mode null, and phase-locked quadrature deviation. TA21 converts this control logic into a concrete operational protocol specifying preparation steps, measurement procedure, fitting procedure, and control experiments. The depth coordinate arising in the Q5 framework from the 320-slot transport graph structure does not have a direct laboratory counterpart. The protocol, therefore, works with a structural analog: depth modes of frequency n are simulated by inserting a phase modulator in one arm of a standard two-path interferometer and driving it with a structured phase profile equal to n times a fixed reference profile. The modulation frequency n plays the role of the depth-mode index, with n equal to zero giving a flat constant phase corresponding to the smooth gate-aligned mode, and increasing n giving increasing phase oscillation corresponding to higher depth modes. The protocol tests mode-dependent transport sensitivity rather than literal realization of the depth coordinate. The spectral scaling of leakage with modulation frequency is the object under test, not the geometric origin of that scaling. The measurement procedure scans the global inter-arm phase and records the interference intensity as a function of phase for each modulation frequency. The intensity pattern is fit to a quadrature form, extracting coefficients A_n and B_n. The deviation magnitude is then computed as the square root of the sum of squares of A_n minus one and B_n, providing a scalar observable that can be plotted against n. The TA20 signature is tested by checking three conditions. First, smooth-mode null: the deviation magnitude at n equal to zero should vanish or be negligible. Second, monotone scaling: the deviation magnitude should show a trend-level monotonic increase with n up to experimental uncertainty, with the stronger model-dependent form predicting scaling as eta times n squared. Third, phase-locked quadrature: the deviation must remain in the cosine-phi and sine-phi basis rather than appearing as random fringe drift, amplitude noise, or uncontrolled phase wandering. Three control experiments provide false-positive rejection. Power variation changes beam intensity without changing phase structure and should leave the deviation magnitude unchanged, ruling out intensity artifacts. Noise injection adds random phase noise and should reduce visibility without producing clean n-dependent scaling, ruling out noise-driven drift. Symmetric modulation applies identical phase modulation to both arms and should suppress the deviation to near zero, because the symmetric modulation cancels in interference. The symmetric modulation control is the strongest of the three because it directly tests whether asymmetry is essential to the signal, which matches the leakage architecture. The measurement machinery is standard: two-path interferometry and quadrature extraction are established experimental techniques. The interpretation is not standard and is classified separately. The mapping from phase modulation to depth-mode analog is a structural analogy, not a formal derivation. The TA18 through TA19 depth-leakage model is falsified at the protocol level if the deviation magnitude shows no n-dependence, if the deviation is random or phase-unlocked, or if the symmetric control fails to suppress the deviation. A positive result compatible with all three conditions does not constitute proof of the Q5 mechanism but constitutes a result consistent with it. 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.
This record presents TA20 (Depth-Mode Leakage Signature), part of the Q5 Transport Architecture Series developed under the Zero-Point Hypothesis framework. TA20 is a control theorem rather than a prediction theorem. TA19 established that depth-mode observable deviations scale as eta times n squared in the perturbative low-mode regime. TA20 identifies the diagnostic signature that distinguishes genuine Q5 depth-mode leakage from ordinary experimental noise, systematic offset, or generic drift. It does not add new content to the leakage mechanism but provides the control structure needed for experimental interpretation of TA19. The signature consists of three necessary conditions that must hold simultaneously. The first is monotone scaling: deviation amplitude shows a trend-level monotonic increase with depth-mode frequency in the low-mode regime, up to experimental uncertainty. The second is smooth-mode null: for the gate-aligned mode with n equal to zero, the deviation vanishes or is negligible, because the constant mode has maximal overlap with the gate and minimal leakage. The third is phase-locked quadrature: the deviation appears in the cosine-phi and sine-phi quadrature basis of the T71 interference correction, with amplitude consistent with the n-squared scaling. These are necessary conditions, not sufficient ones. A deviation pattern satisfying all three is compatible with the Q5 depth-mode leakage mechanism but does not constitute proof of it. This distinction is stated explicitly and is one of the central epistemic contributions of the theorem. The three conditions test different layers of the mechanism. Monotone scaling tests spectral dependence. The smooth-mode null tests admissibility alignment and is particularly powerful because it is tied directly to the gate geometry rather than being a free parameter. Phase-locked quadrature tests phase structure and is the strongest discriminator experimentally, since a generic systematic effect would need to accidentally reproduce the specific T71 quadrature decomposition with coherent cosine and sine components and correlated scaling across mode numbers. Ordinary experimental noise generically fails at least one of the three conditions. Generic noise does not respect n-squared scaling in mode frequency. Systematic offsets do not vanish when n is equal to zero. Noise-driven deviations are not generically phase-locked to the T71 quadrature basis. The conjunction of all three conditions, therefore, provides a false-positive rejection criterion. The phase-locking condition includes a caveat: the intrinsic mechanism generates quadrature structure, but observational averaging, decoherence, or detector integration could wash out phase structure in practice. A phase-independent observation is therefore not consistent with the intrinsic mechanism at the signal level, but this must be interpreted with care in the presence of measurement imperfections. TA20 provides the control conditions that TA21 (Experimental Quadrature Extraction Protocol) will operationalize into a concrete experimental procedure. 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.
This note proposes a simple, falsifiable interferometric test for structured deviations from standard two-path interference. In conventional interference, the observed intensity is described by a single effective phase parameter: I(ϕ)=P1+P2+2P1P2cosϕ, and deviations are typically attributed to visibility loss or phase offset. The present protocol tests for a different class of behaviour: phase-locked deviations that depend on the internal structure of the phase profile applied to one arm of the interferometer. A controlled phase modulation of the form ϕ(x)=n⋅f(x)\phi(x) = n \cdot f(x) ϕ(x)=n⋅f(x) is introduced in one arm, where nn n indexes the complexity of the imposed phase structure. The resulting interference signal is measured as a function of the global phase and decomposed into quadrature components: In(ϕ)=Cn+Vn[Ancosϕ+Bnsinϕ]. A deviation metric is defined relative to the baseline (flat-phase) case: Δn=(An−A0)2+(Bn−B0)2. The testable prediction is that, if structured phase-dependent effects are present, then: the baseline satisfies Δ0≈0\Delta_0 \approx 0 Δ0≈0; the deviation Δn\Delta_n Δn increases systematically with the imposed phase structure, and the deviation remains phase-locked in the quadrature basis and cannot be absorbed into a simple phase offset. The protocol includes explicit control conditions to distinguish such behaviour from conventional effects: symmetric modulation of both arms should eliminate the deviation; random phase noise should reduce visibility but not produce systematic scaling in Δn\Delta_n Δn; and phase referencing is required to resolve the quadrature structure. The outcome is binary. If no structured deviation is observed, the hypothesis is falsified. If a phase-locked, structure-dependent deviation is observed, then the standard interference model is incomplete under these conditions. This proposal is intentionally model-independent. While motivated by broader work on structured phase transport in discrete systems, the experimental test does not rely on that framework and can be evaluated directly.